From Text to 'Lived' Resources: Mathematics Curriculum Materials and Teacher Development (Mathematics Teacher Education)

From Text to ‘Lived’ Resources

MATHEMATICS TEACHER EDUCATION VOLUME 7 SERIES EDITOR Andrea Peter-Koop, University of Oldenburg, Germany Patricia Wilson, University of Georgia, United States EDITORIAL BOARD Andy Begg, Auckland University of Technology, New Zealand Chris Breen, University of Cape Town, South Africa Francis Lopez-Real, University of Hong Kong, China Jarmila Novotna, Charles University, Czechoslovakia Jeppe Skott, Danish University of Education, Copenhagen, Denmark Peter Sullivan, Monash University, Monash, Australia Dina Tirosh, Tel Aviv University, Israel SCOPE The Mathematics Teacher Education book series presents relevant research and innovative international developments with respect to the preparation and professional development of mathematics teachers. A better understanding of teachers’ cognitions as well as knowledge about effective models for preservice and inservice teacher education is fundamental for mathematics education at the primary, secondary and tertiary level in the various contexts and cultures across the world. Therefore, considerable research is needed to understand what facilitates and impedes mathematics teachers’ professional learning. The series aims to provide a significant resource for teachers, teacher educators and graduate students by introducing and critically reflecting new ideas, concepts and findings of research in teacher education.

For other titles published in this series, go to http://www.springer.com /series/6327

Ghislaine Gueudet · Birgit Pepin · Luc Trouche Editors

From Text to ‘Lived’ Resources Mathematics Curriculum Materials and Teacher Development

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Editors Ghislaine Gueudet IUFM Bretagne site de Rennes Rue Saint- Malo 153 35043 Rennes CEDEX France [email protected]

Birgit Pepin Sør-Trøndelag University College 7004 Trondheim Norway [email protected]

Luc Trouche Institut français de l’Education École Normale Supérieure de Lyon 15 parvis René-Descartes, BP 7000 69342 Lyon cedex 07 France [email protected]

ISBN 978-94-007-1965-1 e-ISBN 978-94-007-1966-8 DOI 10.1007/978-94-007-1966-8 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011935535 © Springer Science+Business Media B.V. 2012 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Foreword

‘Mathematics Curriculum Material and Teacher Development’ can be read as the title of a dull book on an old fashioned topic from the era of curriculum development in the 1980s, perhaps ‘jazzed up’ by the catchword ‘teacher development’ from the 1990s. So, one might expect a latecomer to research in Mathematics Education. A simple cursory look over the content of the table of contents of this book shows that this is a false assumption. There are at least three major issues investigated in this book, which make it an up-to-date and fascinating contribution to research in Mathematics Education (or Didactics of Mathematics as I would prefer to call it): – ‘Curriculum material’ has definitely not been perceived in the restricted way it had been discussed two or three decades ago. The fact that the authors use the concept ‘curriculum resources’ highlights that beside the traditional curriculum materials, like textbooks and other curricular documents, a whole range of texts and other resources have been taken into account, including software, electronic resources and the Internet. All these resources seem to become increasingly important in expressing and sharing ideas not only on curriculum materials themselves, but also in terms of curriculum development. They also help in terms of teacher education and everyday practice. The inclusion of more modern resources does not deny the most important teacher resource – the textbook. A main message of this book is to place the artefact ‘mathematics textbook’ in a wider, systematic perspective of material resources available for (mathematics) teachers and students. The book also shows that this broadening of the concept of teacher resources is helpful for understanding practices in various contexts. In selected countries, and communities of mathematics teachers, it is a fact that a wide range of ‘resources’, apart from textbooks and traditional curriculum documents, is present and relevant for teachers’ daily practice. Teachers’ professional knowledge, practical constraints (like money and other classroom arrangements) and cultural resources like language, collegiality, organisation and time, amongst others, have to be analysed to comprehensively understand the processes involved in teacher use of resources. In fact, this book opens a perspective on resources, which is not necessarily material.

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– The book supports recent trends in research on teaching and learning mathematics with the help of artefacts: to fully understand the role of curriculum material, it is not sufficient to simply analyse the artefact as such. A comprehensive content analysis of an artefact used by teachers can help to develop deeper knowledge of its functions in mathematics education. Nevertheless, it is only by analysing the use of the artefact that one may be able to adequately judge upon the affordances and constraints of a given artefact. For example ‘instrumental genesis’ (initiated and introduced to Didactics of Mathematics by Rabardel) analyses how an artefact is turned into an ‘instrument’ via the genesis of individual or social utilisation schemes. The research literature claims that a curriculum resource can only be judged by an analysis of its inherent features in addition to an analysis of the ways in which the different agents of the educational process use these resources. In an instrumental genesis approach, this is condensed in the concept of ‘utilisation scheme’, which is also fundamental to the documentation approach described in this book. As a consequence, the documentation approach conveys the notion of an agent having created the ‘document’ for a specific purpose. – In the book the word artefact is used in a broad sense, leaning on Wartofsky’s (1979) notion (XIII: ‘anything which human beings create by the transformation of nature and of themselves’) which differs from the traditional understanding of curriculum resources. The texts in this book are not only analysing material resources, but pay due attention to immaterial sources available to (mathematics) teachers. Beside material resources, a comprehensive analysis of teachers’ resources must also take into account immaterial resources like colleagues and communities of teaching practices. The book discusses ‘collaborative use’, and selected chapters explore the relations between teacher communities of practice, the documents shared in these communities and the consequences for the professional development of teachers from this collaboration. Here, the individual use of resources is adequately complemented by using resources in an environment shared by a community of teachers. Moreover, the book shows under which conditions such collaboration can empower teachers to become active instructional designers. With the broadening of the view from material to immaterial resources, from individual to collective use of resources, methodologies investigating documentation and professional interaction (sharing of knowledge) of teachers also have to be extended beyond the ‘standard’ features of classroom and school research (often done by video-taping and consecutive case study analysis) or large scale statistical research using questionnaires (maybe complemented by interviews and the like). A reader sensitive to research methodology will find a whole range of research methods to explore the diverse phenomena – with various foci according to the different theoretical stances taken by the authors. As a consequence of the innovative character of the book, no consensus on research methodology has been reached yet – and this heterogeneity seems to be appropriate for a newly developed approach and the explorative character of the investigation of resources used by mathematics teachers.

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Having stated this, one characteristic nevertheless stands out, and for the majority of the book’s chapters: nearly all texts heavily rely on case studies. The empirical results point to the necessity of a mix of research methods to better understand teachers’ use of resources. Although the last paragraph typically puts forward an argument, which shows the value and importance of the book for researchers in Mathematics Education, I would like to highlight that the texts in this book can also be very helpful for practising teachers, who could learn about the wide range of resources available for enhancing their teaching practice. Curriculum developers and policy makers may benefit from the book’s reports of investigations, which show once again that implementing change in education and educational reform is not a straightforward, top-down process. Researchers are reminded that having the best available ideas and concepts for change does not imply factual change of teaching. The book shows that sharing artefacts and collectively developing utilisation schemes in collaborative groups of teachers and researchers can be a more effective means to curriculum change. Cooperation around appropriately designed resources – be they material and/or conceptual – can be a way to develop teaching and learning mathematics. Giessen, Germany

Rudolph Straesser

Reference Wartofsky, M. W. (1979). Models. Representation and the scientific understanding. Dordrecht: Reidel Publishing Company.

Introduction Ghislaine Gueudet, Birgit Pepin, and Luc Trouche

The teachers, in their professional activity, interact with a wide range of resources; these interactions and their consequences hold a central place in teachers’ professional development. The purpose of this book is to develop this perspective and to explore it in the field of mathematics education. We consider on the one hand curriculum material. Traditionally, textbooks remain central resources for the teaching of mathematics in most countries. Nevertheless, other kinds of resources, in particular digital resources, and amongst them resources accessible via the Internet, are increasingly used. Understanding the evolutions brought by digital material is a central motivation of our work. On the other hand, the reason for introducing the term ‘resource’ instead of ‘material’ is to broaden the perspective on the elements available for the teachers’ work, and to include in particular interactions with a variety of agents: – Interactions between the teacher and her students constitute central resources for this teacher. Digitisation creates new forms of students’ productions and new modes of communication between students and teachers; but even an expression on a student’s face in class can constitute a resource for the teacher. – Interactions between the teacher and her colleagues seem to hold an increasing place. Teachers can collectively design curriculum plans, lessons, and once again the digital means convey new forms of communication, networking and association. Teachers collect resources, select, transform, share, implement and revise them. Drawing from the French term ‘ingénierie documentaire’, we call these processes ‘documentation’. The literal English translation is ‘to work with documents’, but the meaning it carries is richer. Documentation refers to the complex and interactive

G. Gueudet (B) CREAD, Université de Bretagne Occidentale, IUFM Bretagne site de Rennes, 35043 Rennes Cedex, France e-mail: [email protected]

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ways that teachers work with resources; in-class and out-of-class, individually, but also collectively. We propose a new perspective, considering teachers not as passive users, but as designers, creative ‘users’ and ‘sharers’ of their own resources, and viewing these resources as ‘lived’ resources. Teachers’ professional knowledge influences this design; at the same time, the documentation work extends existing-and generates new-professional knowledge. Working in 12 different countries, the authors develop a variety of perspectives on teacher resources, on their use and on the associated teachers’ professional development, with different foci and theoretical frameworks. The book is organised in four parts. Each is complemented by a reaction, presenting an expert’s view of the whole section. The first part focuses on the different kinds, and nature of, curriculum resources for mathematics teachers from a practical, methodological and theoretical point of view. It examines what is, or is not, available for teachers’ professional activity. It also introduces the question of what kinds of changes are afforded by digital resources: – Jill Adler introduces a conceptualisation of resources as re-sourcing teachers’ professional activity. She focuses on teacher professional knowledge, and provides evidence of different uses, by teachers in class, of knowledge resources. – Ghislaine Gueudet and Luc Trouche propose what they coin as documentational approach of didactics for the study of the teacher’s documentational work. This new theoretical approach emphasises that geneses, documentational geneses as well as professional geneses, are strongly intertwined. The authors also expose a specific methodology for the study of these geneses: the reflexive investigation of teachers’ documentation work; – Maria Alessandra Mariotti and Mirko Marracci consider the question of semiotic mediation initiated by the development of the available digital resources. They explore the semiotic potential of an artefact for teacher use in their classrooms. – Gérard Sensevy focuses on didactical intentions, for individual teachers and for different kinds of teacher groups. He studies the influence of resources on teachers’ pedagogical intentions. Furthermore, he considers teacher action in class in terms of joint actions, and which include student actions, where the students’ contributions constitute a major resource for teachers. The Reaction to Part I is written by Bill Barton. The second part of the book focuses on the characteristics of curriculum material. The articles raise questions about the design of curriculum materials, and about their integration, appropriation and transformation by teachers in and for their everyday teaching. Is the teacher use of curriculum materials aligned with the use envisioned by curriculum designers? What are the consequences of teacher transactions with resources for teacher professional development? The various factors shaping the nature of the resources, their design and their use, are examined here, with a

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specific focus on sociocultural factors and how these influence the development of curriculum materials. – Kenneth Ruthven investigates the use and integration of technology in mathematics classroom practice. He identifies five structuring features, of different natures, that shape the incorporation of new technologies into teachers’ practice: working environment, resource system, activity format, curriculum script and time economy; – Janine Remillard considers different modes of engagement which teachers develop vis-a-vis curriculum resources and how they develop. She argues that teachers are often positioned, or position themselves, as passive users; and her overarching aim is to reframe the teacher–curriculum relationship such that teachers are positioned as partners and collaborators with curriculum resources. – Birgit Pepin investigates the role of resources, more precisely a task analysis schedule, as catalyst for teacher learning. She explores the different forms of feedback resulting from developing and working with a ‘tool’ designed to analyse mathematical tasks/curriculum materials for instruction. Her results provide deeper insights, at one level, into the processes of teacher learning with the help of analytic tools and the feedback these may afford, and at another, how a tool or artefact may change into a catalytic tool at the interface between task design and enactment. – William Schmidt describes the development of a textbook content metric that can be used in longitudinal studies to map and measure the curricular experience of individual students. Teachers and schools, sometimes districts, choose textbooks, and teachers in turn decide on the ‘coverage’ of those textbooks. This, in turn, has implications for student exposure to these curriculum materials, and the ways of working with them. – Christine Proust proposes a historical perspective on the nature of ‘school documentation’, in terms of design and use of mathematical texts in the scribal schools of Mesopotamia about 4,000 years ago. She observes patterns of this documentation across different schools indicating strong institutional conditioning. The Reaction to Part II is proposed by Malcolm Swan. The third part focuses on the use of resources by teachers and students, in-class and out-of-class, and includes studies that explore the influence of the resources’ characteristics on teacher and student activity. Furthermore, the articles in this part consider the interactions between the various educational agents, and the effects of these interactions on the development and design of resources: – Carolyn Kieran, Denis Tanguay and Armando Solares study the ‘how’ and the ‘why’ of teachers adapting researcher-designed resources, and in the context of integration of computer algebra system (CAS) technologies. They claim that the whole adaptation process, from its beginning (how teacher engage with a resource designed by researchers) to the changes made in class during the implementation, rests on teacher knowledge and beliefs.

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– Using classroom videos, Dominique Forest and Alain Mercier analyse how teachers can organise their pedagogic practice and student interventions drawing on language and gestures. They show how classroom videos can become resources for teacher professional development and research. – Sebastian Rezat focuses on textbooks, considering teachers’ and students’ use of textbooks. He establishes links between teacher’s use of mathematics textbooks effecting students, and vice versa, and argues that students’ use of resources must be considered as an important aspect within teachers’ documentation work. – Maria Trigueros and Maria-Dolores Lozano study documentational geneses of teachers working within Enciclomedia, a national project in Mexico that offers a particular online resource. They identify developments in terms of teacher documentation systems and of teacher pedagogic practice, which includes the use of the digital means offered and supported by traditional textbooks. – Paul Drijvers uses and further develops the concept of orchestration. He argues for a specific focus on what happens in class, the didactical performance, and identifies types of orchestrations. Survey results suggest that teachers’ intentions may differ from their actual teaching. He investigates factors leading teachers to retain a given type, and conditions for evolutions and development. The Reaction to Part III is proposed by Luis Radford. The fourth part of the book focuses on the collaborative aspects of teacher documentation, considering that teachers are in contact, and work, with various groups and communities in their professional lives. In this part concepts are introduced that illuminate the influence of the nature of groups and communities, the particularities of the processes of documentation within groups, and individual–collective relationships. The articles in this part identify various potential roles and interventions of collaborative teacher documentation in mathematics teacher education. – Carl Winsløw proposes a comparative study of two kinds of teacher collectives: lesson studies as a means for professional development of mathematics teachers in Japan; and Danish high-school teachers’ collaboration in the setting of multidisciplinary modules. He introduces, and provides evidence for, the importance of didactic infrastructures, their constraints and affordances in terms of teachers’ collaborative work in preparing, observing and evaluating their teaching. – Ghislaine Gueudet and Luc Trouche extend the documentational approach of mathematics didactics presented in chapter two by emphasising the importance of collective aspects in teachers’ documentation work. Drawing on the notion of ‘communities of practice’, they introduce the notions of community genesis and community documentation genesis, and study the relationships between the different kinds of geneses. – Jana Visnovska, Chrystal Dean and Paul Cobb problematise the rhetoric of teachers as instructional designers. They argue that all teachers engage in documentation work; but the ability of designing coherent instructional sequences requires specific support and appropriate professional development structures.

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The Reaction to Part IV is proposed by Barbara Jaworski. Deborah Ball offers a general view on all contributions; the conclusion section presents a synthesis of the book’s main results. The authors in this book provide different lenses to view the interactions between teachers and teaching resources, and the implications for teacher professional development. These different views come together in the book, resulting in the emergence of a new theorisation of teacher documentation work, and a new perspective on teachers’ resources.

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Contents

Part I

Teacher Resources

1 Knowledge Resources in and for School Mathematics Teaching . . Jill Adler

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2 Teachers’ Work with Resources: Documentational Geneses and Professional Geneses . . . . . . . . . . . . . . . . . . . . . . . Ghislaine Gueudet and Luc Trouche

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3 Patterns of Didactic Intentions, Thought Collective and Documentation Work . . . . . . . . . . . . . . . . . . . . . . . Gérard Sensevy

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4 Resources for the Teacher from a Semiotic Mediation Perspective . Maria Alessandra Mariotti and Mirko Maracci

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Reaction to Part I: Resources Can Be the User’s Core . . . . . . . Bill Barton

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Part II

Text and Curriculum Resources

5 Constituting Digital Tools and Materials as Classroom Resources: The Example of Dynamic Geometry . . . . . . . . . . . Kenneth Ruthven

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6 Modes of Engagement: Understanding Teachers’ Transactions with Mathematics Curriculum Resources . . . . . . . Janine T. Remillard

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7 Task Analysis as “Catalytic Tool” for Feedback and Teacher Learning: Working with Teachers on Mathematics Curriculum Materials . . . . . . . . . . . . . . . . . . . . . . . . . Birgit Pepin 8 Measuring Content Through Textbooks: The Cumulative Effect of Middle-School Tracking . . . . . . . . . . . . . . . . . . . William H. Schmidt

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9 Masters’ Writings and Students’ Writings: School Material in Mesopotamia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christine Proust Reaction to Part II: Some Reactions of a Design Researcher . . . . Malcolm Swan Part III 10

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Researcher-Designed Resources and Their Adaptation Within Classroom Teaching Practice: Shaping Both the Implicit and the Explicit . . . . . . . . . . . . . . . . . . . . . . . . Carolyn Kieran, Denis Tanguay, and Armando Solares

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Classroom Video Data and Resources for Teaching: Some Thoughts on Teacher Education . . . . . . . . . . . . . . . . Dominique Forest and Alain Mercier

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Interactions of Teachers’ and Students’ Use of Mathematics Textbooks . . . . . . . . . . . . . . . . . . . . Sebastian Rezat

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Teachers Teaching Mathematics with Enciclomedia: A Study of Documentational Genesis . . . . . . . . . . . . . . . . . Maria Trigueros and Maria-Dolores Lozano

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Teachers Transforming Resources into Orchestrations . . . . . . . Paul Drijvers

Part IV

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Use of Resources

Reaction to Part III: On the Cognitive, Epistemic, and Ontological Roles of Artifacts . . . . . . . . . . . . . . . . . . . Luis Radford

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Collaborative Use

A Comparative Perspective on Teacher Collaboration: The Cases of Lesson Study in Japan and of Multidisciplinary Teaching in Denmark . . . . . . . . . . . . . . . . . . . . . . . . . . Carl Winsløw

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Communities, Documents and Professional Geneses: Interrelated Stories . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghislaine Gueudet and Luc Trouche

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Mathematics Teachers as Instructional Designers: What Does It Take? . . . . . . . . . . . . . . . . . . . . . . . . . . . Jana Visnovska, Paul Cobb, and Chrystal Dean

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Reaction to Part IV: Teacher Agency: Bringing Personhood and Identity to Teaching Development . . . . . . . . . . . . . . . . Barbara Jaworski

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Afterword: Using and Designing Resources for Practice . . . . . . . . . Deborah Loewenberg Ball

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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ghislaine Gueudet, Birgit Pepin, and Luc Trouche

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contributors

Jill Adler School of Education, University of the Witwatersrand, 2050 Johannesburg, South Africa; King’s College London, London, UK, [email protected] Deborah Loewenberg Ball School of Education, University of Michigan, Ann Arbor, MI 48109-1259, USA, [email protected] Bill Barton Department of Mathematics, University of Auckland, Auckland 1142, New Zealand, [email protected] Paul Cobb Peabody College, Vanderbilt University, Nashville, TN, USA, [email protected] Chrystal Dean Department of Curriculum and Instruction, Appalachian State University, Boone, NC, USA, [email protected] Paul Drijvers Freudenthal Institute, Utrecht University, PO Box 85170, 3508 AD Utrecht, The Netherlands, [email protected] Dominique Forest IUFM de Bretagne, Université de Bretagne Occidentale, Rennes Cedex, France, [email protected] Ghislaine Gueudet CREAD, Université de Bretagne Occidentale, IUFM Bretagne site de Rennes, 35043 Rennes Cedex, France, [email protected] Barbara Jaworski Mathematics Education Centre, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK, [email protected] Carolyn Kieran Département de Mathématiques, Université du Québec à Montréal, Montréal, QC, Canada H3C 3P8, [email protected] Maria-Dolores Lozano Instituto Tecnológico Autónomo de México, CP 1000, México City, Mexico, [email protected] Mirko Maracci Department of Mathematics, University of Pavia, Pavia, Italy, [email protected]

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Maria Alessandra Mariotti Department of Mathematics and Computer Science, University of Siena, Siena, Italy, [email protected] Alain Mercier ADEF: Université de Provence, ENS-Lyon IFE, 32 rue Eugène Cas, 13004 Marseille, France, [email protected] Birgit Pepin Faculty of Teacher and Interpreter Education, Sør-Trøndelag University College, 7004 Trondheim, Norway, [email protected] Christine Proust Laboratoire SPHERE (CNRS & University Paris-Diderot), Paris, France, [email protected] Luis Radford École des sciences de l’éducation, Laurentian University, Sudbury, ON, Canada P3E 2C6, [email protected] Janine T. Remillard University of Pennsylvania, Philadelphia, PA 19146, USA, [email protected] Sebastian Rezat Institut für Didaktik der Mathematik, Justus-Liebig-University Giessen, 35394 Giessen, Germany, [email protected] Kenneth Ruthven Faculty of Education, University of Cambridge, Cambridge CB2 8PQ, UK, [email protected] William H. Schmidt Michigan State University, East Lansing, MI 48824, USA, [email protected] Gérard Sensevy Brittany Institute of Education, University of Western Brittany, France, [email protected] Armando Solares Universidad Pedagógica Nacional, México City, México, [email protected] Malcolm Swan Centre for Research in Mathematics Education, School of Education, University of Nottingham, Nottingham NG8 1BB, UK, [email protected] Denis Tanguay Université du Québec à Montréal, Montréal, QC, Canada, [email protected] Maria Trigueros Instituto Tecnológico Autónomo de México, CP 1000 México City, Mexico, [email protected] Luc Trouche Institut français de l’Education, École Normale Supérieure de Lyon, 15 parvis René-Descartes, BP 7000, 69342 Lyon cedex 07, France, [email protected] Jana Visnovska School of Education, The University of Queensland, St Lucia, QLD 4072, Australia, [email protected] Carl Winsløw Department of Science Education, University of Copenhagen, 1350 København K, Denmark, [email protected]

Part I

Teacher Resources

Chapter 1

Knowledge Resources in and for School Mathematics Teaching Jill Adler

1.1 Introduction This book, and the range of chapters within it, take as its starting point the role of curriculum resources in mathematics teaching and its evolution. Teachers draw on a wide range of resources as they do their work, using and adapting these in various ways for the purposes of teaching and learning. At the same time, this documentation work (as it is referred to by Gueudet and Trouche, Chapter 2) acts back on the teacher and his or her professional knowledge. Documentation work is a function of the characteristics of the material resources, teaching activity, the teachers’ knowledge and beliefs, and the curriculum context. The chapters that follow explore and elaborate this complexity. An underlying assumption across chapters is an increasing range of textual resources for teaching and wide availability of digital resources. The empirical work that informs this chapter took place in mathematics classrooms with limited textual and digital resources, and it is this kind of context that gave rise to a broad conceptualisation of resources in mathematics teaching that included the teacher and her professional knowledge, together with material and cultural resources, like language and time. In Adler (2000) I describe this broad conceptualisation, theorising material and cultural resources in use in practice in mathematics teaching in South Africa. The discourse used is of a teacher ‘re-sourcing’ her practice – a discourse with strong resonances in documentation work. This chapter builds on that work, foregrounding and conceptualising professional knowledge as a resource in school mathematics teaching. I begin by locating our concern with knowledge resources, a discussion that leads on to the methodology

J. Adler (B) School of Education, University of the Witwatersrand, 2050 Johannesburg, South Africa King’s College London, London, UK e-mail: [email protected]

G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_1,  C Springer Science+Business Media B.V. 2012

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we have developed in the QUANTUM1 research project to adequately describe their use in mathematics teaching. This current research has as its major question, what and how mathematics comes to be constituted in pedagogic practice? Professional knowledge in use in practice, and how this shapes what is made available for learning, come into focus. The methodology we have developed is then illustrated through recent empirical work in two secondary mathematics classrooms in South Africa. These illustrations add force to the argument for foregrounding knowledges in use in descriptions of classroom practice and teachers’ interactions with resources. Moreover, while the methodological tools offered here emerge in response to a particular context, related data and theoretical gaze, they are, I propose, useful for studying the evolution of knowledge resources in use in teaching across contexts.

1.2 Locating the Study of Knowledge Resources QUANTUM has its research roots in a study of teachers’ ‘take-up’ from an upgrading in-service teacher education programme in mathematics, science and English language teaching in South Africa (Adler & Reed, 2002). By ‘take-up’ we mean what and how teachers appropriated various aspects of the programme, using these in and for their teaching. The notion of ‘take-up’ enabled us to describe the diverse and unexpected ways teachers in the programme engaged with selections from the courses offered and how these selections were recontextualised in their own teaching. We were able to describe teachers’ agency in their selections and use, and illuminate potential effects. Amongst other aspects of teaching, we were interested in resources in use. We problematised these specifically in school mathematics practice (Adler, 2000), where I argued for a broader notion of resources in use that includes additional human resources like teachers’ professional knowledge (as opposed to their mere formal qualifications), additional material resources like geoboards which have been specifically made for school mathematics, everyday resources like money as well as social and cultural resources like language, collegiality and time. I also argued for the verbalisation of resource as ‘re-source’. In line with ‘take-up’, I posited that this discursive move shifts attention off resources per se and refocuses it on teachers working with resources, on teachers re-sourcing their practice. In focus were selected material (e.g. chalkboards) and cultural resources (language, time). Theoretical resources were drawn from social practice theory, leading to an elaborated categorisation of resources, supported by examples of their use in practice in terms of their ‘transparency’ (Lave & Wenger, 1991). These combined

1 QUANTUM is a Research and Development project on mathematical education for teachers in South Africa. Its development arm focused on qualifications for teachers underqualified in mathematics (hence the name) and completed its tasks in 2003. QUANTUM continues as a collaborative research project.

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to illustrate that what matters for teaching and learning is not simply what resources are available and what teachers recruit, but more significantly how various resources can and need to be both visible (seen/available and so possible to use) and invisible (seen through to the mathematical object intended in a particular material or verbal representation), if their use is to enable access to mathematics. Out of focus in this work were human resources: teachers themselves, their professional knowledge base and knowledges in use. The teachers in our study were studying courses in mathematics and mathematics education. We were thus interested in their ‘take-up’ from these courses. However, we had difficulty ‘grasping’2 teachers’ take-up with respect to mathematical content knowledge in particular. Our analysis of interviews, together with observations in teachers’ classrooms over 3 years, suggested correlations between teachers’ articulation of the mathematical purposes of their teaching and the ways in which they made substantive use of ‘new’ material and cultural resources (language in particular). These results are in line with a range of research that has shown how curriculum materials are mediated by the teacher (e.g. Cohen, Raudenbush & Ball, 2003). Remillard (2005) describes the interaction between a teacher and the curriculum materials he or she uses as relational, and thus co-constitutive. A relational orientation to teachers and resources serves as a starting point for a number of chapters in this volume (see Chapters 5 and 7). Our analysis, in addition, pointed to unintentional deepening of inequality. The ‘new’ curriculum texts selected by teachers from their coursework and recontextualised in their classroom practice appeared most problematic when teachers’ professional knowledge base was weak. Typically, this occurred in the poorest schools (Adler, 2001). These claims are necessarily tentative. Our methodology did not enable us to probe teachers’ take-up with respect to mathematics content knowledge over time. Moreover, as we attempted to explore professional knowledge in practice in the study, we appreciated the non-trivial nature of the elaboration of the domains of mathematical knowledge, knowledge about teaching and the didactics of mathematics in the construction of teacher education – a point emphasised recently by Chevallard and Cirade (in Gueudet & Trouche, 2010). In a context where contestation over selections from knowledge domains into mathematics teacher education continues, the importance of pursuing knowledge in use in teaching through systematic study was evident. Mathematical knowledge for and in teaching, what it is and how it might be ‘grasped’ became the focus in the QUANTUM study that followed. The methodology we have developed makes visible the criteria teachers transmit for what counts as mathematics, and through these, the domains of knowledge teachers recruit to ground mathematics in their classroom practice. It is this conceptualisation that has enabled an elaboration of knowledge resources in use in mathematics teaching.

2 I use ‘grasp’ here in a technical sense to convey the message that knowledge in use in practice is not unproblematically ‘visible’, but is made so through the deployment of specific methodological tools and analytic resources.

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1.3 Conceptualising Knowledge Resources In Adler (2000), and as discussed above, I argued for a conceptualisation of ‘resource’ as both a noun and a verb, for thinking about resource as ‘the verb “re-source,” to source again or differently where “source” implies origin, that place from which a thing comes or is acquired’. Here too, ‘resource’ is both noun and verb – ‘knowledge resources’ refers to domains of knowledge – the objects, processes and practices within these – that teachers recruit as they go about the work of teaching. This conceptualisation of knowledge as resource coheres with the orientation to the notion of ‘lived resources’ that underpins this volume. While my focus is domains of knowledge (not curriculum material), I am similarly concerned with what is selected, transformed and used in practice, and what is produced as a result. Selecting from domains of knowledge and transforming these in use for teaching is simultaneously the work of teaching and its outcome, that is that which comes to be legitimated as mathematical knowledge in a particular practice. Teachers recruit (or appeal to) knowledge resources to legitimate what counts as mathematics in a school classroom context. We work with a social epistemology, and thus understand that what comes to count as mathematics in any pedagogical practice (such as in school) is a function of the inner workings of pedagogic discourse (Bernstein, 1996). In other words, mathematical knowledge is shaped by the institutions of schooling and curriculum and by the activity of teaching within these. In this sense, professional knowledge in use in practice needs to be understood as shaped by pedagogic discourse. Consequently, a methodology for ‘seeing’ knowledges in use in teaching requires a theory of pedagogic discourse. An underlying assumption in QUANTUM, following Davis (2001), is that pedagogic discourse (in both teacher education and school) proceeds through the operation of pedagogic judgement. As teachers and learners interact, criteria will be transmitted of what counts as the object of learning (e.g. what an ‘equation’ is in mathematics) and how the solving of problems related to this object is to be demonstrated (what are legitimate ways of knowing, working with and talking about equations). As teachers provide opportunities for learners to engage with the intended object, at every step they make judgements as to how to respond to learners, what to offer next and how long to pursue a particular activity. As Davis argues, all pedagogic judgements transmit criteria for what counts as mathematics. For example, in many South African classrooms, learners can be heard describing the steps in solving a linear equation as follows: to ‘solve for x’ in 3x − 7 = 5x + 11, learners say ‘We transpose or take the xs to one side and the numbers to the other side’. The teacher in this case could judge this expression as adequate, as reflecting shared procedural meaning in the classroom; alternatively, the teacher could judge the description as unclear; the language used does not refer adequately to the objects (algebraic terms) being operated on and also potentially misleading from a mathematical point of view. The teacher could then question the learner as to the specific meaning of ‘transpose’ or ‘take’ as the learner is using it, probing so as to transmit more mathematical criteria for the transformation of the

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equation, and in particular, the operation of adding additive inverses. In this latter case, through responses learners provide, and further questioning, the teacher then negates (even if only implicitly) the first description by legitimating mathematically justified steps offered. In this interaction process, the criteria transmitted are that steps for solving equations require mathematical justification. In QUANTUM we describe these moments of judgements as appeals, arguing that teachers appeal to varying domains of knowledge to legitimate what count as valid knowledge in their classrooms. What comes to count as valid is never neutral (Bernstein, 1996).3 Pedagogic discourse necessarily delocates and relocates knowledges and discourses, and recontextualisation (transformation) creates a gap wherein ideology is always at play. What teachers recruit is thus no simple reflection of what they know. An underlying assumption here is that the demands of teaching in general, and the particular demands following changes in the mathematics curriculum in South Africa, bring a range of domains of knowledge outside of mathematics into use. A range of mathematical orientations are discernable in the new South African National Curriculum, including mathematics as a disciplinary practice, thus including activity such as conjecturing, defining and proof; mathematics as relevant and practical, hence a modelling and problem-solving tool; mathematics as an established body of knowledge and skills, thus requiring mastery of conventions, skills and algorithms; and mathematics as preparation for critical democratic citizenship, and hence a use of mathematics in everyday activity (Graven, 2002; Parker, 2006). What mathematical and other knowledge resources teachers select and use, and how these are shaped in pedagogical discourse, are important to understand. In our case studies of school mathematics teaching, we are studying what and how teachers recruit mathematical and other knowledge resources in their classroom practice so as to be able to describe what comes to function as ground in their practice, how and why. Five case studies of mathematics teaching in a secondary classroom have been completed, each involving a different topic and unit of work.4 We pursued a range of questions, the first of which was, from what domains of knowledge does the teacher recruit knowledge resources in her teaching? I focus here on this question, and its elaboration in two of the five case studies, cognisant that as knowledge in use come into focus, so other resources, as well as details on other aspects of teaching, go out of focus.

3 In this chapter I do not explore the ideological or political in the constitution of mathematics in and for teaching. We have done this elsewhere, particularly in our reporting of the constitution of mathematics for teaching in teacher education (see Adler & Davis, 2011). 4 Studies in school classrooms have been undertaken by master’s students and a postdoctoral fellow at the University of the Witwatersrand, working in QUANTUM. I acknowledge here the significant contribution of Mercy Kazima, Vasen Pillay, Talasi Tatolo, Shiela Naidoo and Sharon Govender and their studies to the overall work in QUANTUM, and specifically to this chapter.

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1.4 Evaluative Events, Criteria at Work and Knowledge Resources in Use As is described in more detail elsewhere (Adler, 2009; Adler & Davis, 2006; Davis, Adler, & Parker, 2007), our methodology is inspired by the theory of pedagogic discourse developed by Basil Bernstein, and its illumination of the ‘inner logic of pedagogic discourse and its practices’ (Bernstein, 1996, p. 18). Any pedagogic practice, either implicitly or explicitly, ‘transmits criteria’; indeed this is its major purpose. What is constituted as mathematics in any practice will be reflected through evaluation, through what and how criteria come to work.5 How then are these criteria to be ‘seen’? The general methodology draws from Davis (2005) and the proposition that in pedagogic practice, in order for something to be learned, to become ‘known’, it has to be announced in some form. Initial orientation to the object, then, is through some (re)presented form. Pedagogic interaction then produces a field of possibilities for the object. Through related judgements made on what is and is not the object, possibilities (potential meanings) are generated (or not) for/with learners. All judgement, hence all evaluation, necessarily appeals to some or other locus of legitimation to ground itself, even if only implicitly. An examination of what is appealed to and how appeals are made (i.e. how ground is functioning) delivers up insights into knowledge resources in use in a particular pedagogic practice.6 Following the linear equation example above, if the teacher probes for or indeed inserts the notion of additive inverses, then he or she is appealing to mathematical discourse and recruiting resources from the mathematical domain. If, however, the teacher proceeds with everyday terms such as move, take over or transpose, then the grounds functioning are non-mathematical. Where appeals to the everyday dominate, and the sensible comes to overshadow the intelligible, potential mathematical meanings for learners might well be constrained (see Davis et al., 2007). Of course, what teachers appeal to is an empirical question. Our analysis to date has revealed four broad domains of knowledge to which the teachers across all cases appealed (though in different ways and with different emphases) in their work: mathematical knowledge, everyday knowledge, professional knowledge7 and curriculum knowledge. Teachers, in interaction with learners, appealed to the domain of mathematics itself, and more particularly school mathematics. We have described, a posteriori, four categories of such mathematical knowledge and/or activity that, in turn, are resonant of the multiple mathematical orientations in the current South African curriculum as discussed above: 5 It is important to note this specific use of ‘evaluation’ in Bernstein’s work. It does not refer to assessment nor to an everyday use of judgement. Rather it is a concept for capturing the workings of criteria for legitimation of knowledge and knowing in pedagogical practice. 6 This set of propositions is elaborated in Davis et al. (2003), as these emerged through collaborative work in QUANTUM. 7 In Adler (2009), everyday knowledge and professional knowledge are collapsed, both viewed as knowledge from practical experience. The separation comes from the development of this chapter.

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• mathematical objects have properties, mathematical activity follows conventions (e.g. in an ordered pair, we write the x co-ordinate first); • mathematical knowledge includes knowledge of (justifiable) procedures, mathematical activity is following rehearsed procedures (e.g. the first step to add two proper fractions is finding a common denominator); • mathematical justification can be empirical (e.g. testing whether a mathematical statement is true by examining an instance – substituting particular numbers or generating a particular visual display); • mathematical argument or justification involves generalising and proving (e.g. examining whether a statement is always true). The second domain of knowledge to which teachers appealed was nonmathematical and is most aptly described as everyday knowledge and/or practice. Across the data, teachers appealed to sensible, that is practical or experiential, knowledge to legitimate or ground the object being attended to.8 For example, the likelihood of events was discussed in relation to the state lottery, or obtaining a ‘6’ when throwing dice; simplifying algebraic expressions (e.g. 2x + 3y − 3x + 2y) was exemplified by grouping similar material objects (two apples, three bananas, etc.); in a task that required students to cut up a fraction wall containing a whole, halves, thirds, quarters, fifths, etc., up to tenths, and then reorganise/mix the fraction pieces and make wholes from different unit fractions, some students pasted pieces that together formed more than a whole. The teacher’s explanation as to why this was inappropriate was grounded in the way bricks are cemented to form walls. Connecting, or attempting to connect, mathematical ideas to everyday knowledge and experience is a topic of considerable interest, indeed concern in mathematics education in South Africa, where the goals of application, modelling and critical citizenship in the curriculum have produced a prevalence of such discourse in many classrooms. What is critical, of course, is that whatever is recruited extra-mathematically needs to connect with learners’ meaning-making while simultaneously holding the integrity of the intended mathematical idea. A third domain is teachers’ own professional knowledge and experience: what they have learned in and from practice. For example, all five teachers called on their knowledge from practice of the kinds of errors learners make and built on these in their teaching. Knowing about student thinking and misconceptions is a central part of what Shulman (1986) termed pedagogic content knowledge (PCK), and its centrality in teachers’ practice is well described in Margolinas (in Gueudet & Trouche, 2010). There are two inter-related sources for practice-based knowledge: the teacher’s own personal experience and the accumulated knowledge from research in mathematics education, that is from research on practice beyond the individual teacher. In this chapter I refer only to the former, which we have called experiential knowledge.

8 In our description of ground, we are not concerned with their mathematical correctness or whether they are appropriate. Our task is to describe what teachers recruit, whatever this is.

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Teachers’ appeals extended beyond the three domains discussed above to include what we still rather loosely call curriculum knowledge. In all our cases, and in some cases this was a significant resource for the teacher, the teacher appealed to the official curriculum, recontexualised in, for example, a textbook or an examination question. In other words, what counted as legitimate was based on exemplification or description in a textbook or what would count for marks in an examination (e.g. the definition of a polygon is that which is found in the textbook; the justification for why it is important to label axes and points on a graph is that these attract marks in an examination). Of interest is whether and how this legitimation is integrated with or isolated from any mathematical rationale. In the remainder of this chapter, I present two of the five cases to illustrate our methodology and to illuminate the knowledge resources in use in mathematics teaching.

1.5 Knowledge Resources in Use in School Mathematics Teaching The five case studies noted above have been described in detail elsewhere (Adler & Pillay, 2007; Kazima, Pillay, & Adler, 2008). The two selected for discussion here are telling: they present different approaches to learning and teaching mathematics, together with similar and different knowledge resources in use. In so doing, and akin to material resources, they problematise notions of professional knowledge that are divorced from practice and context, opening up questions for mathematics teacher education.

1.5.1 Case 1. Procedural Mathematics, Justified Empirically, Sensibly and Officially9 Nash,10 is an experienced and qualified mathematics teacher. He teaches across Grades 8–12 in a public school where learners come from a range of socioeconomic backgrounds. He has access to and uses curriculum documents issued by the National Department of Education (DoE), a selection of mathematics textbooks, a chalkboard and an overhead projector. He collaborates with other mathematics teachers in the school, particularly for planning teaching and assessment. He is well respected and regarded as a successful teacher in his school and in the district. In this case study, Nash was observed teaching linear functions to a Grade 10 class. His approach to teaching can be typically described: he gave explanations from the chalkboard; learners were then required to complete an exercise sheet he

9

For a detailed account of this study, see Pillay (2006) and Adler and Pillay (2007). This is a pseudonym.

10

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prepared. He did not use a textbook nor did he refer his learners to any textbook during the lessons observed. A six-page handout containing notes (e.g. parallel lines have equal gradients), methods (steps to follow in solving a problem) and questions (resembling that of a typical textbook) formed the support materials used. This handout was developed by Nash in collaboration with his Grade 10 teaching colleagues.11 In the eight lessons observed, Nash dealt with the notion of dependent and independent variables, the gradient and y-intercept method for sketching a line, the dual intercept method, parallel and perpendicular lines, determining equations of straight lines when information about the line is given in words and also in the form of a graph and solving simultaneous linear equations graphically. He completed the unit with a class test. The overall pass rate was 94%, class average was 65% and 34% obtained over 80%. Of course, success is relative to the nature of the test and the pedagogy of which it forms part. The test questions were a replica of questions in the handout given to learners and so a reproduction of what had been dealt with in class. In the first two lessons, Nash dealt with drawing the graph of a linear equation first from a table of values, and then using the gradient and y-intercept method. In Lesson 3, he moved on to demonstrate how to draw the graph of the function 3x – 2y = 6, using the dual intercept method. The extract below is from the discussion that followed. It illustrates an evaluative event, the operation of pedagogic judgement in this practice and the kinds of knowledge resources Nash recruited to ground, and as grounds for, the dual-intercept method for graphing a linear function. The beginning of the event – the (re)presentation of the equation 3x − 2y = 6 – is not included here. Extract 1 picks up from where Nash is demonstrating what to do. The appeals – moments of judgement – are underlined, and related grounds described. Judgments in this extract emerge in the interactions between Nash and four learners who ask questions of clarification, thus requiring Nash to recruit resources to ground and legitimate what counts as mathematical activity and so mathematical knowledge in this class. Learners’ questions were of clarification on what to do, suggesting they too were working with procedural grounds. There were possibilities for mathematical justification and engagement, for example why only two points are needed to draw the graph and how the direction of the graph is determined. However, these are not taken up and the grounds offered remain empirical – in what can be ‘seen’. Here the dual-intercept method is the simplest because it is accurate. It avoids errors that come with changing the equation into ‘standard form, that is y = mx + c. To ‘do’ the dual-intercept method, you use the intercepts on the axes, that is when x = 0 and when y = 0. You need only these two points. They determine the shape of the graph.

11

This documentation practice, unfortunately in the light of this book, was not in focus in our research.

You don’t need all the other parts? You don’t have to put down the other parts . . . its useless having −6 on the top there (points to the y axis) what does the −6 tell us about the graph? It doesn’t tell us much about the graph. What’s important features of this graph . . . we can work out . . . from here (points to the graph drawn) we can see what the gradient is . . . is this graph a positive or a negative?

(chorus) positive. it’s a positive gradient . . .we can see there’s our y-intercept, there’s our x-intercept (points to the points (0;−3) and (2;0) respectively)

Lr 1: Nash:

Lrs: Nash:

(in the next minute, a learner asks about labelling of points, and Nash responds with emphasis on the marks such labelling attracts in examinations

. . . first make your x equal to zero . . . that gives me my y-intercept. Then the y equal to zero gives me my x-intercept. Put down the two points . . . we only need two points to draw the graph

Nash:

Extract 1. Lesson 3, Case 1. (Lr = learner)

Mathematical conventions are official – those expected in the examination

Grounds: curriculum knowledge.

Mathematical activity is procedural and properties justified empirically

Grounds: empirical Important features of a graph are what can be ‘seen’

There is no justification for only needing two points. Nash might understand the geometry theory here, but this is simply asserted The assertion is questioned by L1, and the theory not followed. Rather, an empirical explanation is given

Grounds: procedural. Steps to follow are described, and justified mathematically.

Knowledge resources in use

12 J. Adler

Nash: Learner 4: Nash:

Lr 2:

Nash: Lr 2: Nash:

The simplest method and the most accurate . . . Compared to which one? Compared to that one (points to the calculation of the previous question where the gradient and y-intercept method was used) because here if you make an error trying to write it in y form . . . that means it now affects your graph . . . whereas here (points to the calculations he has just done on the dual intercept method) you can go and check again . . . you can substitute . . . if I substitute for 2 in there (points to the x in 3x – 2y = 6) I should end up with 0

Mathematical activity demands accuracy and is error free

Grounds: avoiding error

Mathematics is procedural

Explanation focuses on how you get the correct gradient by following the steps. Grounds: procedural

Grounds: procedural

Further procedural question

Lr 2: Lr 3:

Sir, is this the simplest method sir? How do you identify which side must it go, whether it’s the right hand side (Nash interrupts) (response to Lr 2) You just join the two dots That’s it? Yeah . . . the dots will automatically . . . if it was a positive gradient it will automatically . . . if this was (refers to the line just drawn) negative . . . that means this dot (points the x-intercept) will be on that side (points to the negative x axis) . . . because if the gradient was negative, how could it cut on that side? (points to the positive x axis). Is this the simplest method sir?

Knowledge resources in use

Extract 1. Lesson 3, Case 1. (Lr = learner)

1 Knowledge Resources in and for School Mathematics Teaching 13

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In this event, Nash’s responses were about what to do. Legitimation was provided by steps to follow or what could be ‘seen’. Appeals were to procedural knowledge, to some empirical feature of the object being discussed or to curriculum knowledge (what counts in the examination). This event, and the operation of pedagogic judgement, is typical of how Nash conducted his teaching of this particular set of lessons. Table 1.1 summarises the full set of 65 events across the eight lessons, and the knowledge resources Nash recruited. As indicated above and in the numbers in the table, more than one kind of knowledge resource could be called on within one event. Nash’s appeals to everyday knowledge and his professional experience were not evidenced in this event. Briefly, his recruiting of everyday knowledge, which were to add meaning for learners, was often problematic from a mathematical point of view. For example, he attempted to explain independent and dependent variables by referring to a marriage, husband and wife and expressed amusement and concern when discussing this in his postlesson interview! Table 1.1 Case 1, linear functions, grade 10 Total occurrences

% Occurred

Events

65

Appeals/knowledge resources Mathematics Empirical Procedures/conventions

24 43

37 66

18 14

28 22

6 7

9 11

Experience

Professional Everyday

Curriculum

Examinations/tests Text book

In overview, mathematical ground in this set of lessons was procedural, with justification empirical, sensible and official. Nash recruited from the domains of mathematical, professional and curriculum knowledge. That these latter are key in Nash’s practice were reflected in his post-lesson interview. Nash talked at length about how he plans his teaching, key to which is a practice he calls ‘backwards chaining’. First and foremost when you look[ing] at the topic/my preferred method is . . . backwards chaining. [which] means the end product. What type of questions do I see in the exam, how does this relate to the [Gr 12] exams, similar questions that relate to further exams and then work backwards from there . . . what leads up to completing a complicated question or solving a particular problem and then breaking it down till you come to the most elementary skills that are involved; and then you begin with these particular skills for a period of time till you come to a stage where you’re able to incorporate all these skills to solve a problem or the final goal that you had.

He also illuminated how his experience factors into his planning and teaching, and his attention to error-free mathematics. Learners’ misconceptions and errors are a teaching device – and in the context of the perspective of this book – a resource in his teaching. They are not a feature of what it means to be mathematical.

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You see in a classroom situation . . . you actually learn more from misconceptions and errors . . . than by actually doing the right thing. If you put a sum on the board and everybody gets it right, you realise after a while the sum itself doesn’t have any meaning to it, but once they make errors and you make them aware of their errors or . . . misconceptions – you realise that your lessons progress much more effectively . . . correcting these deficiencies . . . these errors and misconceptions.

1.5.2 Case 2. Mathematical Activity as Conjecture, Counterexample and Proof12 Ken13 is also an experienced and qualified mathematics teacher. He has a 4-year higher diploma in education majoring in mathematics, an honours degree in Mathematics Education and at the time of the data collection was studying for his master’s degree. He has thus had opportunity to learn from the field of mathematics education research. He has 11 years’ secondary teaching experience across Grades 8–12. The conditions in his school are similar to those in Nash’s school, and gradelevel teachers similarly prepare support materials and assessments for units of work. Ken too is well respected and successful in his school. Ken prepared and presented a week’s work focused on polygons; the relationship between its sides, vertices and diagonals; generalisation and proof to his Grade 10 class. He described his plans for the lessons as a set of ‘different’ activities to ‘revise’ and enable learners to reflect more deeply on geometry. The five lessons were organised around two complex, extended tasks. The first involved the relationship between the number of sides of a polygon and its diagonals. The second was an applied problem requiring learners to interpret a situation and recognise the need for using knowledge of equal areas of parallelograms on the same base and with the same height to solve the problem. The extract below is from the first of the five lessons and the initial work on the first task: learners were to find the number of diagonals in a 700-sided polygon, a sufficiently large number to require reasoning and generalising activity. The extract captures an evaluative event, with the presentation of the task marking the beginning of the event. It continues for 14 min as the teacher and learners interact on what and how they could make a conjecture towards the solution to the problem. Some progress is made, as learners are pushed to reflect on specific empirical cases. As with extract 1, the underlined utterances illustrate the kinds of appeals and so knowledge resources Ken recruits in his practice. All judgements towards the object – a justified account of the relationship between the number of sides and diagonals in a polygon – emerge from utterances of either or both learners and the teacher.

12 13

For detailed account of this study, see Naidoo (2008). This is a pseudonym.

Ok! Guys, time’s up. Five minutes is over. Who of you thinks they solved the problem? . . . . I just divided 700 by 2. You just divided 700 by 2. Sir, one of the side’s have, like a corner. Yes . . . (inaudible), because of the diagonals. Therefore two of the sides makes like a corner. So I just divided by two . . . (Inaudible). So you just divide the 700 by 2. And what do you base that on? So what do you base that on because there’s 700 sides. So how many corners will there be if there’s, 700 sides?

Lr2:

Ken:

Ken: Lr2:

Let’s hear somebody else opinion Sir what I’ve done sir is . . .First 700 is too many sides to draw. So if there is four sides how will I do that sir? Then I figure that the four sides must be divided by two. Four divided by two equals two diagonals. So take 700, divide by two will give you the answer. So that’s the answer . . . So you say that, there’s too many sides to draw. If I can just hear you clearly; . . . that 700 sides are too many sides, too big a polygon to draw. Let me get it clear. So you took a smaller polygon of four sides and drew the diagonals in there. So how many diagonals you get? In a four sided shape sir, I got two

[. . .] there is discussion about 700 sides and corners, whether there are 350 or 175 diagonals

Ken:

Lr 1: Ken: Lr 1:

Ken:

The class begins with Ken (standing in the front of the class), placing the following problem onto the Overhead Projector: How many diagonals are there in a 700-sided polygon? The students are asked to work on it for 5 min. After 7 min, Ken calls the class’ attention, and the interaction below follows:

Extract 2. Lesson 1, Case 2

L2 grounds his conjecture empirically, pragmatically and procedurally

L1s response is procedural. Following a challenge from the teacher, the grounds extend to include perceived properties of the mathematical object. Again this is challenged by the teacher

Knowledge resources in use

16 J. Adler

Yes, I don’t want to confuse myself So you don’t want to confuse yourself. So you’re happy with that solution, having tested only one polygon? Inaudible response . . . What about you Lr4? You said you agree. He makes sense. (referring to Lr1). . .He proved it. . . . He used a square. He used a square? Are you convinced by using a square that he is right? But sir, here on my page I also did the same thing. I made a six-sided shape and saw the same thing. Because a six thing has six corners and has three diagonals.

Lr2: Ken:

Interaction continues. Ken intervenes as he hears some confusion between polygon and pentagon, and turns the class’ attention to definitions of various polygons having learners look up meanings in their mathematics dictionaries

So what about a five-sided shape? Then sir

What about a five-sided shape? You think it would have five corners? How many diagonals?

Lr1:

Ken:

Lr2: Ken: Lr4: Ken: Lr5:

These grounds are again challenged by Ken

Two. So you deduced from that one example that you should divide the 700 by two as well? So you only went as far as a four sided shape? You didn’t test anything else?

Ken:

Mathematical activity involves reasoning; providing examples and counterexamples Mathematical objects have properties and are defined

Grounds functioning in this interaction remain empirical and include counterexamples

Learners first confirm with an additional example – six sides, then ask about five sides, and Ken picks up on this additional empirical case and counterexample

Learners ground responses in the empirical and sensible Challenge to the empirical ground and single case

Ken challenges the empirical ground and single case

Knowledge resources in use

Extract 2. Lesson 1, Case 2

1 Knowledge Resources in and for School Mathematics Teaching 17

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The discussion and clarification of different polygons continued for some time, after which Ken brought the focus back on to the problem of finding the number of diagonals in a 700-sided figure, and work on this continues through the rest of this lesson and the next two lessons. It is interesting to note that in all the discussion on the 700-sided figure, the empirical instances discussed, and the diagrams made public, a polygon is assumed to be regular and convex. Properties discussed focus on the number of sides and related number of angles in a polygon (again regular and convex), and a diagonal is defined as a line connecting two non-consecutive corners. One route to solving the problem – noticing a relationship between the number of corners and the number of diagonals from each corner – and so the possibility of a general formula becomes dominant. It is interesting too that the term ‘vertex’ is not used, and the everyday word ‘corner’ persists in the discussion. Ken’s focus throughout the two lessons is on conjecture, justification, counterexample and proof as mathematical processes. A shared understanding of the mathematical object itself – a polygon and its diagonals as defined geometrically – through which these processes are to be learned is assumed. Judgements in this extract flow in interaction between Learners 1, 2, 4, 5 and the teacher. The knowledge resources called in fit within the broad category of mathematics. In particular, the ground for the teacher is reflected in his insistence on mathematical justification. However, these grounds are distinctive. The first appeal (Lr1) is to the empirical, a particular case that can be ‘seen’ (two of the sides makes like a corner) and a related procedure (I just divided by 2), followed by Ken’s challenge through an appeal to properties of a 700-sided polygon. The appeal of Lr2 is also to the empirical, to a special case (four sides), and this is supported by Lr4, and then by Lr5 (who did ‘the same thing’ with six sides). It is interesting to reflect here on what possible notion of diagonal is being used by Lr5. While there has been discussion on diagonals as connecting non-consecutive corners, it is possible Lr5 is considering only those that pass through the centre of the polygon. Ken does not probe this response, rather picking up on Lr1’s suggestion of a counterexample (what about a five-sided shape?), which is also an empirical case. The appeals by the teacher, as he interacts with, revoices and responds to learner suggestions, are to the meta-mathematical domain, and so providing the criterion that the justifications provided are not yet mathematically adequate – they do not go beyond specific cases. The grounds that came to function over the five lessons are summarised in Table 1.2. In sum, a range of mathematical grounds (with empirical dominant, and including appeals to mathematics as generalising activity) overshadowed curriculum knowledge, with everyday knowledge barely present. In the pre-observation interview, Ken explained that his intention with the lessons he had planned was to focus on the understanding of proofs. He wanted them to see proof as ‘a way of doing maths, getting a deeper understanding and communicating that maths to others’. In the postlessons interview, interestingly, Ken explained that these lessons were not part of his normal teaching. He used the research project to do what he thought was important, but otherwise did not have time for. He nevertheless justified this inclusion in terms of the new curriculum, which had a strong emphasis on proof, on ‘how to prove and

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Table 1.2 Case 2, geometric thinking, grade 10 Total occurrences

% Occurred

Events

37

Appeals/knowledge resources Mathematics Empirical General Procedures/conventions

23 14 8

64 36 23

0 2

0 5

11 0

32 0

Experience

Professional Everyday

Curriculum

Examinations/tests Text book

what makes a proof’. When probed as to why he did not do this kind of lesson in his ‘normal’ teaching, he explained that there was shared preparation for each grade, and ‘because of time constraints and assessments, you follow the prep and do it, even if you don’t agree’.

1.6 The Significance of Knowledge Resources in Use in Practice In the introductory sections of this chapter, I argued that the knowledge resources teachers recruit in their practice are important. Earlier research has suggested that teachers’ professional knowledge was a significant factor in the relationship between teachers and curriculum materials, and particularly so in contexts of poverty. Where curriculum resources are minimal, the insertion of new texts critically depends on what and how teachers are able to use mathematics and other knowledge domains appropriately for their teaching. By implication, a study of curriculum text as ‘lived’ needs to foreground knowledge resources in use. This chapter has offered a methodology – structured by evaluative events and criteria in use to ground objects of learning and teaching – for illuminating knowledges in use. It contributes to the overall perspective offered in this book – a perspective that problematises the interactions between teachers and the resources drawn on in their professional activity. The methodology was put to work in two classrooms, enabling a description of the knowledge resources two teachers who were teaching different topics recruited to ground the mathematics they were teaching. Together with the mathematical domain, and particularly procedural mathematical knowledge, Nash drew on extra-mathematical domains of knowledge, particularly curriculum knowledge and everyday knowledge. Ken drew largely from the meta-mathematical domain. The knowledge resources that sourced the work of these two teachers were substantively different, and so too was the mathematics that came to be legitimated in these classrooms.

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As he explained, Nash backward chained from valued school knowledge reflected in national examinations and built in teaching strategies to elicit errors from learners that he could then correct, and he did this by focusing on procedural knowledge and what is empirically verifiable. This practice produces student ‘success’, though, in Ruthven’s terms, he could be described as following a mathematically constrained script and activity format (see Chapter 5). Ken, on the other hand, uses mathematics in extended ways to engage learners in reasoning practices like conjecturing leading to proof. What is not available here, of course, is the knowledge resources Ken might recruit if he were teaching linear functions, and similarly whether the script in Nash’s class is uniform across topics. We could surmise from Ken’s interview and his ‘confession’ that the observed lessons were done outside of his normal teaching, that grounds different to what we have seen in this episode might well function in his ‘normal’ classes. These teachers’ intentions, and what else they might do, are not at issue here. The object of QUANTUM’s research is not on what a particular teacher does or does not do, in some decontextualised sense, but rather on what comes to be used, and thus how mathematics is constituted in specific practices. Through the cases in this chapter, we see that observing teachers in practice is a window into the varying knowledge resources in use within a particular curriculum practice and set of institutional constraints. These insights were ‘revealed’ through the notion of ‘ground’ as that which is recruited to legitimate what counts as mathematics in teaching. The methodological tools developed in the QUANTUM project probe beneath surface features of pedagogic practice to reveal substantive differences in the way teachers recruit and ground knowledge objects as they go about their mathematical work, and so into how knowledges become ‘lived’ resources.

1.7 In Conclusion: Some Questions for Professional Development Activity In this chapter we have described two teachers’ practices in their mathematics classrooms. Nash and Ken teach in similarly resourced schools, and in a similar policy context. They recruited different knowledge resources, and thus different opportunities for learning mathematics were opened up in their classrooms. The methodology we have used enables us to understand and think about what might support expansion of the potential meanings these two teachers open up in their classrooms. Nash’s practice and his talk about this in his interview reveal the value he places on the high status official curriculum. This suggests possibilities for productive work and reflection with Nash on his privileging of the official curriculum, and how this shapes the ground functioning in his classroom in his teaching reported here. Ken, on the other hand, might benefit more from an investigation of the integration of meta-mathematical knowledge into his teaching more generally.14

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This challenge for teacher education is explored more directly elsewhere (see Adler, 2010) where I problematise the teaching of mathematical reasoning, and its implications for teacher education.

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In QUANTUM, our overall goal has been to ‘see’ across sites of practice (teacher education and school). We have studied pedagogic discourse and the constitution of mathematics for teaching in teacher education sites as well as the school classrooms illuminated in this chapter. For, if we are to improve mathematics teacher education, we need to understand what potential meanings are opened and closed in and across these sites, and how those emerging in teacher education relate to those emerging as dominant school practices. In the introductory section of this chapter, I asserted that the methodology described would be useful for studying the evolution of knowledge resources in use across contexts, and that this was particularly important in contexts of limited material resources. It is certainly useful in our current work where we are studying teachers’ practices over time, with an interest in whether and how professional development interventions focused on aspects of content knowledge in and for teaching relate to knowledges and other resources in use in practice. Acknowledgements This chapter emerges from the QUANTUM research project on Mathematics for Teaching, directed by Jill Adler, at the University of the Witwatersrand (Wits) with Dr Zain Davis, University of Cape Town, as co-investigator. The methodology described here was developed through joint work in mathematics teacher education. The elaboration into classroom teaching was enabled by the work of master’s students at Wits. This material is based upon work supported by the National Research Foundation (NRF), Grant number FA2006031800003. Any opinion, findings, conclusions or recommendations expressed here are those of the author and do not necessarily reflect the views of the NRF.

References Adler, J. (2000). Conceptualising resources as a theme for teacher education. Journal of Mathematics Teacher Education, 3, 205–224. Adler, J. (2001). Re-sourcing practice and equity: A dual challenge for mathematics education. In B. Atweh, H. Forgasz, & B. Nebres (Eds.), Sociocultural research in mathematics education: An international perspective (pp. 185–200). Mahwah, NJ: Lawrence Erlbaum Associates. Adler, J. (2009). A methodology for studying mathematics for teaching. Researchers en Didactique des Mathematiques, 29(1), 33–57. Adler, J. (2010). Mathematics for teaching matters. Education as Change, 14(2), 123–135. Adler, J., & Davis, Z. (2006). Opening another black box: Researching mathematics for teaching in mathematics teacher education. Journal for Research in Mathematics Education, 37, 270–296. Adler, J., & Davis, Z. (2011). Modelling teaching in mathematics teacher education and the constitution of mathematics for teaching. In K. Ruthven, & T. Rowland (Eds.), Mathematical knowledge in teaching (pp. 139–160). Dordrecht: Springer. Adler, J., & Pillay, V. (2007). An investigation into mathematics for teaching: Insights from a case. African Journal of Research in SMT Education, 11(2), 87–108. Adler, J., & Reed, Y. (Eds.). (2002). Challenges of teacher development: An investigation of takeup in South Africa. Pretoria: Van Schaik. Bernstein, B. (1996). Pedagogy, symbolic control and identity: Theory, research and critique. London: Taylor and Francis. Cohen, D. K., Raudenbush, S. W., & Ball, D. L. (2003). Resources, instruction, and research. Educational Evaluation and Policy Analysis, 25, 119–142. Davis, Z. (2001). Measure for measure: Evaluative judgement in school mathematics pedagogic texts. Pythagoras, 56, 2–11.

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Davis, Z. (2005). Pleasure and pedagogic discourse in school mathematics: A case study of a problem-centred pedagogic modality, Unpublished PhD dissertation, University of Cape Town, Cape Town. Davis, Z., Adler, J., Parker, D., & Long, C. (2003). Elements of the language of description for the production of data. QUANTUM Research Project, Working paper #2, Johannesburg: University of the Witwatersrand. Davis, Z., Adler, J., & Parker, D. (2007). Identification with images of the teacher and teaching in formalized in-service mathematics teacher education and the constitution of mathematics for teaching. Journal of Education, 42, 33–60. Graven, M. (2002). Coping with new mathematics teacher roles in a contradictory context of curriculum change. Mathematics Educator, 12(2), 21–27. Gueudet, G., & Trouche, L. (Eds.). (2010). Ressources vives. Le travail documentaire des professeurs en mathématiques. Lyon: Presses Universitaires de Rennes & INRP. Kazima, M., Pillay, V., & Adler, J. (2008). Mathematics for teaching: Observations from two case studies. South African Journal of Education, 28(2), 283–299. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press. Naidoo, S. (2008). Mathematical knowledge for teaching geometry to Grade 10 learners. Unpublished Masters Research Report. Johannesburg: University of the Witwatersrand. Parker, D. (2006). Grade 10–12 mathematics curriculum reform in South Africa: A textual analysis of new national curriculum Statements. African Journal of Research in SMT Education, 10(2), 59–73. Pillay, V. (2006). An investigation into mathematics for teaching: The kind of mathematical problem-solving teachers do as they go about their work. Unpublished Masters Research Report. Johannesburg: University of the Witwatersrand. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

Chapter 2

Teachers’ Work with Resources: Documentational Geneses and Professional Geneses Ghislaine Gueudet and Luc Trouche

Chapters 2 and 16 constitute two connected components of the presentation of a theoretical approach focusing on phenomena central in this book: the interactions between mathematics teachers and resources, and their consequences for professional growth. We name it documentational approach of didactics (Gueudet & Trouche, 2009). We begin (Section 2.1) with a discussion of the elementary concepts of this approach: documentation work, teachers’ documentation, resource/document dialectics and documentational genesis. We then elaborate (Section 2.2) the methodology we use for studying teachers’ documentation and the data we have collected using this methodology over 2 years. In Section 2.3, we detail a case study, extracted from these data. Finally, we present the perspective on teachers’ professional growth yielded by this approach (Section 2.4). We pay particular attention to digital resources, the constituting factors of major evolutions.

2.1 The Documentational Approach of Didactics We begin our discussion of the elementary concepts of the approach we develop by explaining its theoretical roots.

2.1.1 Teachers’ Professional Activity and Professional Growth The approach we present here has a specific orientation to studying mathematics teachers’ activity and development. Informed by activity theory, firstly introduced by Vygotski (1978) and developed by Leont’ev (1979), we consider that the teacher’s activity is oriented by goals (the object of the activity). Moreover, it

G. Gueudet (B) CREAD, Université de Bretagne Occidentale, IUFM Bretagne site de Rennes, 35043 Rennes Cedex, France e-mail: [email protected]

G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_2,  C Springer Science+Business Media B.V. 2012

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must be studied as a social activity, which leads us to pay attention to its context: institution and different social groups. We are interested in the evolution, and factors of evolution, of the teachers’ professional activity. The studies about practising mathematics teachers sometimes separate their practice, their knowledge and their beliefs (Perrin-Glorian, DeBlois, & Robert, 2008). We consider here teachers’ professional growth as a joint evolution of these three aspects. Conceptualising the way the practice articulates with knowledge and beliefs is one of the aims of the theory we expose here. We do not separate knowledge and beliefs, because the boundary between both is often unclear; we use the expression of professional knowledge to refer to both and focus particularly on knowledge related to mathematical content. The reference to activity theory is also directly connected with our interest in mediation and mediating artefacts. However, we refer to resources rather than artefacts and discuss the reasons for this choice in the next section.

2.1.2 Resources and Documentation Work Adler (2000, Chapter 1) proposes a conceptualisation of resources, emphasising the variety and the broadness of the range of resources intervening in teachers’ professional activity. We retain here a similar conceptualisation and perspective: ‘It is possible to think about resource as the verb re-source, to source again or differently’ (Adler, 2000, p. 205). With this perspective, a resource can be an artefact, i.e. an outcome of human activity, elaborated for a human activity with a precise aim (Rabardel, 1995, Chapter 4). But resources exceed artefacts: For a teacher who draws on them in her activity, the reaction of a student, a wooden stick on the floor can also constitute resources. The teacher interacts with resources, selects them and works on them (adapting, revising, reorganising, etc.) within processes where design and enacting are intertwined. The expression documentation work encompasses all these interactions. We consider that documentation work is central in teachers’ professional activity. It pertains to all the facets of this activity: all the places, all the groups teachers are involved in. We also use the word documentation, which means, for us, both this work and its outcomes. Retaining a wide perspective on resources does not mean ignoring the specificities of different kinds of resources. The work presented here originates in an interest in digital resources and their consequences for the teaching of mathematics. Teachers download lesson plans and exercises texts on websites; they modify, combine several files, elaborate their own texts, share them, etc. Digital resources evidence the documentation work. More generally, material resources have a particular status, at least for the researcher, from a methodological point of view. The interaction between teachers and material resources can indeed be visible in some cases: written notes on a book, an answer to an email, modifications in a file, etc. Non-material resources are more difficult to capture; some of them are nevertheless determining, like interactions in class with students. Several chapters in this book emphasise the importance of these interactions: chapters about the use of

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resources (Chapter 11 evidences the importance of non-verbal interactions) as well as chapters considering the joint action of teachers and students (Chapters 3 and 11), the teacher–student interactions in the use of textbooks in particular (Chapter 12). These interactions constitute a specific kind of resource, in particular because they are likely to modify other resources, or the relation between the teacher and a given resource (an exercise text can be modified, because of a student’s reaction, for example). They are both resources and a part of the social dimension of the teachers’ professional activity.

2.1.3 The Resource/Document Dialectics and the Documentational Geneses The documentational approach draws on the instrumental approach, developed by Rabardel (1995) in cognitive ergonomics and then integrated into mathematics didactics (Guin, Ruthven, & Trouche, 2005). Rabardel distinguishes between an artefact, available for a given user, and an instrument, which is developed by the user, starting from this artefact, in the course of his/her situated action. These development processes, the instrumental geneses, are grounded, for a given subject, in the appropriation and the transformation of the artefact, to solve a given problem, through a variety of usage contexts. Through this variety of contexts, utilisation schemes of the artefact are constituted. A scheme (Vergnaud, 1998) is an invariant organisation of the targeted activity, which is structured by operational invariants, developed in various contexts met for the same class of situations. This approach also distinguishes, within the instrumental geneses, two intertwined processes: instrumentation (constitution of the schemes of utilisation of the artefacts) and instrumentalisation (by which the subject shapes the artefacts). The instrumental approach has mostly been used to study the consequences of technology-rich environment for the student learning, despite a growing interest for teachers in the educational research about technology in mathematics (Hoyles & Lagrange, 2010). We propose here a theoretical approach extending the scope of the instrumental approach. Moreover, it borrows from other research studies, about document management (Pédauque, 2006), which enlightens the evolutions brought by digital resources, and about curriculum material (Remillard, 2005, Chapter 6). The teacher, in her documentation work, for a given class of situations, draws on a set of resources of various nature. Introducing a new vocabulary, we consider that this set of resources bears, for this class of situations, a document, within a documentational genesis (Fig. 2.1). The documentational genesis jointly develops a new resource (made up of a set of resources selected, modified and recombined) and a scheme of utilisation of this resource. We can represent this process, in a very simplified way, by the following equation: Document = Resources + Scheme of utilisation The document can be considered as an outcome of the subject’s activity. This static equation must nevertheless not obscure the fact that geneses are dynamic;

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Fig. 2.1 Schematic representation of a documentational genesis

they are ongoing processes: a document comprises resources, which can be associated with others and involved in the development of other documents. A scheme of utilisation is an invariant organisation of the activity to achieve a type of task; however, it can evolve in the course of the documentation work. It can be adapted to take into account new features of the context; several schemes can be associated, etc. We illustrate our model with a first short example, drawing on a previous study. Sarah has taught mathematics from grade 6 to grade 9, in France, for 10 years. An important objective assigned by the official curriculum is to introduce students to rigorous proofs in the context of geometry. For the class of situations ‘designing and setting up the introduction to proof in geometry’, Sarah selects exercises in the textbook where the figures are coded (equality of lengths and right angles). She uses dynamic geometry software and with it elaborates coded figures. Her students write in their workbook ‘a property of a figure cannot be claimed from mere observation, if there is no coding symbolising this property’. She declares, in an interview, that her long experience in grade 9 classes has led her to pay attention to the difficulties raised by proof in geometry, especially difficulties linked with the use of figures. In this case, we consider that the teacher, in the course of her work and over several years, developed a document, comprising recombined resources: extract of the textbook, dynamic geometry software, etc. This document also entails a scheme of utilisation of these resources, with operational invariants like ‘the proof of a result in geometry must be associated with a coded figure’ and ‘a coded figure helps to identify the relevant properties for the proof’. We share with other authors in this book [in particular Adler (Chapter 1), Remillard (Chapter 6), and Pepin (Chapter 7)] a perspective considering that teachers ‘learn’ when choosing, transforming resources, implementing them, revising them, etc. The documentational approach proposes a specific conceptualisation of this learning, in terms of genesis. Documentation being present in all aspects of the teacher’s work, it yields a perspective on teachers’ professional growth as a complex set of documentational geneses. Understanding this growth requires a holistic view on these geneses, by considering all the documents developed by the teacher: her documentation system.

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2.1.4 Resource System and Documentation System Studying teachers’ documentational geneses evidences articulations between different documents. Naturally, the class of situations ‘designing and setting up the assessment corresponding to the cosine lesson’ is connected with ‘designing and setting up the introduction of cosine’: the objective of the teacher’s activity is different, but the mathematical content being the same, the same operational invariants are likely to intervene in the documents developed for each class. However, the mathematical dimension is not the only element accounting for articulation between documents. ‘Designing and setting up an assessment’ can also be considered as a class of situations, corresponding to a general aim of the teacher’s activity, connected with ‘managing the class and following the students’, another general aim. Within the framework of the instrumental approach, Rabardel and Bourmaud (2003) consider systems of instruments, whose structure depends on the structure of the subject’s professional activity. The classes of situations are articulated and organised, because the various aims can be more or less similar or linked. Drawing on this conceptualisation, we consider that the documents of a teacher are articulated in a structured documentation system. The resource system of the teacher constitutes the ‘resource’ part of her documentation system (i.e. without the scheme part of the documents). Ruthven (Chapter 5) also introduces a concept of resource system, belonging to the five key structuring features of classroom practice he identifies. What we consider here as resource system does not fully coincide with Ruthven’s definition, because of the broader meaning of resources we retained. The resource system comprises material elements, but also other elements that are more difficult to collect, like conversations between teachers. We have presented here the theoretical construct framing our research. The complex object we study also requires a specific methodology, connected with this theory.

2.2 Studying Documentation Work: Reflective Investigation We elaborated the theory and an associated methodology simultaneously in the development process of the documentational approach. We briefly present here this methodology that we named the reflective investigation of the teacher’s documentation work.

2.2.1 Methodological Principles The main principles grounding this methodology are as follows: – A principle of long-term follow-up. Geneses are ongoing processes and schemes develop over long periods of time. This indicates the need for long-term observation, within practical constraints.

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– A principle of in- and out-of-class follow-up. The classroom is an important place where the teaching elaborated is implemented. As mentioned above, these direct interactions with students are crucial resources for the teacher. They bring adaptations, revisions and improvisations, as Drijvers (Chapter 14) emphasises, distinguishing between an exploitation mode of a didactical configuration, planned by the teacher, and the didactical performance she realises in class. However, an important part of teachers’ work takes place beyond the students’ presence – at school, at home, in teacher development programs, etc. We pay attention to all these different locations. – A principle of broad collection of the material resources used and produced in the documentation work, throughout the follow-up. – A principle of reflective follow-up of the documentation work. We closely involve the teacher in the collection of data, with the pragmatic aim of broad collection and in-class and out-of-class follow-up previously discussed. The active involvement of the teacher yields a reflective stance (Schön, 1983). We built a data collection device, presented in the following section, corresponding to these principles.

2.2.2 Data Collection Tools The data collection we propose is planned to last several years; a teacher is followed at least 3 weeks each year. We detail here the schedule and the tools used. Figure 2.2 presents the overall agenda of the yearly follow-up. During the first year, the teacher fills in a logbook over at least 3 weeks, describing her activity relative to one of the classes she teaches. The researcher visits the teacher three times at home for interviews and collection of resources. He/she asks (during the first interview) the teacher to draw a schematic representation of the structure of the resources she uses. We call it a schematic representation of the resource system (SRRS). An example of an SRRS is displayed in Fig. 2.4. During the following years, the teacher is still followed in a class of the same level, for the same mathematical content. The overall structure remains the same (Fig. 2.2), but the focus is much more on developments: the teacher is asked to bring the necessary modifications, to explain the changes, compared to the previous year about the questionnaire, the SRRS and during the first interview. We focus in this book on two teachers whom we followed for 2 years: 2008–2009 and 2009–2010.

2.2.3 Choice of Two Teachers The two teachers we followed teach in middle school (from grade 6 to 9). They have been selected with very different profiles, according to several dimensions that we assume as crucial for the topic of our study: they differ in terms of ICT

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First encounter:

First visit:

Second visit:

- presentation of the methodology, its spirit and its tools.

- about the resources in general;

- about the lesson observed.

Classroom observation

Third visit: - About the lesson observed; - Complements about the chapter and resources.

- about the chapter followed.

Tools:

Tools:

Tools:

Tools:

Tools:

- Schedule; - Questionnaire;

- Interview guidelines;

- Interview guidelines

- Observation guidelines

- Interview guidelines;

- Logbook.

- SRRS

- SRRS; - Collection of resources.

Week 1

Week 2

Week 3

Filling the logbook

Fig. 2.2 Agenda of the follow-up

degree of integration (Assude, 2007), of participation in collectives and of institutional responsibilities (Gueudet & Trouche, 2009). These teachers were neither at the beginning nor at the end of their career. Myriam (50 years in 2009) has a strong degree of ICT integration; she regularly takes part in in-service training; she took part in several IREM1 groups. Pierre (35 years in 2009) has a strong degree of ICT integration; he is responsible for ICT in his school and member of Sésamath2 association. This association gathers mathematics teachers; we describe it in more detail in Chapter 16. We mention here only several of the resources designed by the association which are accessible on its website, in particular the Sésamath textbook (which exists in digital and paper versions). As explained above, we followed each teacher in only one class. Moreover, we chose that class to capture phases of reorganisation of the documentation work. We thus followed Pierre in one grade 6 class with a data-processing speciality which brings an opening towards new forms of work (better equipped students, motivated 1 2

IREM, Institute for Research on Mathematics Teaching. http://www.sesamath.net/

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for out-of-class interactions). We followed Myriam in grade 9 on a chapter (functions) corresponding to a change in the official curriculum. In 2008–2009, it was the first time that she taught that topic. Myriam and Pierre are not chosen as ‘exemplar’ teachers. We do not aim at describing ‘good’ documentation work. Myriam and Pierre share a strong professional involvement; they spend probably more time on their out-of-class work than do the average mathematics teacher in France. This longer time is likely to evidence better the phenomena that we want to capture; however, we consider that these phenomena take place for every teacher. In this chapter, we detail the case of Myriam; the case of Pierre is studied in Chapter 16.

2.2.4 Analysing the Data Collected This data collection is followed by a data exploitation device, which comprises various aspects. We carry out a quantitative treatment of the logbook: length of the out-of-class and in-class work, places for this work, number of occurrences of a given activity, length, number of uses of a given resource, length, nature and number of collective work moments, list of implied participants, etc. For the interviews, we note in the same way the occurrences of the types of activity, resources and persons mentioned. The questionnaire provides us with concrete information about the teacher’s career and her current working environment. We also gain access, through the questionnaire, to elements of her professional and personal history, in particular in terms of family environment and collective involvements. We complement these first treatments with the SRRS to evidence elements of structure of the teacher’s activity and of her resources, systematically identifying moreover collective dimensions (Chapter 16). We identify, in the logbook and the interviews, all the elements relating to the lesson observed in class. We observe in the lesson’s transcript how the interactions between teacher, students and knowledge lead to adaptations of resources, during or after the lesson. We conduct a systematic comparison of the first-year and second-year data, quantitatively and qualitatively. In the next section, we present a case where we applied this data collection and analysis device.

2.3 A Case Study In this section we study the case of Myriam and of her teaching about functions in grade 9 to illustrate the concepts presented in Section 2.1.

2.3.1 Synthetic Description of Myriam’s Activity The synthetic description of Myriam’s activity during the follow-up (Fig. 2.3) corresponds to the year 2009–2010.

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Myriam introducing functions in grade 9 Myriam has one grade 9 class, with 20 students. The theme of functions was introduced into the grade 9 curriculum in 2008–2009. The official curriculum is divided between ‘core content of knowledge’, which every student should learn, and other contents. Functions do not belong to the core content. The students must obtain a diploma, ‘diplôme de brevet des collèges’, at the end of the year. This diploma comprises a computer certification ‘brevet informatique et internet’ (shortened as B2i*); Myriam is responsible for ensuring some of the corresponding skills.

Preparing the lesson about functions For the preparation of her lesson, Myriam uses several websites: Sésamath, and institutional websites. She also uses the classroom textbook and her personal notebook from the previous year. She finds on an institutional website an activity ‘the box’, where the students are asked to compute (with their calculator and then with a spreadsheet) the volume of a rectangular box for several values of the side x of squares, withdrawn on each corner to build the box.

She retains this activity and takes rough paper to propose to the students to build their own boxes. Introduction of functions and graphics She implements the ‘box’ problem in class and uses it to introduce the vocabulary and notations: function, image, antecedent, f(1)= 8, f: 1

8. The whole activity, with the spreadsheet, and the course synthesis last 3 h

(H1–H3). Myriam has observed, in 2008–2009, that many students failed to place a point given by its coordinates in a Cartesian coordinate system. Thus she presents during the fourth hour a mere placement activity, before introducing the notion of the graph of a function, during the fifth hour (H5), and presenting exercises about graphs (H6).

Fig. 2.3 Synthetic description of Myriam’s activity, introducing functions in grade 9, 2009–2010. ∗ http://www.educnet.education.fr/formation/certifications/b2i

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Work on exercises After the introduction of the different notions, vocabulary and representations, she presents her students a sheet with five exercises, coming from several sheets downloaded on the Sésamath website (H7). One exercise concerns rectangle areas; all the others are situated within extra-mathematical contexts. The students are organised in homogeneous groups. They have to write their solutions on a slide (this session is filmed and observed). Myriam expects everybody to succeed the two first exercises, which actually happens; she observes only some difficulties in the notation and vocabulary in the students’ productions, which are discussed and corrected the following day (H8). Other exercises are presented in H9 with e-exercises video projected and solved by the whole class, and in H10 with the calculator. Snow, email and spreadsheet The eleventh hour of the lesson was planned for Wednesday, January 6. But the snow begins to fall, and Myriam is blocked at home. On January 7, she comes to school, but only three students managed to reach it. She starts an exercise with them of the textbook about graphs, which requires the use of a spreadsheet. She is concerned about the following days, because school transport is cancelled. She decides to send the exercise by email to the students and asks them to solve the exercise and to send back the graph, drawn with the spreadsheet. Fifteen students send it, and it is later corrected in class. In February, Myriam gives a short test on functions. She is not very satisfied with the results: some students still use incorrect notations or are unable to properly read a table of values. She presents them with additional work with the spreadsheet that the students have to send to her by email.

Fig. 2.3 (continued)

In this description for the class of situations ‘designing and setting up the introduction of functions’, we observe that Myriam uses many material resources of various kinds; digital and non-digital resources are strongly intertwined. We give below examples of geneses which occurred in the course of this activity.

2.3.2 Resources, Documents and Geneses: The Case of Myriam We have selected, amongst all the geneses we can infer from our data, examples involving knowledge and resources which seem to be of particular importance for Myriam (mentioned on several occasions in the logbook, during the interviews, etc.). We also retained the example of a genesis corresponding to a new class of situations recently introduced in Myriam’s system of activity for material reasons.

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2.3.2.1 Mathematics as a Tool for Other Topics: Influence of an Operational Invariant Myriam’s professional knowledge and beliefs strongly influence her documentation work: her choice of resources, the way she associates them, etc. For example, Myriam is convinced that mathematics is a tool useful for other scientific topics: biology, physics, etc. An important factor for this belief is that Myriam is married to a physics teacher. The discussions with her husband are resources coming from a specific community, which re-source her practice. In the Sésamath exercises, as in the class textbook, she chooses many exercises related to biology or physics: this is an instrumentalisation process; her knowledge and beliefs guide the choice of resources and drive the teacher’s agency. In the exercises she presents, many different letters are used to symbolise functions and variables: not only f(x) but also h(t), d(v), etc. We consider that she has developed an operational invariant like ‘the students must be able to manipulate functions with different names, because they will be asked to do so in physics and biology’. 2.3.2.2 Official Texts as Resources Myriam cares a lot about official recommendations (national, in France). Every Friday, she reads the ‘official publication of national education’3 which presents the official curriculum, announces the dates of the exams, etc. She intervenes as an in-service teacher trainer,4 discusses with the regional inspectors, etc. She has read a lot of texts about the ‘core-content’ reform. The organisation she chooses for the exercise sessions we observed, with a homogeneous group, is directly related to this reform. One of the official texts that Myriam often uses describes such an organisation. This text is an important resource for Myriam (coming from the ‘official’ institution); it frames her choices in an instrumentation process. We separated here, somewhat artificially, the associated processes of instrumentation and instrumentalisation for the sake of clarity; both processes are nevertheless strongly intertwined, as illustrated in the following examples. 2.3.2.3 Students’ Productions as Resources Students’ productions constitute essential resources for teachers. Many of Myriam’s choices are grounded in observations of difficulties encountered by students in 2008–2009. She changes her introductory activity, as she tells us in the third interview: ‘I think that my starting activity this year was easier, more concrete [. . .] Choosing a good starting activity is very important, it determines the mood for the whole lesson [. . .] Last year the students rejected the notion of function, they said 3

http://www.education.gouv.fr/pid285/le-bulletin-officiel.html In France every teacher can propose in-service training, on a given subject, usually for 1 or 2 days. The regional inspectors select some of the propositions to constitute the yearly ‘training offer’. 4

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it was useless’. The starting activity of 2008–2009 was situated within the frame of geometry (perimeter of a parallelogram inscribed in a right-angled triangle) and required delicate modelling work to determine a rather simple formula. Moreover, all the students conjectured the formula without calculation using a GeoGebra dynamic figure, which contributed to their reluctance towards a formal proof. It led Myriam to change her introductory activity; we consider that it also produced an evolution of her operational invariants (instrumentation). 2.3.2.4 Distant Work with the Students and Development of a Document In 2008–2009, Myriam participated for the first time in the assessment of the computer certification (B2i, Fig. 2.3). For this reason, she had to ask the students to send her emails with attached files. She created a special email address (we can consider this process as instrumentation). In 2009–2010, the heavy snowfalls prevented the students from coming to school for almost 1 week. Myriam used a students’ mailing list to present homework in an instrumentalisation movement. She is only starting with such requests, so she does not give much attention to being precise about the name of the file to send back or its format. She did not yet develop a stable orchestration (Trouche, 2004; see also Chapter 14) for such situations. The students sent back files with non-significant names; some sent spreadsheet files, while others copied their graphs in a word-processing file – in this case, Myriam cannot see how they built their graphs. We consider that Myriam is developing a document for the class of situations ‘designing and setting up distant work about the graphs of functions’. The document has a ‘resource’ part, associating in particular the classroom textbook, a spreadsheet, email addresses for the teacher and the students, amongst others. Our observation took place at a moment of important evolutions, for this class of situations, linked with new digital means. We hypothesise that Myriam starts to develop operational invariants like ‘when asking the students to send spreadsheets productions, it is necessary to precise that the spreadsheet file itself must be sent, and not copied into a word processing document’ and ‘correcting spreadsheet productions requires access to the formula written in the spreadsheet’. With the data we gathered, we cannot claim that Myriam actually developed these operational invariants; further observations are necessary to confirm this hypothesis. We consider it nevertheless as consistent, being connected with a more general operational invariant: ‘correcting the students’ mathematical exercises requires access to their procedures’, which seems to intervene in many documents developed by Myriam. Beyond these examples, we are interested in capturing more generally Myriam’s (and other teachers’) professional growth; for this purpose, we need to consider the documentation system as a whole.

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2.4 Documentation Systems and Professional Geneses We discussed in Section 2.3 several examples of documentational geneses of instrumentation and instrumentalisation processes in the case of Myriam. In this section we first go back to this case, trying to capture the structure of Myriam’s documents, their evolutions and factors of evolution. Then we address more generally the issue of documentation systems and professional geneses, focusing especially on the evolutions brought by digital resources.

2.4.1 Myriam’s Documentation System and Its Evolutions Figure 2.4 presents Myriam’s representation of her resource system, drawn in 2009–2010. Myriam represents herself, and specific aspects of her documentation work (in particular her teaching project), in the centre of a space organised in four zones: – On the left and in the middle, her work at home. This does not mean individual work: for example she is inscribed in mailing lists and receives at home information from these lists. She also works at home with resources given by colleagues. – Up right, her work at school without students. Meeting with colleagues, entering the students’ marks in the special software (Pronote5 ), which builds the school report at the end of each term. – In the middle of the right side, her work at school in the classroom with her students. Myriam has her own classroom equipped with a computer, a video projector and an overhead projector (which she intends to replace soon with a webcam). – Down right, in-service training collectives. Myriam is involved in two such groups. One group gathers some mathematics teachers of nearby schools (five teachers); they exchange exercises and discuss changes in the curriculum. They meet once a month. This group is not officially recognised by the institution. Myriam is also a member of an ‘official’ group, where a regional inspector participates. This group works on problems and investigations in mathematics. These articulated zones correspond to a structure of Myriam’s professional activity. This confirms the relevance of our ‘global’ positioning: the teacher’s resources are structured according to her activity. We emphasise here central features of Myriam’s professional activity and related characteristics, and evolutions of her documentation system: – Myriam develops real agency in the elaboration of her courses linked with the available digital means: she collects exercises from different downloaded files 5

http://www.index-education.com/fr/telecharger-profnote.php, ‘Pronote’ means ‘professional marks’.

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Fig. 2.4 Schematic representation of the resource system in 2009–2010, Myriam. The original SRRS was handmade and naturally in French. We translated this one and typed it. ∗ Online exercises, http://matoumatheux.ac-rennes.fr/accueilniveaux/accueilFrance.htm

to build one exercise sheet; she integrates new software that she does not fully master, etc. Her preparation work at home is represented on more than half of the SRRS. This corresponds perhaps more to the importance of this work in her opinion than to the time actually spent. She filled in the logbook for over 27 days (about the work with her grade 9 class; this includes Saturdays and Sundays). She mentions about 14 h of work in class with the students, 10 h at school for other purposes (several kinds of meetings) and 12 h at home. She can certainly be considered as an expert teacher. We consider that this characteristic acts as a lens, evidencing phenomena that happen for all teachers. – She is very concerned about official instructions. She follows them and even anticipates further institutional requirements. Her involvement in the assessment of the ‘B2i’ leads her to send work by email and to develop professional knowledge linked with these email exchanges with the students.

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– Myriam is involved in many collectives, which strongly influence her documentation (we detail such processes in Chapter 16). As discussed above, the collective she forms with her students (Section 2.3) and her interactions with students are central resources. She also discusses with colleagues at school and within the in-service training groups. These groups provide her with new ideas of exercises and develop her interest in problems and in investigation situations. This is probably one of the reasons accounting for her choice of introductory activity (‘the box’, Fig. 2.3). We have also mentioned the discussions with her husband. Myriam evokes in fact many discussions with her family, which intervene in her documentation work: not only her husband, who teaches physics, but also her sister, who teaches maths, and her daughter, who is now in grade 11. In H10 (see Fig. 2.3), she devotes a whole hour to the work around the calculator, much more than what was done in 2008–2009, because she realised the importance of the calculator at high school by observing the work of her daughter. – She considers that mathematics needs to provide tools for other scientific topics, physics and biology in particular. This feature is linked with the issue of collectives: her husband is a physics teacher, and at school her discussions are especially with the physics and biology colleagues. It strongly influences her choice of exercises and problems, which are often connected to these scientific topics. The whole of Fig. 2.4 could be interpreted as a description of Myriam’s documentation work: on the left, the resources she draws on; in the centre, her own ‘creations’; on the right, the implementation which supplies new resources – back to the left (kept in the ‘binder with the lessons of previous years’). The presence, in the centre, of ‘ME’ written in capital letters emphasises that this work and, according to our perspective, the associated geneses deeply influence the teacher.

2.4.2 Professional Geneses and Integration of Technology In this section, we address more generally the question of documentational geneses for teachers and the consequences of the use of digital resources, drawing on observations realised with several teachers (including Myriam and Pierre). 2.4.2.1 A Specific Perspective on Professional Growth The documentational approach offers a specific perspective on teachers’ professional growth. The documentation systems articulate professional knowledge and the teacher’s resource system. Therefore, considering teacher’s documentation systems leads in particular to identify structuring elements in their professional knowledge. These elements include what Ruthven (Chapter 5) names the curriculum script (model of goals and actions guiding the teaching of a particular topic). We consider nevertheless

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that this script is mostly adapted to enlighten the decisions that the teacher takes in class; the documentational approach aims at presenting a more holistic view of the teachers’ activity. It can naturally be used to study technology integration phenomena and more generally to understand the professional evolutions resulting from the generalised availability of digital resources. 2.4.2.2 Using Digital Resources: Consequences We notice here different types of such evolutions: – The balance evolves between what is limited to the group formed by the professor and her students, and what is more largely accessible. In particular, Myriam and Pierre use the Pronote software, which was retained by the administration in each of their schools. It confers a public dimension to the marking, making the marks immediately accessible to the other teachers from the same class and to the administrative staff. – The spatial organisation in class includes new forms of display. The two teachers observed use a video projector; Pierre, moreover, has an interactive whiteboard that he combines with his traditional whiteboard (Chapter 16). Myriam also uses an overhead projector to exploit two forms of display as well. This leads to raise the question of new forms of ostension (the teacher showing the content to be learned, Salin, 1999) associated with these new displays, which would require a complementary study. – Using digital files allows an immediate modification of these files as soon as the teacher observes a problem during the implementation in class. The impact of the interactions with the pupils thus seems increasingly important for the teacher’s resources. – The email allows fast and flexible exchanges of files between the teachers and permits out-of-class exchanges between students and teachers.

2.5 Conclusion The stake of a documentational approach of didactics is not limited to the analysis, in terms of professional genesis, of the consequences for teachers of their interactions with resources (Cohen, Raudenbush, & Ball, 2003). It constitutes a change of perspective and an invitation to see documentation work as central in the teachers’ activity and documentational geneses as the components of a complex professional genesis. The expression documentational approach of didactics aims at emphasising that the objective is to not only propose a didactical analysis of the teacher’s documents but also consider the documents as central within the didactic phenomena, and in particular within teachers ongoing professional development. This perspective has already been discussed in Gueudet and Trouche (2009), where we introduced the resource/document dialectics and the concept of documentational genesis. The specific methodology that we have implemented here enabled

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us to investigate further, taking advantage of the long-term evolution, over several years. Complementary studies are needed; the theoretical aspects of the approach have to be refined, in particular by confronting the documentational approach and other theories used for the study of teachers’ growth, of professional activity and of mediations in/for this activity. The consequences of this approach for the design and use of resources, in particular with an objective of professional development (Chapters 7 and 17), require a specific attention. It is also an aspect of our work in progress, in particular about innovative teacher training programs grounded in collaborative documentation work (Gueudet, Soury-Lavergne, & Trouche, 2009). Acknowledgements The authors warmly thank Jill Adler for her comments, and in particular for her revision of the English language in this chapter.

5.00–5.15 (P.M.)

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She informs me that my webcam has been delivered

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References Adler, J. (2000). Conceptualising resources as a theme for teacher education. Journal of Mathematics Teacher Education, 3, 205–224. Assude, T. (2007). ‘Teachers’ practices and degree of ICT integration’. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp. 1339–1348). Larnaca, Cyprus: CERME 5. Retrieved May 2011, from http://ermeweb.free.fr/CERME5b/ Cohen, D. K., Raudenbush, S. W., & Ball, D. L. (2003). Resources, instruction and research. Educational Evaluation and Policy Analysis, 25(2), 119–142. Gueudet, G., Soury-Lavergne, S., & Trouche, L. (2009). Soutenir l intégration des TICE: quels assistants méthodologiques pour le développement de la documentation collective des professeurs? Exemples du SFoDEM et du dispositif Pairform@nce. In C. Ouvrier-Buffet & M.-J. Perrin-Glorian (Eds.), Approches plurielles en didactique des mathématiques (pp. 161–173). Paris: Laboratoire de didactique André Revuz, Université Paris Diderot. Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71, 199–218. Guin, D., Ruthven, K., & Trouche, L. (Eds.). (2005). The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument. New York: Springer. Hoyles, C., & Lagrange, J.-B. (Eds.). (2010). Mathematics education and technology – Rethinking the Terrain. The 17th ICMI Study. New York: Springer. Leont’ev, A. N. (1979). The problem of activity in psychology. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 37–71). New York: M.E. Sharpe. Pédauque, R. T. (Coll.) (2006). Le document à la lumière du numérique. Caen: C & F éditions. Perrin-Glorian, M.-J., DeBlois, L., & Robert, A. (2008). Individual practising mathematics teachers. Studies on their professional growth. In K. Krainer & T. Wood (Eds.), Participants in mathematics teacher education (pp. 35–59). Rotterdam, The Netherlands: Sense Publishers. Rabardel, P. (1995). Les hommes et les technologies, approche cognitive des instruments contemporains. Paris: Armand Colin (English version at http://ergoserv.psy.univ-paris8.fr/Site/default. asp?Act_group=1). Rabardel, P., & Bourmaud, G. (2003). From computer to instrument system: A developmental perspective. Interacting with Computers, 15(5), 665–691. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Salin, M.-H. (1999). Pratiques ostensives des enseignants. In G. Lemoyne & F. Conne (Dir.) (Eds.), Le cognitif en didactique des mathématiques (pp. 327–352). Montréal: Les presses de l’Université de Montréal. Schön, D. (1983). The reflective practitioner: How professionals think in action. London: Temple Smith. Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9, 281–307. Vergnaud, G. (1998). Toward a cognitive theory of practice. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity (pp. 227–241). Dordrecht, The Netherlands: Kluwer. Vygotski, L. (1978). Mind in society. Cambridge, MA: Harvard University Press.

Chapter 3

Patterns of Didactic Intentions, Thought Collective and Documentation Work Gérard Sensevy

Understanding someone’s action requires, in particular, understanding his intention. Understanding an intention, or a system of intentions, does not necessarily mean understanding the whole action, but at least an essential part of it. This statement, however, could be understood as a form of solipsism (the intentions are specific to the individual) and mentalism (intentions are in the head). On the contrary, this text defends and illustrates an alternative conception, in which intentions are regarded as more or less shared and more or less external to the individual (Duranti, 2006). This chapter participates in the general project of this book in underlining the essential dialectics between the documentation work and the shaping of intentions. In this perspective, it aims to demonstrate how intentions are formed in a system of resources (Chapter 2). Intentions are therefore understood, through the documentational genesis process as resulting largely from a documentation work (Chapter 2) performed by the teacher. This contribution falls within the scope of the Joint Action Theory in Didactics (JATD) (Amade-Escot & Venturini, 2009; Ligozat, 2008; Sensevy, in press; Sensevy & Mercier, 2007; Schubauer-Leoni, Leutenegger, Ligozat, & Flückiger, 2007), a theory situated in the general paradigm of joint action (Blumer, 2004; Clark, 1996; Eilan, Hoert, Mc Cormack, & Roessler, 2005; Mead, 1934; Sebanz, Bekkering, & Knoblich, 2006; Tomasello, 2008). In this framework, human activity is seen as grounded in the recognition of signs founded in others’ behaviors. It is viewed as a social game (Bourdieu, 1990, 1992; Bourdieu & Wacquant, 1992). The didactic activity is modelized as occurring in a didactic game that can be described, in particular, with the concepts of didactic contract and milieu (Brousseau, 1997; Sensevy, in press; Sensevy & Mercier, 2007; Sensevy, Mercier, Schubauer-Leoni, Ligozat, & Perrot, 2005) and their relation. According to this theory, the joint work of teacher and students can be seen under the description of two articulated moments: (1) didactic activity in situ, in which the

G. Sensevy (B) Brittany Institute of Education, University of Western Brittany, France e-mail: [email protected]

G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_3,  C Springer Science+Business Media B.V. 2012

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teacher makes the students play the didactic game; (2) the preparation of this activity when the teacher builds the game he will implement. We argue that the teacher’s intentions are shaped in his documentation work. The morphogenesis of intentions, in the documentation work, is thus the link between the building of the game and the actual play that the teacher institutes. In this chapter, we rely on practical descriptions of teachers’ and students’ practices, but our first objective is theoretical. We propose conceptual elements with a three-fold purpose. We try to achieve a better understanding of (1) the relations between intentions and didactical action; (2) the relations between classroom preparation and the actual implementation; and (3) how these relations unfold in a collective that can in some cases produce a specific thought style (Fleck, 1979, p. 99), a system of categories shared in this collective, that ultimately produces ‘the readiness for directed perception and appropriate perception of what has been perceived.’ We then propose a description of the elaboration process of the game, supported by three related assertions we work out in this chapter. First, the resources system that the teacher mobilizes (in the process conceptualized by Gueudet and Trouche, Chapters 2 and 16) is a key source of his action. Second, the teacher’s prior didactic intentions do not have to be found ‘in his head’ or ‘in the situation,’ but in the dialectical relationship between resources or documents, and the way he anticipates the progress of the game in situ. The didactic intentions in action stem from the dialectical relationship between prior intentions and the game as it is enacted in didactic transactions. Third, the process that connects documents, prior intentions, and intentions in action is rooted in the inclusion of the action of individuals in a collective structure. In the first part of this chapter, elaborating on Baxandall’s Patterns of Intention (1985), we develop a framework to understand intentions from a generic viewpoint. We argue that prior intentions function as strategic rules that drive the teacher’s game. The second part is devoted to the study of two empirical examples, which may illustrate the above framework. In particular, we show how prior intentions, as strategic rules (Hintikka and Sandhu, 2006), are drawn from the documentation work, and how the strategies they enact depend on the structure of the milieus suitable for the didactic action in situ. In the third part of the chapter, we briefly summarize our findings.

3.1 Patterns of (Didactic) Intentions In his book (1985), Baxandall formulates a system of descriptions of the intentions of certain artists (e.g., Picasso) in relation to specific paintings (e.g., the Portrait of Kahnweiler). For this purpose, he first built a generic framework for studying how an English engineer, Benjamin Baker, built in the east of Scotland a bridge over the Forth River. To summarize Baxandall’s conceptions, we can look at the following quotation:

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The intention to which I am committed is not an actual, particular psychological state or even a historical set of mental events inside the heads of Benjamin Baker or Picasso, in the light of which – if I knew them – I would interpret the Forth Bridge or the Portrait of Kahnweiler. Rather, it is primarily a general condition of rational human action which I posit in the course of arranging my circumstantial facts or moving about the triangle of re-enactment. This can be referred to as ‘intentionality’, no doubt. One assumes purposefulness – or intent or, as it were, ‘intentiveness’ – in the historical actor but even more in the historical objects themselves. Intentionality in this sense is taken to be characteristic of both. Intention is the forward-leaning look of things. It is not a reconstituted historical state of mind, then, but a relation between the object and its circumstances (Baxandall, 1985, pp. 41–42).

One can notice the importance of the so-called ‘triangle of re-enactment’: there is a situation (first term), a problem arising from this situation (second term) and the solution-object (third term). Let us see how Baxandall summarizes his investigation into the Forth Bridge: One came first to the general Charge that the agent, Benjamin Baker, would be responding to, and noted that while it could be terse – ‘Bridge!’ – it was a rubric for performance that contained within it various general terms of the problem – spanning, providing a way, not falling down. From this one moved on to specific terms of the problem, which I called the Brief, though the name does not matter . . . Together Charge and Brief seemed to constitute a problem to which we might see the bridge as a solution (Baxandall, 1985, p. 35).

To finish summarizing the framework provided by Baxandall, I will address the issue raised by the ‘relationship between object and its circumstances.’ Baxandall argues in the following way: Some of the voluntary causes I adduce may have been implicit in institutions to which the actor unreflectively acquiesced: others may have been dispositions acquired through a history of behavior in which reflection once but no longer has a part. Genres are often a case of the first and skills are often a case of the second (Baxandall, 1985, p. 42).

We can now put forth a first formal framework for the description of intentions we will project on the description of didactic intentions. 1. The objects (and actions) can be described as solutions to a particular problem. To understand an object or action, it is worth asking the question of the problem they are supposed to respond to, and, in some way, which shaped them. One can see a close relationship between this way of conceiving things and the background epistemology in Dewey’s (1922) and Brousseau’s (1997) works, both of which focused on the notion of a situation. 2. Intentions are inherent to physical objects and environments in which these objects (and actions) are located. This view is obviously opposed to mentalistic or psychologizing conceptions of intentions. To understand the intent of an agent in a situation, even before questioning him or eliciting his rationale, we have to understand how the symbolic and physical milieus within which he is acting will lead to such or such intention. In this perspective, material objects themselves (e.g., tools) are purveyors of intentions and plans (Suchman, 1987),

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for the use of people playing the appropriate social game. The concept of ‘affordance’ enables us to understand how objects may be viewed as purveyors of intentions: ‘what we perceive when we look at objects are their affordances, not their qualities. We can discriminate the dimensions of difference if required to do so in an experiment, but what the object affords is what we normally pay attention to’ (Gibson, 1979, p. 134). More broadly, it is the symbolic milieu (e.g., the meanings associated to a specific genre), and therefore the identification of the games that the agents are expected to play in specific situations that may give access to the intentions. One of the fundamental aspects of this milieu is that symbolism is not confined to action in situ and the here and now. Most of our actions are prepared. 3. It is useful and relevant to consider these intentions at various levels of granularity (specificity). In this respect, Baxandall distinguishes the ‘Charge’ that can ‘summarize’ the general intention specific to a particular action and the ‘Briefs’ that characterize these intentions locally. It is interesting to notice that these scale levels call for a differential description of the action. In this perspective, one may usefully appeal to Searle’s (1983) distinction between ‘prior intentions’ and ‘intentions in action’ to figure out how the prior intentions are redesigned as intentions in action in the current action at stake.1 4. The intentions have to be thought about in a broader framework than that fixed by the common epistemology. We saw in particular how Baxandall seeks to extend the meaning of the word ‘intent’ to both institutional practices (including genres) and skills. One can therefore read the intentions in the categories of perception and action that are provided by the institutions, and in skills inherent in the ‘handling’ of a particular object. The four dimensions of the framework presented above can and should be specified in didactic action, and more specifically to the situation of the teacher who ‘prepares the classroom.’ We must be aware of the specificity of this situation. In the intentional part of the documentation work, the teacher uses the resources of a given milieu to organize them into a document. Following Gueudet and Trouche (Chapter 2), we can consider such devices as artefacts monitored by a scheme of use. We have to acknowledge the intentional structure specific to the documentation work. The teacher, related more or less to a group, selects resources according to certain intentions. The arrangement he produces from these resources in turn redefines the system of intentions, which will be further reorganized in the effective course of action. As a ‘historical object’ (Baxandall, 1985, p. 42), a document embeds purposefulness. Intentional structure and actional structure codetermine the other in the document. In some ways, this is both the condition and the effect of this codetermination.

1

In this respect, Pacherie’s recent work (2008) may be also of some help.

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The second part of this chapter will be devoted to the empirical study of some elements of this process.

3.2 Didactic Intentions: An Empirical Study The short empirical study that follows will allow us a first use of the theoretical framework above. For this, we will refer an example in mathematics in elementary school, in which we try to understand what is going on when the teaching intentions are designed in a specific collective, within an implementation process of a new version of a given instructional design. The collective we talk about is a group of teacher educators and researchers. The teacher educators are half the time in their primary school classroom and so able to implement the instructional sequences designed by the group. The collective work we describe has been conducted for 2 years and is inscribed in an ongoing process. The instructional sequence is called Treasures Game. It is a months-long didactical sequence, designed for Kindergarten by Brousseau and his team at the beginning of the 1980s, as documented in Pérès (1984). Brousseau (2004) has presented strong theorization of this research design, which he considered as a fundamental situation for the notion of a representation. The Treasures Game consists of producing a list of objects to be remembered and communicated. The didactic device takes place over a long period (about 45 sessions, of variable duration), which thus becomes a ritual time, but one where the rules change as the game progresses. There are four main stages in the game (Schubauer-Leoni, Leutenegger, Ligozat, Flückiger and Thevenaz-Christens, 20102 ): In Stage 1, the teacher presents two or three small new objects belonging to the world of children to all the children. The objects are passed from hand to hand, the teachers ask the students to name them and then she puts them in a gray box (the treasure chest). She then asks: what’s in my box? A student then calls out the name of an object, the teacher pulls it out of the box and places it in full view of everyone. ‘Is my box empty?’ she asks, and if not, the game continues, and so on. The game takes place every morning. Every two or three days, new objects (two or three objects) appear and are added to the previous ones. By the end of 1 month, the whole class has emptied a box of 40 objects seemingly disparate but carefully chosen. This stage is played out with the entire group of students and focuses on the creation of a verbal system of reference for the objects in the treasure chest. Stage 2 starts the individual memory game, as each pupil must remember the two objects that are hidden daily in the treasure chest. All the children understand the game and are able to succeed, which means, memorize short lists of items from the morning to the evening. 2

I would like to thank Florence Ligozat for sharing this text with me.

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Stage 3 is an individual game aimed at making lists whose production is driven by an important change in the rule of the game (jump from 2 to 10 or 12 hidden objects): the informational leap that finishes Stage 2. Stage 4 is aimed at collectively developing a common code and is driven by a communication game between pupils. The purpose is to offer children new to school life, an opportunity to experience the necessity to rely on a graphical code (drawn or written) to remember a set of objects and to communicate about them with others. It is the very basis of the representation process that is triggered through this game.

3.2.1 Designing New Versions of the Treasures Game: The Building of Intentions as a Collective Strategic System We now focus on Stage 3 of the instructional sequence at the end of Stage 2. It is an important moment, in that the students are confronted with what Brousseau coined as an ‘informational leap.’ The epistemic strategies that enable the students to memorize the right objects (by relying on their ‘internal’ memory) are invalidated by the large number of objects they have to retrieve. For students, it is impossible to memorize 10 or 12 objects without an external (public) representation of these objects. Therefore, introducing this informational gap aims to foster the students’ passage to an external representation, or inscription system. In the following, we compare three ways of thinking about this crucial moment, in the two studies we mentioned in the previous sections, and within our collective. The first way of thinking is presented by Brousseau (2004, p. 256): The passage from 3 to 10 represents a considerable complexification of the situation. The unruffled teacher notices failures, but remains encouraging. ‘Think, we’ll get it’ . . . No child of that age can invent or even conceive the answer all of a sudden, by making a list of objects designed with small drawings of these objects, because the process can succeed only if one controls together all the components. On the other hand, the project can be meaningful only if the children consider, at the outset, specific means to carry it out. The situation appears to be blocked, which causes teachers’ anguish. Yet, we observed that each year, drawings and lists appear. For Brousseau, the adoption of a list of written codes, even if it ‘causes teacher’s anguish,’ is not really problematic. ‘Elements of solution appear and spread in the classroom’ and ‘The method of making lists of drawings is quickly adopted.’ Schubauer-Leoni et al. (2010), in the ‘second generation’ of the implementation of the Treasures Game, consider this issue as follows: ‘This is the trickiest moment and one should not expect the pupils to put in place the relevant strategies straight away . . . The problem faced by the pupils is that they must feel empowered to shift to a remembering process based on inscriptions. T cannot suggest drawing as this would be too strong a command for the pupils and

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it would prevent them from feeling the need for a list. It is in the discussion between the pupils that this idea can come to light.’ We can acknowledge a similar conception of the way the teacher and the students have to deal with the informational leap and the necessity to adopt a ‘remembering process based on inscriptions.’ As ‘Brousseau’s teacher’ must stay ‘unruffled’ and trust the students’ invention, ‘Schubauer-Leoni and coworkers’ teacher’ ‘cannot suggest drawing,’and the necessity of designing writing codes (the inscription process) has to stem from ‘discussion between the pupils.’ In the same light, in the two texts, one can find that the teacher’s role is to encourage the students, in particular by assuring them there is a way to win the game. If we now look at the way our collective dealt with this issue, we have to keep in mind the following points: First, Brousseau’s and Schubauer-Leoni and coworkers’ conceptions were well-known by the collective, given that their papers have been studied before implementing the teaching sequence, and discussed throughout the implementation process. From this viewpoint, the collective documentational genesis (Chapter 16) encompasses the elaboration of these texts, in relation to the actual implementation. The collective was thus sure that the teacher had to stand to the side, and leave the students to figure out how to solve the informational leap problem. One can notice that such a perspective is consistent with the roots of the theory of didactic situations (Brousseau, 1997) as it is usually understood. According to this theory, an essential purpose of the didactical process consists of enabling students to build a first-hand relationship to a given piece of knowledge. To reach that goal, the teacher has to monitor this process by making sure that the students experience the mathematical necessity (in this case, the power of public representations). Nevertheless, at the end of the Stage 2, when the informational leap had to be realized in the classroom, a discussion unfolded in the collective about this issue, initiated by the teacher who has the responsibility to carry out the lesson. Indeed, the collective habit of this group was to anticipate as precisely as possible students’ actual participation and the range of didactic behaviors that students might produce in the didactical situations. In doing so, the collective tried to identify a link between the milieu and the teacher’s action, and the students’ behaviors. When trying to fulfil this a priori analysis pattern, in the case of the session in which the informational leap was presented, the collective was not able to anticipate by what concrete means students would be able to figure out the necessity of using inscriptions. In this respect, it was the teacher’s responsibility to manage the situation by improvising on the basis of the conceptual background that was at the root of the collective’s work. One can thus consider how the collective work on available resources (from Brousseau’s team and Schubauer-Leoni’s team) provides a specific strategic system that one can describe as follows: confronted with the inevitable failure of his students, the teacher had to let the students know that the game can be played with success (encourage the students by giving them the assurance they can win); she had to stand to the side to allow the students to experience the necessity of the inscription system (let them find they can make a list); she knew that she was going

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to face uncertainty stemming from her ignorance of the students’ possible moves to find a ‘solution’ (she is prepared to use any opportunity to guide the students’ learning trajectory). Let us now consider the actual implementation of this part of the situation, which means, according to our theoretical framework, how the system of these strategic rules is enacted in actual strategies.

3.2.2 Implementing Instructional Sequences Within a Collective: A Teacher’s Rational Improvisation This part of the instructional sequence has been videotaped and transcribed.3 Several months after the sequence was carried out, an auto-analysis interview was conducted between the teacher and another member of the research team. The studied episode took place at the end of Stage 2. This session occurred in a workshop gathering five students. It was the first day of a two-day process, in three phases (Day 1: morning; Day 1: afternoon; Day 2: morning). In the following, we focus on one of the crucial moments of the Treasures Game, in describing how the joint action of the teacher and the students fosters the emergence of ‘making a list.’ 3.2.2.1 Day 1: Morning The teacher presented the 10 objects that had to be remembered for the afternoon. She handed the items in the bag (the equivalent of the ‘treasure chest’), and stressed the goal of the game: the students had to remember, and ‘each child will be on his own to remember all things this afternoon.’ At the end of this episode, a significant dialogue takes place between a student (Ima) and the teacher: You have to write down. . . I have to write down what, my dear (inaudible) I have to write down all the children who did the Treasures Game? I have to write down all the objects? What do I have to write? Ima: You have to write down (inaudible) of Treasures Game.

Ima: T: Ima: T:

In fact, at this moment, Ima wanted the teacher to write down the list of the students who played the game on that day. This is a generic classroom habit, specified to the Treasures Game situations. In the classroom, it is important to write down who has done the activity, to know who hasn’t. 3 This part of the chapter has been written on the basis of data collected by two members of the collective, Dominique Forest and Anne Le Roux-Garrec. I would like to thank them. I am grateful to Dominique Forest for the fruitful discussions we had about the interpretation of these data.

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So the teacher takes the list from a table near the students and points to the students’ names: T: Ok, I know that’s you, the other ones will do it afterwards . . . It is interesting to notice a kind of uncertainty in the teacher’s behavior. Ima referred to a list. Even though after the teacher understood that the list Ima was talking about was not the type of list she was waiting for, this behavior could be considered as an opportunity for the teacher to give the students an incentive to think about making a list, for instance, by revoicing Ima’ s proposal. The teacher comments on this event as follows: I know I’ll have trouble, finally I am afraid I find it hard to make them think about the written record. Oh, I know that, I know that because we talked a lot about it in the group, and I know that at a point the written record must appear, and I do not see how it will appear. I do not know what I was thinking then, but when Ima said ‘write down,’ then I said to myself there is something, something that I must keep under my sleeve, because the idea of writing record, if it does not emerge after, at this moment there are traces, which emerge now, traces I will be able to rely on. We can understand how the teacher was able to reenact her intentions in the dialogue focused on her videotaped practice. It is possible to recognize the strategic system we mentioned above. In particular, she knew she must let the students ‘find by themselves’ that they can make a list. This is one of the core constituents of the strategic system elaborated within the collective (‘we talked a lot about in the group’). One may say that this strategic rule stems from the ‘thought style’ (Fleck, 1979) inherent in the work of this collective.4 There are some fundamental relations and properties that are impossible to challenge in a thought style, a kind of ‘bedrock’ (Wittgenstein, 1997), which turns ‘individual thought over to an automatic pilot’ (Douglas, 1987, p. 63). We argue that the ‘let the students find by themselves they can make a list’ strategic rule is such a core principle in the collective thought style. Nevertheless, it is worth noticing that this strategic rule is not easily converted to an actual strategy. As we put it above, it is in some ways contradictory to the habit of thought, elaborated in the collective, which consists of drawing a precise a priori analysis to anticipate the students’ learning behavior. This uncertainty is obvious in the teacher’s comments (‘I know that at a point the written record must appear, and I do not see how it will appear’), which seems to mirror Brousseau’s and Schubauer-Leoni and coworkers’ statements we quoted above. Thus, it is perfectly understandable that the teacher be tempted to use all the

4 We argue that one can consider the educational process as the slow elaboration of a thought style (Sensevy et al., 2008).

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opportunities she could find in the students’ utterances, even though there is a risk of misunderstanding. The following part of the teacher’s auto-analysis sheds light on this topic: I think I try to bring out small things, because it can be reused, maybe there are seeds that are sowed there. On the excerpt in which the teacher shows students the list of children: This is the list of children, but it’s true, I do not present it without purpose. This is a sample of a list, which . . . So there is an emergence of something, I thought, if it is difficult for them to achieve the written record, perhaps I will be able to build on it. The interviewer asks: But you do not go any further at this time? No, because it is not the right time, it’s not the game. And here it is about a list, which describes the children who played the game, it is not at all the idea of a written trace, which keeps a permanent memory for later. In the list of children Ima refers to, we deal with a written trace, which allows us to validate: has each child played? So it’s not at all the same approach. In this excerpt, we can understand how the teacher’s action (the teacher’s game on the student’s game, in the theoretical sense of the JATD) surrounding the ‘question of the list’ consists of reducing uncertainty by ‘sowing seeds,’ that is, by paying attention to the student’s mention of the list of children, to be able to reuse this meaning later on. But it is interesting to note that this behavior does not entail a Jourdain effect5 (Brousseau, 1997). Indeed, the teacher explains that the strong conceptual difference between the two types of lists (the ‘list of children’ and the ‘Treasures Game list’) prevents her from relying too firmly on the student’s designation of the list of children. One can acknowledge from the teacher’s declaration how two fundamental aspects of the didactic game are at stake. First, the chronogenesis (the genesis of time) constraint (No, because it is not the right time, it’s not the game) that explains that the teacher has to wait for the right time, the kairos, as the ancient Greeks said, to engage the classroom discussion on the issue of the list. Second, the mesogenesis constraint, which is closely linked to the chronogenetic one. In this episode, the mesogenesis (the genesis of milieu) necessity refers to the need, for the teacher, to introduce some specific meanings to create common ground and upon which she will be able to elaborate in order to help the students figure out how to produce a remembering process. 3.2.2.2 Day 1: Afternoon The teacher asked each student to recall the names of objects, without success. Although some of them recalled more than others, nobody was able to recall the ten objects (carefully chosen, in kind and number.) The teacher asked the students to name and count the objects, and she made them acknowledge their failure. At a point in this session, the teacher insisted: ‘I would like us to succeed, because we did

5 A Jourdain Effect occurs when the teacher pretends to acknowledge a specific piece of knowledge in an ordinary student’s behavior.

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not win’. It is important to note how the teacher emphasizes the students’ failure, as an impossibility to retrieve the ten objects. But one has to identify, in the teacher’s speech, the use of the pronoun ‘we,’ which means the teacher includes herself in the failure and the necessary subsequent research. For her, it is a way to deal with the difficult uncertainty students could feel. In doing so, she tries to enact the two main strategic rules in this part of the Treasures Game: (1) the students have to experience the limits of internal memory; (2) this failure must not alter their commitment in the enquiry. It is interesting to focus on a slight move, in the teacher’s game, which occurred at the end of the session. Out of the blue, the teacher first intended to show the ten objects to the students, then she changed her mind: ‘Well, I am going to show you the objects [the teacher takes the bag]. I intended to let you see, but before that, I would like us to succeed, cause we did not win.’ To understand this point, the interviewer asks the teacher a question: I: You said ‘I am going to show you the objects’ and you didn’t do that . . . T: No, because the problem is elsewhere, I would like to get it, I would like it to emerge, and I am afraid that, I tell myself that, by showing them the objects, they think ‘oh, it’s easy, I could have done it,’ they could be in trouble. In fact, I want to leave them feeling the failure, I want to leave them telling themselves ‘the hidden objects that I can’t see yet, what is the representation I could give them.’ In my opinion, it’s the point. During the following interactions, the teacher went on by underlining several times the reality of the failure, and its inescapability: ‘you will not succeed, it’s too difficult.’ In the same time, she diffuses the idea of a possible solution: ‘we should find a means.’ She characterized this means as a ‘little means,’ thus signifying that every student had the possibility to find a solution. Even though the teacher tries to help the students, they do not provide a solution. One could identify a kind of fatigue among the students. So the teacher introduces in the milieu the ‘meaning of writing’ in the following way: S1: T: S2: T:

And if you tell us? Oh, me, I won’t say anything. Ah, ah, ah, she tells us nothing because it’s the Treasures Game. Yes indeed you are playing, but Ima, what did she say she wanted to do in order to remember in the evening? S3: Write!

The teacher gave a clear incentive, by focusing student’s attention on Ima’s word about the possibility of writing, and the students acknowledge this reminder. It is interesting to consider the teacher’s analysis of these moves. Here I am cheating, I am cheating, because what Ima wanted to do in the morning was keep a trace of the students’ participation in the game to be sure they have participated. On my side, it meant keep a memory to remember . . .

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The teacher critiques her own behavior, but after having emphasized again her recognition of the difference between the student’s viewpoint and her own, she reconsiders her previous analysis: After all, it’s sure that the idea of a list, the idea of ticking the students’ participation to be sure all the students have played, this idea is in the same spirit to keep a memory, it’s what I reactivate here. Even though they do not have this anticipatory idea of keeping a trace as a representation to use it later on . . . So, it’s not really cheating, it’s, umm, bridging the gap from my behavior to a behavior that they can adopt in their personal approach. One can see this latter assertion as witnessing the complexity of the didactical practice. If one follows the teacher’s justification, one can say that to reach her goal (enabling the students to refer to writing), the teacher admits a kind of minor misunderstanding of the nature of the remembering process. One can raise the hypothesis that to the extent to which the failure of the internal memory has been acknowledged by the students, the production of the ‘solution’ (writing) is not a major stake. The crucial point is that the students commit themselves to the writing process, given that the teacher’s monitoring of this process will enable them to understand the very nature of the remembering process, and thus to correct the initial minor misunderstanding that will have allowed the joint process to proceed. 3.2.2.3 Day Two: Morning The day after, after having reminded the students of their difficulty to retrieve the ten objects, the teacher reactivated the writing solution: T: And I, I still wanted 10 objects, even though it was a lot. So, this evening, will it be easy to remember them? S1: No! T: No, so what could we do to remember this evening? Ima: We write. T: You, you would like to write, so you need a sheet of paper. So, go ahead (the teacher gives Ima a sheet of paper and a pencil), for me, it’s alright. T: But she does that for herself, OK. To remember on her own. You, if you want to remember you have to do something too? S2: Yes, me, I want to write, too (the teacher gives him a sheet of paper and a pencil). S3: Me too. It’s that, we all are going to write down (the teacher gives a sheet of paper and a pencil to every student). Eventually, the instructional sequence continued. The students started to produce some inscriptions as a means to remember the objects, and the Treasures Game proceeded.

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3.3 Conclusion In this chapter, we first focused on the issue of intentions, on the basis of Baxandall’s work. Within the framework of the JATD, we consider human practices as social games. In this respect, we argue that to understand people’s actions, we have to identify what we modelize as the game they play. Thus, people’s intentions are to be drawn from these games, and we consider intentional systems as strategic systems. In doing so, we highlighted a conception of intentions in which intentions are public, found in the milieu of the action. In that sense, an intention is more or less always collective, not necessarily in the sense that it stems from a collective, but in that it has to be viewed as the expression of an institutional thought style that stems from the social game at play. This thought style plays a prominent role in the orchestration process (Chapter 14) that teachers enact. In the empirical study we outlined in this chapter, the teacher’s intentions were collective, in the first sense of the term that we acknowledge below. The teachers work in a particular institution, broadly speaking, a didactic institution, which brings them to a specific thought style. For example, a teacher has to enable the students to establish more or less a first-hand relationship to a given piece of knowledge, and one who wants to understand the dynamics of the teaching--learning process has to take this general feature into account when identifying the teacher’s intentions. In this chapter, the case study allows us to understand how the didactic intentions lie in the documents designed by the teachers and in the relationship the teacher has built with these documents. The case at stake is interesting in that it shows the nature of the teacher’s intention, about the necessity, for the students, to experience the failure of internal memory, and the consequent adoption of a writing strategy. This system of intentions is not an individual’s system, but the result of a collective documentation work, which is based on the study of the previous versions of the Treasures Game. But in the texts presenting these previous versions, as we saw, not enough was said about the way of dealing with the necessity of the list, even though the researchers present this necessity as critical in the teaching process. In this respect, we have shown how the teacher’s strategic system, as a system of prior intentions, is designed to achieve her two-fold purpose (failure of internal memory, necessity of a writing strategy), by standing to the side. We argue that it is impossible to understand the joint action of the teacher and the students, in this classroom, without acknowledging this two-fold purpose, which is purpose of the collective. But taking into account this collective purpose is not sufficient. We try to show that it is necessary to document the way the teacher, against this common ground, puts in place actual strategies that concretize the strategic rules that monitor his behavior. To understand the concrete action of a teacher, even though it has been designed in a collective documentation work, one has to acknowledge the teacher’s ‘feel for the game’ (Bourdieu, 1990) that enables her to rationally improvise, and to reach the collective goals beyond the collective preparation. In this respect, teachers could be seen as ‘instructional designers’ (Chapter 17) to the extent to the results of their improvisation modify the research design.

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In this perspective, a thought style, conceived of ‘the readiness for directed perception and appropriate perception of what has been perceived’ is a precious support for people’s practices, but it does not provide people with all the ‘solutions’ of the practice. In this study, the classroom concretization of prior intentions, as a strategic system, rests on the teacher’s capacities to enact a particular way of ‘standing to the side’ within the joint action. In the Treasures Game situation, as it was implemented here, we have to acknowledge that this enactment is not easy. It seems that a major reason for this difficulty could be the ‘lack of intentiveness,’ to use Baxandall’s neologism, of the scientific texts the collective was using. The resources and documents embed purposefulness, but in some cases, not enough.

References Amade-Escot, C., & Venturini, P. (2009). Analyse de situations didactiques: Perspectives comparatistes. Dossiers des Sciences de L’éducation. Numéro Spécial, 20. Baxandall, M. (1985). Patterns of intention: On the historical explanation of pictures. New Haven, CT: Yale University Press. Blumer, H. (2004). George Herbert Mead and human conduct. Walnut Creek, CA: AltaMira Press. Bourdieu, P. (1990). The logic of practice. Cambridge: Polity Press. Bourdieu, P. (1992). Language and symbolic power. Cambridge: Polity Press. Bourdieu, P., & Wacquant, L. (1992). An invitation to reflexive sociology. Cambridge: Polity Press. Brousseau, G. (1997). The theory of didactic situations in mathematics. Dordrecht, The Netherlands: Kluwer. Brousseau, G. (2004). Les représentations: étude en théorie des situations didactiques. Revue des sciences de l’éducation, 30(2), 241–277. Clark, H. (1996). Using language. Cambridge: Cambridge University Press. Dewey, J. (1922). Human nature and conduct. New York: Modern Library. Douglas, M. (1987). How institutions think. London: Routledge. Duranti, A. (2006). The social ontology of intentions. Discourse Studies, 8(1), 31–40. Eilan, N., Hoert, C., Mc Cormack, T., & Roessler, J. (2005). Joint attention: Communication and other minds: Issues in philosophy and psychology. Oxford: Oxford University Press. Fleck L. (1979). Genesis and development of a scientific fact. Chicago: The University of Chicago Press. Gibson, J. J. (1979). The ecological approach to visual perception. Hillsdale, NJ: Lawrence Erlbaum Associates. Hintikka, J., & Sandu, G. (2006). What is logic? In D. M. Gabbay, P. Thagard, & J. Woods (Eds.), Handbook of the Philosophy of Science. Volume 5, 20: Philosophy of Logic. London: Elsevier. Ligozat, F. (2008). Un point de vue de didactique comparée sur la classe de mathématiques. Etude de l’action conjointe du professeur et des élèves à propos de l’enseignement/apprentissage de la mesure des grandeurs dans des classes françaises et suisses romandes. Thèse de Sciences de l’Education, Université de Genève et Université d’Aix-Marseille. Mead, H. G. (1934). Mind, self, and society. Chicago: University of Chicago Press. Pacherie, E. (2008). The phenomenology of action: A conceptual framework. Cognition, 107, 179–217. Pérès, J. (1984). Use of the theory of situations with a view to identify didactic phenomena during a period of school learning. PhD thesis, University of Bordeaux II. Schubauer-Leoni, M.-L., Leutenegger, F., Ligozat, F., & Flückiger, A. (2007). Un modèle de l’action conjointe professeur-élèves: les phénomènes qu’il peut/doit traiter. In D. G. Sensevy &

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A. Mercier (Eds.), Agir Ensemble. L’action didactique conjointe du professeur et des élèves dans la classe (pp. 52–91). Rennes, France: PUR. Schubauer-Leoni, M.-L., Leutenegger, F., Ligozat, F., Flückiger, A., & Thevenaz-Christens, Th. (2010). Producing lists of objects to be remembered and communicated. The « treasure game » with 4 and 5 year old children. Fapse Genève University, Translated from French by N. Letzelter & F. Ligozat Searle, J. R. (1983). Intentionality. Cambridge: Cambridge University Press. Sebanz, N., Bekkering, H., & Knoblich, G. (2006). Joint action: Bodies and minds moving together. Trends in Cognitive Science, 10(2), 70–76. Sensevy, G. (in press). Overcoming fragmentation: Towards a joint action theory in didactics. In B. Hudson & M. A. Meyer (Eds.), Beyond fragmentation: Didactics, learning, and teaching. Leverkusen, Germany: Barbara Budrich Publishers. Sensevy, G., & Mercier, A. (2007). Agir ensemble. L’action didactique conjointe du professeur et des élèves. Rennes, France: PUR. Sensevy, G., Mercier, A., Schubauer-Leoni, M.-L., Ligozat, F., & Perrot, G. (2005). An attempt to model the teacher’s action in mathematics. Educational Studies in mathematics, 59(1), 153–181. Sensevy, G., Tiberghien, A., Santini, J., Laubé, S., & Griggs, P. (2008). Modelling, an epistemological approach: Cases studies and implications for science teaching. Science Education, 92, 424–446. Suchman, L. A. (1987). Plans and situated actions: The problem of human-machine communication. New York: Cambridge University Press. Tomasello, M. (2008). Origins of human communication. Cambridge: MIT Press. Wittgenstein, L. (1997). Philosophical investigations. Oxford: Blackwell.

Chapter 4

Resources for the Teacher from a Semiotic Mediation Perspective Maria Alessandra Mariotti and Mirko Maracci

4.1 Introduction The potentialities of ICT tools for learning have been extensively studied with a main focus on the their possible use by the students and the consequent benefits for them, but there has been the tendency to underestimate the complexity of the teacher’s role in exploiting these potentialities. In this chapter, assuming a semiotic mediation perspective (Bartolini Bussi & Mariotti, 2008), we will discuss different kinds of artefacts that are offered to the teachers to enhance the teaching–learning activity in the classroom. Thus, as Adler (Chapter 1) suggests, we shift “attention off resources per se, and refocus(es) it on teachers working with resources; on teachers re-sourcing their practice. Teachers’ “re-sourcing practice” can be viewed in at least two ways: firstly, it may be interpreted as exploiting resources and developing professional activity (e.g. professional growth), as explained and discussed by Gueudet & Trouche in the theoretical frame of documentational approach (Chapter 2). Secondly, it may be interpreted as exploiting resources in the classroom to achieve a specific educational goal. This latter sense opens the research direction that we followed in our study, assuming the specific frame of the Theory of Semiotic Mediation (TSM) (Bartolini Bussi & Mariotti, 2008). Such theoretical approach explicitly considers the role of the teacher and describes how she can exploit the use of an artefact, managing different didactical situations to make the expected semiotic process happen. Following Bartolini Bussi (1998), we describe the teacher’s action making use of the metaphor of orchestration. As argued in Mariotti & Maracci (2010), the term orchestration here can be related to what is labelled the didactical performance component of the instrumental orchestration within an instrumental approach (Chapter 14), but the objectives are different. In fact, the objective of the teacher’s orchestration within a semiotic mediation approach is not that of guiding students’ instrumental

M.A. Mariotti (B) Department of Mathematics and Computer Science, University of Siena, Siena, Italy e-mail: [email protected]

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geneses, but that of developing shared meanings, having an explicit formulation, de-contextualized from the artefact use, recognizable and acceptable by the mathematicians’ community. In the following, we want to go further in the description of the “use of an artefact”, discussing the use of different kinds of resources related to its functioning as a tool of semiotic mediation (Bartolini Bussi & Mariotti, 2008, p. 754).

4.2 Mediation and Teaching–Learning According to a Semiotic Approach The TSM is centred around the seminal idea of semiotic mediation introduced by Vygotsky (1978) and it aims to describe and explain the process that starts with the student’s use of an artefact and leads to the student’s appropriation of a particular mathematical content. The TSM addresses this issue combining a semiotic and an educational perspective, and elaborating on the notion of mediation while considering the crucial role of human mediation (Kozulin, 2003, p. 19) in the teaching–learning process. Taking a semiotic perspective means to acknowledge the central role of signs in the teaching–learning activity. The use of the term “sign” is inspired by Pierce. We assume an indissoluble relationship between signified and signifier. In the stream of other researchers (Arzarello, 2006; Radford, 2003) we developed the idea of meaning that originates in the intricate interplay of signs (Bartolini Bussi & Mariotti, 2008). Consequently, specific attention is paid to the processes of production of signs and of their transformation, which in turn is considered as evidence of learning. Fostering or guiding this process is a crucial issue and a demanding task for the teacher. In the following sections, we outline how one can organize a teaching–learning sequence by integrating the use of an artefact. Such description is developed around the key notions of semiotic potential of an artefact and of didactic cycle. Within this frame we describe different resources1 which can support the teacher in exploiting the semiotic potential of a given artefact.

4.3 The Semiotic Potential of an Artefact and the Didactical Cycle Following Hoyles (1993), one can speak about the relationship between artefact and knowledge as evoked knowledge. For experts the artefact may evoke specific knowledge corresponding to what is mobilized to solve specific problems.

1 Assuming a semiotic mediation perspective, Mariotti and Maracci (2010) address the issue of how an ICT tool can be a resource for the teacher.

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Fig. 4.1 Copy of a Roman abacus (1st century AD)

For example, the positional notation of numbers may be evoked by an abacus (Fig. 4.1). Similarly, a Dynamic Geometry System may evoke the classic “rule and compass” geometry. However, there is the need to distinguish between meanings emerging from the practice based on the use of the artifact and the mathematics knowledge evoked in the expert’s mind. The notion of semiotic potential of an artifact is meant to capture that distinction and to make it explicit. By semiotic potential of an artefact we mean the double semiotic link which may occur between an artefact and the personal meanings emerging from its use to accomplish a task and at the same time the mathematical meanings evoked by its use and recognizable as mathematics by an expert.2 Thus, taking a semiotic perspective we will focus on the semiotic processes occurring in the classroom when the teacher manages the use of an artefact according to specific didactical goals. According to the TSM, the teaching–learning process starts with the emergence of students’ personal meanings in relation to the use of the artefact to the accomplishment of a task. The emergence is witnessed by the appearance of specific personal signs – the unfolding of the semiotic potential. The process of semiotic mediation develops in the collective construction of shared signs, related to both the use of the artefact and to the mathematics to be learnt (Fig. 4.2). The evolution of signs can be promoted through the iteration of didactic cycles (Fig. 4.3) where different categories of activities take place, each of them contributing differently but complementarily to develop the complex process of semiotic mediation: (a) activities with the artefact on the basis of the tasks purposefully designed for promoting the emergence of signs referred to artefact-use; (b) activities

2 The distinction between personal meanings and mathematical meanings may remind of Brousseau’s distinction between knowing (in French: connaissance) and knowledge (in French: savoir) (Brousseau, 1997). Even if they are not in antithesis, the two perspectives cannot be reduced to one another: the former stresses the semiotic dimension of the teaching--learning processes, which is in the shadow for the latter.

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Personal meanings Artefact signs

Use of an artefact for accomplishing a task

Mathematical meanings Mathematical signs

Teacher’s mediation

Fig. 4.2 Semiotic potential of an artefact and teacher’s mediation

of individual writing involving students in semiotic activities concerning written productions. For instance, students might be asked to write individual reports on the previous activity with the artefact, reflecting on their own experience, and raising possible doubts or questions; (c) classroom discussions which constitute the core of the semiotic process. According to the idea of mathematical discussion (Bartolini Bussi, 1998), the teacher’s main objective is to exploit the semiotic potentialities of individual contributions that move towards mathematical meanings. In other words, assuming a semiotic mediation perspective calls for the establishment of a specific activity format (Chapter 5) which consists of the iteration of didactical cycles. Fig. 4.3 The didactical cycle Activities with the artefact

Individual production of signs

Collective production of signs

Though the ICT tool can be considered the key resource, the use of which is not limited to the initial phase (Mariotti & Maracci, 2010), other resources may support the teacher’s actions throughout all the didactical cycle. In this contribution we will consider the specific resource provided by written texts. According to their different potentials, the teachers may exploit different types of texts – either internal or external to the mathematics class community: the texts produced by the teacher to describe the task, those produced by students in the different moments of the teaching–learning sequence, or the texts provided by a historical source. To show the potentialities of such texts as resources for the teacher, we will focus on how the teacher can effectively use them in the classroom for triggering and

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sustaining semiotic mediation, and hence on the semiotic processes activated by the students and prompted by the use of those texts.

4.4 Texts as Resources for the Semiotic Mediation Process We consider the term “text” in a broad sense including any kind of organized set of signs, also belonging to different semiotic systems, although in the following we will limit ourselves to considering mostly written verbal texts. A text provides a number of signs organized in a stable structure that may become object of reflection and discussion, and for this very reason the text has the potential of triggering the production of new signs. To understand the specific types of resource we intend to discuss, we can explicitly refer to Wartofsky’s (1979) classification of cultural artefacts in primary, secondary and tertiary artefacts. As Wartofsky explains: What constitutes a distinctively human form of action is the creation and use of artefacts, as tools, in the production of the means of existence and in the reproduction of the species. [. . .] Primary artefacts are those directly used in this production; secondary artefacts are those used in the preservation and transmission of the acquired skills or modes of action or praxis by which this production is carried out (1979, p. 202).

Secondary artefacts are therefore representations of modes of actions with artefacts. When representations of modes of actions are drawn on systems of signs, the element of convention comes to play a large role generating new semiotic systems. The relationship between the abacus and the positional system of number representation is a paradigmatic example. There is also another class of artefacts that Wartofsky calls tertiary artefacts and may be often an evolution of secondary artefacts. [. . .] which can come to constitute a relatively autonomous ‘world’, in which the rules, conventions and outcomes no longer appear directly practical, or which, indeed, seem to constitute an arena of non-practical, or ‘free’ play or game activity (1979, p. 202).

Examples of tertiary artefacts are the mathematical theories which organize the mathematical models constructed as secondary artefacts. Assuming such a perspective, Bartolini Bussi, Mariotti, & Ferri (2005) discuss the semiotic potential emerging from combining the use of a primary artefact, a perspectograph, and secondary artifacts, texts drawn from ancient treatises of painting, to form together the base of the development of tertiary artefacts. The semiotic potential of such combination of intertwined elements was based on the potential of different artefacts of evoking each other. The polysemy or multi-voicedness of cultural artefacts make them useful as teacher resources to foster mathematical discussions in the classroom (Bartolini Bussi, 1998): because of its evocative potential, a text may be used to fuel the evolution of signs, objective of a semiotic mediation process. A condition for a text for being potentially useful is that of being interpretable by the students in terms of their experience with the primary artefact in play as well as with the mathematics. In our teaching experiments, the teacher utilized texts of different types:

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(a) written texts produced by a pair of students and (b) written text drawn from a historic source. Each type of text has different potentialities with respect to the semiotic mediation process. Consider the case (a). A text written by a classmate may assume the status of the simulation of a pair interaction, where one of the interlocutors expresses herself through a written text. In the other cases (b), an asymmetry appears immediately between the reader and the voice expressed by the text whose authority may come from the well-known reputation of the author or from official reference to the community of mathematicians. Generally speaking, reading an original source is a specific activity on the basis of an hermeneutic effort referring to the tension between the meaning of the text in the perspective of the author and the meaning for the reader in her personal perspective (Jahnke, Arcavi, Barbin, Bekken, Furinghetti, Idrissi, da Silva, & Weeks, 2000). In the following sections, we will show how all the resources described above can be used in synergy by the teacher. We start by illustrating the key role of the text that describes the task (the formulation of the task) in fostering the unfolding of the semiotic potential; then we discuss examples concerning the teacher’s utilization of written texts: texts produced by students and a text drawn from a historic source. The examples are drawn from the same teaching experiment centred on the use of Cabri (Laborde & Bellemain, 1995), which involved Italian and French 10th grade classes (for details, see Falcade, 2006; Falcade, Laborde, & Mariotti, 2007).

4.5 The Teaching Experiment The educational goal was to use Cabri for introducing students to the idea of function as co-variation. The design of the sequence of activities was consistent with the structure of the didactic cycle. Students’ productions and audio-recordings of classroom activities were collected and analysed. The idea of function was introduced within a geometrical setting, as a relation between points of the plane which are linked through a geometrical construction. One can recognize the possibility of establishing a rich system of connections between certain components of Cabri and their use – such as basic points and points obtained through a construction, the dragging tool and its effect on the different kinds of points, the trace tool and the macro tool – and the mathematical notion of function and all the related notions – such as that of independent and dependent variables, parameter, domain, image and graph. The first activity proposed to the students concerned the exploration of the effect of a macro-construction. They had to explore systematically the effect of dragging a point, experiencing both the free and the conditioned movement. The aim was to introduce the notion of variation and co-variation as a base for a definition of function. The trace tool was extensively used, and the study of the trajectories of the different points contributed to the appropriation of rich meanings embedding a

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Fig. 4.4 What appears on the screen activating the trace tool

dynamic component, for the notion of domain and image of a function (Fig. 4.4). Collective discussions were orchestrated by the teacher with the aim of formulating shared mathematical definitions of the notions of function, domain and image. Later on, after the introduction of numerical functions, the students were assigned the problem of providing a geometric representation of a numerical function. Once obtained the solutions to this problem, the students were asked to interpret an excerpt of a text by Euler addressing the same question, and to compare their own answer with the method described by Euler. The ensuing collective discussion had the aim of sharing the individual interpretations and promoting the evolution of personal meanings towards the mathematical meaning of graph of a function.

4.6 The Text Formulating the Task The design of the starting activity was intended to foster the students’ production of personal signs related to the use of the dragging tool that could subsequently evolve towards the desired mathematical signs. Specific attention is put in the formulation of the task, specifically in the choice of the words referring to different aspects to be focussed on. Task. Displace all the points you can. Observe what moves and what does not. Explore systematically, that is, displace one point at time and note which points

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move and which do not. Summarize the results of your exploration in the table below.

Points which can be dragged Points which move Points which do not move

The expressions “displace”, “move” and “drag” are present, and are used with different meanings. “Displace” and “drag” are used as nearly synonymous to refer to the direct action made by the user upon the points. There is a slight difference between the two words, as the first is a word of “natural language”, while the second is a word of “Cabri language”: “move” is used to refer to the movement of a point as a result of direct or indirect action upon it. This difference is not made explicit. It is left to the pupils to make sense of this difference through their exploration in Cabri. The transcripts of students’ conversations show interchangeable use of the expressions “move” and “displace”, until the students realize and express the distinction between what moves and what can be displaced. We can therefore say that as an effect of working on the task, the students produced and shared two distinct signs, “displace” and “move”. These two signs directly refer to the activity with Cabri, but they have the potential of being related to the mathematical signs of independent variable (point that can be moved) and dependent variable (points that move but cannot be moved). The following exchange between two students is a good example of what can be expected during the solution of this task. 30. Egi: I wanted to ask . . . points that can be displaced, in what sense . . . that every time move. 31. Mar: Can be displaced . . . I told you its hard . . . all of them move but you can displace only three of them. H moves under the action of A, B and P. A semiotic perspective introduces a specific dimension in the design of the task: the production of certain signs can be considered the effect of the specific task, but what is crucial is not only what is requested to be done, but also how such request is worded.

4.7 Written Texts Produced by Students The second example concerns an episode occurred during a collective discussion designed with the twofold aim of clarifying and systematizing the ideas that emerged during the first phase of activities with the artefact, and finally, expressing these ideas in a “mathematical statement”: the definition of function. This kind of

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activity was not new for the students, who were accustomed to engaging in defining processes. The discussion developed over three lessons (approximately 5 h). The first phase started when the teacher recalled the recent activities with Cabri: both the pupils and the teacher referred to Cabri tools and phenomena experienced during those activities. Different elements in play were highlighted by the students and explicitly related to the corresponding mathematical ideas of (independent and dependent) variable, domain and image. Then a crucial point arose: the students realized that characterizing a function implies determining common features and this corresponds to determining when two functions can be said to be “equal”. Grasping the opportunity, the teacher shifted the focus of the discussion and asked the students, working in pairs, to try and formulate a “definition of equal functions”. Cabri was available, and students were prompted to check different examples to test their conjectured definitions. Finally, students were asked to express through a written text a “definition of equal function” that took into account the ideas from the activity in Cabri. The following definitions were proposed. And–Ale: “Two functions are equal if they have the same domain and the same image for all the domains subsets of the original domain which defines the functions.” Gio–Fed: “Two functions are equal if they have the same number of variables, the same domain, and the same procedure (in the construction of the macro).” Mar–Gab: “Two functions are equal when they have the same image and (when) the same domain is fixed (for both).” Tiz–Seb: “In our opinion two functions are equal if having the same domain and the same definition procedure they have the same image. If either the domain, or the definition procedure, or the image are not equal, neither the functions are not equal.” Almost all the definitions mentioned the main elements in play: the domain, the procedure and the image. The first definition presents a characterization in which the domain is conceived in terms of subsets and uses a quantifier (“for all”). This is a static definition in terms of sets: no reference to variation is made. Though it may appear quite strange, nevertheless this characterization originated from the pupils’ previous experience, specifically from the relation built between the idea of image of a function and that of trajectory coming from the use of the trace tool. This will emerge from the collective discussion.

4.7.1 Comparison of Texts for Sharing a Definition All the produced texts above highlight clear potentialities with respect to developing a relationship between meanings emerging from the Cabri experience and the mathematical meanings at stake. Hence, these texts may be considered artefacts, secondary artefacts with respect to the primary artefact Cabri. The teacher

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decided to exploit the semiotic potential of these secondary artefacts and organized a collective discussion centred on the comparison between the four produced texts. She provided the students with a copy of the produced texts, left a few minutes for reading them, and opened the discussion clarifying the aim: to formulate a shared definition of “equal functions” starting from the four given produced texts. In the following, we report some excerpts from the transcript of the discussion. Excerpt 1 1. T (teacher): [. . .] we must find an agreement on a definition, which can be one of these, or an improvement of one of these, or the fusion of these . . . We must decide. 2. And: According to me, Gab’s and Mar’s definition is wrong. 3. T: So, And, according to you, Gab’s and Mar’s definition is wrong. Let’s read it again (she reads again) “two functions are equal when they have the same image and (when) the same domain is fixed for both”. 4. And: Because to get to the same image, someone could pass through . . . we could have several journeys; in fact, if there were a subset of the domain . . . we can’t say that the functions are. . . 5. T: . . . Tiz, could you try to explain it? 6. Tiz: Yesterday, we saw that we can, by doing the same domain, we can create the same image and this, with different functions (procedures). After declaring the main goals of the activity, the teacher moderated the interventions focussing on one of the produced texts. The intervention of And (4) made explicit the origin of his and Ale’s definition: the equality of function is related to the coincidence of the trajectories for each subset of the domain. Still, the reference to the use of the trace tool was not explicit; rather it was introduced by the metaphor of journeys (4). Realizing that perhaps some students could not share And’s way of reinvesting the experience with the trace tool, the teacher prompted an explanation (5). Tiz intervenes referring to previous work in Cabri and raised the issue of considering explicitly the procedure that realizes a function. In the following, other interventions focussed on this same issue until the teacher redirected the discussion to the comparison between the definitions and asked to go back to And and Ale’s text. The re-formulation of this sentence in terms of procedure was collectively achieved. Excerpt 2 44. T: Let’s read the text. You say that if they have the same domain and the same image for each subset of the domain. . . 45. Tiz: But, here it’s like to have the same procedure. 46. T: Hummm, and why it’s like to have the same procedure? 47. Several voices: . . . Because . . .

4

48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

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Gab: . . . As we go further, the subsets of the domain and vice versa . . . T: Do you agree, And? Gio: The domain is the plane, then you have the straight line, then a segment . . . T: What are these? And: The domain can be whatever. Gio: They are subsets. T: And then, the procedure, what does it do? That is to say, I . . .. Where does it start from? And: The domain can be one point too . . . if we want! T: The subset of the domain can be one point too. Oh! And: For whatever point, we get the same point of the image. T: And this gives the idea to say that . . . Gio: I’m doing the same procedure. And (together with Gio): I’m doing the same procedure. T: I’m doing the same procedure. Therefore, for whatever point of what? And: For each point of the domain we have the same . . . as the result of the function, the same point of the image. T: Do you agree? (referring to Tiz) Perplexed silences. The teacher writes on the blackboard and reads: “For each point of the domain, we have as the result of the function, the same point as the image”.

At the beginning, students seemed to accept that “to have the same domain and the same image for each subset of the domain” it’s like “to have the same domain and the same procedure” (45, 48 and 50). The agreement with And was based on previous experiences in Cabri, when students’ actions with the tools generated different phenomena according to which And and Ale’s definition appeared sensible. Nevertheless, when the teacher asked for an explicit agreed-upon statement, students remained silent and perplexed (64). In fact, the conclusion that was written on the blackboard by the teacher does not explicitly recognize the key role of the procedure in the identification of a function. In addition, it requires a conceptual move from an experience-based definition, tightly tied to Cabri activities, to a purely mathematical definition, where any reference to moving points and procedures disappears. Further discussion was needed to reach the acceptance of comparing functions point by point. In summary, relying on the potential of students’ produced texts for developing a relationship between experience-based meanings and mathematical meanings, the teacher decided to exploit these produced texts launching a collective discussion based exactly on their comparison. During the discussion she guides the semiotic process towards the inter-subjective construction of a specific mathematical meaning which may be quite different from the students’ personal meaning. We claim that texts produced by the students provide a powerful resource for the teacher.

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4.8 Texts Drawn from History In this section, we discuss the potentialities of a text drawn from a historical source: an excerpt of a text by Euler dealing with the problem of providing a geometric representation of a numerical function: Introduction in Analysis Infinitorum, Tomus secundus, Theoriam Linearum curvarum (1748).3 The students were presented with the excerpt4 and were asked to interpret it. The text analysis shows the potential of evoking the mathematical meanings of graph of a function, and also actions and meanings related to the use of Cabri. For this reason, we consider it as a resource to be exploited to develop a semiotic mediation process. As mentioned earlier, because of its belonging to the shared cultural background, a text drawn from a historical source brings the voice of mathematics to the classroom, through the voice of a famous mathematician. In addition, to the extent to which it can evoke to the students their experience with Cabri, it can contribute to establishing a connection between the students’ personal meanings and the mathematical meanings. The text presented to the students contained Euler’s description of the main steps of the graph construction. For the reader’s convenience, we organize the text into a sequence of steps and mark the missing paragraphs. First Step: Representing the independent variable x as a variable segment AP on a straight line RS: 1. A variable quantity is a magnitude considered in general, and for this reason, it contains all determined quantities. Likewise in geometry a variable quantity is most conveniently represented by a straight line RS of indefinite length [. . .] Since in a line of indefinite length we can cut off any determined magnitude, the line can be associated in the mind with the variable quantity. First we choose a point A in the line RS, and associate with any determined quantity an interval of that magnitude which begins at A. Thus a determined portion of the line, AP, represents the determined value contained in the variable quantity. 2. Let x be a variable quantity which is represented by the line RS, then it is clear that any determined value of x which is real can be represented by an interval of the line RS. For instance, if P is identical with the point A then the interval AP vanishes and represents the value x=0. The farther removed from A the point P is, the greater the definite value of x represented by the interval AP. The interval AP is called the abscissa. The abscissas manifest the determined values of x.

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Introduction to Analysis of the Infinite, Book II. In the text presented to the students some parts were omitted and diagrams and graphs were removed. 4

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Second Step: Representing the dependent variable as a segment PM on the line perpendicular to RS and passing through P. 3. [. . .] 4. Since the indefinite straight line represents the variable x, we would like to see how a function of x can be most conveniently represented. Let y be any function of x, so that y takes on a determined value when a determined value is assigned to x. After having taken a straight line RAS to denote the values of x, for any determined value of x we take the corresponding interval AP and erect a perpendicular interval PM corresponding to the value of y [. . .] Third Step: Associating functions with curves. 5. [. . .] 6. For all determined values x of the line RS, at the point P we erect the perpendicular PM corresponding to the value of y, with different M’s for different P’s [. . .]. All the extremities, M, of the perpendiculars form a line that may be straight or curved. Thus, any function of x is translated into geometry and determines a line, either straight or curved, whose nature is dependent on the nature of the function. 7. In this way, the curve which results from the function y is completely known, since each of its points is determined by the function y. At each point P, the perpendicular PM is determined, and the point M lies on the curve. [. . .] Although Euler certainly had no idea of such a technical drawing device as Cabri, the dynamic description of the graph that he provides is highly consistent with what could be obtained using the Cabri tools. The variation of the independent variable x is represented by the variation of the segment AP (1 and 2), and the variation of the dependent variable PM is implied by the variation of the independent variable. Co-variation is made explicit by the direct link between the two segments. The reference to a point P, moving on a line in Cabri is immediate: in its motion, P “drags” the segment PM, whose length changes in function of the position of P. Thus, the metaphors used in the description of the graph may be directly related to Cabri tools (line, point on a line, dragging . . .). Finally, the potential reference to the trace tool is also very clear (all the extremities, M, of the perpendiculars form a line which may be straight or curved).

4.8.1 The Activities Centred on Euler’s Text Before the activity with Euler’s text the students were introduced through a lab activity to the geometric representation of a function, that is the representation of the variation of a numerical variable and the co-variation of two numerical variables

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linked through a given function. This is the same problem that is at stake in the text by Euler that shortly after the students were asked to interpret. In the following session, the students were asked to read and try to make sense of Euler’s text. Though the activity took place in the computer lab, the students were not explicitly asked to use Cabri. As homework, the students had to explain what they had understood about the method proposed by Euler and to compare it with the method they elaborated for representing a numerical function geometrically through Cabri. A final discussion aimed at sharing a definition of graph of a function – as a geometric representation of a numerical function – and a method for realizing such representation.

4.8.2 Unfolding of the Polysemy of the Text The analysis of the students’ written texts reveals that almost all of them accomplished the interpretation task by providing a paraphrase of the text. The paraphrases are characterized by the use of the mathematical terms previously introduced, such as function, independent or dependent variable. It seems that, the use of these terms fostered the interpretation of the text, that is using these terms helped the students to penetrate the text. Egi: [. . .] the values that are enclosed by AP represent the independent variable x, called abscissa. Thus, to represent a function of x [Euler] decided to take a line perpendicular to x, let’s call it y. To represent any value of the line RS, so that according to the variation of AP (independent variable) the numbers of PM vary too. Besides the use of the specific terminology, many students referred, more or less directly, to the possible use of Cabri. Fed: “Euler describes the function y with independent variable x, whose domain is an unlimited straight line. [. . .] The magnitude of AP, which varies on the straight line according to the movement of P, is called abscissa.” Mar: “The distance of P from A varies as x varies, that [P], as a consequence drags with itself the line MP, perpendicular to RS. This line has been called y; in this way one can say that when x varies y varies, so that a function is created.” The use of Cabri is evoked through metaphors linked to the idea of movement as variation, which is at the core of any DGE. Below is an example where the trace tool is explicitly referred to: Fil:

“[. . .] Let us consider the function y(x), the dependent variable will change according to the variation of the values given to x; this function can be represented geometrically, drawing a straight line RAS which represents the

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values of x, for each value of the variable x assigned to AP, one will draw a perpendicular PM to RAS, such that PM is equal to y. Now if we apply the trace tool at the point M, we find all the points of the function of x [. . .]” Finally, it is worthwhile noticing that some students autonomously decided to use Cabri to make sense of some parts of the text of Euler (we remind the reader that no diagrams were available in the text): Gab: “[. . .] I have no perplexities, though I met some difficulties in understanding the paragraphs 4 and 6, until I used Cabri and reproduce . . ..” Because of its evocative power, Euler’s text works as a secondary artefact related to the primary artefact Cabri: the dynamic description of variables, function and graph provided is consistent with what could be experienced in Cabri. At the same time, realizing Euler’s construction within Cabri may help making sense of the text itself. Hence, the two artefacts have the potential of evoking each other. The articulation of the world of Cabri and the mathematical world evoked by the text can offer the teacher a resource to be exploited in the collective discussion: a multiple perspective relating the activities in Cabri to the mathematical meanings of graph. Indeed, all this happened in the subsequent classroom discussion that was orchestrated by the teacher, with the aim of sharing a definition of graph of a function and a method for producing it. In that discussion, after a first phase in which the students shared their understanding about the method described in Euler’s text, the teacher decided to ask for a drawing illustrating Euler’s method with the temporarily-notdeclared aim of soliciting the reference to Cabri (40. T: “Would anybody be able to make a drawing [. . .] there was not even a drawing [in Euler’s text] [. . .]”). The need of a direct experience of executing the operations to understand the text was also recognized by students: at first the students and the teacher alternated at the blackboard, trying to produce together a suitable drawing, then the class agreed to use Cabri to illustrate Euler’s method: “now we have Cabri which can help us a little. As a matter of fact, what we are going to do is to try to construct all this stuff within Cabri” (182). Thus, Cabri is given the role of contributing to clarify the text. The intertwinement between the text (secondary artefact) and Cabri (primary artefact) is evident.

4.9 Conclusions Within the frame offered by the TSM the use of an artefact has a twofold nature: on the one hand, it is directly used by the students as a means to accomplish a task; on the other hand, it is indirectly used by the teacher as a means to achieve specific educational goals. In this sense, a specific ICT tool can be considered a fundamental resource for the teacher. Nevertheless, according to the model of the teaching action

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provided by the TSM, other types of resources meant to foster and enhance the semiotic mediation process, can be outlined. Asking students to work in pairs at the computer is expected to foster social exchange, accompanied by production of signs related to the use of the artefact, words, sketches, gestures and the like. In this respect not only the specificity of the task but also in particular its formulation constitutes basic resources to trigger the unfolding of the semiotic potential provided by the artefact. Moreover, students may be involved individually in different semiotic activities concerning written productions. All these activities are centred on semiotic processes leading to the production and elaboration of signs, related to the previous activities with tools. Wartofsky’s (1979) classification into primary and secondary artefacts helped us make explicit the synergy between Cabri (primary artefact) and different kinds of written texts related to it (secondary artefacts). Such synergy made these artefacts resources for the teacher to exploit according to her didactic goals. In summary, the realization of the evocative potential of the primary and the secondary artefacts may feed the semiotic mediation process and, thus, foster the evolution towards the mathematical meanings at stake (that is the “tertiary artefacts” which can frame and organize what has been constructed in relation to the use of – primary and secondary – artefacts). As a final remark it seems important to stress a particular contribution offered by this study with respect to teachers’ education. Beside the theoretical contribution given by this study for the development of the TSM, there are interesting implications concerning teachers’ classroom practice and teacher education in general. The awareness of the semiotic potential of written texts and the capacity of selecting and exploiting them in the classroom could become an educational aim for teacher education. The functioning of a text as a resource for developing a semiotic mediation process depends on the possibility of triggering an interpretative process. This may happen either through an explicit or through an implicit request, such as a request of comparison or elaboration. Interpreting concerns both meaning making and expressing, and consequently producing and elaborating signs. In exploiting the polysemy of a text the teacher intentionally articulates meanings coming from the experience with a primary artefact and meanings emerging from a secondary artefact.

References Arzarello, F. (2006). Semiosis as a multimodal process. Relime V1, Especial, 267–299. Bartolini Bussi, M. G. (1998). Verbal interaction in mathematics classroom: A Vygotskian analysis. In H. Steinbring, M. G. Bartolini Bussi, & A. Sierpinska (Eds.), Language and communication in mathematics classroom (pp. 65–84). Reston, VA: NCTM. Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 750–787). Mahwah, NJ: LEA. Bartolini Bussi, M. G., Mariotti, M. A., & Ferri, F. (2005). Semiotic mediation in the primary school: Dürer glass. In M. H. G. Hoffmann, J. Lenhard, & F. Seeger (Eds.), Activity and

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sign – Grounding mathematics education. Festschrift for Michael Otte (pp. 77–90). New York: Springer. Brousseau, G. (1997). Theory of didactical, situations in mathematics. Dordrecht, The Netherlands: Kluwer. Falcade, R. (2006). Théorie des Situations, médiation sémiotique et discussions collective, dans des séquences d’enseignement avec Cabri-Géomètre pour la construction des notions de fonction et graphe de fonction. Grenoble: Université J. Fourier, unpublished doctoral dissertation. Falcade, R., Laborde, C., & Mariotti, M. A. (2007). Approaching functions: Cabri tools as instruments of semiotic mediation. Educational Studies in Mathematics, 66(3), 317–333. Hoyles, C. (1993). Microworlds/schoolworlds: The transformation of an innovation. In C. Keitel & K. Ruthven (Eds.), Learning from computers: Mathematics education and technology (pp. 1–17). NATO ASI Series. Berlin: Springer. Jahnke, H. N., Arcavi, A., Barbin, E., Bekken, O., Furinghetti, F., Idrissi, A., et al. (2000). The use of original sources in mathematics classroom. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education. The ICMI study (pp. 291–328). Dordrecht, The Netherlands: Kluwer. Kozulin, A. (2003). Psychological tools and mediated learning. In A. Kozulin, B. Gindis, V. S. Ageyev, & S. M. Miller (Eds.), Vygotsky’s educational theory in cultural context (pp. 15–38). Cambridge: Cambridge University Press. Laborde J.-M., & Bellemain, F. (1995). Cabri-géomètre II and Cabri-géomètre II plus [computer program]. Dallas, TX: Texas Instruments and Grenoble/France: Cabrilog. Mariotti, M. A., & Maracci, M. (2010). Un artefact comme outil de médiation sémiotique: une ressource pour le professeur. In G. Gueudet & L. Trouche (Eds.), Ressources vives. Le travail documentaire des professeurs en mathématiques (pp. 91–107). Rennes, France: Presses Universitaires de Rennes et INRP. Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70. Vygotsky, L. S. (1978). Mind in society. The development of higher psychological processes. Cambridge: Harvard University Press. Wartofsky, M. (1979). Perception, representation, and the forms of action: Towards an historical epistemology. In M. Wartofsky (Ed.), Models, representation and the scientific understanding (pp. 188–209). Dordrecht: D. Reidel Publishing Company.

Reaction to Part I Resources Can Be the User’s Core Bill Barton

How does a resource become “lived”? If we may play on the etymology for a moment, becoming “lived” means enlivened, to get full of life, to become, to be born. The four chapters of this part tell us how a resource enters the world “mewling and puking in the nurse’s arms” (Shakespeare, As You Like It, Act II, Scene 7). The bard gets it right, again. The carer of the resource is responsible for nurturing and shaping its potential, helping it to grow, ignoring the unseemly squeaking, and clearing up the spilt milk. On the one hand, Adler shows us how the nurse invests herself into the new life: how does teacher knowledge emerge during events in the classroom? Gueudet and Trouche want us to focus on the nurse’s actions in caring for the baby: how do resources become transformed in a particular teacher’s hands? Sensevy, on the other hand, wants us to look at the nurse’s aims: how are the teacher’s actions driven by developing intentions? Mariotti and Maracci ask us to watch the baby itself as it interacts with the nurse and others in the world: how can resources change the way people think and act? The common stance is one of mediation, the transformation of resources by teachers as they are reborn from a prior, relatively fixed state to a new dynamic existence in action in the classroom. I am tempted to play with etymology yet again. Mediation does not derive from media, but in this section we are being asked to pay attention to media. Famously, “the medium is the message” (McLuhan, 1964), or, more appropriately for this context, The Medium is the Massage (McLuhan and Fiore, 1967). Not only the resource but also the form of the resource alter the way it can be used and transformed by a teacher. We are presented with four different ways to conceive of resource mediation. The authors draw heavily on established theory, modifying it for their purpose, and we are left with a strongly grounded feeling. What do the four perspectives offer us? Adler draws on social practice theory to present us with an integrated view. Teacher’s knowledge, their access to texts, the classroom environment, the language

B. Barton (B) Department of Mathematics, University of Auckland, Auckland 1142, New Zealand e-mail: [email protected]

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resources available, and the pre-defined curriculum merge through the act of teaching to legitimise a particular view of mathematics. Teachers, whether they like it or not, whether they are prepared for it or not, are central in this process and bear its responsibility. I understand immediately the research-based wisdom that teachers are the most important factor in learning. As mathematics educators we are asked to pay attention to how we, in teacher education, open or close opportunities for particular mathematical perceptions. Gueudet and Trouche introduce documentational genesis, the evolution of teaching materials in the hands of a teacher drawing on several resources for a particular classroom outcome. On the basis of activity theory and an instrumental approach, the focus on documents changes the way we look at teaching. Tracing documentary evolution enables us to see, physically, the teacher’s moves in the game of instruction, and also the development of a teacher’s ideas, intentions and pedagogic orientation over a long time. For me, the importance of this perspective is the way it highlights continual change. I believe that many teachers would regard their practice as relatively stable – and many developers and education researchers comment on teachers’ resistance to change. A documentational genesis is likely to prove the lie to such statements, and thereby challenges us all to think again about the way development can be influenced. For example, it will reveal constant but gradual change – the antithesis of many programmes of teacher development. Sensevy also relies on documentation, and follows Bourdieu’s idea of a social game and Brousseau’s didactic contract. He asks us to pay attention to the way a teacher sets up the game (or contract), embedding explicit pedagogic intentions in both the resources and the elaboration of the game. The research data forces me to consider the ways the process goes wrong: during classroom interaction the response to the resources can diverge from the intention. This creates a didactic moment, a decision point, a phenomenon investigated by many researchers. Mason (1999, 2010) also focuses on teaching moments, and Schoenfeld (1987, 2008) persists in his analysis of classroom decisions. Schoenfeld’s KOG analysis (knowledge, orientations and goals) of teacher behaviour resonates with Sensevy’s work. My response to this perspective is to wonder anew how to prepare for such moments. The very act of Sensevy’s research sensitised his teachers to their predicament. They knew that they would be questioned on their actions at the critical moments, and it was almost as if that knowledge altered the decision they made. Can heightened awareness be a mode of professional development? How could we bring this about? Mariotti and Maracci turn our attention to the learner to learn about the teacher’s mediation of resources. Semiotic mediation of artefacts require us to investigate the meaning given to a resource, and how that meaning changes (or can be changed) with teacher action. A key word I take from the chapter is “invoking”. Meaning is invoked; learning does not reside in the resource, it is invoked by it. My reaction, then, is to think about the invoking power of a resource. This gives us, for example, a way of investigating technology: does modern technology have a greater power to invoke, perhaps because it is interactive and dynamic compared to texts. Are recorded lessons to be seen in the same way? Mariotti and Maracci note in their last paragraph that written texts have the advantage (over spoken words and gestures)

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of permanence and reproducibility. No longer! Video recordings, where the richness of gesture and articulation are preserved, may have more invoking power than conventional texts. In several places Mariotti and Maracci themselves invoke the multiple roles of resources: the double semiotic link, Wartofsky’s triple classification of artefacts and Winsløw’s pragmatic and didactic roles. Artigue (2002) distinguishes between three “values” when discussing the role of technology in mathematics education. The pragmatic value or productivity of the technology: how it helps us in the mathematical action we are currently undertaking. The epistemic value: how technology helps students understand the mathematical objects they are dealing with. The heuristic value: how technology contributes to understanding future or more advanced concepts. Hence, not only is the mediation of the resource transforming a generalised object into an object-in-action, but the mediation occurs on several levels simultaneously. To what extent are teachers aware of this in general, and in the moment? Taken collectively, the four chapters raise the issue of teacher awareness of their mediation role with respect to resources. Assuming the analyses are well-founded (and I have argued that indeed they are), we must ask ourselves how teachers come to know to transform as well as how to transform. We must also ask about developing both the confidence (to undertake mediation of resources) and the habit (to do so). If nothing else, these chapters emphasise the importance of such tasks. But, seated as they are in well-tried theoretical frames, we might expect that the four chapters will illuminate familiar phenomena within classroom experiences. Do they do this? The familiar phenomenon of teacher resistance to change has already been mentioned, and we are asked to re-evaluate this perception in the light of evidence that teacher change evolves over long time periods. University lecturers’ reluctant weaning from blackboards and slow adoption of technology can be better understood as being wedded not to the practice of chalk dust and dusters, but to the particular construction of mathematics that blackboards activate. Similarly, the apparently wasteful teacher habit of writing and rewriting mathematical notes that are readily available in neatly formatted and triple-checked textbook form can be explained by documentational genesis, and the need to personally transform ones’ pedagogical intentions through the resource. What about the research-verified phenomenon of teachers excitedly engaging in mathematical content whether or not it relates to their teaching? Their enthusiasm is not just a product of an inordinate love of their subject. Teachers’ mathematical knowledge also goes out of date, and they are aware of it. They are also aware that subject knowledge is one of the most powerful sources of effective teaching – and it needs to be re-sourced. We would think twice about using a 30-year-old text, but many teachers are still using their 30-year-old mathematics. No wonder that a judgement-free opportunity to re-source is welcomed with open arms. And finally, what does the analysis tell us about the powerful effect of communities of professional teachers? Why is teacher development so much better when done in a community? Because resource mediation is a social practice. Explicitly, in

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Sensevy’s thought collective but also implicitly. An unmediated resource is someone else’s voice; the teacher, through mediation, has a conversation with its originator. How much more powerful it is when the dialogue is a discussion. Teaching resources, like their fuel namesakes, must be mined. Often the extraction is an expensive business, requiring an investment of time and money (e.g. software or textbook production). Sometimes there is significant pollution and waste (e.g. travesties of repetitive exercises masquerading as mathematics), corruption (e.g. false claims for technology) and a carbon footprint that needs to be compensated (wasted teacher time in top-down workshops). Nevertheless, when their energy is released by the internal combustion of a teacher in action, the results can be explosive and, as the anagram of the title to this reaction suggests, become the core of the teacher’s task.

References Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245–274. Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention. Journal of Mathematics Teachers Education, I, 243–267. Mason, J. (2010). Attention and intention in learning about teaching through teaching. In R. Leikin & R. Zazkis (Eds.), Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice (pp. 23–47). New York: Springer. McLuhan, M. (1964). Understanding media: The extensions of man. New York: Mentor. McLuhan, M., & Fiore, Q. (1967). The medium is the massage. New York: Random House. Schoenfeld, A. (1987). What’s all the fuss about metacognition. In A. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189–215). Hillsdale, NJ: Lawrence Earlbaum. Schoenfeld, A. (2008). On modelling teachers’ in-the-moment decision-making. In A. Schoenfeld (Ed.), A study of teaching: Multiple lenses, multiple views (pp. 45–96). Reston, VA: National Council of Teachers of Mathematics.

Part II

Text and Curriculum Resources

Chapter 5

Constituting Digital Tools and Materials as Classroom Resources: The Example of Dynamic Geometry Kenneth Ruthven

5.1 Introduction This chapter examines the often unrecognised challenges that teachers face in seeking to make effective use of new mathematical tools and representational media in the classroom, highlighting several key facets of professional learning associated with overcoming these challenges. It focuses on the appropriation of digital tools and media as resources for the mainstream practice of secondary-school mathematics teaching, taking the particular example of dynamic geometry to illustrate this process. First, the chapter demonstrates the interpretative flexibility surrounding a resource and the way in which wider educational orientations influence conceptions of its use. It does so by showing how pedagogical conceptions of dynamic geometry have shifted between pioneering advocates and mainstream adopters; and how such conceptions vary across adopters according to their wider approaches to teaching mathematics. Second, the chapter outlines a conceptual framework intended to make visible and analysable the way in which certain structuring features shape the incorporation of new technologies into classroom practice. This conceptual framework is then used to examine the case of a teacher leading what – for him – is an innovative lesson involving dynamic geometry, and specifically to identify how his professional knowledge is being adapted and extended. This shows how the effective integration of new technologies into everyday teaching depends on a more fundamental and wide-ranging adaptation and extension of teachers’ professional knowledge than has generally been appreciated.

K. Ruthven (B) Faculty of Education, University of Cambridge, Cambridge CB2 8PQ, UK e-mail: [email protected]

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5.2 The Interpretative Flexibility of Educational Resources Studies of the social shaping of technology have drawn attention to the ‘interpretative flexibility’ through which the function and operation of a tool remain open to adaptation (MacKenzie & Wajcman, 1999). In particular, conceptions of a technology influence its non-adoption by potential users, or its appropriation by them in the light of their interests and circumstances; indeed, technologies may be taken up in ways which, in terms of the speculative intentions of their designers, appear as something of a misappropriation. The concept of ‘innofusion’, then, blurs the conventional technocratic model of development in proposing that innovation carries on throughout the process of diffusion, as a technology and its modalities of use become aligned with user concerns and adapted to use settings (Williams & Edge, 1996). Contemporary educational studies adopt a similar perspective on curriculum materials and pedagogical guidance. Such resources have long provided a staple approach to influencing classroom practice. However, attempts to ‘teacher proof’ them, and the recurring failure of these efforts even more so, testify that teachers act as interpreters and mediators of them. This reflects a broader pattern in which the unfolding of innovation in education is shaped by the sense-making of the agents involved (Spillane, Reiser, & Reimer, 2002). Teachers typically select, combine and adapt resources, and they necessarily incorporate them into wider systems of classroom practice (Ball & Cohen, 1996). Accordingly, conceptualisations of how resources are used have developed from rather limited views of teachers simply following or subverting them, to more sophisticated perspectives encompassing teacher interpretation of, and participation with, them (Remillard, 2005). Interpretative flexibility became very apparent during the early development of geometry software. Originally intended to provide computer-supported analogues to established manual processes for the construction of figures, geometric software underwent a significant evolution with the recognition that, on a computer screen, such figures could be made dynamic, changing shape in response to the dragging of points or segments, but preserving their defining properties (Scher, 2000). Although the dragging operation rapidly became a defining feature of dynamic geometry software, its functional versatility and corresponding complexity were not anticipated, and are still in the process of being established (Arzarello, Olivero, Paola, & Robutti, 2002; Laborde, 2001). Equally, although dynamic geometry systems were developed with educational purposes in view, they were not initially devised with a particular pedagogical approach in mind (Scher, 2000). However, pioneering work quickly associated dynamic geometry with a pedagogical orientation in which such software served ‘to create experimental environments where collaborative learning and student exploration are encouraged’ (Chazan & Yerushalmy, 1998, p. 8), so that ‘mathematics becomes an investigation of interesting phenomena, and the role of the mathematics student becomes that of the scientist’ (Olive, 2002, p. 17). Nevertheless, evidence about how dynamic geometry has actually been taken up in schools offers an enigmatic picture. For example a national survey conducted in the United States found

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an association between teachers nominating dynamic geometry as their most valued software and reporting skill-development as their main objective for computer use (Becker, Ravitz & Wong, 1999).

5.3 An English Study of Teacher Constructions of Dynamic Geometry A recent English study has thrown further light on the use of dynamic geometry in mainstream practice (Ruthven, Hennessy, & Deaney, 2008). Much of the pioneering development of dynamic geometry systems has taken place in countries – notably France and the United States – which comparative studies show to have retained a strongly Euclidean spirit within their school geometry curriculum, resulting in greater attention to formalisation and systematisation, including an emphasis on proof (Hoyles, Foxman, & Küchemann, 2001). The Euclidean lineage of dynamic geometry might be expected to fit poorly with a national curriculum which refers – as does the English one framing the practice studied – not to ‘geometry’ but to ‘shape, space and measures’. However, the scope to employ the software as a means of supporting observation, measurement and calculation resonates with the empirical style of English school mathematics, and such modalities of reasoning were found to be prevalent when dynamic geometry was used. The study found echoes of the exploratory rhetoric of the software’s advocates in teachers’ suggestions that dynamic figures helped students to ‘find out how it works without us telling them’, or ‘tell you the rule instead of you having to tell them’, so that students were ‘more or less discovering for themselves’ and could ‘feel that they’ve got ownership of what’s going on’, even if teachers might have to ‘structure’, ‘hint’, ‘guide’ or ‘steer’ students towards the intended mathematical conclusion. Case studies identified a range of practical expressions of this idea. One case involved a strongly teacher-led, whole-class approach, in which dynamic presentation by the teacher was used to make it easier for students to ‘spot the rule’ so that ‘you’re not just telling them a fact, you’re allowing them to sort of deduce it and interact with what’s going on’. In the other cases, the classroom approaches involved more devolution to students, through investigations structured towards similarly preconceived mathematical results, with the teacher ‘drawing attention to’, ‘flagging up’ and ‘prompting’ them. On the issue of students themselves making use of the software, classroom approaches were found to be based variously on avoiding, minimising or capitalising on the demands of using dynamic geometry. In the first case referred to above, the software was used only for teacher presentation on the grounds that ‘it would take a long time. . . for [students] to master the package’ and ‘the return from the time investment. . . would be fairly small’, so that ‘the cost benefit doesn’t pay’. In two further cases, the normal pattern was ‘to structure the work so [students] just have to move points [on a prepared figure]’, so that ‘they don’t have to be complicated by that, they really can just focus on what’s happening mathematically’. In the final

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case, getting students to construct their own dynamic figures was seen as a vehicle for developing and disciplining their geometrical thinking; using dynamic geometry was introduced to them in terms of: ‘It’s not just drawing, it’s drawing using mathematical rules’. Thus, the degree to which students were expected to make use of dynamic geometry was influenced by the extent to which this was conceived as promoting mathematically productive activity. A related issue concerned handling the apparent mathematical anomalies which arise when dynamic figures are dragged to positions where an angle becomes reflex (with the associated problem of measurement), or where rounded values obscure an arithmetical relationship between measures (as featured in Fig. 5.1). The potential for such situations to arise was considerable in the type of topic most widely reported as suited to dynamic geometry: the study of angle properties. For example two of the case studies included a lesson on the angle sum of polygons (both employing a figure of the type shown in Fig. 5.1). In the first case, the teacher took great care to avoid exposing students to apparent anomalies of these types, through vigilant dragging to avoid ‘possibilities where students may become confused, or things that might cloud the issue’. In the other case, the teacher actively wanted students to encounter such difficulties so as to learn ‘that you can’t assume that what you’ve got in front of you is actually what you want, and you have to look at it . . . and question it’; equally, resolving such situations was seen as serving ‘to draw attention to . . . how the software measures the smaller angle, thus reinforcing that there are two angles at a point and [that students] needed to work out the other’. Thus, approaches to handling these apparent mathematical anomalies were influenced by whether they were seen as providing opportunities to develop students’ mathematical understanding, in line with a more fundamental pedagogical orientation that saw analysis of discrepancies as supporting learning. This study, then, highlights several noteworthy aspects of the interpretative flexibility of dynamic geometry. First it shows that the forms of guided discovery that dynamic geometry is typically used to support in English classroom practice, as well as the empirical and arithmetical modes of reasoning associated with them, are very different from the types of mathematical enquiry and modes of mathematical reasoning envisaged by the original proponents of the software. Equally, it shows

72.0°

79.0° 107.0°

76.0°

Fig. 5.1 Dynamic geometry figure for establishing the angle sum of a pentagon

60.0°

Angle sum = 394.1°

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how differing approaches to staging guided discovery, and organising the associated software use, reflect varied interpretations of the functionality for students of dynamic geometry, shaped by contrasting conceptions of what it means for students to learn mathematics. These case studies were carried out in mathematics departments that were professionally well regarded for their use of digital technologies. Even in these departments, the exposure of any one class to dynamic geometry was of the order of a handful of lessons each year. Moreover, when the software was used, teachers largely sought to minimise disruption to customary patterns of classroom activity. Indeed, research on how teachers make use of the interactive whiteboards now widely available in English classrooms reports that software such as dynamic geometry is generally rejected as over-complex or used only in limited ways (Miller & Glover, 2006). Such observations suggest that it is not just the way in which teachers conceptualise dynamic geometry as a teaching resource that influences their response to it, but more basic concerns about how to realise its incorporation within a viable classroom practice.

5.4 Structuring Features of Classroom Practice Such concerns are often overlooked in educational reform, and with them the craft knowledge that underpins everyday classroom practice (Brown & McIntyre, 1993; Leinhardt, 1988). In particular, much proposed innovation entails modification of the largely reflex system of powerful schemes, routines and heuristics that teachers bring to their classroom work, often tailored to their particular circumstances. The conceptual framework that I will now develop focuses, then, on the functional organisation of a system of (often tacit) pedagogical craft knowledge required to accomplish concrete professional tasks (consequently this framework does not directly consider the subject disciplinary knowledge required of the teacher, although this too plays a part). This section will introduce five key structuring features of classroom practice and show how they relate to the constitution of digital tools and materials as classroom resources: working environment, resource system, activity format, curriculum script and time economy.

5.4.1 Working Environment Making use of computer-based tools and materials in teaching often involves changes in the working environment in which lessons are conducted; namely, the physical surroundings where lessons take place, their general technical infrastructure, and the social organisation associated with them. In many schools, lessons have to be relocated from the normal classroom to a dedicated computer suite so as to make machines available in sufficient numbers for students to work with them. Such use entails disruption to normal working practices

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and makes additional organisational demands on the teacher (Jenson & Rose, 2006; Ruthven, Hennessy, & Deaney, 2005). Well-established routines which help lessons to start, proceed and close in a timely and purposeful manner in the regular classroom (Leinhardt, Weidman, & Hammond, 1987) have to be adapted to the computer suite. The alternative of providing sets of handheld devices or laptop computers in the ordinary classroom raises similar organisational issues. For example teachers report having to develop classroom layouts that assist them to monitor students’ computer screens, as well as classroom routines to forestall distraction, such as having students push down the screens of their laptops during whole-class lesson segments (Zucker & McGhee, 2005). More recently, there has been a trend towards provision of digital projection facilities or interactive whiteboards in ordinary classrooms. Their attraction to many teachers is that they require fewer modifications to the customary working environment of lessons (Jewitt, Moss, & Cardini, 2007; Miller & Glover, 2006). Such facilities can be treated as a convenient enhancement of a range of earlier display and projection devices, and allow a single classroom computer to be managed by teachers on behalf of the whole class.

5.4.2 Resource System New technologies have broadened the types of subject- and topic-specific resources available to support school mathematics. Educational suppliers now market textbook schemes alongside exercise and revision courseware, concrete apparatus alongside computer micro-worlds and environments, manual instruments alongside digital tools. The collection of mathematical tools and materials in classroom use constitutes a resource system which depends for its successful functioning on their being used in a co-ordinated way aligned with educational goals (Amarel, 1983). Studies of the classroom use of computer-assisted instructional packages have attributed strong take-up of particular materials to their close fit with the regular curriculum and their flexibility of usage (Morgan, 1990). Equally, teachers report that they would be much more likely to use technology if ready-to-use resources were readily available to them and clearly mapped to their scheme of work (Crisan, Lerman, & Winbourne, 2007). An important factor here is the limited scope that many digital materials offer for the teacher customisation characteristic of the use of other resources, and recognition of this has encouraged developers to offer greater flexibility to teachers. However, whatever the medium employed, teachers need to acquire knowledge in depth of materials so as to make effective use of them and to integrate them successfully with other classroom activity (Abboud-Blanchard, Cazes, & Vandebrouck, 2007; Bueno-Ravel & Gueudet, 2007). Something close to the textbook – even if taking a digitised form – remains at the heart of the resource system in many classrooms, valued for establishing a complete and coherent framework within which material is introduced in an organised and controlled way, appropriate to the intended audience. Indeed, one common use of

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interactive whiteboards in classrooms is to project and annotate textbook pages or similar presentations (Miller & Glover, 2006). More broadly, educational publishers are seeking increasingly to bundle digital materials with printed textbooks, often in the form of presentations and exercises linked to each section of the text, or applets providing demonstrations and interactivities. Such materials are attractive to many teachers because they promise a relatively straightforward and immediately productive integration of old and new technologies. Textbook treatments of mathematical topics necessarily make assumptions about what kinds of tools will be available in the classroom. Nowadays, it is increasingly assumed that some kind of calculator will be available to students. If well designed, textbooks explicitly develop the calculator techniques required and establish some form of mathematical framing for them. However, it is rare to find them taking account of other digital mathematical tools. Here, textbook developers face the same problems as classroom teachers. In the face of a proliferation of available tools, which should be prioritised? And given the currently fragmentary knowledge about bringing these tools to bear on curricular topics, how can a coherent use and development be achieved? Such issues are exacerbated when tools are imported into education from the commercial and technical world. Often, their intended functions, operating procedures, and representational conventions are not well matched to the needs of the school curriculum.

5.4.3 Activity Format The processes of classroom teaching and learning are played out within recurring patterns of teacher and student activity. Classroom lessons can be segmented according to recognisable activity formats: generic templates for action and interaction which frame the contributions of teacher and students to particular types of lesson segment (Burns & Anderson, 1987; Burns & Lash, 1986). The crafting of lessons around a succession of familiar activity formats and their supporting classroom routines helps to make them flow smoothly in a focused, predictable and fluid way (Leinhardt, Weidman, & Hammond, 1987), permitting the creation of prototypical activity structures or activity cycles for lessons as a whole. Monaghan (2004) studied secondary teachers who had made a commitment to move from making little use of ICT in their mathematics classes to making significant use. For each participating teacher, a ‘non-technology’ lesson was observed at the start of the project, and further ‘technology’ lessons over the course of the year. Monaghan found that technology lessons tended to have a quite different activity structure. In all the observed non-technology lessons, teacher-led exposition including the working-through of examples was followed by student work on related textbook exercises. Of the observed technology lessons, only those which took place in the regular classroom using graphic calculators displayed this type of structure. Most of the technology lessons focused on more open tasks, often in the form of investigations. These featured an activity structure consisting typically of a short introduction to the task by the teacher, followed by student work at computers

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over most of the session. Both types of technology lesson observed by Monaghan appear, then, to have adapted an existing form of activity structure: less commonly that of the exposition-and-practice lesson; more commonly that of the investigation lesson. Other studies describe classroom uses of new technologies that involve more radical change in activity formats, and call for new classroom routines. For example to provide an efficient mechanism through which the teacher can shape and regulate methods of tool use, Trouche (2005) introduces the role of ‘sherpa student’, taken on by a different student in each lesson. The sherpa student becomes responsible for managing the calculator or computer that is being publicly projected during wholeclass activity; what is distinctive about this activity format is the way in which it is organised around the teacher guiding the actions of the sherpa student, or opening them up for comment and discussion by the remainder of the class; the particular function it serves is in providing a mechanism by which the teacher can manage the collective development of techniques for using the tool. A new activity format of this type calls, then, for the establishment of new classroom norms for participation and the adaptation of existing classroom routines to support its smooth functioning.

5.4.4 Curriculum Script In planning to teach a particular topic, and in conducting lessons on it, teachers draw on (evolving) knowledge gained in the course of their own experience of learning and teaching that topic, or gleaned from available curriculum materials. Such knowledge is organised as a curriculum script, where ‘script’ is used in the psychological sense of a form of event-structured organisation: a loosely ordered model of relevant goals and actions that guides teachers’ handling of the topic, and includes variant expectancies of a situation and alternative courses of action (Leinhardt, Putnam, Stein, & Baxter, 1991). A curriculum script interweaves ideas to be developed, tasks to be undertaken, representations to be employed and difficulties to be anticipated in the course of teaching that topic, and links these to relevant aspects of working environment, resource system and activity structure. Teachers frequently talk about the use of new technologies in terms which appear to involve the adaptation and extension of established curriculum scripts (Ruthven & Hennessy, 2002). For example they talk about a new technology as a means of improving existing approaches to a topic, suggesting that it serves as a more convenient and efficient tool for supporting specific mathematical processes, or provides a more vivid and dynamic presentation of particular mathematical properties. Nevertheless, it is easy to underestimate the host of small but nuanced refinements which existing curriculum scripts require so as even to assimilate a new technology, let alone adapt the approach taken to a mathematical topic in the light of fresh insights gained from using the technology to mediate it. When teachers participate in development projects, they experience pressure (often self-administered) to use technology more innovatively. Monaghan (2004) reports, for example, that teachers had difficulty in finding resources to help them

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devise and conduct technology lessons on an investigative model. Consequently, they were obliged to plan such lessons at length and in detail, and then found themselves teaching rather inflexibly. The extent and complexity of such adoption is still greater when ‘imported’ technologies need to be aligned with the school curriculum. Monaghan compares, for example, the relative ease with which new lessons could be devised around the use of graphware specifically devised for educational use, with the much greater demands of appropriating ‘imported’ computer algebra systems to curricular purposes. These challenges become particularly severe in an educational culture, such as the French one, which emphasises a rigorous articulation of mathematical ideas and arguments (Artigue, 2002; Ruthven, 2002).

5.4.5 Time Economy The concept of time economy (Assude, 2005) focuses on how teachers seek to manage the ‘rate’ at which the physical time available for classroom activity is converted into a ‘didactic time’ measured in terms of the advance of knowledge. Although new tools and materials are sometimes represented as displacing old to generate a time bonus, it is more common to find a double instrumentation in operation, in which old technologies remain in use alongside new. In particular, old technologies may make an epistemic, knowledge-building contribution as much as a pragmatic, task-effecting one (Artigue, 2002). This double instrumentation means that new technologies often give rise to cost additions rather than to cost substitutions with respect to time. Thus, a critical concern of teachers is to fine-tune resource systems, activity structures and curriculum scripts to optimise the rate of didactic return on the time investment (Bauer & Kenton, 2005; Crisan et al., 2007; Smerdon, Cronen, Lanahan, Anderson, Iannotti, Angeles, & Greene, 2000). A critical issue is what teachers perceive as the mathematical learning that results from students using new tools. As noted in the earlier discussion of dynamic geometry, teachers are cautious about new tools which require substantial investment, and alert for modes of use which reduce such investment or increase rates of return. These concerns to maximise the time explicitly devoted to recognised mathematical learning are further evidenced in the trend to equip classrooms with interactive whiteboards, popularised as a technology for increasing the pace and efficiency of lesson delivery, as well as harnessing multimodal resources and enhancing classroom interaction (Jewitt et al., 2007). Evaluating the developing use of interactive whiteboards in secondary mathematics classrooms, Miller & Glover (2006) found that teachers progressed from initial teaching approaches in which the board was used only as a visual support for the lesson, to approaches where it was used more deliberately to demonstrate concepts and stimulate responses from pupils. In the course of this development, there was a marked shift away from pupils copying down material from the board towards use ‘at a lively pace to support stimulating lessons which minimise pupil behaviour problems’ (p. 4). However, in terms of the type of mathematical resource used with the board, there was little progression beyond textbook type sources and prepared presentation files; more generic

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mathematical resources such as spreadsheet, graphing and geometry programs were rejected by teachers as over-complex or used by them only in limited ways.

5.5 Practitioner Thinking and Professional Learning in an Innovative Lesson The conceptual framework sketched in the last section will now be used to analyse the practitioner thinking and professional learning surrounding one of the lessons from the earlier study of classroom practice incorporating dynamic geometry use (Ruthven et al., 2008). In the original study, this specific nomination was followedup not only because the teacher concerned had talked lucidly about his experience of teaching such a lesson for the first time, but also because he displayed particular awareness of the potential of dynamic geometry for developing visuospatial and linguistic aspects of students’ geometrical thinking. Thus, this case was chosen for investigation as a prospectively interesting outlier where a teacher appeared to be developing a form of classroom practice more consonant with the style of dynamic geometry use envisaged by its protagonists. Because the teacher was unusually expansive in interview, touching on a range of aspects of practitioner thinking and professional learning, this case was also particularly suited to further analysis in terms of the structuring features identified in the conceptual framework outlined in the previous section. Nevertheless, it should be borne in mind that the original study was not designed or conducted with this conceptual framework in mind; rather, it has provided a subsequently convenient means of exploring application of the framework to a concrete example.

5.5.1 Orientation to the Lesson When initially nominating a recent lesson as an example of successful practice, the teacher explained that it had been developed in response to improved technology provision in the mathematics department, notably the installation of interactive whiteboards in ordinary classrooms. He reported that the lesson (with a class in the early stages of secondary education) had started with him explicitly constructing a triangle, and then the perpendicular bisectors of its edges. The focus of the investigation which ensued had been on employing dragging to examine the idea that this construction might identify the ‘centre’ of a triangle (Fig. 5.2). According to the teacher, one particularly successful aspect of the lesson had been the extent to which students actively participated in the investigation. Indeed, because of the interest and engagement shown by students, the teacher had decided to extend the lesson into a second session, held in a computer room to allow students to work individually at a computer. For the teacher, the ready recall by students in this second session of ideas from the earlier session was another striking aspect of the lesson’s success. In terms of the specific contribution of dynamic geometry to

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Fig. 5.2 The basic dynamic figure employed in the investigative lesson

this success, the teacher noted how the software supported exploration of different cases, and overcame the manipulative difficulties which students encountered in using classical tools to attempt such an investigation by hand. But the teacher saw the contribution of the software as going beyond ease and accuracy; using it required properties to be formulated precisely in geometrical terms. These, then, were the terms in which this earlier lesson was nominated as an example of successful practice. We followed up this nomination by studying a lesson along similar lines, conducted over two 45-minute sessions on consecutive days with a Year 7 class of students (aged 11–12) in their first year of secondary education.

5.5.2 Working Environment Each session of the observed lesson started in the normal classroom and then moved to a nearby computer suite where it was possible for students to work individually at a machine. This movement between rooms allowed the teacher to follow an activity cycle in which working environment was shifted to match changing activity format. Even though the computer suite was, like the teacher’s own classroom, equipped with a projectable computer, starting sessions in the classroom was expedient as doing so avoided disruption to the established routines underpinning the smooth launch of lessons. Moreover, the classroom provided an environment more conducive to sustaining effective communication during whole-class activity and to maintaining the attention of students. Whereas in the computer suite each student was seated behind a sizeable monitor, blocking lines of sight and placing diversion at students’ fingertips, in the classroom the teacher could introduce the lesson ‘without the distraction of computers in front of each of them’. It was only recently that the classroom had been refurbished and equipped, and a neighbouring computer suite established for the exclusive use of the mathematics department. The teacher contrasted this new arrangement favourably in terms of the easier and more regular access to technology that it afforded, and the consequent

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increase in the fluency of students’ use. New routines were being established for students opening a workstation, logging on to the school network, using shortcuts to access resources and maximising the document window. Likewise, routines were being developed for closing computer sessions. Towards the end of each session, the teacher prompted students to plan to save their files and print out their work, advising them that he’d ‘rather have a small amount that you understand well than loads and loads of pages printed out that you haven’t even read’. He asked students to avoid rushing to print their work at the end of the lesson, and explained how they could adjust their output to try to fit it onto a single page; he reminded them to give their file a name that indicated its contents, and to put their name on their document to make it easy to identify amongst all the output from the single shared printer.

5.5.3 Resource System The department had its own ‘schemes of work’ (a term used in English schools for a written schedule of topics to be taught to particular year-groups, that usually includes suggestions for resources to be used) with teachers encouraged to explore new possibilities and report to colleagues. This meant that teachers were accustomed to integrating material from different sources into a common scheme of work. However, so wide was the range of computer-based resources currently being trialled that our informant (who was head of department) expressed concern about incorporating them effectively into departmental schemes, and about the demands of familiarising staff and students with such a variety of tools. In terms of coordinating use of old and new technologies, work with dynamic geometry was seen as complementing established work on construction with classical manual tools, by strengthening attention to the related geometric properties. Nevertheless, the teacher felt that old and new tools lacked congruence, because certain manual techniques appeared to lack computer counterparts. Accordingly, old and new were viewed as involving different methods and having distinct functions. While ruler and compass were seen as tools for classical constructions, dynamic software was ‘a way of exploring the geometry’. Equally, some features of computer tools were not wholly welcome. For example the teacher noted that students could be deflected from the mathematical focus of a task, spending too much time on cosmetic aspects of presentation. During the lesson the teacher had tried out a new technique for managing this, by briefly projecting a prepared example to show students the kind of report that they were expected to produce, and to illustrate appropriate use of colour coding. In effect, by showing students to what degree, and for what purpose, he regarded it as legitimate for them to ‘slightly adjust the font and change the colours a little bit, to emphasise the maths, not to make it just look pretty’, the teacher was developing sociomathematical norms (Yackel & Cobb, 1996) for using the new technology, and developing a classroom strategy for establishing these norms.

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5.5.4 Activity Format Each session of the observed lesson followed a similar activity cycle, starting with teacher-led activity in the normal classroom, followed by student activity at individual computers in the nearby computer suite, with this change of rooms during sessions serving to match working environment to activity format. Indeed, when the teacher had first nominated this lesson, he had remarked on how it combined a range of activity formats – ‘a bit of whole class, a bit of individual work and some exploration’ – to create a promising lesson structure; one that he would ‘like to pursue because it was the first time [he]’d done something that involved quite all those different aspects’. In discussing the observed lesson, however, the teacher highlighted one aspect of the model which had not functioned as well as he would have liked: the fostering of discussion during individual student activity. He identified a need for further consideration of the balance between opportunities for individual exploration and for productive discussion, through exploring having students work in pairs. At the same time, the teacher noted a number of ways in which the computer environment helped to support his own interactions with students within an activity format of individual working. Such opportunities arose from helping students to identify and resolve bugs in their dynamic geometry constructions. Equally, the teacher was developing ideas about the pedagogical affordances of text-boxes, realising that they created conditions under which students might be more willing to consider revising their written comments because this could be done with ease and without his interventions being seen by students as ‘ruin[ing] their work’ by spoiling its presentation. This was helping him to achieve his goal of developing students’ capacity to express themselves clearly and precisely in geometrical terms through refining their statements of properties.

5.5.5 Curriculum Script The observed lesson followed on from earlier ones in which the class had undertaken simple classical constructions with manual tools: in particular, using compasses to construct the perpendicular bisector of a line segment. Further evidence that the teacher’s curriculum script for this topic originated prior to the availability of dynamic geometry was his reference to the practical difficulties which students encountered in working by hand to accurately construct the perpendicular bisectors of a triangle. His evolving script now included not only the knowledge of ‘unusual’ and ‘awkward’ aspects of software operation liable to ‘cause[] a bit of confusion’ amongst students, but also of how such difficulties might be turned to advantage in reinforcing the mathematical focus of the task so that ‘sometimes the mistakes actually helped’. Equally, the teacher’s curriculum script anticipated that students might not appreciate the geometrical significance of the concurrence of perpendicular bisectors, and incorporated strategies for addressing this, such as trying ‘to get them to see that

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. . . three random lines, what was the chance of them all meeting at a point’. This initial line of argument was one already applicable in a pencil and paper environment. Later in the interview, however, the teacher made reference to another strategy which brought the distinctive affordances of dragging the dynamic figure to bear on this issue: ‘When I talked about meeting at a point, they were able to move it around’. Likewise, his extended curriculum script depended on exploiting the distinctive affordance of the dynamic tool to explore how dragging the triangle affected the position of this ‘centre’. This suggests that the teacher’s curriculum script was evolving through experience of teaching the lesson with dynamic geometry, incorporating new mathematical knowledge specifically linked to mediation by the software. Indeed, he drew attention to a striking example of this which had arisen from his question to the class about the position of the ‘centre’ when the triangle was dragged to become right angled. The lesson transcript recorded: Teacher: What’s happening to the [centre] point as I drag towards 90 degrees? What do you think is going to happen to the point when it’s at 90?. . . Student: The centre’s going to be on the same point as the midpoint of the line. Teacher [with surprise]: Does it always have to be at the midpoint? [Dragging the figure] Yes, it is! Look at that! It’s always going to be on the midpoint of that side. . . . Brilliant! Reviewing the lesson, the teacher commented that this property hadn’t occurred to him; he ‘was just expecting them to say it was on the line’. Reacting to the student response he reported that he looked at the figure and ‘saw it was exactly on that centre point’, and then ‘moved it and thought . . . of course it is!’. What we witness here, then, is an episode of reflection-in-action through which the teacher’s curriculum script for this topic has been elaborated.

5.5.6 Time Economy In respect of the time economy, a very basic consideration of physical time for the teacher in this study was related to the proximity of the new computer suite to his normal classroom. However, a more fundamental feature of this case was the degree to which the teacher measured didactic time in terms of progression towards securing student learning rather than pace in covering a curriculum. At the end of the first session, he linked his management of time to what he considered to be key stages of the investigation: ‘the process of exploring something, then discussing it in a quite focused way as a group, and then writing it up’, in which students moved from being ‘vaguely aware of different properties’ to being able to ‘actually write down what they think they’ve learned’.

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A further crucial consideration within the time economy is investment in developing students’ capacity to make use of a tool. As noted in the larger study from which this case derives, teachers were willing to invest time in developing students’ knowledge of dynamic geometry only to the extent that they saw this as promoting their mathematical learning. This teacher was unusual in the degree to which he saw working with the software as engaging students in disciplined interaction with a geometric system. Consequently, he was willing to spend time to make them aware of the construction process underlying the dynamic figures used in lessons, by ‘actually put[ting] it together in front of the students so they can see where it’s coming from’. Equally, this perspective underpinned his willingness to invest time in familiarising students with the software, recognising that it was possible to capitalise on earlier investment in using classical tools in which ‘doing the constructions by hand first’ was a way of ‘getting all the key words out of the way’. As this recognition of a productive interaction between learning to use old and new technologies indicates, this teacher took an integrative perspective on the double instrumentation entailed. Indeed, this was demonstrated earlier in his concern with the complementarity of old and new as components of a coherent resource system.

5.6 Discussion Although only employing a dataset conveniently available from earlier research, the case study presented in this chapter starts to illuminate the professional adaptation on which the constitution of digital tools and materials as classroom resources depends. While the status of the conceptual framework that has been used to identify structuring features of classroom practice must remain tentative, it prioritises and organises previously disparate constructs developed in earlier research, and has proved a useful tool for analysis of already available case-records. It has the potential to be employed not just in relation to secondary mathematics teaching, but also to other school phases and curricular areas, and to other types of resource; indeed, much of the earlier research from which the various central concepts have been drawn has such a range. At the same time, however, the differing provenance of the five central constructs raises some issues of coherence. The original construct of curriculum script, for example, is very clearly psychologically based, focusing on individual knowledge schemes. One might also add that the term ‘script’ (originating in a psychological metaphor for memory structures) risks failing to convey the sense intended here of an organised repertoire of potential actions and interactions for teaching a topic as opposed to a specific sequence. By contrast, the construct of working environment may appear to refer to a material situation independent of the teacher. However (as suggested by Adler in Chapter 1), a more adequate theorisation takes a structuring feature as being constituted not just by an existing system of contextual constraints but by teachers’ interpretation of these and adaptation to them. Moreover, this co-constitution takes place on the social plane as well as the individual; indeed,

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these planes interact inasmuch as individual adaptation to such constraints is subject to a degree of socialisation, while the corresponding social norms evolve by virtue of a wider cultural appropriation of what originated as innovative micro-genetic adaptations on a very local scale. Thus, while each of these structuring features of working environment, resource system, activity format, curriculum script, and time economy are anchored in a particular form of constraint under which the work of teaching takes place, these constraints do not wholly condition practice, but interact and afford some degree of adaptation. For example in the case study detailed in this chapter, the proximity of the teacher’s normal classroom to the computer classroom afforded him the option to move between them as the location for the lesson. Moreover, the way in which he exercised this option was guided by his assessment of the suitability of the two locations for different activity formats. This, in turn, permitted the teacher to develop a new type of activity structure covering each session as a whole, efficient in terms of time economy, and providing what he considered a promising structure for an investigative lesson to capitalise on student use of digital resources. In terms of the specific digital resource in play, dynamic geometry, the teacher established a resource system in which this software fulfilled complementary functions to classical tools, each supporting particular aspects of students’ learning of mathematics, and so justifiable in terms of time economy. Finally, the teacher’s curriculum script for the topic was evolving, through adaptation and extension of an investigative task previously carried out without digital tools, the associated activity formats and corresponding refinement of his knowledge about supporting the interactive development of mathematical ideas. Acknowledging the concern of some chapters of this book with the collaborative use of resources, the collective role of the school department in fostering teachers’ professional learning was not a focus of this case study. In this department, however, it was clear that the internally developed schemes of work provided a key means of prompting the spread between teachers of new teaching ideas, often supported by self-devised materials. Nevertheless, the teacher had not yet reached the point at which this particular teaching sequence could be incorporated in the relevant departmental scheme. Indeed this case illustrates the bricolage which typifies the process of appropriating a new tool in the absence of well-established professional practice; a bricolage which, in the English educational system at least, is often left to the individual teacher rather than organised collectively. Likewise, the teaching sequence studied in this chapter was far from being captured in documented form. Although he had prepared a worksheet to remind his students of certain pieces of advice for their work, the teacher was generally rather sceptical of the value of such aids: ‘I don’t like pre-prepared worksheets’; ‘Normally I don’t use worksheets very much at all’. This arose from his strong valorisation of the explicit collective (re)construction of mathematical situations: ‘I always like to start with a blank page and actually put it together in front of the students so they can see where it’s coming from’. For him, it was this interactivation of a teaching sequence (guided by his curriculum script) that lay at the centre of his teaching. Under these conditions, then, this new teaching sequence might be expected to eventually be shared with

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colleagues more through observation or simulation of a lesson than by the reading or following of a documentary reification of this professional knowledge. This prompts comparison of the ideas developed in this chapter with those of other chapters in this book, notably those chapters that focus on the integration of digital tools and materials into everyday mathematics teaching. In terms of the core idea of ‘resource’ itself, following the concrete sense in which that term is widely used within the teaching profession, the focus of this chapter is on material ‘resources’ in classroom use, whereas, as Gueudet and Trouche note in Chapter 2, they use the term more loosely to cover any teacher resource, material and nonmaterial. Another significant contrast between the conceptual framework used in this chapter and that of Gueudet and Trouche lies in the central metaphor employed to capture the organisation, retrieval and exchange of professional knowledge. For Gueudet and Trouche, this is the ‘document’; in the conceptual framework employed in this chapter it is the ‘script’. Although neither Gueudet and Trouche nor myself are entirely happy with our respective metaphors, they do point to an important contrast in modalities of memory and thought, similar to that discussed by Proust in Chapter 9. This may well reflect divergences of professional practice and values between educational systems, notably as these bear on the planning of lessons. Such divergences might be linked, for example, to differing types of evidence used for professional accountability (lesson planning, for example, as against student progression) and models of lesson process (establishing disciplinary narrative, for example, as against ensuring curricular coverage), as well as intensity of work (with contrasting expectations as regards lesson preparation reflecting very different volumes of teaching and other duties required of teachers). Relatedly, although Gueudet and Trouche note in Chapter 2 how a teacher’s curriculum script serves particularly to guide the decisions that the teacher takes in class, it is important to emphasise that this script also plays a crucial part in preactive planning of a lesson agenda, and in post-active reflection on (and learning from) a lesson (Leinhardt et al., 1991). Indeed, I would hypothesise that every ‘document’ expresses elements of some underlying ‘script’. Nevertheless, it is important to acknowledge the part that the use and adaptation of documentary materials may play in supporting and developing the personal curriculum scripts of teachers, particularly those whose subject knowledge is modest (as noted by Pepin in Chapter 7). In Chapter 14, Drijvers raises the question of how the conceptual framework used in this chapter relates to the construct of instrumental orchestration. In terms of the concrete instrumental orchestrations that Drijvers describes, the answer is simple: each corresponds to a particular type of activity format centred on a specific use of one or more tools. More broadly, as described by Drijvers, didactical configuration and exploitation mode are features of what is commonly referred to within research on teacher thinking and planning as pre-active teaching, and didactical performance is likewise an aspect of interactive teaching (Clark & Peterson, 1986). In terms of the structuring features of classroom practice identified by the conceptual framework employed in this chapter, didactical configuration concerns organisation

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of the working environment as well as some more generic aspects of the functioning of the resource system; exploitation mode relates to more topic-specific aspects of the functioning of the resource system as well as to the tool mediation of processes within the curriculum script; and didactical performance relates to the way in which the curriculum script guides interactive teaching. Drijvers notes that the conceptual framework presented in this chapter is a more generic one, not specifically tied to the integration of technological resources in the way that the orchestration framework is. Arguably, these qualities are complementary. Indeed, an important conceptual weakness, both of advocacy for technology integration and research into it, has been lack of attention to the broader situation in which ordinary teachers find themselves (Lagrange, 2008; Ruthven & Hennessy, 2002). It is in this spirit that the conceptual framework used in this chapter has been developed by synthesising observations from recent studies of technology use, particularly in school mathematics, in the light of earlier conceptualisations of classroom teaching and situated teacher expertise. Turning to future development of the conceptual framework presented in this chapter, other insights have already been gained through a parallel analysis of mathematics teachers’ appropriation of graphing software (Ruthven, Deaney, & Hennessy, 2009). However, further studies are now required in which both data collection and analysis are guided by the conceptual framework, so that it can be subjected to fuller testing and corresponding elaboration and refinement. If they are to adequately address issues of professional learning, such studies need to be longitudinal as well as cross-sectional, and to focus on teachers’ work outside as well as inside the classroom. Likewise, the current reach of this conceptual framework is deliberately modest; it simply seeks to make visible and analysable certain crucial aspects of the incorporation of new technologies into classroom practice which other conceptual frameworks largely overlook. By providing a system of constructs closer to the lived world of teacher experience and classroom practice, it may prove able to fulfil an important mediating function, allowing insights from more decontextualised theories to be translated into classroom action, and serving to draw attention to practical issues neglected in such theories. Acknowledgements Particular thanks are due to the teacher colleague featured in the case study; to Rosemary Deaney who carried out the fieldwork for it; and to the UK Economic and Social Research Council which funded the associated research project. This chapter draws on and develops ideas and material from two earlier publications (Ruthven, 2009, 2010). These publications drew, in turn, on papers discussed at the CERME conferences in 2007 and 2009, in the RME and TACTL SIGs at the AERA conference in 2009, and at the CAL conference in 2009.

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Williams, R., & Edge, D. (1996). The social shaping of information and communications technologies. Research Policy, 25(6), 856–899. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. Zucker, A., & McGhee, R. (2005). A study of one-to-one computer use in mathematics and science instruction at the secondary level in Henrico County public schools. Menlo Park, CA: SRI International. Retrieved 31, January, 2011, from ubiqcomputing.org/FinalReport.pdf

Chapter 6

Modes of Engagement: Understanding Teachers’ Transactions with Mathematics Curriculum Resources Janine T. Remillard

6.1 Introduction The last decade has seen considerable progress in theory building related to teachers’ use of mathematics curriculum resources (Adler, 2000; Brown, 2009; Gueudet & Trouche, 2009; Remillard, 2005). Scholars agree that the process of using a curriculum resource is not one of straightforward implementation; rather curriculum use involves an interaction between the teacher and the resource. A number of studies, including many in this volume, have documented a variety of personal, professional, and classroom-based results from teachers using curriculum resources as tools (Chapters 7, 10, and 14; Gueudet & Trouche, 2010; Remillard, Herbel-Eisenmann, & Lloyd, 2009). In this chapter, I explore and theorize the relationships that teachers develop with curriculum resources as they use them. I focus, in particular, on mathematics curriculum texts produced to guide teachers in the design of daily instruction. In the United States, these resources tend to be published in print format most commonly by commercial companies. The chapter is informed by research on elementary and middle school teachers in the United States, where the nature of mathematics curriculum resources have undergone substantial change since the publication of the NCTM Standards in 1989. Traditionally, the primary focus of mathematics textbooks was student exercises and practice problems with minimal attention paid to pedagogy. Because the Standards targeted both the kinds of mathematical tasks students are asked to do and the nature of instruction around these tasks, new curriculum materials place a great deal of emphasis on pedagogical guidance for the teacher. As a result, teacher’s guides now provide teachers with new kinds of information to read and suggest different kinds of teaching practices to enact. Researchers are finding that using these changed resources, often referred to as Standards-based curriculum materials, presents challenges for many teachers and requires considerable reorientation (Drake & Sherin, 2009; Lloyd, 1999; Remillard, 2000); moreover, many teachers use them in ways J.T. Remillard (B) University of Pennsylvania, Philadelphia, PA 19146, USA e-mail: [email protected]

G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_6,  C Springer Science+Business Media B.V. 2012

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not intended by the designers (Collopy, 2003; Remillard & Bryans, 2004). My aim in this chapter is to examine these teacher–curriculum interactions from a conceptual perspective and offer a framework for characterizing them. I argue that teachers are positioned by and through their encounters with curriculum materials as particular kinds of users of them. I explore how this positioning happens, including how teachers participate in it, as well as its implications for teachers and for the prospect of reform in mathematics teaching and learning. In her book, entitled Teaching Positions, Ellsworth (1997) draws on film studies to argue that whenever two or more people engage in an interaction, be it spoken, textual, in film, or pedagogical, the speaker always makes assumptions about the audience. “Films, like letters, books, or television commercials are for someone . . . Most decisions about a film’s narrative structure, ‘look’, and packaging are in light of conscious and unconscious assumptions about ‘who’ its audience ‘is’, what they want, how they read films” (p. 23, emphasis in original). This concept, known in film studies as mode of address, Ellsworth argues, involves positioning the audience in particular ways that are the necessary starting place for interaction. This starting place is where the viewer (or hearer, or reader) enters a relationship with the story or ideas in the text. However, this positioning is also problematic in its shaping of the relationship around power and authority in the interaction. Using Ellsworth’s perspective as an analytical lens, I examine the relationships teachers enter into when they use curriculum resources, how they are positioned by the materials, and how they position themselves as readers and users of texts. In doing so, I build on the idea of mode of address and Rosenblatt’s (1980, 1982) theory of transactions with text to offer a model of how these interactions are shaped and how they shape the role of curriculum materials in teaching. I argue that, in addition to having a mode of address, curriculum materials have forms of address, particular “looks” or formats that reflect and reinforce the mode of address. Moreover, teachers interact with curriculum resources through an identifiable stance or mode of engagement. Like the modes of address, modes of engagement have particular forms. To interrogate current patterns in how teachers use curriculum resources or to imagine alternatives, it is necessary to understand these constructs and their interrelationships.

6.2 Modes of Address and Curriculum Resources In film studies, a mode of address captures who the film’s designers think the audience is, what they want, and how they read. All films (or texts) have an intended audience and are written to capture and appeal to, to speak to that audience. Ellsworth uses the metaphor of seats in a theater to explain. There is one seat or “position” from which the film does its best work or looks its best. Building on this idea of position, she argues: There is a “position” within power relations and interests, within gender and racial constructions, within knowledge, to which the film’s story and visual pleasure is addressed. It’s from that “subject position” that the film’s assumptions about who the audience is work with the least effort, contractions, or slippage (p. 24).

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In other words, to work as imagined, films, and I would add texts, need the intended audience to be who the mode of address assumes they are. Thus, part of the work of a mode of address is to affirm that position, to keep them wanting what they should want, to enlist the viewer in reinforcing her positioning or epistemological stance. It is in this way that modes of address do not merely speak to an intended audience, but actually seek to assert control over that audience or to enlist a particular kind of participation. Modes of address are not neutral. Moreover, the drive behind the mode of address is not merely literary or artistic in nature (i.e., like the relationship between author and reader); it is commercial. Films and texts need the intended audience to be who they (the designers) think they (the audience) are to sell. In systems with strong commercial publishing industries, the need for a text to sell is particularly important, but even in school systems that use a single, state sponsored textbook, the need for it to appeal to and be usable by teachers remains present. The same is true for materials developed noncommercially. Even though the designers may not be interested in “selling” the curriculum, per se, they need the teacher to “buy in” to the orientation set out in the materials. Curriculum materials, then, are written with particular teachers (readers) in mind – teachers who exist in the minds of the writers. They are written to both appeal to those readers’ needs and desires and to affirm them to keep the text–teacher relationship intact. Later, I discuss how this affirmation occurs through the specific forms of address used most frequently in curriculum materials in the United States. Naturally, texts have multiple modes of address or, as Ellsworth (1997) suggests, multiple entry points. A film might use different characters to draw in audience members outside of the primary intended audience. A film written with white, adolescent, males as the primary audience, Ellsworth explains, uses other characters, such as a strong, intelligent woman, to capture other audience members. The concept of multiple entry points applies to curriculum materials as well. For example, many curriculum designers assume their audience desires or needs day-by-day guidance for teaching mathematics lessons and they design their materials attentive to this desire. However, they may be cognizant that some teachers seek challenging activities that they can pick up and use without making a substantial investment in the structure of the curriculum. These designers make such activities accessible and visible while addressing the needs of their primary audience. The concept of multiple entry points as it applies to mathematics texts and their use is illustrated in findings from a qualitative study I undertook in the early 1990s (Remillard, 1996). The study included a document analysis of a commercially developed elementary mathematics textbook published in the wake of the Standards and a concurrent year-long study of two fourth-grade teachers using this text for the first time. Through regular classroom observations (18 distributed across the year), I wrote field notes that described the classroom interactions and use of the textbook for each teacher and then used audiotaped interviews to triangulate observations and uncover the teachers’ approach to textbook use. My analysis of the text revealed two distinct entry points in the text. The basic, core program consisted of 13 traditionally titled chapters that emphasized the mastery of computational procedures.

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At the same time, the text included language and work practices associated with the NCTM Standards, such as exploration, group work, and manipulatives, along with a number of highly visible but auxiliary options, including a daily problem to solve and exploratory activities. From this presentation, one might infer that the intended audience was the typical elementary teacher at the time, someone seeking a mainstream, procedurally focused mathematics curriculum, but who was open to intermittently incorporating problem solving, manipulative work, and partner activities into lessons. These Standards-aligned activities themselves, however, did offer a second entry point. The possibility of these two entry points is further illustrated by the two teachers in the study. One of the teachers, Ms. McKeen, fits this description of the intended audience. Her use of the textbook focused on the routine practice problems it provided for students to complete. She also used some of the optional, reform-oriented activities available in the text when time permitted. The other teacher in the study, Ms. Yarnell, was attracted by another entry point into the text. She focused on the supplemental exploration activities found in the teacher’s guide (but not the student text), the manipulative-based instruction described in the margins of the teacher’s guide that surrounded the picture of the student’s page, and the pages of daily problems available at the back of the teacher’s guide. In her use of the textbook, Yarnell drew primarily from these resources and used the student practice pages infrequently (see Remillard, 1999, for more details.) I return to this example when discussing teachers’ modes of engagement.

6.3 Forms of Address and Curriculum Materials In my analyses of teachers using curriculum materials, I have come to appreciate the significance of form and look. When speaking of film, Ellsworth (1997) argues that narrative structure, look, and packaging all represent the mode of address – who the film makers think the audience is. I argue that when it comes to curriculum materials, form takes on its own significance and meaning. And while it is a critical component of the mode of address, form deserves a particular analytic focus. The form of address of a curriculum resource refers to the physical, visual, and substantive forms it takes up, the nature and presentation of its contents, the means through which it addresses teachers. The form of address is what teachers actually see, examine, and interact with when using a curriculum resource. The form of address is akin to what Otte (1986) referred to as a text’s objectively given structure. In his exploration of the concept of text and textbook, Otte argued that one must consider the text as both an “objectively given structure of information” (the physical form that the text takes), and a “subjective scheme” (how it is understood or perceived). Drawing on Otte’s writing, Love and Pimm (1996) referred to objectively given structures as, “what can be seen when looking at such materials” (p. 379). As the following discussion suggests, the form of a curriculum resource includes, but goes beyond, what is seen. Naturally, what is

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seen and encountered in a resource is inextricably linked to the subjective schemes that surround it, the tradition, meanings, and expectations that mediate the reader’s interpretation of the objective structure. The form of address is multi-faceted and includes all aspects of a resource. Not all forms of address have been examined in studies of teachers and curriculum materials, however. I use the following descriptions to illustrate the multifaceted nature of these forms. In the 1980s, elementary mathematics textbooks used in the United States tended to look alike. Developed and marketed by commercial publishers, the typical text consisted of 12 or 13 chapters that placed primary emphasis on computational processes, such as addition, subtraction or two-digit multiplication. These volumes also devoted chapters to several noncomputational topics, including measurement and geometry. The chapters were arranged into two-page modules; each on a different skill, around which each day’s lesson was shaped. The pages were glossy, colorful, and included photographs of children happily engaged in activities that employed math skills, like cooking, selling, or constructing. There might be an example of the particular skill at the top of the first page, followed by a series of similar exercises. One would likely find a set of story problems at the bottom of the second page. The large, spiral-bound teacher’s guide contained pictures of the student pages surrounded by wide margins that provided instructional suggestions, including questions to ask or even scripts to follow, and answers to all the exercises and questions on the students’ page. Teachers could teach a lesson from this two-page spread with minimal preparation. This example illustrates how a particular form can become a cultural convention, an accepted and expected package. In the mid-1990s, a number of curriculum developers began to challenge that cultural convention. Working outside of the commercial publishing market, they designed new materials intended to reflect the vision put forth by the NCTM Standards (NCTM, 1989). These Standards-based materials offered an instructional approach and set of mathematical goals and activities that differed substantially from the typical textbook described above. At the same time, curriculum authors (sometimes steered by pressure from publishers1 ) made different choices about the form of address their materials took. Some curriculum developers adopted forms familiar to teachers and presented their unfamiliar curriculum designs within the familiar curriculum package described above. At first glance, Everyday Mathematics, an elementary school program first published in the early 1990s by the University of Chicago School Mathematics project, looked very much like conventional textbooks. The teacher’s guide was a large, spiral-bound book with glossy pages packed with images of the student’s pages and teaching suggestions along the margins. Other designers opted for forms that appeared radically foreign to the textbook market. The elementary program, Investigations in Number, Data, and Space, developed by TERC and first published in 1998, for example, presented its program in a set of modules, each bound separately. The pages were printed with blank ink on

1 Even though they were not designed by commercial publishers, the Standards-based curriculum programs are published and marketed by commercial publishers.

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matte paper and contained descriptions of the teacher’s role in setting up and directing the lesson in addition to a good bit of blank space. Small pictures of the students’ work page, if applicable, were placed off to the side. The periodic pictures were pen and ink sketches of children engaged in the activities described in the lesson.2 These brief descriptions illustrate some of the various forms of address that print curriculum resources take. There are unlimited possibilities and variations. Increasingly, designers are making curriculum resources available in the form of webpages with links to different kinds of support and guidance that do not stop at mathematical and pedagogical suggestions. Some resources include video clips of classrooms using suggested activities and live discussion forums where teachers may seek insights from others. (See Chapters 2 and 5 for examples of discussions of how electronic curriculum resources influence teachers’ use of them.) In my examinations of mathematics curriculum resources and my studies of teachers using them, I have found that the large number of characteristics that make-up the form of address can be loosely classified into five interrelated categories: structure, look, voice, medium, and genre. Some of these characteristics tend to be given more attention in discussions of curriculum resources than others. I contend that each category is relevant to how teachers engage and utilize resources. Moreover, each category represents a set of design considerations and decisions that are not always made explicitly.

6.3.1 Structure Structure is the feature most commonly examined in curriculum resources. It refers to how the resource is organized and what it contains. Some refer to structure as the nature and organization of the content of the curriculum, the particular mathematical concepts and goals, and the underlying pedagogical assumptions. The components of structure can be parsed in a variety of ways. On the basis of his analysis of science curriculum resources, Brown (2009) identified three basic facets of curriculum resources that comprise their structure: (a) representations of concepts specific to the domain, (b) representations of tasks or procedures that students are expected to undertake, and (c) physical objects and representations of physical objects that are intended to support students’ work on the tasks and understanding of the concepts. These three facets, Brown argues, “encompass the most fundamental aspects of the curriculum’s content and structure: its core ideas, the activities undertaken in their explorations, and the objects that support such activities” (p. 27). Brown’s use of the term “representations of” signals the notion that most curriculum resources represent concepts (like the Pythagorean theorem) and tasks (like “Find the area of the triangle”). In some cases, the resource provides physical objects (like rulers) to be

2 It is worth noting that the second edition of the Investigations (TERC, 2008) has a physical appearance more in line with conventional mathematics teacher’s guides than the first edition, although it continues to be organized in modules.

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used by students. Often, the materials represent these and other objects pictorially (like a picture of a bridge in which triangles are used). As I discuss later, these representations makingup the structure of the curriculum are then read and interpreted by teachers. Analyses of the structure of curriculum resources tend to consider how the various components are organized, the mathematics content included or excluded through the representations, and the valence or emphasis of the content, including how the content is represented. Organization refers to how the features in the curriculum resource are packaged. Earlier, I described some structural elements of the typical commercial textbook published in the United States before the mid 1990s. These curriculum resources generally contained work pages for the student, answers to the student problems, guidance or even scripts to use during instruction, auxiliary activities, orienting resources (such as the table of contents, scope and sequence chart, and other resources that might help teacher structure the curriculum). In my analysis of Standards-based curriculum resources, I have found that many of these organizational elements are present, however, they may be packaged differently. For example, when it was first published, Investigations in Numbers, Data, and Space (TERC, 1998) consists of individual lessons or multi-lesson sessions (grouped according to a larger idea or investigation), but each session is not organized around the students’ work pages. Rather, the sessions are typically organized around a number of activities that are intended to occur in the class, some of which have associated student pages.3

6.3.2 Look Look refers to the purely visual appearance of the resource – what teachers see when they look at it. In the United States, cultural and institutional traditions exist, which influence the designed look of curriculum resources, even those designed by different publishers. Many of the commercially designed curriculum resources, for example, have a decidedly commercial look. They are printed on glossy pages, contain colorful photographs of smiling children, and include pages that read like advertisements for the materials. Colors and fonts are used in such a way that particular words seem to jump out at the reader. A number of the noncommercially published materials I have reviewed have a look that appears subdued when compared to those just described. Look is the result of a number of design choices, and is also influenced by the structure of the program. For example, a resource that represents reasoning and problem solving as central components of mathematics will have a different look than a resource that places primary emphasis on mastery of discrete skills.

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6.3.3 Voice Voice refers to how the authors or designers are represented and how they communicate with the teacher (Love & Pimm, 1996). In the case of most curriculum resources I have examined, the authors are invisible and little information is provided about who they are or what their experience is. The invisibility of the author may be a device to depersonalize the text and increase its authority. It may well be a tradition that has evolved over time. Despite the invisibility of the authors, curriculum resources have a voice that is manifested through the way they communicate with the teacher. Most curriculum resources place primary emphasis on what the teacher should do. I think of this as talking through teachers (Remillard, 2000). That is, the authors communicate their intent through the actions they suggest the teacher takes. Few resources speak to the teacher by communicating with teachers about the central ideas in the curriculum. Some researchers have argued, however, that speaking to teachers is one way that curriculum resources can be designed to be educative for teachers (Davis & Krajcik, 2005; Schneider & Krajcik, 2002). Davis and Krajcik identified a set of design heuristics that curriculum designers might follow to make their resources explicit for teacher learning. Offering transparent and direct guidance related to reasons and purposes underlying task selections or anticipating students’ responses to tasks are two such examples. In their analysis of two elementary mathematics programs, Stein and Kim (2009) found differences in how designers communicated with teachers. The teacher’s guide of one program spoke primarily through the teacher. It offered pedagogical guidance, but few explanations. The other program included a number of efforts to speak to the teacher, including elaborations of reasons underlying pedagogical recommendations, notes to the teacher about common student errors or developmental learning trajectories, and example student dialogue. Another curriculum resource I examined, which was designed for teacher educators, included a journal written by a fictitious facilitator of the program. The journal was intended to provide facilitators using the resource with insights into the decision-making processes a facilitator might go through when using it with a group of teachers. In this sense, voice is related to structure because it is the inclusion or exclusion and placement of particular structural elements that shape the resource’s voice. The voice of curriculum resources is also evident in the language used. HerbelEisenmann used discourse analysis tools drawn from Morgan (1996) to analyze the voice of the student text of a Standards-based middle school curriculum, focusing on how the authoritative structures in the writing constructed the author, the reader, and mathematical reasoning. She noted an absence of first person pronouns-a common approach taken in student texts–and suggested that this tendency concealed the presence of human beings in the design of the text. She also suggested that the authors’ frequent use of second person pronouns in conjunction with objects in statements such as “the graph shows you,” obscures the authority of the authors and gives inanimate objects power to perform animate activities.

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6.3.4 Medium Medium refers to the form of delivery of the resource and has particular relevance with the increased availability of electronic instructional resources. Currently, the majority of curriculum resources are print based, a medium familiar to most teachers. However, as the use of electronic media and access to computer and networking technologies are becoming common, more teachers are using electronic and webbased resources (Gueudet & Trouche, 2009; Chapters 2 and 5). This evolution brings to the fore the need to consider medium in the examination of teachers’ interactions with these resources. Unlike those that are text-based, electronic resources allow for and often assume a nonlinear path through their offerings, giving the user a degree of navigational decision-making control not as apparent with print medium. On the other hand, as Gueudet and Trouche (2010) argue that the notion of author and authorship is often less transparent in online sources than in printed texts.

6.3.5 Genre The final category of form is genre. Unlike the other four categories, which reflect authors’ or publishers’ decisions, genre reflects what a curriculum guide is within a larger classification of written material for teachers. The curriculum guide is designed to offer a package that will aid in the construction of curriculum. In essence, it is meant to guide action and in this sense, it is more like a cookbook or manual than a novel. For this reason, there are elements of form that curriculum resources cannot completely transcend, despite designers’ efforts. The notion that a curriculum resource is a particular kind of artifact connects to Otte’s (1986) suggestion that texts have both objectively given structures (what can be seen) and subjective schemes (ways of being understood or expectations upheld about them). The curriculum-text genre signals particular subjective schemes among teachers who are familiar with them. Genre is important because it has implications for the expectations teachers bring to a curriculum resource that influence the way they engage it. I take this discussion up in the section that follows. The role of genre in meaning making is elaborated by Ongstad (2006) in his semiotic analysis of communication in mathematics and mathematics education. “Genre,” Ongstad explains, “precisely presupposes much of what can be expected in the kind of communication in question” (p. 262). Its familiarity conjures a “zone of expectation” and aids in how one makes sense of any form of communication, textual, or discursive. Naturally, any form of communication is likely to contain unfamiliar elements as well. Ongstad uses the term “rheme” to identify the unfamiliar or new. Learning or making meaning necessarily involves an interaction between the familiar (the theme) and the new (the rheme) in which the theme contextualizes and aids in the interpretation of the rheme. “Particular genres such as textbooks,

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definitions, explanations, and proofs for instance, will often have an implicit regime for balancing theme-rheme (or given and new) that we learn to use and recognise” (p. 263). An examination of the recent expansion in the development of curriculum resources, in response to both curriculum reform efforts and new digital technologies, raises questions about whether there exists a single curriculum-text genre. Even though all curriculum resources share a purpose or theme that distinguishes them from other forms of text, categories of resources have emerged within this genre. Standards-based curriculum materials, for example, which are researcher-developed as opposed to commercially developed, for many teachers, have become a genre within this broader class of resources. In other words, they provoke a particular theme that is distinct from conventional resources. That said, the examples described in Section 6.5, suggest that the broad curriculum-text genre can be powerful in the interpretive process for many teachers.

6.3.6 Why Forms Matter Forms of address are powerful mediators of teachers’ engagement with a particular curriculum resource. My position is influenced by sociocultural theory, which explains how artifacts mediate human activity (e.g., Cole & Engeström, 1993; Pea, 1993; Vygotsky, 1978; Wartofsky, 1973). From this perspective, curriculum resources are artifacts or tools that are part of the material world made and used by humans to accomplish goal-directed activity. They have material dimensions, but as constructions of culture, they also have social and cultural meaning. Indeed, they are “products of sociocultural evolution” (Wertsch, 1998) and are both shaped by and have the power to shape human action through their affordances and constraints. From this perspective, curriculum resources have the potential to enable, extend, or constrain human activity. My understanding of the power of forms of address has also been influenced by Rosenblatt’s (1980, 1982) work in literacy theory on transactions with text. In her writing about children’s interactions with literature, Rosenblatt argued that reading involves a transactional process between the text and the reader in a particular context. By framing this transactional relationship between the reader and the text, she makes an important distinction that is easily missed: the reader’s relationship is with the text and not the author. This is not to say that the author’s presence in the text cannot be detected in its design nor that subjective schemes or the genre is not at play in this transaction. But the reader engages and interacts with the designed artifact, not the author. When speaking of literature, Rosenblatt (1982) explains that although the author may have a particular plan for a book, he or she cannot predict what the reader will make of it. Similarly, when teachers engage with curriculum resources, they interact with the designed artifact and not the author’s intentions. This is why form matters. Form denotes certain meanings to the reader, a zone of expectation (Ongstad, 2006), which influence how teachers engage the resource.

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6.4 Modes of Engagement I use the term modes of engagement to refer to how a teacher interacts with the forms of address of the text. According to Rosenblatt (1980), readers enter into a transaction with the text they are reading and “there ensues the adoption, either consciously or unconsciously, of a predominant attitude or stance” (p. 388). This stance focuses the reader’s attention. Similarly, mode of engagement refers to what a teacher does in her transactions with a particular curriculum resource, how she engages, infuses meaning, and makes sense of its offerings. A teacher’s mode of engagement reflects her beliefs and epistemological stance. The notion of mode of engagement is also related to positioning theory (Harré & Langenhove, 1999). People use narrative, communication, or what Harré and Langenhove call “storylines” to position or locate themselves with respect to the scripts or roles made available in a particular context. As discussed earlier, these roles emerge from the teacher’s engagement with the forms of address. And as Ongstad (2006) suggests, the zone of expectation is influential in this interaction. When examining teachers’ use of curriculum resources, it helps to account for modes of engagement because the act of reading curriculum guides is different from the act of reading many other texts. This is where forms of address, and particularly genre, come into play. The forms used to address the reader signal to the reader what to expect, offering possible scripts. When a reader sees a poem written on a page, she sees many elements of its physical form – the blank page surrounding it, the uneven margins, the placement of the author’s name – all of which indicate to her that it is a poem and this indication prompts a particular mode of engagement that is unique to that reader in relation to the poem. As I described earlier, curriculum guides represent a particular genre of text that contain predictable elements or themes and these elements engender a response or mode of engagement from the teacher.

6.5 Forms of Engagement Just as the mode of address of a resource can be seen in its forms, a teacher’s mode of engagement can be understood through the forms that engagement takes up. In my research, I have found that a teacher’s mode of engaging a curriculum resource includes four primary forms or kinds of reading: what she reads for; which parts she reads; when she reads; and who she is as a reader. These forms of engagement overlap with several kinds of readings described by Sherin and Drake (2009) in their research. Earlier, I contrasted two teachers, Ms. McKeen and Ms. Yarnell, who read the same textbook in different ways. Specifically, they read different parts of the teacher’s guide and they read looking for different kinds of guidance. McKeen tended to read the routine lessons and focused on what the text had designed for students to do. Yarnell, in contrast, tended to read the auxiliary exploratory components of the text and read for the big mathematical understandings students were intended to develop (see Remillard, 1999, for details). These examples illustrate the first two

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forms of engagement. Sherin and Drake referred to distinction in what teachers read for as reading for activities or reading for big ideas. Few researchers have examined the different parts of the textbook read by teachers. This form of engagement seems particularly important in the United States where curriculum programs at all levels are becoming increasingly more laden with supplementary and alternative offerings, increasing the number of entry points, to invoke Ellsworth’s term. A third form of engagement is when a teacher reads the text. In their study of 10 elementary teachers’ curriculum strategies, Sherin and Drake (2009) found that teachers read their curriculum guides differently and at different times relative to instruction – before, during, and after. When teachers’ read is related to what they are reading for and their particular stance toward curriculum materials in teaching, discussed below. The fourth primary form of engagement is who a teacher is as a reader. In other work (Remillard, 2005; Remillard & Bryans, 2004), I refer to this positioning as stance or orientation. Teachers generally have a stance toward curriculum materials that is influenced by their views about mathematics, teaching, and the role that curriculum resources can and should play in the process of teaching mathematics. It is also influenced by their view of the particular resource. In my research, I have found teachers’ orientation toward curriculum materials in general to be strikingly influential in what they read, what they read for, and when they read their particular teacher’s guide. Rosenblatt (1982) used the term “reader’s stance” in a similar way: The reader may be seeking information, as in a textbook; he may want direction for action, as in a driver’s manual . . . In all such readings he will narrow his attention to building up the meanings, the ideas, the directions, to be retained; attention focuses on accumulating what is to be carried away at the end of the reading (1982, p. 269).

Clearly, forms of engagement are open to revision as a result of further reading and experience. Moreover, the claims made earlier about the role of the forms of address in signaling to the reader what to expect, might lead one to conclude that different forms of address, would prompt a change in how teachers initially engage new curriculum resources. Findings from a number of studies, however, suggest otherwise. At least initially, teachers are inclined to engage a new curriculum resource in similar ways to their interactions with previously used resources (Collopy, 2003; Lloyd, 1999; Remillard, 1991; Remillard & Bryans, 2004; Sherin & Drake, 2009). I posit that this tendency illustrates the two-way transaction between the teacher and the resources described by Rosenblatt (1982). Even though the forms of the resource contribute significantly to the reading, the reader does as well. Moreover, the genre figures substantially in the transaction: The words in their particular pattern stir up elements of memory, activate areas of consciousness. The reader, bringing past experience of language and of the world to the task, sets up tentative notions of a subject, of some framework into which to fit the ideas as the words unfurl (Rosenblatt, 1982, p. 268).

I believe that the genre of a curriculum guide – what it is and what it represents – provokes a mode of engagement that is particular to the teacher and, consequently, shapes the teacher–curriculum transaction. Thus, despite other elements of form, the

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genre, for many teachers, seems to trigger a “zone of expectation” (Ongstad, 2006) and assumptions early in the encounter.

6.6 Examples of Modes of Engagement What follows are several brief illustrative examples of what I mean by modes of engagement. They are all drawn from the first two years of a qualitative study of 14 elementary teachers from two different schools. Both schools had recently adopted the first edition of Investigations in Numbers, Data, and Space (TERC, 1998), an elementary program published in the United States. A brief description follows. Although it fit the genre of a curriculum guide, the Investigations curriculum was different from conventional materials in structure, voice, and look. It was designed to reflect the vision of NCTM Standards (NCTM, 1989, 2000), which was evident in its mathematical and pedagogical structures. The program emphasized conceptual understanding and reasoning about mathematical ideas. The majority of the recommended student activities involved collaborative exploration and problem solving, followed by class discussions. The program included worksheets that students were to complete, but they were minimal and generally intended to be integrated into the interactive part of the lesson. The program also included features designed to be educative for teachers (Davis & Krajcik, 2005). These features included information for the teacher in the form of mathematical explanations, rationales for the mathematical decisions made, examples of student work and talk, summaries of relevant research, and suggestions for assessment. The authors included the following statement in the introduction to each unit guide: ‘Because we believe strongly that a new curriculum must help teachers think in new ways about mathematics and about their students’ thinking processes, we have included a great deal of materials to help you learn more about both’ (p. 6). In this way, the program spoke in a different voice than was typically the case. Finally, the layout of the program offered an unfamiliar look. A great deal of text was devoted to describing the teacher’s role and what she might do and look for during the lesson’s activities. The following descriptions illustrate different modes teachers took when engaging this resource. They also illustrate how these modes interacted with the various forms of address, including the multiple entry points.

6.6.1 Mr. Jackson: Reading for Worksheets Mr. Jackson, a fourth grade teacher, had been teaching for 30 years when his school adopted the Investigations program. He was familiar with and a dedicated user of commercially published mathematics textbooks. In fact, although the school had officially adopted Investigations as its primary mathematics program, he had a set of older textbooks he distributed to his students and used along with Investigations. He used both programs in strikingly similar ways.

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During our observations of Jackson’s interactions with the curriculum, we noticed that he went first to the student pages to see what written work was part of the lesson. Even though these worksheets were not necessarily designed to be central to the lesson, he gave them central focus. Next, he would get a general sense of the structure of the lesson – what students were to do in what order. We have no evidence that he read the description of the teacher’s role provided with each lesson. In the lessons we observed, he moved fairly quickly to assigning independent student work and seemed to be most comfortable with this sort of teaching format. He orchestrated these lessons using the same teaching practices he used when using the commercial textbook. He asked questions and accepted answers, rarely asking for an explanation. When students worked at their desks on the assigned task, he sat at his desk and graded papers rather than interacting with them about the work. He often reminded students to “do your own work” and infrequently followed individual work with whole-class discussions. Often he ended class by asking students to hand in their papers after being sure their names were on them. When asked how math was going mid-November, he said: “Well we’ve followed along in the math books that we’re using, and the um, with Investigations, we’re basically right where we’re supposed to be, according to the schedule that we set up at the beginning of the year.” We identified Mr. Jackson as reading for worksheets because that is what he looked for and read in the curriculum. Although many aspects of the form of address of Investigations differed from the familiar text, he found the worksheets familiar and they dominated his reading of the curriculum. Mr. Jackson’s focus on the worksheets illustrates one way that multiple entry points work in a curriculum resource.

6.6.2 Ms. Hatcher: Reading for the Script Ms. Hatcher, a second grade teacher, had been teaching for 20 years when her school started using Investigations. Like Mr. Jackson, she had a long history using curriculum resources faithfully, but she avoided commercially published textbooks, opting for alternative resources that focused on problem solving and conceptual understanding. She was a careful reader of these curriculum guides and tried to follow them as best she could. For her, following the Investigations curriculum meant doing exactly what the authors suggested. Sometimes, this presented a challenge for her because, the teacher’s guide did not always tell her exactly what to do. As this quote suggests, Ms. Hatcher used the curriculum to create a script for herself. When asked how she used the curriculum to plan, she said: I reread the curriculum, reread whatever it is in the Investigations book we’re using . . . I am really following the teacher’s guide. If I’m having trouble understanding it, I will sometimes script it out; otherwise I highlight or maybe mark what I want to make sure I touch on.

When reading the teacher’s guide she focused on the lesson description, making careful notes in the margins. She also consulted the book frequently during each

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lesson. Indeed, we observed that Ms. Hatcher had her class follow each step of the lesson provided in the guide. She too tried to fulfill her role in the script by asking students to explain their answers and interacting with and challenging students during small group work. Ms. Hatcher’s stance on curriculum resources was that they provided the teacher with a script. When reading curriculum materials, which she did with great care, she read as much as she could, looking for a script. Even though she did not always find as much detail as she would have liked, she used what was available to create her own script.

6.6.3 Ms. Jordan: Reading for Big Ideas Ms. Jordan taught third grade and had been teaching for 4 years when she began using Investigations. She had limited experience with curriculum resources and clear ideas about the kind of mathematical understanding she wanted to foster in her teaching. She was attracted to Investigations because of its structure – the mathematical ideas it offered. She engaged the resource through these ideas. When reading the text, rather than wanting to know what to do, she wanted to find the important mathematical ideas. She then used them to shape her use of the lesson descriptions. When she first looked at the book, she went to the section that described the mathematical emphasis. She described her planning this way: I look at the mathematical emphasis first to see, first of all, what it is that I’m trying to get from them by the end. I’ll look at the teacher notes as they come up within the actual script of the lesson. I read the lessons a lot of times over because there are certain components I want to say, but I don’t want it to be scripted. . . . It requires rereading to make sure I have the mathematical emphasis down and I know what I’m trying to get. So even if the lesson leads a different way, I know the math aim I’m going for that day and I try to stick to that even if we have to veer off somewhere.

Ms. Jordan was the only teacher we studied who talked about using the mathematical emphasis and she was one of two who read the support pages at the beginning of each unit. We observed many instances of Ms. Jordan veering off her plans during a lesson. This happened most often when she felt her students were not getting the important ideas. When this happened, she often inserted an improvized review session of what she saw as the important ideas or made explicit connection to the previous day’s activity. We identified Ms. Jordan as reading for ideas, because it was the mathematical idea that most guided her decisions when using the curriculum. Like the others, her mode of engagement illustrates Rosenblatt’s (1980) assertion that when entering into a transaction with a text, readers adopt a predominant attitude or stance. Ms. Jordan’s stance was shaped by her goals in teaching mathematics and her view of how a curriculum guide could support those goals. Ms. Jordan was one of two teachers in the study (n = 14) who engaged the Investigations materials in ways that aligned with their dominant mode of address.

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They were the only teachers focused on the mathematical goals and emphases and who read the additional information for the teacher, using it to inform their teaching decisions and understanding of student learning.

6.7 Possibilities of Shifting Modes of Engagement The examples above illustrate how past experience with and assumptions about curriculum resources in teaching are drawn into the teachers’ transactions with new resources. Returning to Ellsworth’s notion of (1997) positioning, we see that curriculum materials have particular modes of address – ways of communicating with teachers – and these modes of address prescribe roles for teachers that position them as certain kinds of readers. But teachers enter this transaction with their own expectations, beliefs, and routines that shape their modes of engaging. Most often, these modes of engagement are formed in response to past experience with curriculum resources. In this way, teachers are positioned by their own encounters with curriculum materials as well as by the materials themselves. Margolinas and Wozniak (2010) used the term “generating documents” to refer to encounters with curriculum documents early in the career that have a forming or generative role in teachers’ future practice and, I argue here, future modes of engagement. My interest in how teachers’ modes of engagement are shaped and their tendency to be resilient over time is influenced by the design of nonconventional materials that potentially offer a new genre of resource and anticipate a different kind of teacher use. Many Standards-based programs seek to alter conventional teaching and curriculum-material practices. Like all materials, they are designed with a target audience in mind. These materials work best when the audience is who the designers intend and behaves as the designers expect it to. To be successful, the materials must enlist the teacher in being part of that target audience. Doing so necessarily involves contributing to a shift in the way teachers use these resources. As the evidence in this chapter and others (e.g., Chapter 10) suggests, achieving this shift requires more than a simple change in modes of address. The possibility that these modes can develop over time gives research an important focus to consider. Gueudet and Trouche (2009) found that the process of constructing and reconstructing resources leads to substantial change in teachers’ approaches. Moreover, Drake and Sherin (2009) found that, over time, teachers using a Standards-based program developed greater trust in the curriculum along with a clearer vision of its purpose. I see the domain of modes of engagement, how teachers engage and read curriculum materials, to be a fruitful place for learning to take place and an ideal focus for professional development of teachers. These learning opportunities would also offer valuable sites for inquiry. At this point, we understand little about the processes through which teachers might learn to engage with curriculum resources in substantially new ways and position themselves as partners with them.

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Acknowledgements The author is grateful to the insightful and substantive feedback provided by Carolyn Kieran and the three editors of this volume. This chapter develops concepts and analyzes presented in several earlier publications (Remillard, 1999, 2010; Remillard & Bryans, 2004). This research described within was funded by the National Science Foundation (Grant nos. REC-9875739; ESI-9153834) and the Pew Charitable Trust (Grant no. 91-0434-000). The views expressed within are those of the authors and are not necessarily shared by the grantors.

References Adler, J. (2000). Conceptualising resources as a theme for teacher education. Journal of Mathematics Teacher Education, 3, 205–224. Brown, M. W. (2009). The teacher-tool relationship: Theorizing the design and use of curriculum materials. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction, (pp. 17–36). New York: Routledge. Cole, M., & Engeström, Y. (1993). A cultural-historical approach to distributed cognition. In G. Salomon (Ed.), Distributed cognitions: Psychological and educational considerations (pp. 1–46). Cambridge: Cambridge University Press. Collopy, R. (2003). Curriculum materials as a professional development tool: How a mathematics textbook affected two teachers’ learning. Elementary School Journal, 103(3), 287. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14. Drake, C., & Sherin, M.G. (2009). Developing curriculum vision and trust: Changes in teachers’ curriculum strategies. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 321–337). New York: Routledge. Ellsworth, E. (1997). Teaching positions: Difference, pedagogy, and the power of address. New York: Teachers College. Gueudet, G., & Trouche, L. (2009).Towards new documentation systems for mathematics teachers. Educational Studies in Mathematics, 71(3), 199–218. Gueudet, G., & Trouche, L. (2010). Des ressources aux documents, travail du professeur et genèses documentaires. (From resources to documents, teacher’s work and documentational geneses). In G. Gueudet & L. Trouche (Eds.), Ressources vives. Le travail documentaire des professeurs en mathématiques (pp. 57–74). Rennes, France: Presses Universitaires de Rennes et INRP. Harré, R., & van Langenhove, L. (1999). Positioning theory. Oxford, MA: Blackwell. Lloyd, G. M. (1999). Two teachers’ conceptions of a reform-oriented curriculum: Implications for mathematics teacher development. Journal of Mathematics Teacher Education, 2(3), 227–252. Love, E., & Pimm, D. (1996). ‘this is so’: A text on texts. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics, part 1 (pp. 371–409). Boston: Kluwer. Margolinas, C., & Wozniak, F. (2010). Rôle de la documentation scolaire dans la situation du professeur: le cas de l’enseignement des mathématiques à l’école élémentaire (Role of teaching documentation inthe teacher’s situation: the case of mathematics at elementary school). In G. Gueudet & L. Trouche (Eds.), Ressources vives. Le travail documentaire des professeurs en mathématiques (pp. 183–199). Rennes, France: Presses Universitaires de Rennes et INRP. National Council of Teachers of Mathematics (NCTM). (1989). The curriculum and evaluation standards for school mathematics. Reston, VA: Author. NCTM. (2000). The principles and standards for school mathematics. Reston, VA: Author. Ongstad, S. (2006). Mathematics and mathematics education as triadic communication? A semiotic framework exemplified. Educational Studies in Mathematics, 61, 247–277. Otte, M. (1986). What is a text? In B. Christiansen, A. G. Howson, M. Otte (Eds.), Perspectives on math education (pp. 173–202). Dordrecht, The Netherlands: Kluwer.

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Pea, R. (1993). Practices of distributed intelligence and designs for education. In G. Salomon (Ed.), Distributed cognition: Psychological and educational considerations (pp. 47–87). Cambridge, MA: Cambridge University Press. Remillard, J. T. (1991). Abdicating authority for knowing: A teacher’s use of an innovative mathematics curriculum. (Elementary Subjects Center Series No. 42). East Lansing, MI: Michigan State University, Institute for Research on Teaching, Center for the Learning and Teaching of Elementary Subjects. Remillard, J. T. (1996). Changing texts, teachers, and teaching: The role of textbooks in reform in mathematics education. Unpublished doctoral dissertation, Michigan State University, East Lansing, MI. Remillard, J. T. (1999). Curriculum materials in mathematics education reform: A framework for examining teachers’ curriculum development. Curriculum Inquiry, 29(3), 315–342. Remillard, J. T. (2000). Can curriculum materials support teachers’ learning? Two fourth grade teachers’ use of new mathematics text. Elementary School Journal, 100, 331–350. Remillard, J. T. (2005). Examining key concepts of research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Remillard, J. T. (2010). Modes d’engagement: Comprendre les transactions des professeurs avec les ressources curriculaires en mathématiques (Modes of engagement: Understanding teachers’ transactions with mathematics curriculum resources). In G. Gueudet & L. Trouche (Eds.), Ressources vives, le travail documentaire des professeurs, le cas des mathématiques (pp. 201–216). Rennes, France: Presses Universitaires de Rennes et INRP. Remillard, J. T., & Bryans, M. B. (2004). Teachers’ orientations toward mathematics curriculum materials: Implications for teacher learning. Journal for Research in Mathematics Education, 35, 352–388. Remillard, J. T., Herbel-Eisenmann, B. A., & Lloyd, G. M. (Eds.). (2009). Mathematics teachers at work: Connecting curriculum materials and classroom instruction. New York: Routledge. Rosenblatt, L. M. (1980). What facts does this poem teach you? Language Arts, 57(4), 386–394. Rosenblatt, L. M. (1982). The literary transaction: Evocation and response. Theory into Practice, 21(4), 268–277. Schneider, R., & Krajcik, J. (2002). Supporting science teacher learning: The role of educative curriculum materials. Journal of Science Teacher Education, 13(3), 221–245. Sherin, M. G., & Drake, C. (2009). Curriculum strategy framework: Investigating patterns in teachers’ use of a reform-based elementary mathematics curriculum. Journal of Curriculum Studies, 41(4), 467–500. Stein, M. K., & Kim, G. (2009). The role of mathematics curriculum materials in large-scale urban reform: An analysis of demands and opportunities for teacher learning. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 37–55). New York: Routledge. TERC. (1998). Investigations in numbers, data, and space. Menlo Park, CA: Dale Seymour. Vygotsky, L. S. (1978). Mind in society. Cambridge, MA: Harvard University Press. Wartofsky, M. (1973). Models. Dordrecht, The Netherlands: D. Reidel. Wertsch, J. V. (1998). Mind as action. New York: Oxford University Press.

Chapter 7

Task Analysis as “Catalytic Tool” for Feedback and Teacher Learning: Working with Teachers on Mathematics Curriculum Materials Birgit Pepin

7.1 Background 7.1.1 Mathematics Education and Curriculum Materials The field of teachers and mathematics curriculum materials is rapidly growing. Much of its growth, particularly in the United States, is due to the current explosion of curriculum development projects in response to particular standards (e.g. NCTM, 1989), in addition to an increasingly widespread practice of mandating the use of a single curriculum to regulate mathematics teaching. In educational research, more broadly, some earlier work has focussed on teachers’ use of texts. Ben-Peretz (1984) distinguished between the curriculum “proposed” by materials and the curriculum enacted by the teacher, and she argues that teachers draw on their professional experience (and beliefs) to “assign meaning to the curriculum materials they use daily in their classrooms” (p. 71). She used the term “curriculum development” and argued for two phases of curriculum development, the second being when the teachers work with the materials to make them suitable for their students, and she describes these actions as “uncovering the potential of curriculum materials” for use in the classroom. More recent research in mathematics education views curriculum use as “participation with the text” (Remillard, 2005) indicating the dynamic interrelationship between teachers and curriculum materials. These studies (e.g. Chapter 6) not only develop insights into the use of curriculum materials, but also how teachers learn from their use. This view of dynamic interaction between teacher and curriculum, agent and tool, is also reflected in Lloyd’s study (1999) arguing that “curriculum implementation consists of a dynamic relation between teachers and particular curricular features.” (p. 244). Remillard’s (2000) study of teachers’ use of textbook as a response to Standards-based curricula highlights the importance of considering different perspectives in the field. B. Pepin (B) Faculty of Teacher and Interpreter Education, Sør-Trøndelag University College, 7004 Trondheim, Norway e-mail: [email protected] G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_7,  C Springer Science+Business Media B.V. 2012

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Furthermore, and innovatively, Gueudet and Trouche (2009) coined the term “documentation work” indicating teachers’ work with materials: “looking for resources, selecting/designing mathematical tasks, planning their succession, managing artefacts, etc.” (p. 199). This paved the way for a “new” and creative thinking about mathematics curriculum material in connection with teacher “use” of those materials. This also provides a new perspective, viewing teachers not as passive users, but as designers of their own resources, and there are interesting and interrelational dependencies between teachers’ professional knowledge and curriculum design, each influencing each other in the process (Chapter 3). Interestingly, and somewhat in contrast, Ruthven (Chapter 5) developed a conceptual framework on the basis of five constructs, and amongst them the “curriculum script”, as compared to Gueudet & Trouche’s “document” (Chapters 2 and 16). Thus, it is evident that in mathematics education there is a growing body of scholarship and research that places teachers at the centre of the “teaching enterprise” raising questions about the effects of curriculum materials on classroom instruction and pupil learning. What happens when teachers use particular curriculum programmes (e.g. reform programmes), and why? An underlying assumption is that teachers are central players in the process of transforming curriculum ideals, captured in the form of mathematical tasks, lesson plans and pedagogical recommendations, into real classroom events. What they do with curriculum resources matters (Lloyd, Remillard, & Herbel-Eisenmann, 2009).

Thus, what teachers do with mathematics curriculum materials, how they “mediate” them (Chapter 4) and why, how they choose particular mathematical tasks, and how this complex net of choices influences classroom activity, is crucial for understanding not only the “implementation” of curricular programmes, but also for informing the work on the development of new programmes. Moreover, it is important for understanding how students interact and work with the curriculum materials (Chapter 12), and how they may learn in turn (Chapter 8). In line with researchers working in this field (e.g. Davis & Krajcik, 2005; Remillard, 2005), I use the term “curriculum materials”, and sometimes “textbook” materials, to refer to printed and often published materials designed to be used by mathematics teachers and pupils during classroom instruction.

7.1.2 Curriculum Materials, Teacher Use of Materials and Teacher Knowledge Underpinning the study reported in this chapter is the assumption that teacher learning involves teacher autonomy and agency when analysing, choosing, changing and transforming materials, devising alternatives, and “enacting” the materials (Ben-Peretz, 1984). Paris (1993) emphasises that teacher agency in curriculum matters involves “the creation or critique of curriculum, an awareness of alternatives to establish curriculum practices, the autonomy to make informed curriculum choices, an investment in self, and ongoing interaction with others.” (p. 16)

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Moreover, seminal work by Ball and Cohen (1999) discusses the role of curriculum materials, in particular textbooks, with respect to teacher learning. They assert that Curriculum materials could only become central to teacher learning, if the traditional boundaries between texts’ presentation of content and teachers’ teaching were redrawn to make central the work of enacting curriculum. (p. 7)

In terms of improving instruction, materials are often seen to offer resources for teachers’ work with their students, and not designed to encourage teachers’ investigations of and work with the material. Sadly, it is claimed, teachers must often learn alone “with few resources to assist them”. Thus, they call for the creation of curricula that would help teachers to better enact curriculum in practice. If the boundaries of curriculum design and development were reconsidered and redrawn, curriculum materials could offer teachers more opportunities to learn in and from their work. (p. 8)

Furthermore, in the United Kingdom, the recent Williams Review (Williams, 2008) drew attention to the need, and challenge, of strengthening mathematical knowledge for teaching system-wide, and recognised the scale of professional development initiatives required to secure that knowledge. As new mathematics curriculum materials are being and have been developed in many countries (e.g. US: NCTM, 1989; UK: DfES, Standards Unit, 2005), teacher learning is considered an important aspect and part of the “enactment” of these materials: the design and teaching of new materials as a potential place for teacher learning (e.g. Remillard & Bryans, 2004). For example, recent research in science education advocates educative curricular resources as both a tool for teacher learning and as a support for teachers to become curriculum designers (e.g. Davis & Krajcik, 2005). Ruthven, Laborde, Leach, and Tiberghien (2009) argue that “the availability of design tools capable of identifying and addressing specific aspects of the situation under design can support both the initial formulation of a design and its subsequent refinement in the light of implementation” (p. 329). It appears that there is large potential for curricular and pedagogical resources to be designed, or existing materials to be amended and enriched, so as to fulfil an educative function for the teacher (Ruthven, 2008). The rationale for the work with teachers was based on findings from TIMSS (Hiebert, Gallimore, Garnier, Givvin, Hollingsworth, Jacobs, et al., 2003) which report that high achieving countries engage students more frequently in rich mathematical activities, (and more rigorous reasoning) than lower achieving countries. In particular, it is claimed, students (in high achieving countries) are presented with more rich and open mathematical problems that require them to make connections between mathematical ideas. There are likely to be several reasons that may explain “poorer results”, amongst them the following: (a) mathematics teachers do not possess a deep understanding of the mathematics they are asked to teach and (b) teaching mathematics with attention to conceptual underpinnings is difficult, and time consuming, and unless it is given priority, seems not consistent with the tradition of school mathematics. Thus, although most policy makers would probably

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agree that making “learning mathematics with understanding” has become a shared objective, teachers seldom have opportunities, and time, to develop mathematical tasks and teaching sequences, where “richness of tasks” and “learning mathematics with understanding” are emphasised. With these two hypotheses in mind, I embarked on a professional development programme to include opportunities for teachers to deepen their own understanding of selected key concepts of curriculum they were likely to teach, to improve their knowledge of ways students may understand the content, and learn about analyzing, selecting and enriching mathematical activities, and subsequently learn about how to teach these in their classrooms. The approach to teacher learning centres around the analysis and enrichment/amendment of mathematical tasks, that is curriculum materials (work sheets, textbooks, etc.) that teachers typically use for their teaching. Teaching is seen as a dynamic process that goes beyond what happens in the classroom to include analysis and selection of curriculum materials as part of planning and reflection (Chapter 2). The literature claims that improvements in planning and reflection have great potential for improving teaching (Ball & Cohen, 1999; Fernandez, 2002; Hiebert et al., 2003). Moreover, it is in agreement with research on teacher learning which claims that effective professional development must provide opportunities for teachers to work together, analyse and discuss curriculum materials in connection with classroom practice (e.g. Whitehurst, 2002; Chapters 15–17). Teacher learning is seen here in the widest sense, as teachers work together or on their own and with materials that help them to develop their knowledge in and for teaching.

7.2 Tools for Reflection and Feedback In this study, I explore the role and nature of feedback resulting from the development and use of a tool designed to help teachers develop further understandings of characteristics of mathematical tasks, their selection, amendment, enrichment and potential use with their students. This fits largely within the studies on feedback in professional learning (e.g. Hargreaves, 2000) and that teachers can learn from feedback (student feedback in the case of Hargreaves, 2000). There is also a large body of research of teachers’ experiences of learning through enquiry and collaborative projects (e.g. Fennema, Carpenter, & Franke, 1996; Greeno & Goldman, 1998; Chapters 15 and 17), amongst them those that highlight the importance of tools (e.g. Baumfield, Hall, Higgins, & Wall, 2009; Chapter 16). A “tool” can be viewed in different ways. Boydston (1986) claimed A tool is a mode of language, for it says something to those that understand it, about the operations of use and their consequences . . . in the present cultural settings, these objects are so intimately bound up with intentions, occupations and purposes that they have an eloquent voice. (p. 98)

Thus, whilst a tool may have different forms, using a tool in the context of pedagogic practice, it is likely to re-frame teachers’ experiences (this is also

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acknowledged by the instrumental approach, introducing the tool/instrument dialectics, Chapter 2). For example, a new tool is likely to add something to the repertoire; equally it may disrupt participants’ practice and take something away (Baumfield, 2006). This reflects the tool’s catalytic quality: it may change participants’ practice and environment. The individual agency of the teacher rests with the decisions s/he takes as a result of feedback from the use (of the tool), thus her/his reactions to the feedback. Literature on learning through inquiry and feedback in professional learning (e.g. Hargreaves, 2000) emphasises how teachers can learn from student feedback (to bring about learning and/or change in classrooms). There is also a large amount of literature linking student achievement and feedback (e.g. Butler & Winne, 1995) where feedback is conceptualised as “information with which a learner can confirm, add to, overwrite, tune, or restructure information” (p. 275). In their study on the importance of feedback Hattie and Timperley (2007) claim that “feedback is one of the most powerful influences on learning and achievement” (p. 81). They conceptualise feedback as information provided by an agent, may it be a teacher, peer, book, self, experience or curriculum materials, regarding aspects of participant’s performance or understanding. To understand the interaction of enquiry and feedback in teacher learning, Baumfield et al. (2009) investigated how a tool designed for student awareness of their learning also supported teacher professional development. There is ample evidence (e.g. Hattie & Jaeger, 1998) that the presence of feedback (in whichever form it may be) increases the likelihood that learning will occur. In this study I conceptualise feedback not as feedback from students, but from other sources, such as the analysis schedule, curriculum materials, or peers, for example. Moreover, and leaning on research by Hattie and Timperley (2007), I distinguish between four levels of feedback: the task level (how well the tasks are understood/performed); the process level (the main process(es) needed to understand/perform tasks); the self monitoring level (directing and regulating actions); and the personal evaluation level (personal evaluation and affect) (p. 87). Winne and Butler (1994) also claim that feedback can have different sources: external (e.g. provided by contexts or other participants); and internal (e.g. self-generated such as monitoring their actions). Using this theoretical frame I seek to develop deeper understandings, and theorise, “feedback” in connection with “tools” – and where tools are perceived in different ways than previously done – and investigate and relate their connected power to teacher learning.

7.3 The Study The study built on previous work with teachers, textbooks and other curriculum materials (e.g. Haggarty & Pepin, 2002; Pepin, 2008, 2009). Supported by a grant from the National Centre for Excellence in the Teaching of Mathematics (NCETM1 ) 1

https://www.ncetm.org.uk/

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and whilst based in the United Kingdom, I worked with teachers over a period of 9 months. The work with teachers consisted of two phases: 1. Curriculum exploration and transformation (deepening teacher knowledge of mathematics concepts and teaching mathematics “with understanding”, development of analysis schedule for curriculum materials, analysing mathematical tasks, selection of tasks and activities for lessons); 2. Link to practice and lesson analysis (“enacting” the curriculum materials, reflection on enactment-videoed lesson, sharing of work and discussions). Teachers participated on a voluntary basis. Originally, five teachers of the school’s mathematics department intended to participate, but due to illness and private circumstances this number was reduced to two: John and Paul. John was a relatively experienced teacher (who had been educated at the university department), whereas Paul was a trainee teacher on a “Graduate teacher programme” (i.e. learning to teach “on-the-job” whilst working in and being employed by a school). In addition, two other newly qualified teachers joined the group for a few months at the beginning of the project. In this chapter, I will mainly refer to John’s and Paul’s participation. It is important to distinguish between two levels of study here: the professional development work with teachers; and the research study that drew on the work with teachers. The main aim of the professional development project was to work with these teachers to develop their mathematical knowledge in/for teaching, that is (1) to assist teachers to analyse, select and amend mathematically rich problems; and (2) to assist teachers to “implement” their analyzed/amended activities into their pedagogical practice. Here the author acted as “trainer” and “critical friend” to guide and help teachers in their endeavour to develop their pedagogic practice. The main aim of the research study was to develop a deeper understanding of how curriculum materials, and working with teachers on curriculum materials, can support teacher learning and professional knowledge (Chapter 17). After funding was obtained, six meetings were scheduled over the course of 9 months. In terms of professional development, during those meetings the focus was on different activities (see phases and activities, Table 7.1). In terms of research and data collection the following data were collected over the 9 months period: • Observation reports (descriptions) of what went on during each of the six sessions. • Written evaluation/feedback after each of three whole day university sessions. • Lesson observations (Paul and John observing each other’s lesson, also the teacher educator). • Group interview after the lessons (and based on the observed lessons). • Group interview as evaluation of the whole project.

Reading, discussing and presenting literature on “learning mathematics with understanding”

Development/amendment/ re-shaping of mathematical task analysis tool for task analysis (on the basis of Pepin, 2008). Analysis of “different” mathematical tasks (e.g. LEMA). Auditing/collection of resources available in the school’s mathematics department. Use of “tool” to analyse/amend/ enrich mathematical tasks.

Use of “tool” for analysis of assessment tasks/test questions. Linking assessment, mathematical tasks and National Curriculum. Preparing tasks for teacher use in lessons. Learning Walk (lesson observation: teacher educator and each teacher observing the other’s lesson). Observing each other’s lessons. Discussion of lessons (play back) and feedback to each other.

Literature review

Development of task analysis schedule

Assessment, task analysis & National Curriculum

Learning walks

Task analysis

Activity

Phase

Reflective (feedback on practice: suggest alternative strategies & trial out different tasks and practices)

Diagnostic

Diagnostic (focused on learning and skills).

Diagnostic, reflective

Reflective (focussed on knowledge/learning)

Types of feedback

Table 7.1 Types of feedback with respect to activity

“Creative” enactment of materials. Making learning more explicit. Opening to critical enquiry helps to build autonomy.

Creative questioning with respect to the purpose and value of mathematical tasks. Alternative strategies of devising a mathematical task. Selection of appropriate tasks for instruction. Confidence in amending and enriching materials for particular purposes. Linking context (assessment and National Curriculum) and content.

Develop insights beyond the immediate context. View mathematics/learning mathematics in different ways How to analyze mathematical tasks. Clarify ideas on the purpose of tasks and on what mathematical tasks may inherently possess.

Teacher knowledge/learning

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In terms of analysis, a procedure involving the analysis of themes similar to that described by Woods (1986) and by Burgess (1984) was adopted and using the “constant comparative approach”. Moreover, I tried at one level to maintain the coherence of each teacher’s responses over the different sessions, by analyzing the data with respect to observations and with respect to the different types of responses (e.g. different interviews, evaluation feedback, etc.); at another level I analyzed across the two cases and using the different concepts of “tools” and testing the hypotheses offered by the literature, and building explanations and theorizations anchored in the data. On a third level, I looked for similarities and differences of teacher responses with respect to what the literature claims about feedback, tools and teacher learning. In theoretical terms the analysis focuses on the types and role of feedback, stimulated through the use of the “tool”, in teacher’s learning, and I use the constructs outlined in the previous section to develop a deeper understanding the interaction of enquiry and feedback in teacher learning. However, due to the small number of cases, it was important to address the potential difficulties with respect to validity of the findings. In terms of validity checks, both teachers were invited to comment on the observation reports, also in the final evaluation interview at the end of the project. It was also important to locate and understand teachers’ classroom practices and classroom cultures in context, that is in the particular school environment in which they were working: an inner city comprehensive school in the North-West of England. Here, it was useful to draw on knowledge gained from previous work with at least one of the two teachers and the partnership relationship with the school. In addition, National Curriculum documents, and the school’s guidelines and curricular texts (including textbooks) were analyzed to study the contextual background of mathematics instruction and the potential influences of these texts on teachers’ perceptions and pedagogic practice.

7.4 Findings In this section I discuss the different types of activities undertaken with respect to information gained by and feedback provided to teachers, and in turn its potential for teacher learning. During the 9 months research and development period different kinds of “activities” were undertaken which can be categories under five different “phases” (see Table 7.1): Phase 1: reading, discussion and presentation of the literature on “learning mathematics with understanding” Phase 2: development/amendment/re-shaping of mathematical task analysis tool for task analysis; Phase 3: use of “tool” to analyse/amend/enrich selected mathematical tasks; Phase 4: preparation of tasks for classroom instruction; Phase 5: “learning walks”. During these phases, and in the different activities, the “task analysis schedule (the ‘tool’)” was developed and used, inside and outside class and in very

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different ways, which in turn influenced its nature. This is the focus of discussion, and illustrated in the following. (1) In the first phase reading, discussion and presentation of the literature on “learning mathematics with understanding” helped teachers to view mathematics in a different way. Lively discussions centred around the issue of what it may mean to learn mathematics with understanding, and one of the key issues identified here was related to “making connections”, and in different ways (e.g. to familiar situations, or to previously taught mathematics). Understanding in general is linked to “something”, . . . experiences perhaps. To understand mathematics we must “connect” it to something relevant or “of meaning” to an individual [pupil]. . . . In my experience too many people are concerned with “how to get the answer”. My perception of learning maths with understanding looks at “why” the answer works. (John, Session 1 evaluation)

The emerging discussion on the literature was also seen as a valuable activity to enhance teacher learning in terms of bringing together theory and practice. . . . the discussions have promoted the deep links between the literature and those aspects [identified earlier as individual aspects of teaching mathematics] . . . the overall process of creating a dialectical fusion between theory and practice has become clearer. I feel more able to “read”. (Paul, Session 1 evaluation)

Thus, this activity appeared to help teachers to develop insights beyond the immediate context and “next-day-lesson” and view mathematics learning in different ways (e.g. “to link something to something else”, connecting theory and practice). The type of feedback likely to be “produced” by the activity was reflective and focussed on knowledge and learning. (2) In the subsequent phase, and in subsequent sessions, these notions helped to identify what kinds of characteristics a mathematical task may/should have so that learning with understanding is more likely to happen. On the basis of a “skeleton” task analysis schedule (e.g. Pepin, 2008) teachers developed, amended and re-shaped this for their own analysis of activities and according to their own understandings. This meant that teachers added or changed categories according to what they regarded important in mathematical tasks. For example, and linking to the task analysis schedule (see Appendix), the category of “connections through mathematics” was further re-defined, to differentiate between and include “connections within mathematics” as well as “connections across other subjects”. As another example, several categories under “processes” were further developed, in particular the category of “analysing” in order to differentiate between “reasoning” and “procedural”. As a third example, the whole category of “familiarity” was introduced, and it was apparent that teachers drew on their repertoire of practical experiences here, in order to identify what was important for them and their teaching (evidence: observation reports). One of my reasons for participation on this project was to extend pupils’ knowledge and experience of mathematics beyond the procedural and technical. I believe that the tasks/activities carried out by students play a major part in this experience. The analysis tool which we are developing exposes aspects of the mathematical process, language demands

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and symbolism held within the structure of a task. This exposure has formalised, for me, necessary parameters which I can address individually within my teaching. (Paul, Session 1 evaluation)

In terms of characteristics of a task, particular notions were highlighted, amongst them the following: • “the need for the task to have a purpose – what are we hoping to develop with the task”; • “tasks that incorporate a range of concepts and cognitive demands [are] ‘richer, and therefore of more benefit”; • “communicating mathematically in a range of ways is very important in cementing a pupil’s understanding”; In the light of the discussions and re-designing the task analysis tool, teachers saw the need to re-think their pedagogic practice, and they also realised the difficulties and efforts connected with this. I would like to increase pupils’ exposure to mathematical language and symbolism to allow the possibility of rich dialogue. I would like to use this opportunity to improve the connections within mathematics and between mathematics and (1) other subjects, (2) “real world”. I would like to raise the level of thinking required by pupils. (Paul, Session 1 evaluation)

Thus, in short the development/amendment of the task analysis schedule appeared to help teachers to clarify ideas on the purpose of tasks and what a mathematical task may inherently possess (e.g. in terms of “aspects of mathematical process”, “language demands” and “symbolism”). There is evidence from the data that the type of feedback likely to be “produced” by the activity was diagnostic and reflective. (3) In the third phase, the task analysis itself helped teachers to see alternative strategies of devising a task and what this may mean for classroom practices/processes. After auditing their own school mathematics department resources, teachers brought to the sessions a range of tasks they used in their lessons (e.g. worksheet “House & Garden” that included mathematical tasks on designing a house and garden, decorating the house’s surfaces, designing a pool in the garden, etc.). The teacher educator also provided selected tasks, for example modelling/“open” tasks from an EU project (LEMA, 2009) and selected textbook tasks (e.g. from German and French school textbooks). These were then analysed on the basis of the re-designed task analysis schedule. For example, it was suggested to enrich the “area & perimeter” part of the question to include ideas from the Standards Unit (DfES, 2005), to give pupils more opportunities to address potential misconceptions (e.g. “same perimeter- same area”; evidence: observation report 3; Paul, Session 3 evaluation). On the same topic it was also suggested to use more practical material for the task, such as string or straw to show that the same perimeter can hold different areas. Thus, this activity appeared to help teachers to develop alternative strategies of devising a mathematical task with respect to potential misconceptions, select and

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amend appropriate tasks with respect to their developing ideas and intended instruction, and use creative questioning with respect to the purpose and value of tasks. There is evidence from the data that the type of feedback likely to be “produced” by the activity was diagnostic and focussed on learning and skills for their classroom instruction. (4) In the fourth phase teachers identified the need to look at assessment (and tests) in connection with task analysis, and with respect to the National Curriculum (in England). Thus, as a first step the group decided to use the previously analysed “House & Garden” tasks for “assessing pupils working on tasks”. Sensitized by the previous sessions in terms of “what we may assess”, that is considering the categories relating to task features (e.g. content; connections; contexts; familiarity; representations; etc.), these were mapped against how these could be assessed by a teacher (e.g. what kinds of questions may a teacher ask him/herself?), and the different levels and sublevels of the National Curriculum. As a second step, and in terms of linking assessment and tests to task analysis, departmental grade 8 tests (on number, three different achievement levels) were analyzed. Interestingly, some supposedly “lower level” questions were actually considered (by teachers) to have more potential in terms of “openness” and richness than some of the higher level questions that aimed at procedural understanding, and were described by teachers as “numbing”. Linking this to Assessment for Learning (AfL) teachers developed ideas, in particular with respect to peer- and self-assessment, and developing pupils’ awareness of skills they are using during a series of lessons. Reflecting on assessing tasks and what pupils have produced and learnt is as important as analysing the tasks. (Suzanne, Session 3 evaluation) . . . I will think about a mathematical task and evaluate what areas of the NC levels it addresses rather than the converse, i.e. teach to the NC levels. (Bill, Session 3 evaluation) There is scope to begin with open investigative tasks, . . . Reducing the number of tasks, whilst allowing more time on fewer tasks allows for greater detail in what pupils discover/learn. (Paul, Session 3 evaluation)

Thus, the link to assessment and the National Curriculum appeared to have given teachers another view point and they appeared to have become more secure in their knowledge about tasks, what they can afford and how they can be taught and assessed. The assessment activities helped teachers to gain confidence in amending/enriching materials for particular purposes, for example for “creative” assessment. In addition, this activity helped to link assessment tasks/tests to the National Curriculum, and to see them in a different light (e.g. as formative curriculum materials rather than evaluative). There is evidence from the data that the type of feedback likely to be “produced” by the activity was diagnostic and focussed on assessment and instruction. (5) In phase 5 and during the later parts of the project (Learning Walk – lesson observations) selected lessons of John and Paul (one morning with 1 and 2 lessons each) were videoed. For these both teachers had prepared and discussed, amongst each other, what and how they planned to do things. The two videoed lessons (one each) were played back in an afternoon session at university, and the subsequent

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discussion (on the basis of these two lessons) centred around teachers’ use of their developed/prepared curriculum materials. One of the foci of “modification” of tasks, and pedagogic practice, was to instil more discussion into their lessons (see also worksheet). . . . the lesson I did was modified and adapted from a lesson that I had previously done . . . but the fact that we modified it is a result of this [project] . . . (Paul, video recall) . . . we had looked through the activities that we had already and . . . we used our awareness of this tool to modify the wording . . . we both wanted the discussion to take place . . . both in pairs and in groups . . . to modify the tasks so it was more explicit that it was discussion that we went for. (John, Video recall)

It appeared that the thinking about tasks helped to make the processes involved in doing the tasks more evident. I agree, I think the . . . process of going from coming up with the initial idea, estimating, making a guess, throwing the ideas out and then honing into, to improve mathematical symbols. The idea of mathematical language in order to revisit the problem . . . I think that’s . . . quite a powerful . . . thing that came out of this whole process. . . . I think that the tool has enabled us to, to mediate the tasks. [my italics] . . . Both on paper and then, because of our awareness of what we want, or a greater awareness through out classroom . . . communication. (Paul, video recall)

The final discussions centred around their collaboration, working with each other as “sparring partners” in this project. Teachers emphasised the importance to work with someone “to bounce off ideas” and go beyond what one may develop when working alone. Because I, I would say I have quite a lot of lessons like that, that kind of thing, but they’ve never been developed beyond what I thought of myself . . . I’ve never, in a lot of my lessons, I’ve never bounced my ideas off anyone else. . . . Like, [John] and I did with this one. . . . that part of it was enriching . . . (Paul, video recall)

Thus, the “learning walks” provided opportunities for peers to suggest alternative strategies (for classroom practice), and for encouragement to trial out different things and work together in a team. There is evidence from the data that the type of feedback likely to be “produced” by the activity was reflective and focussed on practice. In summary, it can be said that teachers developed their ideas, whilst going through the different stages: from reviewing the literature; to tasks analysis and task enrichment; to “creative” applications and considerations of task analysis; to enactment in the classroom. At each stage (and these are not seen as hierarchical) they carried “residuals” from previous sessions, and appeared to become more confident in terms of how to proceed, what to do next, why this may be useful, what they may have learnt, for example as the project went on. The “analysis” and “enrichment” of mathematical tasks appeared to have become an analysis and enrichment of their pedagogic practice.

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7.5 Discussion and Conclusions There is general agreement that curriculum materials can support teacher learning and professional development (e.g. Ball & Cohen, 1999; Schneider & Krajcik, 2002). However, it is less clear what kinds of materials (e.g. textbooks) should be used, what characteristics the materials should have, whether educative or otherwise, and moreover what kinds of processes and “use” of materials help teacher learning. In terms of teacher learning, the literature suggests that effective professional development should have three crucial elements: it should be linked to teachers’ classroom context (e.g. Borko & Putnam, 1996); teachers should be supported longer term (e.g. Marx, Blumenfeld, & Krajcik, 1998); and teachers need to be given opportunities to build new knowledge (e.g. Borko & Putnam, 1996). Thus, there is the general view that, whilst curriculum materials have educative potential, they may not be effective without additional professional support. Feedback from others is generally seen as promoting reflection on and inquiry into practice (Fenstermacher & Richardson, 1993; Schön, 1983). However, in Collopy’s (2003) study one teacher seems to have developed through the use of textbook and other curriculum materials, and without additional professional support, whereas the other did not. In this study the mathematical tasks teachers worked on with a university teacher educator were integral to their daily work, both in terms of where the materials originated from (e.g. tasks chosen from their departmental resources) as well as their use in their daily instruction. However, the following seemed crucially important: (1) the situation created by the professional development activity and (2) the tools and processes surrounding the tasks, for example analysing the tasks, and providing and developing a tool for analysis. It appeared that the focus of developing reflection and thinking with teachers was less afforded by the tasks themselves, and the tasks/curriculum materials themselves could be educative or otherwise. Teachers appeared to need the necessary “tools” (e.g. task analysis tool, knowledge of how to enrich a task) to stimulate their thinking, and in turn (re-)shape the mathematical tasks for their teaching. Thus, it is legitimate to ask what may count as a “tool”, and what a tool should afford. Considering Table 7.1, and with reference to the findings outlined in the previous section, I thus conceptualize feedback as information not only provided by a person (e.g. the teacher educator, or peer teacher), but by other “agents”, such as curriculum materials, or more particularly in this case a task analysis schedule. Leaning on research by Winne & Butler (1994) feedback can here be re-conceptualized as information – whether it be mathematics domain-knowledge, meta-cognitive knowledge about processes involved in working with the mathematical tasks, beliefs about self and tasks or cognitive tactics and strategies – which teachers can use to confirm, add to or re-structure their knowledge, in short as feedback for learning. They also claim that feedback can have “external” and “internal” sources (Bangert-Drowns, Kulik, Kulik, & Morgan, 1991): external sources can be incidentally or intentionally provided by others (e.g. peer teacher, interaction with the environment, written

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comments) – in this case such sources are most likely provided by the social situation created by the professional development activity; internal sources for feedback are self-generated (e.g. teachers monitoring their activities and engagement with the learning task). The main message from these studies is that the learning context to which feedback is addressed needs to be considered: in this case teacher learning with curriculum materials (as compared to pupils learning in classrooms). Considering the different types of feedback (see Table 7.1), it appears that the development and use of the task analysis schedule (the “tool”) was crucial in teachers’ awareness raising/developing understandings of task characteristics and potential of particular mathematical tasks for teaching, hence in terms of support for teachers’ learning. There is evidence that this tool was the pivotal point around which most other activities centred, or were linked to, and which was mentioned in all discussions and evaluations (see earlier quote by Paul): as it developed, when it was used for the analysis and enrichment of mathematical tasks, and in the “enactment” of the amended tasks during instruction. Considering its perceived importance, and in terms of the associated “documentational genesis” (Chapter 2), the feedback the tool provided can be perceived at the four different levels (outlined by Hattie & Timperley, 2007). At the “task level”, the analysis tool provided feedback to teachers about the characteristics of the actual tasks, how well these were understood. At the process level, the analysis tool provided feedback in terms of what these tasks may, or may not, afford in terms of pupil learning and skills, the processes needed to understand the tasks. At the self-monitoring level, the tool provided feedback in terms of confidence of working with such an analysis tool. Finally, at the personal evaluation level, it provided feedback in terms of confidence to engage in further enquiries of such type. Thinking in terms of internal and external sources for feedback, it can be argued that the “tool” (analysis schedule) was an external source at the outset, but became an internal source of feedback. Whilst provided (by the university teacher educator) for teacher use in “skeleton” format, and as further developed from the literature with teachers, they shaped the task analysis tool and made it “their own” (see also “instrumentalisation” in Chapter 2) according to what they regarded as important characteristics for a mathematical task for their teaching. This process started with reading and discussing the relevant literature, and subsequently it was amended and then used on their chosen tasks. This in turn triggered ideas for amendment/enrichment of tasks and for comparison of characteristics with National Curriculum “features” and with national/departmental “test tasks”. At different stages of development and use of the analysis tool, different kinds of feedback resulting from the tool became apparent. Moreover, it is argued that by participating in the practice of enquiry (Greeno & Goldman, 1998) to analyze/work with mathematical tasks and the task analysis tool, teachers gained access to feedback that stimulated their professional learning and enabled them to become reflective (Schön, 1983). Particular attention is given here to the role of the tool for enquiry. The tool for task analysis became a tool for enquiry in activated feedback loops between (1) the two teachers amongst each

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other, (2) between them and the “tool” and (3) between them and the teacher educator and the “tool”. This, in turn, provided support for teacher learning. In this way, the level of teacher engagement and learning lifted the “tool” beyond its level of artefact, to become an “epistemic object” (Rheinberger, 1997): a knowledge object that is developmental in nature and depends on the place it occupies in teachers’ collaborative practices. An epistemic object is an object that is beyond the agents’ knowledge and understanding, at the time of first use, and at the edge of the epistemic horizon (see also Miettinen & Virkkunen, 2005). At the same time epistemic objects are grounded in historically developed practices. They function as generators of novel understandings, conceptualization and perhaps innovative solutions, as they are not yet known with certainty. The creative nature of the work with epistemic object appears to be characterized by working “at the edge of the unknown”: working with them produces developing conceptualizations and understandings. This view of use of tools is anchored in socio cultural theory, in particular Vygotsky’s (1978) notion of “tool” and mediation of tools. Cultural Historical Activity Theory (CHAT) developed this further (e.g. Engeström, 2001) claiming that learning can be mediated by a range of tools and instruments. At its inception the “tool” (analysis task schedule) was defined and meant to analyze mathematical tasks, a kind of technical object grounded in familiar pedagogic practices. However, over time and with different activities the insights gained (through the work with the “tool”) triggered communication between teachers (and teachers and teacher educator) and feedback at different levels; and provided access to a developing depth of perspective which encouraged teachers to explore further. In short, the original tool developed into something else: it became an epistemic object which challenged previous perceptions (e.g. creative questioning of value and purpose of mathematical tasks, and what they can afford); it produced “novel situations” (e.g. confidence in amending and enriching materials for particular purposes); and generated novel understandings of pedagogic practice (e.g. viewing mathematics/mathematics learning in different ways). It can be argued that the “tool” has developed catalytic potential, in the sense that it helped teachers, and gave them opportunities, to engage in a re-framed experience. Using the tool had aspects of familiarity (since it is grounded in the “territory” of mathematical tasks and learning), and at the same time of novelty as some of their perceptions are likely to be challenged and something being added to their repertoire. This combination of familiarity and novelty is likely to create “positive conditions” and for the teachers to experience “positive dissonance” (Baumfield, 2006) whereby routines and expectations are likely to be challenged, or disrupted, without the teachers feeling vulnerable, and more importantly for new ways of feedback to be opened up. This is claimed to be the tool’s catalytic quality: it can open up new avenues (e.g. of feedback), whilst maintaining stability by not being changed itself. Thus, the tool’s catalytic potential is provided by its intrinsic features, its usefulness in teacher everyday professional lives, and its potential for empowerment in terms of teacher learning.

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The crucial process element of the catalytic tool is the kinds of and the nature of feedback “produced”. The feedback from the task analysis tool is developmental, context-specific and highly relevant to teachers’ professional needs: be they reflective; diagnostic; focussed on knowledge/learning, or on skills. In Table 7.1, an overview is provided to show which kinds of activities (related to the “tool”) afforded which kinds of feedback, and in turn are likely to enhance which kinds of teacher learning. In summary, there is evidence that the project has had positive benefits to: • Teacher knowledge with respect to “pupil learning mathematics with understanding”. • Teacher selection and analysis of mathematics curriculum material. • Teacher confidence of amending and enriching material. • Teacher reflectivity with respect to the enactment of curriculum material. At a practical level results show that this project has helped teachers to spend time on developing their knowledge for/in teaching, by thinking about and analyzing curriculum material (some of it educative), developing the material further, and by “enacting” the material and reflecting on the processes. The project has succeeded in raising teachers’ awareness, and knowledge, of the educative nature of curriculum material, and what that may mean for their pedagogic practice. It is suggested that we need to help teachers learn from and work with all types of curricular materials – whether they are educative and well-designed, or otherwise – as they prepare for their teaching. This goes beyond “curriculum delivery”, and involves developing strategies to use the support offered by the school environment and uncovering “creative” ways to support their learning with the help of available “resources” and “tools”. Teachers benefit from opportunities to analyse, examine, enrich or amend new curriculum materials with their colleagues. This involves a process of “mutual transformation” – transforming the curriculum material, as well as potentially transforming the teachers’ notions of what can be done in the classroom, their pedagogic thinking. Adding to this, new resources, such as digital resources, the web of interaction becomes even more complex (Chapter 16). Further research is needed that takes us away from the dualistic thinking of “teachers and texts”, to more sophisticated processes and forms of analysis that include the working environment, the resource system, the activity format, and the curriculum script (Chapter 5). At the theoretical level it is evident that the process of interacting with “material” is complex, and it is often neither explicit nor public. There is evidence from this study that curriculum materials, more precisely a task analysis schedule, can act as catalyst for teacher learning. As the task analysis “tool” developed, it became a catalytic tool providing feedback which in turn helped teachers to develop deeper understandings. In the process it afforded feedback loops and changed its character, from “tool” as artefact to epistemic object at the interface between task design and enactment (see Fig. 7.1). Different forms of feedback emerged from the work with the “tool”, at four different levels. The results provide deeper insights into the processes of teacher learning with the help of analytic tools and the feedback these may afford.

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Task level (e.g. Mathematical tasks and their characteristics)

Process level (e.g. What do the tasks afford?)

Self monitoring level (e.g. confidence about working with tool and mathematical tasks)

139 Personal evaluation level (e.g. confidence to engage in further enquiries)

Provides feedback at four levels

Task analysis tool - a catalytic tool

As epistemic object at the interface between Design

Enactment

Fig. 7.1 Catalytic tool in relation to feedback levels, task design and enactment

Acknowledgements Particular thanks to the teachers featuring in the study; my colleague Dr Linda Haggarty who helped to develop some of the thinking behind the analysis schedule and to the NCETM who funded this project.

Appendix Task analysis schedule Text source: Content

Grade/year: Domain

Conn. through maths Processes

Proced. fluency

Number Algebra Geometry Measures Statistics and probability Within Across other subjects Representing Analysing – reasoning Analysing – procedural Interpreting Oral communication – implicit/none Oral communication – explicit 1 Step to be carried out 2 Steps to be carried out 3 Steps to be carried out

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Task analysis schedule Text source: Task type

Grade/year: Familiarity

Typically met in programme Some novel aspects Situation not met before

Context

Pure Artificial/contrived Authentic

Conceptual understanding

Implicit Explicit Subordinated

Cognitive demand

Knowledge (write, list, name) Comprehension (describe, summarise) Application (use, solve, apply) Analysis (compare/contrast, analyse) Synthesis (design, invent, develop) Evaluation (critique, justify)

Mathematical repres.

Analogy Pictorial (e.g. charts) Symbolic Numerical

Tools

Calculator Computer Geometric tools (compass, protractor)

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Butler, D. L., & Winne, P. H. (1995). Feedback and self-regulated learning: A theoretical synthesis. Review of Educational Research, 65, 245–281. Collopy, R. (2003). Curriculum materials as a professional development tool: How a mathematics textbook affected two teachers’ learning. Elementary School Journal, 103, 287–311. Davis, E. A., & Krajcik, J. S. (2005). Designing educative curriculum materials to promote teacher learning. Educational Researcher, 34(3), 3–14. Department for Education and Skills. (2005). Improving learning in mathematics. Sheffield: Standards Unit. Engeström, Y. (2001). Expansive learning at work: Towards an activity theory reconceptualisation. Journal of Education and Work, 14, 133–156. Fennema, E., Carpenter, T., & Franke, M. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27, 403–434. Fenstermacher, G., & Richardson, V. (1993). The elicitation and reconstruction of practical arguments in teaching. Journal of Curriculum Studies, 25, 101–114. Fernandez, C. (2002). Learning from Japanese approaches to professional development – The case of lesson study. Journal of Teacher Education, 53, 393–405. Greeno, J. G., & Goldman, S. G. (1998). Thinking practices in mathematics and science learning. Mahwah, NL: Lawrence Erlbaum Associates. Gueudet, G., & Trouche, L. (2009). Towards a new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71, 199–218. Haggarty, L., & Pepin, B. (2002). An investigation of mathematics textbooks and their use in English, French and German Classrooms: Who gets an opportunity to learn what? British Educational Research Journal, 28, 567–590. Hargreaves, D. (2000). Knowledge management in the learning society. Paper presented at the Forum of OECD Education Ministers, Copenhagen. Hattie, J., & Jaeger, R. (1998). Assessment and classroom learning: A deductive approach. Assessment in Education, 5, 111–122. Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77(1), 81–112. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., et al. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 Video Study. NCES 2003-013. Washington, DC: U.S. Department of Education, National Center for Education Statistics. LEMA. (2009). Learning and education in and through modelling and applications. Retrieved January, 2010, from www.lema-project.org Lloyd, G. M. (1999). Two teachers’ conceptions of a reform-oriented curriculum: Implications for mathematics teacher development. Journal of Mathematics Teacher Education, 2(3), 227–252. Lloyd, G. M., Remillard, J. T., & Herbel-Eisenmann, B. A. (2009). Teachers’ use of curriculum materials: An emerging field. In J. T. Remillard, B. A. Herbel-Eisenmann & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 3–14). New York: Routledge. Marx, R. W., Blumenfeld, P. C., & Krajcik, J. S. (1998). New technologies for teacher professional development. Teaching and Teacher Education, 14(1), 33–52. Miettinen, R., & Virkkunen, J. (2005). Epistemic objects, artefacts and organizational change. Organization, 12(3), 437–456. National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. Paris, C. L. (1993). Teacher agency and curriculum making in classrooms. New York: Teachers College Press. Pepin, B. (2008). Making connections and seeking understanding: Mathematical tasks in English, French and German textbooks. Presentation at the Nuffield seminar series MKiT – Mathematical Knowledge in/for Teaching, Loughborough, March 2008 (paper provided).

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Chapter 8

Measuring Content Through Textbooks: The Cumulative Effect of Middle-School Tracking William H. Schmidt

8.1 Introduction Textbooks are ubiquitous in schooling worldwide. While other chapters in this part examine the interaction between textbooks and teachers, we focus on one particular inherent characteristic of textbooks, their potential role in providing opportunities for learning mathematics. How textbooks are designed provides a window into the nature of the mathematics that students are expected to learn. They characterise not only the content but also advocate what students are to be able to do with that content – what mathematical behaviours are to be encouraged. In this way they serve as a bridge between the teacher and the students, translating abstractions into reality. They mediate between instruction and the actual behaviours that the students undertake as a part of learning. As a result, such a characteristic of textbooks can constrain opportunity. Using textbook data from a U.S. nationally representative sample of students, we demonstrate a methodology that characterises textbooks related to the content itself but also to the nature of how it is presented especially with respect to the expected behaviours. We do this for different groups of students (those found in different tracks – courses of study) to illustrate how differences in textbooks and their use can constrain opportunity to learn (OTL). Other chapters in this book deal with the interplay between teacher and textbook. For example, Rezat (Chapter 12) shows the linkage between teacher’s and students’ usage of the textbook, while Remillard (Chapter 6) argues that teachers are themselves passive users of curriculum materials. These are studies at the microclassroom level. The emphasis in this chapter is to describe the cumulative effect of textbook usage at the macro-level, across grades, over a student’s high-school career.

W.H. Schmidt (B) Michigan State University, East Lansing, MI 48824, USA e-mail: [email protected]

G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_8,  C Springer Science+Business Media B.V. 2012

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8.2 Background The mathematics achievement of U.S. middle- and high-school students is not considered strong by international standards (see, e.g., TIMSS, 2003, 2007; PISA, 2003, 2006) (http://timss.bc.edu/, http://www.pisa.oecd.org). This has prompted an emerging policy focus centred on two key aspects of the educational system – the curriculum and teacher quality. We focus on the curriculum, characterising mathematics opportunities as represented in textbook coverage across grades 7 through 12 and relating that to the common practice in the United States of tracking begun in the middle school. Studies have shown that curriculum is related to student achievement (see Floden, Porter, Schmidt, Freeman, & Schwille, 1981; Schmidt, 2003; Suter, 2000 for a review of this work). Some of this work has focused on the amount of mathematics covered (Schmidt, 1983, 2003; Stevenson, Schiller, & Schneider, 1994). Other studies have examined the role that a particular course such as algebra plays, not only in terms of what they know but also in terms of future career opportunities. Still other studies have focused on characterising the actual content students have studied and relating those specific opportunities to student achievement. In fact, this has been a traditional emphasis of international studies such as the Third International Mathematics and Science Study (TIMSS). The concept of OTL was designed to capture the type of topics studied and then to relate this to cross-national differences in achievement (McKnight, Crosswhite, Dossey, Kifer, Swafford, Travers, & Cooney, 1987). In Finland, Törnroos (2005) characterising OTL with textbooks reported a significant relationship between student achievement and textbook content coverage. Törnroos analysed Finish mathematics textbooks from grades 5, 6 and 7 and correlated the coverage with the seventh-grade student performance on the TIMSS 1999 test. He found that a strong positive relationship between student performance and the amount of cumulative coverage in the textbooks at the content topic level. In order words, the more the topics were covered in the textbooks, the better the students’ performance. The common element in both international and national studies is that the curriculum is a significant factor in explaining student achievement. The fact that these relationships have been established at the student, classroom and country level only strengthens the central role of this relationship to why schools matter (Schmidt, McKnight, Houang, Wang, Wiley, Cogan, & Wolfe, 2001). One of the factors related to what content students are exposed to in the United States is the practice of tracking. Although not typically practiced in other countries, at least among those studied in TIMSS, it is commonly practiced in the United States and begins in middle school. One estimate suggests that only about 25% of eighth-grade students attend schools that are not tracked (Cogan, Schmidt, & Wiley, 2001). We define tracking as the practice of having different students at the same grade take different courses that have different content. This is distinct from ability tracking where students in different classes (usually sorted by ability) cover the same

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topics but to different depths and for different amounts of time. Tracking results in different content exposures. Cogan et al. (2001) describes the number and nature of the different tracks typically found in the United States. Many times three to six different courses are offered in middle schools at eighth grade, most often including general mathematics, prealgebra and algebra. Each course presents a substantively different curriculum, and in turn affects students’ achievement differently. Prior studies have highlighted two important ways that a student’s eighth-grade course affects their subsequent mathematics achievement: positional advantages and differential achievement growth (Adelman, 1999; Atanda, 1999; Hoffer, 1992; McFarland, 2006; Schneider et al., 1997; Stevenson et al., 1994). Using data from the Longitudinal Study of U.S. Youth (LSAY), we developed a textbook-based methodology resulting in measures of the amount of demanding or complex mathematics content taken by a student and used them to obtain national estimates of what is typically taken by students in each of grades 7 through 12. These measures not only refer to the content itself, but also to the nature of what student behaviours are expected with respect to that content. These estimates can then be cumulated to reflect total exposure over middle and high schools to the more demanding aspects of mathematics (gauged by a combination of content difficulty and expected behaviours) given their starting point in seventh grade, that is the track into which they were placed in middle school. In that way we explore the cumulative content exposure for different tracks. In the analyses presented here the measures of curriculum are based on the textbooks used by each student in each of the mathematics courses taken.

8.3 Textbook-Based Estimates of Curriculum It is broadly accepted that textbooks are a good reflection of the implemented curriculum in most countries, and that textbooks are a particularly accurate reflection of the implemented curriculum in the United States (Fuson, Stigler, & Bartsch, 1988; Li, 2000; Mayer, Sims, & Tajika, 1995; Nicely, Fiber, & Bobango, 1986; Schmidt, McKnight, & Raizen, 1997a; Stigler, Fuson, Ham, & Kim, 1986). The growing emphasis on national standards and achievement testing are likely to increase teacher reliance on textbooks as the best available reflection of national standards and the intended curriculum (Crawford & Snider, 2000). Despite the centrality of the textbook to the implemented curriculum and classroom practices, there have been relatively few attempts to quantify the content coverage of textbooks (Porter, Floden, Freeman, Schmidt, & Schwille, 1986) and to measure the level of actual student exposure since few teachers cover the entire textbook during a school term or year. The TIMSS recognised the importance of obtaining reliable cross-national measures of the implemented curriculum and devoted substantial time and resources to the development of a content classification system for use with mathematics and science textbooks (Schmidt et al., 1997b; Valverde, Bianchi, Wolfe, Schmidt, &

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Houang, 2002). The application of this system has been described in numerous reports concerning the TIMSS results (Schmidt, McKnight, & Raizen, 1997a; Schmidt et al., 2001), but the initial translation of this classification system into quantitative measures has been limited to the work at the US TIMSS National Research Center at Michigan State University (Schmidt et al., 2001). Although the initial work by Schmidt and his colleagues was designed for international comparisons (Schmidt et al., 2001), the classification system has the potential to provide useful measures of the implemented curriculum at the classroom level and in the estimation of the influence of the implemented curriculum on student achievement. The idea of comparing the content of mathematics textbooks to the expectations and demands of mathematics problems is not new. Nicely (1985) and others have studied the content and form of problems in U.S. mathematics books. An extensive amount of comparative textbook analysis has been undertaken to understand differences in student performance in the United States and other countries (Fuson et al., 1988; Li, 2000; Mayer et al., 1995; Schmidt, McKnight, & Raizen, 1997a; Schmidt et al., 2001; Stigler et al., 1986). In virtually all of these studies, the content of small segments or specific problems has been analysed and classified, but comparisons between mathematics textbooks have not been made on the basis of the full book. Miller and Mercer (1997) argue that some students have difficulty because many mathematics topics are introduced too quickly by teachers who are trying to “get through the book”. No previous study has attempted to take into account the proportion of each mathematics book that is actually covered by various teachers, especially as an indicator of the scope of material actually covered. Porter (2002) has made a strong argument of the need to provide a metric or language to measure the content of the curriculum. Using a survey approach, Porter proposed a two-dimensional measurement technique that would take into account both the content of the implemented curriculum and the level of performance or understanding expected of the students in a given classroom. This general approach is similar to the textbook measurement technique described here.

8.4 TIMSS Mathematics Textbook Classification System The TIMSS textbook classification system is built on a set of content categories developed by groups of mathematicians and mathematics teachers representing numerous countries (Schmidt et al., 1997b). The classification system included 44 mathematics content topics that are commonly found in elementary and lower secondary schools among the participating countries. In TIMSS, mathematics textbooks that were used by the majority of the fourth grade and separately by the eighth grade students were included from each participating country for analysis. Each book was divided into units (often sections within chapters that corresponded to one to three days of instruction) and each unit was divided into blocks that are similar in content (usually a few paragraphs). The

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content of each block was coded using as many as eight content codes and as many as five performance expectation codes to characterise each block. The resulting data set provided thousands of block level codes for content. The challenge was to develop summary variables that translate the thousands of block codes into variables that provide useful information about the content of each mathematics text book. For purposes of cross-national comparison, Schmidt and his colleagues used a measure of the proportion of each book allocated to each of 44 mathematics content categories and compared this measure to the amount of time that each teacher reported that he or she devoted to the teaching of each of these topics and to student achievement outcomes (Schmidt et al., 2001). Another application of the TIMSS content codes is reflected in the International Grade Placement (IGP) index. The IGP is based on the curriculum data collected using the General Topic Trace Mappings (GTTM) from over 40 countries for the TIMSS (Schmidt et al., 1997b). It is a weighted average of the typical grade level at which countries first include a topic in their mathematics curriculum and the typical grade level at which countries focus instruction on that topic. For example, an IGP of 6.6 for the topic integers and their properties indicates that across TIMSS countries, the average of the grade level of typical introduction and the grade level of typical instructional focus is a little more than half way through grade six. We interpret this as indicating that, from an international perspective, this topic is typically covered in countries’ mathematics curriculum in grades six and/or seven. Similarly, an IGP of 9.0 for the topic patterns, relations and functions indicates that the average of countries’ typical introduction and typical focus is a little more than two grades later than the previously mentioned topic. This means that this topic is typically covered in countries’ mathematics curriculum around grade nine (Cogan et al., 2001; Schmidt, 2003). The IGP assumes that, given the hierarchical nature of mathematics, topics focused on in later grades are likely more complex or difficult, building on the topics covered in earlier grades. It is important to recognise that the TIMSS is a set of three cross-sectional studies designed to provide cross-national comparisons. These international comparisons are important and provide useful insights into the commonalities and differences in mathematics instruction and learning throughout the world. Nonetheless, they cannot be used to provide cumulative measurements on the same students.

8.5 Longitudinal Study of U.S. Youth An important application of TIMSS classification and coding approach would be to a longitudinal data set with curriculum measurements at each grade level. The LSAY1 provides a national longitudinal data set with extensive student,

1 This work was supported by NSF grant RED-9909569. All conclusions and findings reflect the views of the principal investigator and co-investigators and do not necessarily reflect the views of the National Science Foundation or its staff.

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parent, teacher and classroom variables and mathematics and science textbook information, including the percentage of the book that the teacher plans to cover during the semester or school year. Initiated in the Fall of 1987, the LSAY selected and followed two cohorts of public school students from 50 public high schools. Cohort Two was a group of 3,116 seventh grade students and served as the basis for the analyses reported here. The group was from 50 public middle schools of the feeder schools to the sampled high schools and was followed for 7 years (see Table 8.1). During each of the six school years, each student was asked to complete a mathematics achievement test and a science achievement test (usually in October), and two extensive attitudinal and activity questionnaires (in October and April). For every mathematics and science course that included one or more LSAY students, the teacher was asked to complete a course questionnaire that collected – among other variables – the name, publisher and year of the textbook used in the course and the percentage of the textbook that the teacher expected to cover during that course. Because each student attitudinal and activity questionnaire requested a full course schedule, including the name of the teacher and the class period in which the course occurred, it was possible to match each student to specific teachers and courses, allowing the linking of teacher-reported course variables to each student’s record. We focus on the coding and classification of mathematics text books here. One of the advantages of using the LSAY data set for this purpose is that it allows the measurement of the cumulative level of textbook (and presumably, curriculum) exposure to the full range of mathematics topics for students in different tracks throughout their middle- and high-school years. To the extent that textbooks can be coded to reflect the implemented curriculum, it will be possible to map the cumulative impact of tracking over a period of years and examine the influence of differential curricular exposure on individual student achievement.

Table 8.1 Longitudinal Study of U.S. Youth (LSAY) cohort two participation rates Grade

Same school

New school

Early graduate

Dropout

Lost

Quit

N

7 8 9 10 11 12

3,116 2,718 2,267 2,038 1,907 1,743 Percent 100 87 73 65 61 56

0 270 649 736 724 672

0 0 0 0 2 27

0 9 49 134 216 334

0 89 48 53 76 110

0 30 104 155 190 230

3,116 3,116 3,116 3,116 3,116 3,116

0 9 21 24 23 22

0 0 0 0 <1 1

0 <1 2 4 7 11

0 3 2 2 3 4

0 1 3 5 6 7

100 100 100 100 100 100

7 8 9 10 11 12

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8.6 Measures of Content Coverage The sheer number of mathematics textbooks used by LSAY students is an interesting indication of the decentralised and diverse character of U.S. secondary education, and it posed a challenge to the coding methods used in the original TIMSS textbook coding. If the original TIMSS procedure of coding every page of every book had been followed, substantially more time and resources would have been needed than were available. In TIMSS it was estimated that about 40 days or 240 h were needed by an expert coder to finish an U.S. eighth-grade textbook (typically about 700 pages); this implies over 5,600 days or over 21 years of coding for LSAY textbooks. To code the significantly larger number of LSAY mathematics textbooks, units were defined as chapters and each of the student problems or exercises at the end of each chapter was treated as a block. A careful examination of a number of mathematics textbooks confirmed that the problems or exercises at the end of each chapter were a comprehensive reflection of the major topics included in the chapter. In addition, the use of the problems and exercises made the coding of performance expectations easier than the original TIMSS technique of reviewing small clusters of paragraphs. As in the original TIMSS coding, each block (problem or exercise) was coded at the level of detail appropriate to the material and as many codes were used for each block as were needed (up to a maximum of eight). In practice, the majority of blocks required only one content code and few blocks required more than three content codes. The implementation of this modified coding strategy reduced the time and resources needed to do the basic coding, and this approach may serve as a model for the coding of additional mathematics textbooks or textbooks in other areas. One methodological issue deserves some discussion prior to turning to the measures reported on here. That issue is the metric with which to measure textbook content. If one algebra book, for example, includes 17 problems (blocks) on solving equations with two unknowns and another algebra book includes 25 (blocks) problems on this topic, what is the appropriate metric? If the number of problems is the metric, then a book that includes a larger number of problems would always be measured as more comprehensive than a book with fewer problems, although the number of problems may be an editorial decision or the result of page limitations. Alternatively, a metric could be used that would reflect whether a specific topic was covered in any block within a specific unit, and the number of units covering the topic could then be summed across all of the units in a book. But this approach has problems too. If the authors of one algebra textbook decide to organise their presentation of first-year algebra into 20 chapters (units) and another set of authors organise their material into 30 chapters (units) and if we sum across units, this approach would classify the book with 30 chapters as having 50% more content or coverage that the book with 20 chapters, although the actual content was virtually the same in the two books. Another approach would be to weight each unit by the number of pages in the unit, thus in the preceding example, assuming that the two books have the same

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number of pages, but different numbers of chapters (units), a weighting by pages would nullify the influence of the number of units per se. This approach, however, assumes that a 300-page algebra book contains more substantive content than a 200page algebra book. Given the generally larger size of U.S. textbooks than Japanese textbooks, for example, this approach would be problematic for cross-national comparisons where the textbooks vary significantly across countries (Valverde et al., 2002). From the preceding discussion, it is clear that there is no single metric with which to measure textbook content or coverage. For a more complete discussion of the issues, see Schmidt et al. (1997b). Recognising these limitations, we explored different approaches to the measurement of mathematics textbook content and coverage and applied to the textbook data from the LSAY. The following sections describe four of the approaches and some of the advantages and disadvantages of each measure. The final section will focus on patterns of change and comparisons among the four measures as well as to relate these measures to achievement.

8.6.1 Measure 1: Student Exposure to Mathematics Topics The first Index is designed to measure the exposure of students to mathematics topics in grades 7 through 12. Using the TIMSS mathematics content codes, each block within each unit was coded for its topic. For each topic, the number of blocks within a unit that covered that topic is summed and a proportion is computed to reflect the number of blocks that cover the topic out of the total number of blocks in the unit. The total number of pages in each unit is multiplied by the proportion of coverage for each topic, which produces a value that can be interpreted as the number of pages devoted to that specific topic in that unit. For each unit, this estimated number of pages of coverage is then summed across all mathematics topics to produce a total number of pages of estimated coverage for that unit. This procedure is substantially the same as the procedure employed in TIMSS, which used the proportion of pages devoted to each subject rather than the number of pages because of the large variation in textbook size among the nations participating in TIMSS. The total number of units covered in any given course is based on each teacher’s report of the percentage of the textbook that he or she covered in that course. The original coding of the blocks and units included the beginning and ending page number for each unit. Since the total number of pages in each book is known, the percentage of textbook coverage is applied to the total number of pages in each mathematics textbook, producing an estimate of the number of pages that will be covered in the course. By applying this number to each unit in sequential order (front to back), the number of units that will be covered is computed. If any part of a unit falls within the estimated number of pages to be covered, the unit is included in the computations.

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This approach allows the computation for each classroom of the number of pages of mathematics topic exposure that each student in that class experienced during that school year. Applied to LSAY population, this approach allows the assignment of a mathematics curriculum exposure index that takes into account the specific textbook used in that class and the percentage of the book covered during that course. This Mathematics Topic Exposure Index is expressed in the number of pages of topic coverage and ranges from 0 to 1,013. Obviously, no single student reads a thousand pages of mathematics text, but the Index reflects the estimated number of pages on which one or more mathematics topics was covered. To see this Index in context, it is useful to look at the seventh grade students in terms of the mathematics course in which they were enrolled. The mean number of topic pages for all seventh grade students was 447, but seventh-grade students in remedial mathematics courses were exposed to an average of 365 pages of materials while seventh-grade students in pre-algebra were exposed to an average of 556 pages (see Table 8.2). The same pattern can be seen among eighth-grade students. Eighth-grade students in a remedial mathematics course were exposed to an average of 419 topic pages compared to 642 topic pages for eighth-grade students enrolled in first-year algebra (see Table 8.2). The major problem with this Index is that it makes no differentiation by the content of the mathematics topic to which a student is exposed. A page of exposure to whole number addition is treated the same as a page of exposure to an advanced geometry topic. We sought an index that would provide some information about the topics covered and the relative difficulty of those topics.

Table 8.2 Mean scores on the Mathematics Topic Exposure Index, by grade and course Grade 7

All students in grade Remedial grade-level mathematics Regular grade-level mathematics Advanced grade-level mathematics Pre-algebra Algebra I Algebra I honours

Grade 8

Mean topic pages

Number of students

Mean topic pages

Number of students

447

2, 887

492

2, 351

365

346

419

258

436

1, 846

400

519

449

255

556 611 −

415 16 −

−

−

491 642 747

1, 123 269 118

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8.6.2 Measure 2: Mathematics Topic Exposure – Weighted by International Grade Placement Index Measure 2 uses the same procedures described for Measure 1, but applies the IGP weight to each topic at the book level. The resulting weighted index was deflated to the same range as the original index, which means that the highest weighted score was set equal to 1,013 (the highest score on the original index) and all lower scores were scaled accordingly. A comparison of the mean scores on Measure 2 and Measure 1 shows that the weighting procedure generally decreased the number of topic pages each year (Table 8.3). The general deflation of Measure 2 indicates that a large proportion of the topics covered by the students were relatively easy topics, which were given lower IGP weights and the summation of these weighted scores led to a reduction in the magnitudes of the index. The decrease in the mean number of topic pages to which students were exposed in grades 11 and 12 is a reflection of a large number of students who do not take a mathematics course in those years and thus are exposed to no mathematics topics. It is important to recognise that the population of students in the middle- and high-school years is changing. By the beginning of high school, some students drop out of formal schooling each year. Other students move to a different school and are Table 8.3 Mean scores on the different mathematics topic exposure indices All students in grade each year M1

M2

M3

M4

Grade

Unweighted index

Weighted index (IGP)

Weighted index (PE)

Weighted index (IGP and PE)

Number of students

7 8 9 10 11 12

447 492 631 747 644 348

276 337 496 622 561 326

373 425 564 707 566 330

191 240 363 480 396 251

2,887 2,351 1,968 1,789 1,758 1,606

Grade 7 8 9 10 11 12

Same studentsa 451 502 648 781 674 392

281 347 515 652 588 368

380 434 583 743 593 374

196 247 379 506 415 285

1,776 1,624 1,602 1,573 1,652 1,394

a The

same student population includes students with course and textbook data for 5 of the 6 years covered by LSAY Cohort 2. Variations in the number of cases reflect the absence of either course or textbook data in a specific year. All of the students in this population stayed in school for the full 6 years.

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lost to the study despite extensive tracking efforts and some students simply refuse to continue to participate in the study. A comparison of the mean scores for the population of students in school each year of the LSAY and the mean scores of a set of students who remained in the study throughout the six middle- and high-school years shows minimal differences in pattern or level (see Table 8.3). Taken together, these results indicate that weighting each mathematics topic by the IGP provides a more realistic measure of student exposure to mathematics topics, but the decline in the mean scores in the last years of high school illustrates the problem of examining these indices by looking at the mean score for all students at each grade level, regardless of whether they are actively taking a mathematics course or not. We will return to this problem in a later section.

8.6.3 Measure 3: Student Exposure to Mathematics Topics – Weighted by Performance Expectations In the original TIMSS coding of textbooks and in the subsequent coding of the mathematics textbooks used by students in the LSAY, each block was also coded for the level of performance expectation. For the coding of the mathematics textbooks used by LSAY students, the basic block code was based on an exercise at the end of each section of the text, and it was possible to categorise the level of performance required to successfully answer or solve each exercise. The TIMSS textbook classification system utilised 26 levels of expected student performance, which based on international data were categorised into five levels reflecting increasingly more complex cognitive demands ranging from use of simple algorithms to proofs. Measure 1 was used as the base and the original code for each block (0 or 1) was multiplied by the TIMSS expected student performance weight for that item (see Table 8.3). To some extent, the TIMSS student expectation level can be viewed as an alternative weight to the IGP because both weights seek to quantify the content difficulty of each item. Using this approach, Measure 3 was constructed. Again, the weighted scores were deflated so that the highest score on this index would be equal to the highest score on Measure 1. The mean scores on Measure 3 are similar in pattern and structure to other two Measures, but tend to be slightly lower than Measure 1 and slightly higher than Measure 2 (see Table 8.3).

8.6.4 Measure 4: Student Exposure to Mathematics Topics – Weighted by Performance Expectations and IGP If, as the preceding analysis suggests, performance expectations and the IGP each measure somewhat different aspects of the mathematics topics to which students are exposed, an alternative approach is to weight each block by both performance expectations and the IGP. Measure 4 begins with Measure 3 and also weights it

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using the IGP. As with the previous weighted indices, the new jointly weighted Mathematics Topics Exposure Index was deflated to the original. Again, the mean scores on Measure 4 are similar in pattern and structure to the other indices, but substantially lower in numeric value than any of the previous indices, reflecting the joint weighting process (see Table 8.3). The explanation for the lower numerical values is the same as for the previous measures, and the effect of the joint weighting process was to combine the factors that differentiate each item by content difficulty and performance expectation.

8.7 A Comparison of the Measures The preceding sections have described the construction of four measures of the mathematics content taken by the typical student in middle and high schools. It is useful to look at the distribution of the mean for all students at each grade level over the 6 years covered by the LSAY (see Fig. 8.1). All of the measures display the same general pattern. Measure 1 (the top line in Fig. 8.1) indicates that the average U.S. student is exposed to an increasing amount of mathematics during the middle- and high-school years through grade 10. The sharp decline after grade 10 in the number of pages reflects declining enrolments in mathematics courses in grades 11 and 12. It is a reflection of the curriculum only in that the curriculum is largely elective in the last 2 years of high school. Compared to the total numbers of pages of mathematics (the top line), the lower lines for the three weighted indices indicate that many of the topics presented are not that demanding either in terms of content or the performance expectation. This is a quantitative confirmation of the frequent criticism that the U.S. mathematics curriculum is a mile wide and an inch deep, at least as experienced by the typical or modal student (Schmidt, McKnight, & Raizen, 1997a).

Number of Concept Pages

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Fig. 8.1 Amount of mathematics studied: a comparison of the measures

8

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This is true for both topic difficulty and level of cognitive demand but more so for the latter. This implies that the amount of mathematics to which the typical student is exposed through the textbook that demands reasoning, proof and conjecture is only a fraction of the total especially for grades seven through 10. Accounting for topic difficulty as well reduces the total amount even further – almost a 50% reduction for grades seven through nine. This undoubtedly reflects the remedial and repetitive nature of the middle-school curriculum (Schmidt, McKnight, Cogan, Jakwerth, & Houang, 1999). These patterns may be helpful in understanding one of the reasons that the scores of U.S. students in grade 12 often appear to decline in TIMSS and other crossnational tests. Many U.S. high-school students have no exposure to mathematics in the last 2 years of high school and may be less competent in algebra or geometry by grade 12 than they were when they completed their required courses in grade 10. Also, they do not in general have exposure to very demanding mathematics either from a content or cognitive demand point of view.

8.8 Cumulative Exposure to Mathematics Topics All of the indices have been described in terms of year by year exposure. They are constructed, however, so that we could consider these measures in terms of the cumulative exposure of each student during the middle- and high-school years. To understand these cumulative experiences, it is essential to take into account that the U.S. mathematics curriculum is tracked from middle through high school. One indicator of current tracking patterns is the year in which a student takes his or her first algebra course.2 The more advanced students take pre-algebra in grade seven and algebra in grade eight, and these students are usually considered the top track of the U.S. mathematics curriculum. Most U.S. students take first-year algebra in grade nine and represent the middle track. Students who take their first algebra course in grades 10 or 11 are often vocational students or general education students in systems that require algebra for high-school graduation, and this is the third track in mathematics. Some students are able to meet the minimum mathematics courses through various arithmetic and business mathematics courses and never take an algebra course in high school. When the amount of cumulative exposure to mathematics is viewed by track, the impact of the current U.S. policy is apparent (see Fig. 8.2). Using the Index of Mathematics Exposure weighted by both content difficulty and performance expectation (Measure 4), it is clear that students in the top mathematics track 2 A very small number of students take pre-algebra in grade six and algebra in grade seven, but in the early 1990s when the LSAY was in the field, this pattern was found in less than 1% of students. In this analysis, students who took their first algebra course in either grade seven or eight were combined into the most advanced track. As the TIMSS results indicate, students in many countries routinely take algebra in grades six and seven (Schmidt, McKight, & Raizen, 1997a).

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Number of Concept Pages

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Fig. 8.2 Cumulative exposure (weighted) to mathematics concepts, by track

are exposed to significantly more challenging mathematics earlier than other students and that this initial advantage grows during the 6 years of middle and high schools. It is important to note that all four of the track groups shown in Fig. 8.2 are reasonably close to each other in grade seven, but that the students who take algebra in grade eight are exposed to more challenging mathematics topics than the other track groups. This initial advantage expands with exposure to geometry in grade nine and second-year algebra in grade 10. By the end of high school, this top track group has been exposed to the weighted equivalent of 2,748 pages of challenging mathematics. For students who begin algebra in grade nine, there is a steady growth in exposure to more demanding mathematics. In weighted topic page terms, this group of students is exposed to an average of 2,126 pages by the end of grade 12. This group is significantly less likely to reach calculus in grade 12, which may account for the lower rate of growth from grade 11 to 12 for this group than the top track group. Students who take their first algebra course after grade nine are exposed to an average of 1,758 pages of mathematics by the end of grade 12. This pattern suggests that some schools and teachers may not see these students as strong candidates for college and provide a less rigorous curriculum. Students in the lowest mathematics track, who take no algebra course in high school, are exposed to some mathematics topics in arithmetic and business mathematics courses, but it is less rigorous than the curriculum that includes algebra and often includes only the minimum number of mathematics courses required by the state. By the end of grade 12, this group was exposed to an average of 1,072 pages of demanding mathematics which is over 1,600 pages less than the advanced track but, perhaps even more telling, some 700 pages less exposure than those who

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were not in the advanced or even “normal” track but those who at least took algebra sometime after ninth grade. The consequences of these decisions are likely large (see Fig. 8.2) and related to student achievement.

8.9 Using Textbook Measures of Curriculum to Predict Mathematics Achievement The curriculum measures described in this chapter demonstrated how content exposure could be measured through a content analysis of textbooks. Here we illustrate how such an index or measure of curricula content could be used to model student achievement. Miller (2004, 2006) reported using a cumulative curriculum measure in a structural path model to examine factors associated with the development of student achievement in mathematics during middle school. The model included variables corresponding to the role of parents (parent education, parent college push), home (home learning resources), teachers (teacher push), a cumulative middle-school mathematics curriculum exposure index (on the basis of Measure 4), student education plans, and student gender. Student reading achievements measured at the beginning of ninth grade was used as a control for prior scholastic achievement. The dependent variable was mathematics achievement in the fall of ninth grade. A measure of mathematics curriculum exposure in the middle school was included in the model (Measure 4). The model had good fit and the combination of the variables accounted for 65% of the total variance in the model with a Root Mean Square Error Approximation (RMSEA) of 0.04. The path model is displayed in Fig. 8.3. As expected, parent and prior scholastic achievement (as measured by the reading score) had a large impact on mathematics achievement (see Fig. 8.4) but followed by the impact of curriculum exposure. The total effect was 0.15, relative to 0.69 for the reading score and 0.27

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Fig. 8.3 A path model to predict student achievement in mathematics at the beginning of 9th grade

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Fig. 8.4 Total effect of independent variables of student achievement in mathematics at the beginning of 9th grade

each for parent education and parent college push. Nonetheless, it was greater than the other variables in the model. In other words, after parent and prior scholastic achievement, middle-school mathematics curriculum exposure accounted for a significant amount of variation in student mathematics achievement at the beginning of ninth grade.

8.10 Summary In this chapter, we described the development of a textbook-based methodology for characterizing mathematics content coverage and illustrated its use in longitudinal studies to map and characterise the cumulative curricular experiences of individual students. The measures not only take into account content, but also expected student behaviours which can be linked to classroom instruction. We related the curriculum measures to student mathematics achievement in a formal structural model. Demonstrating the linkage of the textbook as a source of opportunity to both instruction and performance at the macro level further reinforces the importance of focusing on the interplay between teacher and textbook which is one of the foci of this book. It is our hope that this will foster renewed interest in the measurement of textbook content and in the use of such measures in modelling student achievement and educational opportunity. Acknowledgement I would like to thank my colleagues Jon D. Miller, Richard T. Houang and Linda G. Kimmel who co-authored this chapter.

References Adelman, C. (1999). Answers in the tool box: Academic intensity, attendance patterns, and bachelor’s degree attainment. Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. Atanda, R. (1999). Do gatekeeper courses expand education options? (NCES Report 99-303). Washington, DC: National Center for Education Statistics.

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Cogan, L. S., Schmidt, W. H., & Wiley, D. E. (2001). Who takes what math and in which track? Using TIMSS to characterize U.S. students’ eighth-grade mathematics learning opportunities. Educational Evaluation and Policy Analysis, 23(4), 323–341. Crawford, D. B., & Snider, V. E. (2000). Effective mathematics instruction: The importance of curriculum. Education and Treatment of Children, 23(2), 122–142. Floden, R. E., Porter, A. C., Schmidt, W. H., Freeman, D. J., & Schwille, J. R. (1981). Responses to curriculum pressures: A policy-capturing study of teacher decisions about content. Journal of Educational Psychology, 73(2), 129–141. Fuson, K. C., Stigler, J. W., & Bartsch, K. (1988). Grade placement of addition and subtraction topics in Japan, mainland China, the Soviet Union, Taiwan, and the United States. Journal for Research in Mathematics Education, 19, 449–456. Hoffer, (1992). Middle school ability grouping and student-achievement in science and mathematics. Educational Evaluation and Policy Analysis, 14(3), 205–227. Li, Y. (2000). A comparison of problems that follow selected content presentations in American and Chinese mathematics textbooks. Journal for Research in Mathematics Education, 31(2), 234–241. Mayer, R. E., Sims, V., & Tajika, H. (1995). A comparison of how textbooks teach mathematical problem solving in Japan and the United States. American Educational Research Journal, 32, 443–460. McFarland, D.A. (2006). Curricular flows: Trajectories, turning points, and assignment criteria in high school math careers. Sociology of Education, 79(3), 177–205. McKnight, C. C., Crosswhite, F. J., Dossey, J. A., Kifer, E., Swafford, J. O., Travers, K. J., et al. (1987). The underachieving curriculum: Assessing U.S. school Mathematics from an international perspective. Champaign, IL: Stipes Publishing Company. Miller, J. D. (2004). Student achievement in context: The influence of parent and home factors on student achievement in Mathematics. Paper presented at AERA Annual Meeting, San Diego. Miller, J. D. (2006). Parents, teachers, and curriculum: Partners in student achievement in mathematics. Presentation at the Lappen-Phillps-Fizgerald Endowed Chair Lecture, Michigan State University, October 11, 2006. Miller, S. P., & Mercer, C. D. (1997). Educational aspects of mathematics disabilities. Journal of Learning Disabilities, 30(1), 47–56. Nicely, R. F., Jr. (1985). Higher-order thinking skills in mathematics textbooks. Educational Leadership, 42(7), 26–30. Nicely, R. F., Jr., Fiber, H. R., & Bobango, J. C. (1986). The cognitive content of elementary school mathematics textbooks. Arithmetic Teacher, 34, 60. OECD Programme for International Student Assessment. (2003, 2006). Retrieved on July 12, 2010, from http://www.pisa.oecd.org Porter, A. C. (2002). Measuring the content of instruction: Uses in research and practice. Educational Researcher, 31(7), 3–14. Porter, A. C., Floden, R. E., Freeman, D. J., Schmidt, W. H., & Schwille, J. R. (1986). Content determinants, Research Series No. 179B. Ann Arbor, MI: Institute for Research on Teaching, Michigan State University. Schmidt, W. H. (1983). High school course-taking: Its relationship to achievement. Journal of Curriculum Studies, 15(3), 311–332. Schmidt, W. H. (2003). Too little too late: American High Schools in an international context. Brookings Papers on Education Policy. Washington, DC: The Brookings Institution. Schmidt, W. H., McKnight, C. C., Cogan, L. S., Jakwerth, P. M., & Houang, R. T. (1999). Facing the consequences: Using TIMSS for a closer look at U.S. mathematics and science education. Dordrecht, The Netherlands: Kluwer. Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H. C., Wiley, D. E., Cogan, L. S., et al. (2001). Why schools matter: A cross-national comparison or curriculum and learning. San Francisco: Jossey-Bass. Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1997a). A splintered vision: An investigation of U.S. science and mathematics education. Dordrecht, The Netherlands: Kluwer.

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Schmidt, W. H., McKnight, C. C., Valverde, G. A., Houang, R. T., & Wiley, D. E. (1997b). Many visions, many aims: A cross-national investigation of curricular intentions in school mathematics. Dordrecht, The Netherlands: Kluwer. Schneider, B., Swanson, C. B., & Riegle-Crumb, C. (1997). Opportunities for Learning: Course Sequences and Positional Advantages. Social Psychology of Education, 2(1), 25–53. Stevenson, D. L., Schiller, K. S., & Schneider, B. (1994). Sequences of opportunities of learning. Sociology of Education, 67, 184–198. Stigler, J. W., Fuson, K. C., Ham, M., & Kim, M. S. (1986). An analysis of addition and subtraction word problems in American and Soviet mathematics books. Cognition and Instruction, 3, 153–171. Suter, L. E. (2000). Is student achievement immutable? Evidence from international studies on schooling and student achievement. Review of Educational Research, 70, 529–545. TIMSS & PIRLS International Study Center. (2003, 2007). Retrieved July 12, 2010, from http:// timss.bc.edu/ Törnroos, J. (2005). Mathematics textbooks, opportunity to learn and student achievement. Studies in Educational Evaluation, 31, 315–327. Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Dordrecht, The Netherlands: Kluwer.

Chapter 9

Masters’ Writings and Students’ Writings: School Material in Mesopotamia Christine Proust

9.1 Introduction This chapter offers a reflection on masters’ documentation in the context of the scribal schools that flourished in Mesopotamia about four thousand years ago. The approach is historical but, despite the fact that the Mesopotamian sources are very distant from us, the issues addressed here are similar to some of the phenomena analyzed by other authors of this book. The development of scribal schools in the late third millennium and the early second millennium in Mesopotamia corresponds to a switch in the medium used for the accumulation and transmission of knowledge: from memorization, the medium became essentially written during this period. This switch is in a way symmetrical to the ones described in this section (Chapter 5). The normative function of the curriculum (Chapter 6) is a striking feature of the ancient system of education, and this function explains the fact that that the written artifacts that reached us are highly stereotyped (Section 9.5). Literary texts found in ancient schools give evidence of the emergence of an ideology that legitimates the schools and the stratum of erudite scribes that the schools produce (Section 9.4). The impact of historical and cultural context on the masters’ activities is emphasized in Section 9.4 of this chapter as well as in Chapter 16. The collective aspect of creation and transmission of knowledge is addressed in the fourth part of this book; this dimension appears likewise in Section 9.6. In the following, I intend to show how Mesopotamian sources shed light on the documentation work of the most ancient teachers we know.

9.2 Documentation in Scribal Schools The word ‘documentation’ conveys different meanings according to the communities of scholars who use it. For historians, documentation is generally a set of written artifacts that provide information about a given problem. In the present book, C. Proust (B) Laboratoire SPHERE (CNRS & University Paris-Diderot), Paris, France e-mail: [email protected] G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_9,  C Springer Science+Business Media B.V. 2012

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documentation is rather understood as a process, by which teachers transform available resources into documents for teaching (Chapter 16). To avoid ambiguities, I shall use the word ‘sources’ to designate the written artifacts used by historians, and the word ‘resources’ to designate the knowledge, memorized or written, used by the masters of the scribal schools. What kind of resources were the ancient masters using? To answer this question, historians cannot employ the same investigation methods as observers of presentday realities. Historians are dependent on their sources, that is, on a small body of evidence that illuminates only a limited part of ancient realia. Moreover, the picture they paint could be distorted by the work of interpretation that they have to do to make the data intelligible. However, concerning Mesopotamian scribal schools, the situation is exceptionally favorable due to the huge quantity of school tablets handed down to us (Section 9.3). No other educational system of the past is as well documented as that of Mesopotamia. But the sources that we have provide a truncated view of life in the schools, because it is mainly the production of students that has been preserved.1 Indeed, if the curriculum of elementary education is quite well known, the masters’ activities, their relationship with students or peers, their status, their personality, their sources of income, and their role in the city are few documented. We know even less about the material available to them, their libraries, if any. By combining information from various sources, we can nevertheless shed some light on the work of the masters. Direct information is provided by samples of Sumerian literary compositions written by masters for students (Section 9.4). Indirect information on masters is provided by the writings of their students (Section 9.5). More evidence comes from texts written by masters for other purposes than elementary education (Section 9.6).

9.3 Sources and Historical Context The sources that provide information on education in Mesopotamia are mainly school tablets, that is, clay tablets written by young students during the first stage of their education (or ‘elementary level’).2 These tablets were discovered in many archaeological sites, over a large geographical area, including present-day Iraq, Iran, and Syria. On these tablets, young scribes wrote out exercises for learning cuneiform writing, Sumerian vocabulary and grammar, numbers, measures, and calculations. The future careers of the students were probably quite diversified. Many of them were prepared for administrative tasks in various levels of the administration of 1 The historian of education very rarely has the chance to have access to students’ work (drafts, notebooks, exams), which had no value in the eyes of its authors, and was generally destroyed. 2 Veldhuis (1997). We don’t know how old the students were at the beginning of their scribal education. They were old enough to be able to manipulate clay and “calame” (the cane the scribes used to impress signs on wet clay), but still in the charge of their parents. Moreover, the age of the students could have changed according to the place and the period.

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temples, palaces, or private household. Some of them were probably more specifically prepared to specialized crafts based on written tradition, such as medicine and law, or to ensure the transmission of knowledge. It is difficult to describe with more precision both the future professions of the scribes educated in schools and the scholarly crafts. What is clear is that the schools, particularly those of Nippur (see below), played a key role in training social elites. It is also probable that the dynamic extension of scribal schools in the late third millennium (Neo-Sumerian period) and the early second (Old-Babylonian period) in Mesopotamia and beyond was accompanied by the development of a scholarly milieu linked to these schools. It seems that, at least in important schools such as those of Nippur, education was provided by professional scholars, who were quite specialized (see text ‘Edubba D’ below). These scholars were perhaps at the same time instructors in charge of the young beginners, professors in charge of the advanced students, and creative scientists. They could occupy at the same time high positions in the temple, the city, or the palace hierarchy. Since we ignore the exact nature of the scholars’ charge, I prefer to refer to them as ‘masters’ rather than as ‘teachers,’ a term that could implicitly suggest that teaching at the elementary level was their unique activity. The conservation of the unskilled writings of students is partially accidental. It is due primarily to the nature of the writing medium, the clay, a nearly indestructible material. It also ensues from the reuse of dry and waste tablets as construction material. Trapped in walls, floors or foundations of houses, tablets produced by students and subsequently discarded have escaped other forms of destruction. The context in which the education of the scribes took place is not always well known, and it was probably not the same everywhere, or at every time. On the basis of the sources on which they rely, historians point out that the context of the training was institutional, domestic, professional, or religious. In Ur, a city of southern Mesopotamia, the home of a priest seems to have housed important teaching activities in the Old-Babylonian period (Charpin, 2008). In Sippar, farther north, a school was integrated into a household, according to Tanret (2002). The Assyrian merchants who developed a business over long distances between Mesopotamia and Anatolia transmitted the basics of writing and arithmetic to their children by practicing their craft, using methods similar to apprenticeship (Michel, 2008). In Nippur, the great religious and cultural capital of Mesopotamia, situated a hundred kilometers south of the present-day Baghdad, the school context appears to have been institutional and secular. Education was especially important because of the presence of a high court, the development of an important medical tradition, and the political role of scribes from Nippur in the legitimization of kingship. If we consider the situation a thousand years later in Mesopotamia, the context has changed completely. The practice of cuneiform has declined in favor of alphabetic writing on perishable media (parchment and papyrus, nowadays lost), and has become confined to few families of scholars, related to the clergy. This brief historical overview stresses the diversity of the teaching contexts and of the sources used by historians. Each of these contexts deserves particular analysis, which is not possible in the limited space of this chapter. Concerning elementary education (Sections 9.4 and 9.5), I shall limit myself to the Old-Babylonian sources

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from Nippur, on the one hand because of the abundance of tablets, on the other hand because the education in Nippur was a model throughout the vast area in which cuneiform writing was used.3 The section on masters’ writing (Section 9.6) will rely on sources from various known or unknown provenances.

9.4 Literary Sources Literary sources that contain information on scribal schools (‘House of tablets,’ ‘Edubba’ in Sumerian) are compositions that were used for learning Sumerian vocabulary and grammar. In Nippur the students were taught in Sumerian, a dead language at that time. Akkadian, a Semitic language, had been widespread in Mesopotamia during the third millennium to the detriment of the Sumerian, which had probably disappeared as a mother tongue in the early second millennium. Sumerian, however, remained a scholarly and liturgical language until the disappearance of the practice of cuneiform writing in the beginning of our era. Had the literary texts used in schools been written specifically for teaching? It is likely that school masters created original compositions, but they also reused old material, transmitted by written or oral tradition. Some hymns praising the skills of Neo-Sumerian kings and their Old-Babylonian successors may have been extracted from a hagiographic literature developed outside of the schools. Other literary compositions appear to be mere products of the schools. Vanstiphout (1978, 1979), for example, has shown that the first literary text studied in the school curriculum is a hymn praising the king of Isin Lipit-Eshtar (1934 BCE–1924 BCE). It presents all the characteristics of a text specifically built for the teaching of Sumerian grammar. By its organization, this text allows the systematic learning of a vast repertoire of cuneiform signs, grammatical constructions, and rhetorical patterns. Another interesting aspect of this text is the information it gives on the links between schools and society. We learn, for example, that the scribes are supposed to acquire skills in the fields of accounting, law, and surveying. This text, as well as others of the same kind, reveals the ideological role of this education, and shows that the students, or at least a part of them, were intended to belong to a caste devoted to the king (Michalowski, 1987, p. 63). Compositions named ‘Edubba texts’ by Sumerologists evoke more directly everyday life in schools.4 The following picture of a school, extracted from the

3 Old Babylonian tablets from Nippur and now kept in Istanbul and Jena are published in Proust (2007, 2008a). Photos and informations are available on line at http://cdli.ucla.edu/ (Cuneiform Digital Library Initiative website), by entering Museum number or CDLI number (both information are provided here). Parts of the Philadelphia tablets are published in Robson (2001). Veldhuis’ Ph.D. thesis contains a study of lexical tablets from Nippur, and a detailed reconstruction of Nippur curriculum (Veldhuis, 1997). 4 We know six Edubba texts. The Electronic Text Corpus of Sumerian Literature (ETCSL, http:// etcsl.orinst.ox.ac.uk/) provides the following list: Edubba A or “Schooldays” (Kramer, 1949); Edubba B or “A scribe and his perverse son” (Sjöberg, 1973); Edubba C or “The advice of

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composition ‘Edubba A’ or ‘School days’ (Kramer, 1947, p. 205), gives the impression that the school was an institution where the discipline was harsh, and the staff very numerous and specialized. Who was in charge of [. . .] (said) ‘Why when I was not here did you talk?’ caned me. Who was in charge of the [. . .] (said) ‘Why when I was not here did you not keep your head high?’ caned me. Who was in charge of drawing (said) ‘Why when I was not here did you stand up?’ caned me. Who was in charge of the gate (said) ‘Why when I was not here did you go out?’ caned me. Who was in charge of the [. . .] (said) ‘Why when I was not here did you take the [. . .]?’ caned me. Who was in charge of the Sumerian (said) ‘You spoke [. . .]’ caned me. My teacher (said) ‘Your hand is not good,’ caned me.

Another Edubba composition provides more details on the curriculum. The text ‘Eduba D’ (Civil, 1985) contains a dialogue between two students who are training to speak Sumerian. In turn, they praise their own skills and insult their partner. The text begins as follows (translation Vanstiphout, 1997, p. 592)5 : (Examiner and Student) 1. ‘Young man, [are you a student?’ – ‘Yes, I am a student.’] (Examiner) 2. ‘If you are a student, 3. do you know Sumerian?’ (Student) 4. ‘Yes, I can speak Sumerian.’ (Examiner) 5. ‘You are so young; how is it you can speak so well?’

This extract shows that the Sumerian was not the mother tongue of the young scribes, who had to learn to speak fluently this dead language. The following extracts refer to the curriculum (ibid). 11. ‘The [texts] in Sumerian and Akkadian, from A-A ME-ME 12. [To. . .] I can read and write. 13. All lines from d INANA-TEŠ2 14. Till the ‘beings of the plain’ at the end of LU2 -šu I wrote. 15. I can show you my signs, 16. Their writing and their interpretation; and this is how I pronounce them.’ [. . .] 19. ‘Even if I am assigned LU2 -šu on an exercise tablet 20. I can give the 600 LU2 entries in their correct sequence. [. . .] 26. In a single day, the teacher would give me the same pensum four times.

a supervisor to a younger scribe” (Vanstiphout, 1996, 1997); Edubba D or “Scribal activities” (Civil, 1985) – see below; Edubba E or “Instructions of the ummia”; Edubba R or “Regulations of the Edubba”. A French translation of “Edubba A” by Pascal Attinger, with philological notes, can be found at: http://www.arch.unibe.ch/content/e8254/e8548/e8549/index_ger.html?preview= preview&lang=ger&manage_lang=ger 5 See also Civil (1985, pp. 71–72).

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27. In the final reckoning, what I know of the scribal art will not be taken away! 28. So now I am master of the meaning of tablets, of mathematics, of budgeting, 29. Of the whole scribal art, of the disposition of lines, of evading omissions, of . . . 30. My teacher approved (my) beautiful speech. 31. The companionship (in the school) was a joyful thing. 32. I know my scribal art perfectly; 33. Nothing flusters me. 34. My teacher had to show me a sign only once, 35. And I could add several from memory.’

The text shows the relationship between writing, speaking (lines 12 and 30) and memorization (line 35). The list of school tasks enumerated in lines 11, 13, 14, 19, 20, and 28 reproduces the elementary curriculum that was reconstructed from other sources, mainly school exercises (Veldhuis, 1997). Note the details concerning some technical skills (lines 19 and 29): shaping the ‘exercise tablet,’ drawing the lines between rows and columns, arrange correctly the cuneiform signs, and introducing hyphenation in the right places. As indicated above, it must be kept in mind that these texts were composed and used for educational purposes, and they deliver an idealized picture of the schools. Several historians have insisted that this kind of literature tells us more about the ideology of the scribes than about the realities of teaching (George, 2005). Some details described in ‘Edubba texts’ are nonetheless corroborated by other sources. These compositions are thus sources of information of great value to the historian. To sum up, the literary texts show that the documentary material used for teaching Sumerian was composite. Much of this material escapes us forever, namely, the whole oral tradition. The written material includes compositions specifically created for teaching, pieces of ancient literary heritage elaborated in earlier periods and recomposed. Once fixed in a curriculum and gradually standardized, this set of texts gained some stability, and passed without much change from one generation to another during the Old-Babylonian period and partly beyond. The corpus of literary texts used in education was formed after a complex process mixing original creation, selective reuse of earlier knowledge, and standardization. I will focus further on these processes in the case of mathematics.

9.5 School Mathematical Tablets Let us now examine the students’ writings. The city of Nippur has yielded several thousands of school tablets, and among them more than 900 contain mathematical texts. For the historian trying to grasp the ancient practices, several aspects should be considered: the content of texts, of course, and also how these texts are inscribed on clay tablets (layout, structure, arrangement), the types of tablets (shape, size), physical condition, and some quantitative data. All these observations allow us to reconstruct a fairly accurate picture of the curriculum, the pedagogical methods, and the concepts taught, particularly concerning numbers and calculations. I shall limit myself to a brief summary of the studies on this teaching context (for more details,

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see Proust, 2007; Robson, 2001; Veldhuis, 1997), and stress the information that these sources provide us concerning the resources of the masters, when possible. From the first glance at the school tablets, one is struck by their material aspect. The school tablets can be classified in four clearly recognizable types. It is interesting to note that certain types of tablets have Sumerian names, which means that this classification is not only a convenience of the modern historian, but reproduces the one that the scribes themselves had established. This typology is not quite the same everywhere, which shows that, beyond the uniformity of the content, teaching methods might vary from one school to another. The types of school tablets from Nippur are the following: – Type I tablets are large tablets containing a long text, continuously and densely inscribed on the obverse and on the reverse (see tablet Ist Ni 2733 on the CDLI website, no. P254643). – Type II tablets contain different texts on the obverse and on the reverse. On the obverse, a model was noted in an archaic style by a master, and copied once or twice by a student; the copies were sometimes traced and erased repeatedly.6 On the reverse, a dense text was written by heart by a student. Perhaps the Sumerian term ‘tablet to throw’ (saršuba) is associated with this type of tablet (see tablet HS 1703, CDLI no. P229902, containing a lexical list on the obverse and a metrological7 list on the reverse – Fig. 9.1 below). I shall return later to this type of tablet, particularly important in Nippur. – Type III tablets are small rectangular tablets containing a short extract, often a multiplication table, called ‘long tablets’ (imgidda) in Sumerian (see tablet HS 201a, CDLI no. P254581, containing a multiplication table). – Type IV tablets are small square or round tablets, containing a short exercise, called ‘hand tablet’ (imšu) in Sumerian (see tablet Ist Ni 18, CDLI no. P368708, containing an area calculation – Figs. 9.2 and 9.3 below). Type I, II, and III tablets were used in the elementary level of education, and type IV tablet in a second stage. The vast majority of Nippur tablets is of type II, and was used as a sort of diary notebook. Type II tablets very often contain Sumerian texts on one side and mathematical texts on the other side. It has been shown that the text noted on the reverse had been studied and memorized before the one noted on the obverse (Veldhuis, 1997). A statistical study of correlations between the texts of the obverse and reverse of type II tablets allows a reconstruction of the order in which the various texts were studied, and therefore of the curriculum. These texts are very standardized lists (of signs, Sumerian words, phrases, measurements) and tables (metrological, numerical), which appear on many duplicates. The mathematical curriculum in Nippur,

6 In order to erase signs impressed in wet clay, scribes simply rub them lightly with their finger. Tablets bear often fingerprints and erased signs covered by others. 7 The term “metrological” refers to the measure systems (see the following page).

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Ist Ni 3913, CDLI n°P229593. School type II tablet from Nippur, Istanbul Archaeological Museum (Proust 2007, copy pl. XXI). The obverse contains a Sumerian lexical list, including mathematical terms regarding volume calculations. The reverse contains a list of measures of capacity. The right side of the tablet, which contained student copies, is lost. Note the characteristic appearance of the fracture, which results from the fact that the right columns have been written and erased several times, becoming thinner and forming a ledge.

Fig. 9.1 Type II tablet

Ist Ni 18: type IV school tablet from Nippur de type IV. Archaeological Museum of Istanbul. Proust 2007, pl. I

Fig. 9.2 Surface calculation

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Obverse 4.26.[40] its reciprocal 13.30

Reverse

4.26.40

9

40!

1.30 13.30

Ist Ni 10241, CDLI n° P368962. Type IV school tablet from Nippur. Istanbul Archaeological Museum. Proust 2007, pl. XLVIII

Fig. 9.3 Algorithm for reciprocal calculation

that is, the chronological sequence of different lists and tables that was to be learnt by students, may be summarily described by Table 9.1 below. The main function of theses texts was learning the metrological systems and sexagasimal place value notation (SPVN).8 8 Metrological systems (systems used for noting measures of capacity, weight, volume, surface, and length), were described in “metrological lists”. Metrological tables provided a correspondence between measures and abstract numbers, that is, numbers written in sexagesimals place value notation. SPVN was used in mathematical texts. This notation used 59 digits (1–59), made of two kinds

of signs: ones (vertical wedges

) and tens (oblique wedges

), repeated as many times as nec-

essary. For example, 12 is noted . The numbers are made of sequences of digits following a positional principle in base 60: each sign noted in a given place represents 60 times the same sign noted in the previous place (on its right). SVPN does not specify the magnitude of the numbers. For example, the numbers 1, or 60, or 1/60 are noted in the same way (a vertical wedge ). Initial and final zeros are unnecessary, and indeed, they are not attested in any known cuneiform text. However, the absence of notation for median zero was a weakness of the system, which was corrected in later periods: in the mathematical and astronomical texts from the last centuries before our era, scribes used signs indicating the absence of a power of 60 in the positional numbers. In the transcriptions, digits are noted in the modern decimal system, and separated by dots. For example the numbers 44.26.40 which appears in Table 9.1 is a transcription of the cuneiform number . For more details on place value notation, see Proust (2008b).

170 Table 9.1 Mathematical curriculum in Nippur

C. Proust Metrological lists

Metrological tables

Division/multiplication tables

Tables of roots

Table 9.2 Literary and mathematical curriculum in Nippur

Capacity list Weight list Surface list Length list Capacity table Weight table Surface table Length table Height table Reciprocal table 38 multiplication tables (head numbers 50, 45, 44.26.40, 40, 36, 30, 25, 24, 22.30, 20, 18, 16.40, 16, 15, 12.30, 12, 10, 9, 8.20, 8, 7.30, 7.12, 7, 6.40, 6, 5, 4.30, 4, 3.45, 3.20, 3, 2.30, 2.24, 2, 1.40, 1.30, 1.20, 1.15) Square table Square root table Cubic root table

Writing, Sumerian

Mathematics

Simple sign lists Thematic lexical lists Complex sign lists Contract models Proverbs

Metrological lists Metrological tables numerical tables

This mathematical curriculum was coordinated with the literary curriculum as in Table 9.2: The texts written by students were extracted from very long lists and tables, totaling tens of thousands of items, highly standardized, in a fixed order. The stability of texts allows us to draw up a ‘composite text,’ that is, a text composed of all items found on various tablets, in the same order as they were written and taught. The composite text of lexical lists occupies several volumes of ‘Materials for the Sumerian Lexicon’ (MSL), which represents since the publication of the first volume in 1937, an essential part of Sumerologists efforts to establish the Sumerian lexicon. The composite text of mathematical lists and tables is much smaller (published in Proust, 2007). This composite text has no material existence, since no actual tablet contains it entirely. But it probably represents fairly well what was memorized by the scribes.9 9 Current digital databases permit a simultaneous representation of both composite text and real texts written in available sources. The advantages of digital media over paper to represent the lexical lists in all their dimensions have been noted by Veldhuis in his study of school texts of

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Indeed, several piece of evidence show that these lists were memorized, at least partially, by the students. First, mathematical lists and tables are attested almost only on school tablets. Very few ‘reference texts,’ that is tables used by trained scribes for their professional activities, including teaching, are known.10 The masters knew the tables by heart. Second, some characteristics of lexical lists, especially the logic that guides the sequential order of the items, clearly reflect the constraints of memorization. Many digressions in the lexical lists, where items attract others in a seemingly unexpected way, by association of ideas or homophony, could be explained by memorization processes (Cavigneaux, 1989; Veldhuis, 1997). The numerical tables are also written in a quite unexpected order, since they follow an order different from which we feel natural from an educational point of view. First comes the reciprocal table, then the different multiplication tables in descending order: table of 50, 45, 44.26.40, etc. (see Table 9.1 above). The first tables seem to be the more difficult. The explanation could be the following: analysis of school tablets shows that the first sections of the lists were copied with a frequency much larger than the last, because the copy always starts with the beginning of the list, but continues only rarely until the last item. We can assume that, by placing the ‘difficult’ tables in first position, the masters made sure that they were more frequently copied and recited than the simplest, placed at the end of the series of multiplication tables.11 What exactly did it mean, for a student to learn a list or a table? The typology of tablets helps us to answer this question. In a first step, the students learnt to write short excerpts, reproducing a model on the obverse of type II tablets, then they memorized the pronunciation, they recited the excerpt, and, in the last step, they reproduced by heart a large part of the list by writing it on the reverse of type II tablet. Learning therefore inextricably combined writing and memorization. The lists and tables memorized in elementary education were a set of linguistic and mathematics tools that were subsequently used by the scribes throughout their entire administrative or scholarly careers; these tools include repertoires of signs, dictionaries, grammatical paradigms, systems of measurement, and numerical tables. As noted above, these tools are attested mostly in school tablets and rarely among scholarly writings. Students’ drafts provide evidence that the knowledge of masters included a vast repertoire of numerical results, generally memorized and ready to be mobilized in professional practice or teaching. Thus, school texts are not childish texts, but rather

Veldhuis (1997, Ch. 5). He exploited these advantages in the development of his online database (DCCLT, http://cdl.museum.upenn.edu/dcclt/). 10 One of the rare exceptions is a mathematical prism now kept in the Louvre (AO 8865, CDLI No P254391), of unknown provenance. It is a large prism carefully written and crossed by an axial hole, probably to be easily usable. This prism is a precious object which looks very different from the drafts of students. This prism could indicate that the “composite text” was not always entirely memorized by professional scribes, who needed to consult a reference text. 11 This conclusion is largely based on discussions with Anne-Marie Chartier, at a workshop on education in Mesopotamia (Paris, 15/03/2006).

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reflect the knowledge shared by a community of youth and adults passed through the mold of scribal schools. These texts are ‘elementary’ in the sense that they constitute the basic knowledge needed to take on future scribal charges. Resources for elementary education were largely immaterial since they were mainly memorized by the masters. This knowledge was not limited to the city of Nippur, but widely spread in Mesopotamia and neighboring regions. The same measure systems, the same calculation techniques, and almost the same lexical lists were taught not only in southern Mesopotamia, the cradle of the Sumero-Akkadian culture, but also to the east in Susa (west of Iran), to the north in Mari (middle valley of the Euphrates, the border between Syria and Iraq), and later to the west in Ugarit (Syrian coast). The milieu that disseminated this common knowledge was probably linked to professional scribes involved in education, who possibly circulated from one city to another. This ‘academic’ knowledge that transcends regional boundaries could have been relatively autonomous in relation to local practices. For example, the metrology (i.e., the measure system) taught in the schools of Mari and Ugarit was not the one practiced in the everyday administrative activities of these cities. Thus, we see taking shape a common culture belonging to a specific segment of the population whose members, though few, were mobile and influential over a wide geographical area. Mathematical lists and tables used in the elementary level of education form a highly structured and coherent system, whose function appears when analyzing the exercises used in a more advanced level of education. These exercises, often written on type IV tablets (see Fig. 9.2 below), relate to multiplication, reciprocal, and calculation of area and volume. This enumeration already provides interesting information: the operations that are the subject of school training are limited to the field of multiplication. Addition and subtraction are absent. If we look more carefully at how the numbers and measures are noted in the elementary lists and tables on the one hand, and the calculation exercises on the other hand, we see that some basic principles are consistently applied. The measurements are recorded using numerical signs following an additive principle. Metrological tables provide a correspondence between, on the one hand, these measures, and on the other hand, positional numbers, noted in base 60. The exercises where multiplication and reciprocal are performed use positional numbers, with no mention of measuring units or objects counted. The multiplication and reciprocal operate only on positional number, or ‘abstract numbers’ according to Thureau-Dangin (1930). The exercises on area calculation are particularly interesting because they show how the two types of numbers act at specific and distinct stages in the calculation process. The layout of the area calculation in Ist Ni 18 (see below) shows clearly these two stages of calculation since they are written in two distinct areas of the tablet. School tablets reveal an original conception of numbers, where quantification and calculation fill two dissociated functions, undertaken by two different numerical systems: quantification is made by additive numerical systems (in the lower right

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area of the tablet Ist Ni 18), and calculation is made by a positional system (in the upper left area of the tablet Ist Ni 18). The whole set of school tablets from Nippur, including ‘advanced’ exercises, shows the dynamic use of these tables in the practice of calculation. Through the school tablets, the historians have access to concrete aspects of school training. The layout of the text on the clay tablet, the notation of numbers, and the errors provide particularly valuable clues for the reconstruction of calculation practices. More interestingly, they show us how the masters and the erudite themselves had been trained. Our reading of advanced mathematical texts is thus transformed, since they can be tackled using mathematical tools that were inculcated into young scribes, not using our modern arithmetic and algebraic tools. This sample shows how the layout of a text reflects the concepts taught, and help the nowadays historian to capture these ancient concepts. By noting that the look of a textbook is influenced by the structure of a program, Remillard stresses the same phenomenon in her chapter (Section 6.3.3). The scribal schools in Mesopotamia produced a highly standardized body of texts, shaped by teaching practices. This normative effect of teaching practices is not unique. But the particular context of Mesopotamia, with social demand for unification of metrology (see below) and specific practices of teaching (memorization) produced an original curriculum. This coherent and robust curriculum was enjoying a great success though the Ancient Near East.

9.6 Masters’ Writings From this description of the school tablets from Nippur, a picture of a highly stereotyped education, leaving little room for pedagogical creativity, might emerge. But such an image would be the result of extrapolating too readily from information provided by extremely fragmented sources. As mentioned above, to reconstruct everyday life in the Old-Babylonian schools, we have only the written production of beginners, as well as some literary texts. Little information has reached us concerning the parts of education that do not use writing, such as music, theater, or oral literary tradition. Regarding mathematics, consistent evidence indicates that written artifacts represent only a part of the ancient calculation practices since mental arithmetic and concrete calculation tool played an important role (Proust, 2000). Indeed, the resources of the masters, as defined by Gueudet and Trouche (Chapter 2), might have included a complex system of written texts, memorized texts, calculation devices and various communicational processes, but only the written artifacts reached us. We have then to reconstruct a rich environment from truncated evidence. Furthermore, only the first stages of education are well known. As mentioned, we owe our sources to the recycling practices of the scribes. School tablets have been selectively preserved precisely because they had been thrown out. But what do we know about education at the more advanced levels? The available sources are less numerous and more difficult to interpret. For example, excavations of Nippur have

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yielded hundreds of mathematical school tablets, but only three advanced mathematical texts reached us. Even so, it is quite likely that advanced mathematics were produced at Nippur. In addition to the hazards of excavations, several different reasons may explain this paradox. First, the scholarly tablets were circulating. It is possible that the excavators have unearthed tablets written in Nippur but kept in other cities, without being able to identify their origin. Indeed, the Old-Babylonian mathematical tablets bear no indication of place, date or author name. Another explanation is unfortunately plausible. The mathematical tablets are most of the time of unknown origin because they were bought from dealers by European and American museums as well as by private collectors. It may be that such tablets have been found at Nippur and disappeared into the opaque net of the antiquities trade. What is an ‘advanced mathematical text’? In the foregoing discussion, I used the term ‘advanced’ in a vague sense to designate texts other than elementary school texts. But one could make more subtle distinctions and identify various types of writings, such as the production of advanced students, texts written by masters for teaching purposes, and purely erudite texts. The boundary between these types is difficult to trace, and reliable criteria are often lacking. Identifying the function of a text in relation to teaching practices is possible only on a case by case basis. Since the archaeological context is usually unknown, this analysis is usually based primarily on internal evidence. In the following, I shall provide two samples of such analysis, both related to the algorithm of reciprocal calculation. The first sample is diachronic and aims to highlight that the function of a text, including its use in education, can change over time. The second is synchronic and aims to show how the link between teaching and scholarship is a two-way relationship.12 Reciprocal tables provide a sample of text that was a school text in certain periods, and was not in other periods. The earliest known reciprocal tables are dated from the so called Neo-Sumerian period, that is, the late third millennium B.C. The context is the emergence and consolidation of the first centralized states, which have dominated much of Mesopotamia and neighboring regions.13 The policies of these states were characterized by centralized control on the basis of social and economic standardization of writing, metrology, accounting, etc., accomplished through a series of ‘reforms.’ These reforms are known mainly due to references found in some royal hymns that were used thereafter in education. For example, in the text ‘Shulgi B,’ widely used in Old-Babylonian scribal schools, the Neo-Sumerian king Shulgi is supposed to have standardized metrology and to have developed the scribal schools. The interesting aspect of this story is the link established between standardization and development of schools. Indeed, schools played a major role in the creation of new standards, as well as in their wide dissemination. The reform of weights and measures, with the creation of a single and coherent system for all

12

The first sample is more detailed in Bernard and Proust (2008), and the second in Proust (2011). These states were ruled by two king dynasties in two periods separated by one century: the Akkad dynasty, 2300–2200 (Sargon and successors), and the Ur III dynasty, 2100–2000 (Ur-Nammu, Shulgi and successors).

13

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units of measure, results from this context. It is possible that the place value notation was invented in connection with this standardization process. Anyway, the few Neo-Sumerian reciprocal tables provide the earliest known evidence of place value notation. Furthermore, the clay tablets on which the earliest numerical tables are written are fine objects, showing a mastery of the cuneiform writing that is not that of a young beginner.14 The Neo-Sumerian reciprocal tables reflect the activity of the scholars who developed and implemented royal policies of standardization. These tables, as well as the multiplication tables that seem to have appeared later, subsequently were used for elementary education and integrated into the curriculum. Now let us travel in time by crossing over a millennium and a half. We find again the same reciprocal tables, but in a radically different context. Mathematical tablets dated from the Hellenistic period (ca. 3rd century B.C.) were found among the remains of the great libraries of Uruk and Babylon, belonging to priests. These tables were used in teaching, but no longer in the elementary education of children learning the basics of writing and arithmetic. In this late period when cuneiform writing was disappearing, this kind of table belonged to the specialized training of young scientists already literate in Greek and Aramaic. The reciprocal tables were still school texts, but not in the same way as in the Old-Babylonian context. This example shows how the function of a text may change in history, and thus how it may be used as a resource by masters in different ways. The presence of reciprocal tables in Hellenistic libraries shows the importance that the later scholars who knew cuneiform attached to the preservation of ancient intellectual heritage. To sum up, the tables created at the end of the third millennium in a scholarly environment linked to political power, were not necessarily primarily school texts. Subsequently, their content was widely disseminated through communities linked to scribal schools, and used in elementary education in the Old-Babylonian period. Finally, reciprocal tables were incorporated into a frozen body of writing belonging to a ‘canonical’ scholarly heritage, probably compiled in the early first millennium, and, in the Hellenistic period, these tables were transmitted in quite closed religious circles. The function of the same table was by turn linked to engineering, elementary education, and antiquarism. These changes illustrate that, as noted by Remillard (Section 6.3.6), ‘the reader’s relationship is with the text and not the author.’ Indeed, the relationship between text and reader changes over time. The second sample allows us to grasp another aspect of the relationship between erudite texts and school texts. This sample concerns an algorithm used in the Old-Babylonian period to calculate reciprocals of regular numbers15 that do not

14

See Ist Ni 374, CDLI no. P257557. A regular number in a given base (here in base 60) is a number whose reciprocal can be written with a finite number of digits. These numbers are products of divisors of the base, therefore, in base 60, their decomposition into prime factors does not include factors other than 2, 3 or 5. The ancient Mesopotamian mathematicians certainly knew all the one-place regular numbers (given in standard tables), and probably all two-places regular numbers, as well as a large stock of larger regular numbers (three or more digits). Their algorithms, including the division which was performed by means of multiplication by reciprocal, were mainly based on regular numbers.

15

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belong to standard tables. This algorithm is found in numerous school tablets, such as the following example (Fig. 9.3).16 The algorithm is based on decomposition into regular factors. The regular number for which the reciprocal is sought, here 4.26.40 (in the Mesopotamian base 60 system), is decomposed into factors belonging to known standard tables, here 6.40 and 40, and then the reciprocal of these factors, here 9 and 1.30, are multiplied the one with the other. The result is the sought reciprocal, here 13.30. Note that the factors appear in the trailing part of the number to be factorized. For example, the number 4.26.40 ends with 6.40, thus 4.26.40 is divisible by 6.40.17 The calculation could be summarized as follows: 4.26.40 = 6.40 × 40, thus, recip(4.26.40) = recip(6.40) × recip(40) = 9 × 1.30 = 13.30 Another text of unknown provenance, kept at the University of Philadelphia under the inventory number CBS 1215, contains many calculations of this kind. It is a large multicolumn tablet (three columns on the obverse and three columns on the reverse), divided into 21 sections. The first section begins with the number 2.5, the second with the number 4.10, etc.: each entry is double the previous. Section 8 contains the following calculation: 4.26.40 9 40 1.30 13.30 2 27 2.13.20 4.26.40

Note that the beginning of the calculation is identical to that of the school tablet Ist Ni 10241 described above. However, after finding 13.30, the inverse of 4.26.40, the scribe continues the computation by seeking the reciprocal of 13.30, which provides of course the original number 4.26.40. Each section of the large tablet CBS 1215 contains the calculation of the reciprocal of the entry (direct sequence), followed immediately by the calculation of the reciprocal of the reciprocal (inverse sequence). Almost all the known school exercises of reciprocal calculation, whatever their provenance, legally or illegally excavated, contain one of the calculations contained in the large tablet CBS 1215. So one might think that tablet CBS 16

In translation, the exclamation point after 40 means that indeed the scribe should have written 40 on his tablet, but in fact wrote something else (in this case, he wrote 41 instead of 40). 17 For this reason, Friberg gave the name “trailing part algorithm” to this method (Friberg, 2000, pp. 103–105).

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1215 was a kind of textbook, and was used by masters to prepare exercises for students. However, a closer observation of the text leads to doubt about this too simple explanation. First, the samples found in the school exercises are extracted only from the direct sequences of the large tablet CBS 1215. Second, the choice of regular factors in the decompositions follows fixed rules in direct sequences, but is much freer in reciprocal sequences. These observations suggest that the relationship between the large tablet CBS 1215 and the small exercises is the reverse of that which is generally supposed. In the large tablet, it would appear that the existing school material was compiled, systematized, developed, and reorganized to produce a text whose objectives are not only teaching, but also searching for generalization and justification of the algorithm. These operations on texts (collecting, re-arranging, systematizing, and developing) are analogous to the one implemented by teachers working on resources (Section 2.1.2), but with other purpose than teaching. The result of these operations is a new production, which differs in nature from the original school exercises. The goal, the audience and the expected use is not the same as the pedagogic material it comes from. Such a text would be intended to communicate certain mathematical results to peers rather than to convert knowledge into teaching materials. In fact, it is likely that the two processes, transmission and innovation, were not exclusive and that they interacted with each other in ways to which we do not clearly grasp. This erudite text seems to be the result of teaching practices, where interaction between students and masters played an important role. Rezat, in his chapter (Chapter 12), show how, in a comparable way, a textbook may be the result of interactions between teachers and students. The two cases briefly mentioned above show the difficulty of describing precisely what resources were available to masters (or built by them). The principal reason lies in the nature of our sources, which are fragmentary and do not always permit us to capture the complexity of the relationship between education and scholarship.

9.7 Conclusion What can one say, finally, on the process of making resources for teaching in the Mesopotamian context? Various pieces of evidence show that the knowledge taught at an elementary level constituted a large body that was completely memorized by the experienced scribes, including masters. The knowledge of the master is thus largely embedded in their memory. But a school tablet does not necessarily contain a school text in the sense that this text has not always been specifically elaborated for the purpose of education. A text inscribed on a school tablet may belong to the cultural background of the scribes, and may have been transmitted unchanged, as is the case with dictionaries, multiplication tables, or trigonometric tables today. Sometimes, there is little difference between a text written by a young student and a text in some way belonging to the resources of the master. At a more advanced

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level, we deal with mixed processes of creating and transforming knowledge. The development of exercises is connected with the invention and explanation of new mathematical concepts. These innovations could have emerged from the school activity itself, in the context of a network between scholars. The resources of masters result therefore from a complex and two-way process between learning and scholarship, involving memory, oral communication, writing, and probably material artifacts.

References Bernard, A., & Proust, C. (2008). La question des rapports entre savoir et enseignement dans l’antiquité. In L. Viennot (Ed.), Didactique, épistémologie et histoire des sciences. Penser l’enseignement (pp. 281–302). Paris: PUF, Collection Science, histoire et société (dir. D. Lecourt). Cavigneaux, A. (1989). L’écriture et la réflexion linguistique en Mésopotamie. In S. Auroux (Ed.), Histoire des idées linguistiques (1): La naissance des métalangages en Orient et en Occident. Liège, Bruxelles: Pierre Mardaga. Charpin, D. (2008). Lire et écrire à Babylone. Paris: PUF. Civil, M. (1985). Sur les “livres d écoliers” à l époque paléo-babylonienne. In J.-M. Durand & J.-R. Kupper (Eds.), Miscellanea Babylonica, Mélanges offerts à M. Birot (pp. 67–78). Paris: ERC. Friberg, J. (2000), Mathematics at Ur in the Old Babylonian period. Revue d’Assyriologie, 94, 98–188. George, A. R. (2005). In search of the é.dub.ba.a: The ancient Mesopotamian school in literature and reality. In Y. Sefati, P. Artzi, C. Cohen, B. L. Eichler, & V. A. Hurowitz (Eds.), An experienced scribe who neglects nothing: Ancient Near Eastern studies in honor of Jacob Klein. Bethesda: CDL Press. Kramer, S. N. (1949). Schooldays: A sumerian composition relating to the education of a scribe. Philadelphia, PA: The University Museum. Michalowski, P. (1987). Charisma and control: On continuity and change in early Mesopotamian bureaucratic systems. In M. Gibson & R. D. Biggs (Eds.), The organization of power, aspects of bureaucracy in the Near East (Vol. 46). Chicago: The Oriental Institute of the University of Chicago. Michel, C. (2008). Ecrire et compter chez les marchands assyriens du début du IIe millénaire av. J.-C. In T. Tarhan, A. Tibet, & E. Konyar (Eds.), Mélanges en l’honneur du professeur Muhibbe Darga (pp. 345–364). Istanbul, Turkey: Sadberk Hanim Museum Publications. Proust, C. (2000). La multiplication babylonienne: la part non écrite du calcul. Revue d’histoire des mathematiques, 6, 1001–1011. Proust, C. (2007). Tablettes mathématiques de Nippur. Varia Anatolica (vol. XVIII). Istanbul, Turkey: IFEA, De Boccard. Proust, C. (2008a). Tablettes mathématiques de la collection Hilprecht. Texte und Materialen der Frau Professor Hilprecht Collection (vol. 8). Leipzig, Germany: Harrassowitz. Proust, C. (2008b). Quantifier et calculer: usages des nombres à Nippur. Revue d’Histoire des Mathématiques, 14, 143–209. Proust, C. (2011). Interpretation of reverse algorithms in several Mesopotamian texts. In K. Chemla (Ed.), History of mathematical proof in ancient traditions: The other evidence. Cambridge: Cambridge University Press. Robson, E. (2001). The tablet house: A scribal school in Old Babylonian Nippur. Revue d’assyriologie, 95, 39–66. Sjöberg, A. W. (1973). Der Vater und sein Missratener Sohn. Journal of Cuneiform Studies, 25, 105–169.

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Tanret, M. (2002). Per aspera ad astra. L’apprentissage du cunéiforme à Sippar-Amnanum pendant la période paléo-babylonienne tardive. Mesopotamian History and Environment, Serie III Cuneiform texts (MHET) (Vol. I/2). Ghent, Belgium: Université de Gand. Thureau-Dangin, F. (1930). Nombres concrets et nombres abstraits dans la numération babylonienne, Revue d’assyriologie, 27, 116–119. Vanstiphout, H. L. J. (1978). Lipit-Eshtar Praise in the Edubba. Journal of Cuneiform Studies, 30, 33–61. Vanstiphout, H. L. J. (1979). How did they learn sumerian? Journal of Cuneiform Studies, 31, 118–126. Vanstiphout, H. L. J. (1996). Remarks on “Supervisor and Scribe” (or Dialogue 4, or Edubba C). NABU, 1. Vanstiphout, H. L. J. (1997). School dialogues. In W. W. Hallo (Ed.) The context of scripture, I: Canonical compositions from the biblical world (pp. 588–593). Leiden/New-York/Köln: Brill. Veldhuis, N. (1997). Elementary education at Nippur, the lists of trees and wooden objects. Ph. D. dissertation (Retrieved from http://ls.berkeley.edu/dept/ahma/Faculty/veldhuis.htm), University of Groningen.

Reaction to Part II Some Reactions of a Design Researcher Malcolm Swan

In reading the contributions to this section, I am impressed by their diversity. These pages offer analyses of classroom resources (U.S. textbooks, digital media, Mesopotamian clay tablets); professional development resources; and a number of theoretical frameworks and tools. These artifacts and constructs have arisen from different cultures at different times and have been used in different ways. This is a fascinating collection. Remillard’s chapter notes that each reader enters a particular relationship with the ideas in a text and adopts a particular mode of engagement. In my own case, I read these texts as an educational designer and design-researcher. My area is in the design and development of classroom and professional development resources that equip teachers to transform the experiences of students so that they become more active, creative, and reasoning participants in the learning process. These include multimedia professional development resources and lesson descriptions that enable teachers to become aware of the pedagogical challenges. This has led me to adopt a research approach that belongs to an emerging family of related approaches, known variously as formative research, engineering research (Swan, 2006), developmental research (Gravemeijer, 1998), design experiments (Schoenfeld, 2004), and design research (van den Akker, 1999; van den Akker, Graveemeijer, McKenney, & Nieveen, 2006). From this standpoint, I read these chapters with two questions in mind: What intentions and values are apparent in the materials and tools presented? How can I use these tools to improve my analysis of the ways in which designs are transformed in the hands of teachers and thus improve my own design process? The chapters by Schmidt and Proust both, in different ways, attempt to infer the teachers’ and pupils’ activity and experience of mathematics from an analysis of artifacts. Unlike the remaining chapters, neither calls on direct classroom observation. Schmidt considers the current situation of current mathematical textbooks in the United States, while Proust takes us back 4000 years to school materials used in ancient Mesopotamia. M. Swan (B) Centre for Research in Mathematics Education, School of Education, University of Nottingham, Nottingham NG8 1BB, UK e-mail: [email protected]

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The U.S. educational system is perhaps more textbook dependent than most other cultures. Textbook production is a huge commercial business and its products are weighty tomes (Schmidt notes that the eighth-grade textbooks average 700 pages) that attempt to contain the union of content required by different state and school district adoption systems. Schmidt takes as his starting point the assumption that “textbooks are a particularly accurate reflection of the implemented curriculum in the US” and that they provide a measure of “the curricular experience of individual students”. He then uses weighted page counts to analyze the mathematical content to which students at different grade levels are “exposed”, where these weightings are based on content difficulty. He finds that the number of pages increases to a peak in Grade 10 and declines thereafter; that the content is undemanding and repetitive: “a mile wide and an inch deep”; and that the long-term cumulative effect of tracking widens inequities in terms of “exposure to more challenging mathematics”. Unlike other authors in this section, Schmidt makes no allowance for the qualitatively different ways in which the same textbooks may be interrogated and used. His assumptions, whatever their validity in the United States, would not apply to other cultural situations, such as England, where there is no textbook adoption process and where experienced teachers use them only selectively. In England, schools’ inspectors criticize textbooks for focusing too much on providing practice for examination questions and for failing to promote understanding, connections, and enquiry. They note how effective teachers compensate for this (like Ms. Jordan in Remillard’s chapter) but “in less confident hands” the subject is reduced to “techniques for passing examinations” (Ofsted, 2006). Exposure to textbooks does not correlate with “experiencing mathematics” in my view. In contrast, Proust takes us back 4000 years to school materials in ancient Mesopotamia. The artifacts she considers are clay tablets used in scribal schools – institutions where discipline was harsh. She shows how the tablets produced by students enable her to reconstruct a “fairly accurate picture” of the curriculum, pedagogy, and concepts taught. The activities involved reproducing, memorizing, and practicing measure systems, calculations, and lexical lists. One design feature that I found interesting was the organization of the memorization exercises. Counter-intuitively, the difficult exercises came earlier, presumably so that they were practiced more frequently and students would not miss them. Mathematics curricula have always valued the memorization of facts and fluency in calculation. Internationally, there is a widespread view among educationists that such aspects are over-valued at the expense of understanding concepts and representations, developing strategies for investigation and problem solving, and appreciating the power of mathematics in society (Swan & Lacey, 2008). In the United States, for example, the widely adopted Common Core State Standards for Mathematics (NGA & CCSSO, 2010) include specific references to the importance of developing “Mathematical practices”; in the UK the most recent national curriculum documents emphasize the importance of “Key” mathematical processes (QCA, 2007); across the EU there are calls to include an increased emphasis on “inquiry based learning” (Rocard, 2007). Such aspects, however, remain separated and marginalized in teachers’ practices and textbooks continue to assume a

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pedagogy of the “teacher explanation, demonstration, then student practice” kind where exercises progress gradually from the easier to the harder. There is a considerable mismatch between the values and the content presented in most resources and the aspirations articulated in such national documents. These aspirations cannot become reality without well-designed products and processes that exemplify how they may be interpreted in classrooms and examinations. The chapters by Remillard, Ruthven, and Pepin each considers resources that have a transformational agenda: to replace “traditional” didactic transmission approaches with problem-solving approaches. While Remillard considers the U.S. context, with its agenda of reform through the NCTM Standards (NCTM, 2001), Ruthven and Pepin consider the UK context. The difficulty for designers, as these authors ably show, is that teachers interpret and mutate even the most carefully designed materials in unintended ways. In addition, these authors propose analytical tools that may help us to better understand and anticipate these mutations. As Remillard describes, every curriculum designer has in mind an audience and seeks to enlist a particular kind of participation through a mode of address. Many textbook authors and educational designers prepare materials as if they were writing for themselves, tacitly assuming that teachers will share their values and pedagogical assumptions. Some abdicate from all responsibility for how their materials are used, claiming that teachers must use their own “professional judgment”. Others take their responsibility more seriously and include lesson plans, teaching guides, and resources that attempt to explain the theories that underpin the material. This creates additional complexity, and the materials can become less accessible. Of course it is impossible to create materials that are “teacher proof”, as Ruthven notes, but it is possible to systematically research the range of “interpretative flexibility” that is employed through careful, iterative phases of classroom trialing, observation, and redesign. Classroom trialing is not there to simply “fix” mistakes and omissions in the materials, but it is also intended to evaluate the range of ways in which teachers make use of the materials and to incorporate their wisdom and experience in succeeding versions. In our own materials, we also collect students’ work during trials and then use samples of these as stimulus student material in the next revision. Students are encouraged to analyze alternative approaches, critique and correct the work, and refine their own arguments. This careful process of iterative analysis and redesign is slow and difficult and requires a close interplay of research and development. The value in Remillard’s chapter for me is that she offers a taxonomy for analyzing the structure and form of classroom material and ways of describing the teachers’ engagement with it. This could form a valuable part of the design-research process and alert a designer to aspects that are often ignored. For example, the final form and look of materials is often left to a commercial publisher, yet this is a crucial aspect of task design. In my own work, for example, I found empirically (Swan, 2006) that when I wanted students to discuss and debate some aspect of mathematics, then I needed to design resources to be shared. This required resources in a larger format, making use of posters and cards that could be cut out and moved around, so that students could more easily see them, share ideas and collaborate. We

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learned from trials that teachers were much less likely to use discussion in learning if we restricted the resources to textbooks. Ruthven’s chapter similarly draws attention to often neglected aspects of the context that influence the ways in which teachers use digital resources, in particular dynamic geometry software. Although this software was designed without an explicit pedagogical model in mind, it quickly became valued as a tool for promoting collaborative, “discovery” learning in geometry. When digital resources are observed in use, however, it quickly becomes apparent that well-designed software in the hands of experienced teachers does not necessarily result in mathematical activity. Computer feedback, for example, often encourages trial and improvement rather than the formulation, testing and validation of hypotheses (Joubert Gibbs, 2007). Ruthven examines the appropriation of dynamic geometry software by one teacher using a broad analytical framework that outlines some of the tensions and difficulties that arise. This includes: the working environment (e.g., changing rooms), the resources available (e.g., making links between manual geometrical constructions and computer-based ones), teachers’ patterns of classroom behavior (the tension between individual exploration and productive discussion), the “curriculum script” (the potential actions within the teachers’ repertoire, particularly when surprises occur) and the pressures of time (the tension between covering the curriculum and securing student learning). This framework is very helpful to the design-researcher. It is sobering to realize that dynamic geometry software is now used only rarely in England, and then often as only a demonstration device. This again underlines the importance of designing experiences rather than products. By this, I mean that the typical end-user should be observed using prototypes in realistic circumstances throughout the design process.1 Far from ignoring or abdicating the responsibility for the way our materials are used, we should begin to analyze and describe how they have been used effectively and incorporate these descriptions into the materials themselves. This brings us to the issue of professional development. Pepin notes how new mathematical materials are beginning to recognize the importance of building opportunities for teacher learning into their design (DfES, 2005). Pepin describes her own work with teachers and reflects on the nature of the conceptual tools that teachers need in order to make sense of classroom tasks. The specific tools she describes – her “task analysis tools” – are similar to tools I have also used with teachers (see for example MARS (1999, 2000) and Swan & Crust (1992)). These are powerful in focusing attention on the purpose of classroom tasks and the potential for learning that a task provides. As Pepin notes, they can also serve to enable teachers to audit their curriculum and assessment provision and create tasks for themselves. The context in which Pepin uses these tools is that of a collaborative, mediated, professional development “course”. In my own work we are currently seeking to develop professional development tools for other contexts – the lone

1 As Steve Jobs told his staff at Apple, back in 1997: ‘You’ve got to start with the customer experience and work back to the technology – not the other way around’ (Arthur, 2010, p. 27).

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teacher and the teacher working in school-based groups with no external mediation. The design considerations here are similar to those confronting the designer of classroom materials, but this context is clearly more difficult to observe and is currently under-researched. Such materials equip teachers with a framework and language to critique classroom materials and reflect on their own values for education. This is particularly true if they are presented in a way which they can modify and make their own, as Pepin does. One reason why educational research is generally regarded as neither influential nor useful is that the importance of systematic research-based design is undervalued and its difficulty underestimated (Burkhardt, 2006; Burkhardt & Schoenfeld, 2003). All too often, designers marginalize teachers in the creation of materials and researchers regard classroom contexts and teachers’ practices as so intractable and individual that materials appear almost irrelevant. Educational interventions are (of course) context sensitive and we need more research to understand which contextual factors are critical and which are not and more observational data of materials in action so that we can improve them by building in the productive adaptations made by teachers (Burkhardt & Schoenfeld, 2003). This is labour intensive and requires a collection of well-engineered tools for analysis. I am grateful to these authors, for providing suggestions for tools that will help me to analyze the impact of my own educational designs in a more systematic manner.

References Arthur, C. (2010, October 18). The geeks are about to inherit the earth. The Guardian. London and Manchester, p. 27. Burkhardt, H. (2006). From design research to large scale impact. In J. van den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.), Educational design research (pp. 121–150). Abingdon, Oxon: Routledge. Burkhardt, H., & Schoenfeld, A. (2003). Improving educational research: Toward a more useful, more influential and better-funded enterprise. Educational Researcher, 32(9), 3–14. DfES. (2005). Improving learning in mathematics. London: Standards Unit, Teaching and Learning Division. Gravemeijer, K. (1998). Developmental Research as a Research Method. In A. Sierpinska & J. E. Kilpatrick (Eds.), Mathematics Education as a Research Domain: A search for Identity (pp. 277–295). Kluwer. Joubert Gibbs, M. (2007). Classroom mathematical learning with computers: The mediational effects of the computer, the teacher and the task. Unpublished PhD, University of Bristol, Bristol. MARS. (1999). High school assessment, package 1. White Plains, NY: Dale Seymour. MARS. (2000). High school assessment, package 2. White Plains, NY: Dale Seymour. NCTM. (2001). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. NGA (National Governers Association) and CCSSO (Council for Chief State School Officers). (2010). Common core state standards for mathematics. Retrieved October 10, 2010, from http://www.corestandards.org/ Ofsted. (2006). Evaluating mathematics provision for 14-19-year-olds. London: HMSO. QCA (Qualifications and Curriculum Authority). (2007), The national curriculum. Retrieved October 10, 2010, from http://curriculum.qcda.gov.uk/

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Rocard, M. (2007). EUR22845 – Science education now: A renewed pedagogy for the future of Europe. Retrieved July 18, 2011, from: http://ec.europa.eu/research/science-society/document_ library/pdf_06/report-rocard-on-science-education_en.pdf Schoenfeld, A. (2004). Design experiments. In P. B. Elmore, G. Camilli, & J. Green (Eds.), Handbook of complementary methods in education research (pp. 193–206). Washington, DC: American Educational Research Association. Swan, M. (2006). Collaborative learning in mathematics: A challenge to our beliefs and practices. London: National Institute for Advanced and Continuing Education (NIACE) for the National Research and Development Centre for Adult Literacy and Numeracy (NRDC). Swan, M., & Crust, R. (1992). Mathematics programmes of study: INSET for key stages 3 and 4. New York: National Curriculum Council. Swan, M., & Lacey, P. (2008). Mathematics matters. National Centre for Excellence in Teaching Mathematics. Retrieved January 29, 2009, from: http://www.ncetm.org.uk/files/309231/ van den Akker, J. (1999). Principles and methods of development research. In J. van den Akker, R. Branch, K. Gustafson, N. Nieveen, & T. Plomp (Eds.), Design approaches and tools in education and training (pp. 1–15). Dordrecht: Kluwer. van den Akker, J., Graveemeijer, K., McKenney, S., & Nieveen, N. (Eds.). (2006). Educational design research. London and New York: Routledge.

Part III

Use of Resources

Chapter 10

Researcher-Designed Resources and Their Adaptation Within Classroom Teaching Practice: Shaping Both the Implicit and the Explicit Carolyn Kieran, Denis Tanguay, and Armando Solares

10.1 Introduction Mathematics education research has, over the years, yielded numerous resources, many of which have been both designed with the practitioner in mind and made accessible to them. But little is known about the ways in which teachers take on such research-based resources and adapt them to their own needs. In 2000, Adler proposed that, ‘mathematics teacher education needs to focus more attention on resources, on what they are and how they work as an extension of the teacher in school mathematics practice’ (p. 205). However, the little that exists regarding the research involving researcher-designed resources has focused more on the mathematical design of the resources (e.g., Ainley & Pratt, 2005) or on their impact with respect to student learning (e.g., Hershkowitz, Dreyfus, Ben-Zvi, Friedlander, Hadas, Resnick, Tabach, & Schwarz, 2002), rather than on the ways in which the resources are used by teachers and on why they are used in these ways. The research presented in this chapter centres on the ‘how’ and the ‘why’ of teachers’ adapting of researcher-designed resources.

10.1.1 The Literature that this Research Draws Upon In their Introduction, Gueudet, Pepin, and Trouche state that the aim of this volume is ‘to deepen our understandings of teacher documentation in the field of mathematics education,’ where documentation processes are considered to include the ways in which teachers collect, select, transform, share, implement, and revise resources, as well as the influences upon these processes. Because the teachers whose practices are described in this chapter were participating in a research project that involved

C. Kieran (B) Département de Mathématiques, Université du Québec à Montréal, Montréal, QC, Canada H3C 3P8 e-mail: [email protected]

G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_10,  C Springer Science+Business Media B.V. 2012

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their use of the resources designed by the researchers – resources related to the learning of algebra with Computer Algebra System (CAS) technology – the focus herein is more restrained in that it is oriented specifically to the ways in which the teachers ‘transformed’ these resources in their teaching and to the factors contributing to these transformations. One of the pivotal constructs of the documentational approach of didactics (Gueudet & Trouche, 2009) is documentational genesis, with its dialectical processes involving both the teacher’s shaping of the resource and her teaching practice being shaped by it. Building on a distinction introduced by Rabardel (1995), Gueudet and Trouche (Chapter 2) emphasize that not only does the teacher guide the way the resource is used, but also that the affordances and constraints of the resource influence the teacher’s activity. As they point out, ‘design and enacting are intertwined.’ However, within the framework of the documentational approach, little research has as yet used the design characteristics of given resources as a focal lens for studying the ways in which design might shape teaching practice. Remillard (2005), in her review of the research literature on teachers’ use of mathematical curricula, argues that features of the curriculum matter to curriculum use as much as characteristics of the teacher and that such research is rather unexplored terrain. In the spirit of Remillard, this chapter uses the main features of the researcherdesigned resources as a tool for analyzing the ways in which resources can occasion the shaping of individual teaching practice. However, teachers also shape the way in which resources are used. Robert and Rogalski (2005), for example, have argued that teachers’ personal histories, experience and professional history in a given activity, and knowledge and beliefs about mathematics and teaching, impact on their teaching practice and the ways in which they use curricular materials. In addition, Sensevy, Schubauer-Leoni, Mercier, Ligozat, & Perrot (2005) have noted that the didactic techniques they observed within each of the teachers’ teaching of the same content were ‘produced on a background of beliefs’ (p. 174) that gave rise to a certain consistency in the practice of each teacher. Similarly, Schoenfeld (1998) has described the ways in which teachers’ goals, beliefs, and knowledge interact, accounting for their momentto-moment decision-making and actions. According to Schoenfeld, the practice of teachers, whether it be the activity of the lesson planned by the teacher, or the unscripted activity engendered by unexpected students’ difficulties or responses, is regulated by deep-seated goals, beliefs, and knowledge. In contrast, some of the more recent research related to teaching practice involving computer-technology resources has advanced the argument that models focused on teachers’ established routines are insufficient for analyzing teachers’ activities in technology-based lessons. For example, Lagrange and Monaghan (2010) have found that teachers’ practices in dealing with the complexity of classroom use of technology are far from stable. The literature related to the use of novel teaching materials has also disclosed that different teachers enact the same curriculum materials differently (e.g., Chavez, 2003), and that the same teacher may enact the same curriculum materials differently in different classes (Eisenmann & Even, 2011). Nevertheless, Drijvers, Doorman, Boon, Reed, & Gravemeijer

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(2010), who have combined the constructs of emergent goals (Saxe, 1991) and instrumental orchestration (Trouche, 2004; see also Chapter 14), suggest that the unstable practices of teachers within technology environments and with novel curricula might still be rooted within a system of more stable beliefs and knowledge. These various findings with respect to the tension between, on the one hand, the design forces that can provoke instability and a reshaping of teaching practice, and on the other hand, the force of consistency within a teacher’s existing practice that leads to a teacher’s shaping of given resources, attest to the complexity of the interactions between the dual processes of documentational genesis. One of the aims of this chapter is to better understand the dialectical relation between these two forces.

10.2 Background The study presented herein is part of a larger program of research, the first phase of which was oriented toward student learning: its central objective was to shed light on the co-emergence of algebraic technique and theory within an environment involving novel tasks and a combination of CAS and paper-and-pencil technologies (see Kieran & Drijvers, 2006). The second phase of the program, which was oriented toward teaching practice, included secondary analyses of the video-data from the first phase. From the start, these analyses disclosed specific differences in the manner in which teachers were integrating the researcher-designed tasks into their day-to-day practice. Individual teachers were mediating the technical and theoretical demands of the tasks for their students in quite different ways. These secondary analyses provide the foundation for this chapter.

10.2.1 The Three Teachers and Their Students Of the five teachers participating in our initial study, the three who are featured in this chapter were selected because they all taught in the same city – a large urban metropolis – and thus shared a certain common curricular experience. They shall be named T1, T2, and T3. To help in further maintaining their anonymity, the masculine gender will be used throughout. T1, whose undergraduate degree was in economics, had been teaching mathematics for 5 years, but had not had a great deal of experience with technology use in mathematics teaching, except for the graphing calculator. In observing T1’s teaching prior to the start of the research, the researchers noted that he encouraged his pupils to talk about their mathematics. T1’s class of grade 10 students was considered by the teacher to be of medium-high mathematical ability. They were quite skilled in algebraic manipulation, as was borne out by the results of a pretest we administered. They were used to handling graphing-calculator technology on a regular basis, but had not experienced CAS technology prior to the start of the research.

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T2, who was the most mathematically qualified of the three teachers, had taught mathematics for 16 years, half of this time at the college level, before teaching at the secondary level. He was a leader with respect to the advancement of the use of technology in the school where he was teaching. In our prestudy observations, we noted that his practice tended to be teacher-centred. T2’s students seemed very strong, mathematically speaking, on the basis of the same pretest as was mentioned above, and were experienced with the various capabilities of graphing-calculator technology, but not with CAS. T3, whose undergraduate degree was in the teaching of high-school mathematics, had 5 years of experience in the teaching of mathematics at the secondary level. While he had some prior experience with the use of graphing-calculator technology in his teaching, he had never before used CAS. T3’s students were considered by their teacher to be of average mathematical ability. They had some graphing-calculator experience, but none with CAS. The pretest that we administered indicated that they were weaker in symbol-manipulation ability than the students of the other two classes.

10.2.2 Methodological Aspects At the same time that our research team began to create the task-sequences that would encourage both technical and theoretical development (see also Artigue, 2002; Chevallard, 1999; Lagrange, 2003, for more on task, technique, and theory) in 10th grade algebra students – a creation process that took well over a year – we also made contact with several practicing mathematics teachers to see if they might be interested in collaborating with us. The form of collaboration that we arranged was on several levels. First, the teachers were our practitionerexperts who, within a workshop setting, provided us with feedback regarding the nature of the tasks that we were conceptualizing. They also spent some time learning how to use the CAS technology (hand-held TI-92 Plus calculators – the same devices that would be lent to the students for the entire school year). As well, the week-long workshop included discussions related to the main mathematics-related and technology-related intentions of the researcher-designers. Second, after modifying the task-sequences in the light of the teachers’ feedback, we requested that, at the beginning of the following semester, they integrate all of the task-sequences into their regular mathematics teaching and that they be willing to have us act as observers in their classrooms. Third, throughout the course of our classroom observations, which occurred over a 5-month period in each class, we also offered a form of ongoing support to the participating teachers by being available to discuss with them whatever concerns they might have. In addition, we conducted interviews with some of them immediately after certain lessons that we had perceived to be deserving of further conversation. We observed how the teachers integrated the designed task-sequences into their usual teaching practice, which additional resources they called upon (e.g., the blackboard and the classroom view-screen – a device connected to an

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overhead projector that projects the screen display of the calculator hooked up to it), which ways they adapted the task-sequence materials, and the extent to which the designed resources seemed to be contributing to a reshaping of their practice. In addition to the videotaped observations of each classroom lesson involving our task-sequences, and the follow-up conversations with each teacher, we also observed a couple of lessons of each teacher’s regular teaching practice prior to the start of the research.

10.3 The Researcher-Designed Resources The student version of each task-sequence consisted of a set of activity sheets that presented the task questions and blocked-off spaces for written answers, as well as indications as to when classroom discussion could be expected to occur. The research team also created a teacher/researcher guide to accompany each task-sequence, in addition to a solution key (see the research team’s website for the task-sequences: http://www.math.uqam.ca/~apte/TachesA.html). Thus, while the student task-sequences constituted a central component of the researcherdesigned resources, the resources also included the accompanying teacher guides, the particular CAS tool that was used (along with its guide), and the discussions that were held during the workshop sessions regarding the spirit embedded within the textual materials, as well as any ad hoc conversations that took place during the unfolding of the research.

10.3.1 Three Key Features of the Researcher-Designed Task-Sequences In designing the task-sequences, our intentions revolved around three key aspects: the mathematics, the students, and the technology. Mathematics-wise, all of the task-sequences involved a dialectic between technique and theory within a predominantly exploratory approach, with many openended questions. The mathematics that formed the content of the tasks intersected with, but also extended, the usual fare for grade 10 algebra students. At times, the main mathematical theme of the task-sequence was more technique oriented, as in: factoring the xn −1 family of polynomials for integral values of n, solving systems of linear equations, using factoring to solve equations containing radicals, and exploring the sum and difference of cubes. At other times, the focus was more theoretical in nature, as in the task-sequences relating to the equivalence of algebraic expressions. But in both cases, a combination of technical and theoretical activity related to the mathematics was envisaged. In brief, the intended emphases relating to the mathematics included: (i) coordinating the technical and theoretical aspects of the mathematics, (ii) pattern seeking, inductive reasoning, and development of techniques, (iii) conjecture making and testing, and (iv) deductive reasoning and proof.

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Student-wise, we built into the task-sequences not only questions where the students would be encouraged to reflect on their mathematics, but also indicated moments where they would be expected to talk about their mathematical thinking during whole-class discussions. Tasks that asked students to write about how they were interpreting their mathematical work and the answers produced by the CAS aimed at bringing mathematical notions to the surface, making them objects of explicit reflection and discourse in the classroom. In sum, the intended emphases that related to the students included: (i) encouraging them to be reflective and inquiring into their thinking and (ii) encouraging them to share their ideas, questions, and conjectures during collective discussions. Technology-wise, all of the task-sequences involved technical activity with either the CAS, with paper and pencil, or with both. We viewed the CAS as a mathematical tool that, through the task, stimulates reflection and generates results that are to be coordinated with paper-and-pencil work. The CAS served thus as a confirmation-verification tool and/or a surprise generator (producing results that would, in general, not be expected by the students). Very few CAS commands were required for the task-sequences we designed, simply factor, expand, solve, and the evaluation command; thus, the manipulation of the technological tool itself was not to impede the mathematical thinking encouraged by the task-sequences. Additional technologies that we considered would be used included the view-screen and the blackboard. In sum, the intended emphases that related to the technologies included: (i) taking advantage of the potential of CAS for producing surprising responses that would provoke a rethinking of techniques or theories, for verifying conjectures of a technical or theoretical nature, and for checking paper-and-pencil work and (ii) using the blackboard for rendering public, within class discussions, both teacher explanations and student work.

10.3.2 The Issue of Explicit Versus Implicit Researcher-Designer Intentions The teacher guides, which also contained all of the task questions that were addressed to the students, included many specifics that were addressed to the teacher alone. First, they offered explicit suggestions as to the precise mathematical content that might be addressed within the collective discussions. Second, they presented a few examples that illustrated, pedagogically speaking, how a particular topic might be further explained at the blackboard. The following text from the teacher guide for the first part of the Activity 7 task-sequence illustrates how the researchers made explicit the main mathematical issues for discussion, potential erroneous thinking on the part of the students, and the role of the CAS for the given task questions (see Fig. 10.1). But, in general, the teacher guides did not elaborate on the student- or technology-related intentions of the researcher-designers. For example, the teacher guides did not specify how to conduct the collective discussions – how to encourage

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For discussion: In the course textbook, taking out a common factor is approached without a clear motivation or rationale for its use. Here, the aim of taking out the common factor y − 2 is relatively easy to

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motivate, be it in the expression y − 2 − 10 y − 2 or the expression y − 2

)3 − 10 (y − 2)− y (y −2).

In each case, taking out the common factor enables students to reduce the problem to one of solving a quadratic equation (having solutions: y = 6 and y = –1), whether it be by factoring out ⎞ ⎛ 2 y − 2 on both sides of the equation y − 2 ⎜ y − 2 −10 ⎟ = y y − 2 , or by invoking the zero-product ⎠ ⎝

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)(

)

(

)

⎛ ⎞ 2 theorem in the equation y − 2 ⎜ y − 2 −10 − y⎟ = 0. Moreover, the aim is to orient students to the ⎝ ⎠

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)

possible ‘ taking out of the common factor ’ involving the radical expression in the two subsequent items.

Among those students who take out the common factor y − 2 on both sides of the equation, some are likely to ‘lose’ the solution y = 2. Whether or not this be the case, however, on the basis of this example the teacher should conduct a classroom discussion about what precautions to take before canceling a factor common to both sides of an equation. In effect, for the values of a variable for which the common factor vanishes, this simplification is tantamount to division by zero! Those values of the variable must therefore always be treated (i.e., verified as possible solutions) one by one, before simplification. It is this very simplification, for which the solution y = 2, given by the calculator, is lost, that we hope students will retain.

The teacher can also help students see how to avoid this problem by using the strategy consisting of bringing all terms to one side of the equation:

(y − 2)3 −10(y − 2)− y (y − 2) = 0 and invoking the theorem: ‘a product is zero iff either one of the factors is zero.’

Fig. 10.1 Intentions of the researcher-designers that were rendered explicit within the teacher guide for Activity 7

reflection, how to inquire into student thinking, how to have students share their thinking with their classmates during the collective sessions, how to use the blackboard to help students coordinate their CAS and paper-and-pencil techniques, or how to orchestrate discussions of a theoretical nature.

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The explicitness of the students’ written task-questions was intended, in a sense, to help fill in some of the gaps regarding what was not communicated explicitly to teachers. The written questions that were directed to the students, and the frequent pointers to whole class discussions, were intended to convey to the teachers, albeit in an implicit way, the researcher-designers’ intentions regarding the mathematics, the students, and the technology. For example, task questions such as, ‘Explain why (x + 1) is always a factor of xn − 1 for even valuesof n ≥ 2,’ and  ‘Perform the indicated operation (using paper and pencil): (x − 1) x2 + x + 1 ’ were quite specific in their stress on the use of either theoretical or technical means for approaching the mathematics. Similarly, questions that related to mathematical reflection, such as, ‘Based on your observations with regard to the results in the table above, what do you conjecture would happen if you extended the table to include other values of x?,’ as well as the mention in the Activity sheets that collective discussions were scheduled to follow, were meant to communicate not just the need for student reflection, but also the intention to have students discuss their reflections during the collective sessions. Our technological intentions, especially those regarding the coordinating of paper-and-pencil and CAS techniques, were also explicitly presented in the task questions, for example: ‘Verify the anticipated result above using paper and pencil and then using the calculator,’ and ‘If, for a given row, the results in the left (with paper-and-pencil) and middle (with CAS) columns differ, reconcile the two by using algebraic manipulations in the right-hand column.’ Thus, the teacher guides were a blend of the implicit and the explicit. Explicit within the structure of the task-sequences were the mathematical aims, the issues on which students were expected to reflect, and the ways in which the CAS and paper-and-pencil technologies were to be used. Implicit was the fact that all three of these were to be combined and coordinated, as well as a manner for doing so, within the collective discussions. As will be seen in the upcoming section, teachers adapted both that which had been rendered explicit, as well as that which had been suggested implicitly, within the researcher-designed resources. Before presenting the nature of these adaptations, a few additional remarks are in order with respect to both the implicit and its adaptation. In all reading of text, the reader has a part to play. This notion is discussed in many theoretical writings, including Otte’s (1986) complementarist position on the dialectic between textual structure and human activity, as well as Remillard’s (Chapter 6) view that, ‘the form of a curriculum resource includes, but goes beyond, what is seen.’ Nevertheless, what is unseen can be just as tangible as what is seen, as argued by Helgesson (2002, p. 34): ‘What is implicit, and thus unstated, is not necessarily less clear (or obvious) or less direct than what is explicitly stated; in other words, that an assumption is implicit does not mean that it is hidden and hard to find, or realized to be there only after some reflection.’ Helgesson, who defines implicit as that which is implied, understood, or inferable – tacitly contained but not expressed – points out that the tone and style in which the text is written may also say something about what it is intended to communicate. In keeping with Helgesson, we consider as implicit those unwritten and unspoken aspects of the researcher-designed resources that can be inferred from what was explicitly stated, those aspects that could be said to be

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in the spirit of what was communicated directly. Also in line with Helgesson, we would argue that the implicit does not necessarily require any additional reflective interpretation than that which is called upon for the explicit. Thus, adapting what is implicit should be akin to adapting what is explicit.

10.4 Teachers’ Classroom Adaptations of the Researcher-Designed Resources The two task-sequences that are the focus of this chapter are Activities 6 and 7. Activity 6 was related to the factoring of xn −1, for integral values of n (for a different elaboration of this task, see Mounier & Aldon, 1996, whose work provided the initial inspiration for our task-sequence). Activity 7 dealt with the use of factoring to solve equations with radicals. These task-sequences were selected for two reasons. First, the two of them taken together highlight the duality of the adaptations made by our teacher participants: adaptations dealing with more implicit aspects of the design and with unspecified areas of the researcher-designed resources, and adaptations related to changing or reorganizing an explicit aspect of the design. Second, while the analytical focus documents the ways in which teachers spontaneously transformed the resources, some evidence is also provided of the manner in which teaching practice was being shaped by the nature of the resources. Thus, the fabric of documentational genesis provides the backdrop for an analysis on the basis of the three overlapping, interrelated design features of the resources: the mathematicsrelated, the student-related, and the technology-related, with specific attention to both their implicit and explicit dimensions. The extracts analyzed from Activity 6 bear on adaptations made to the more implicit intentions of the researcher-designers, with examples drawn from the practice of T1 and T2, while Activity 7 focuses on adaptations to the explicit with examples from T3’s practice.

10.4.1 Adaptations Observed During the Unfolding of Activity 6 Our analysis begins with the adaptations made to the implicit, unwritten, and unspoken aspects of the researcher-designed resources. Activity 6, which included the telescoping, reconciling, and proving tasks, aimed at having students discover a general pattern for the factorization of xn −1 and instilling the idea of middle-term cancellation. By working on the reconciliation between CAS and paper-and-pencil factorizations, students were to develop their own factoring abilities and to conjecture and inductively extract some factoring properties. It was intended that they should explore and reflect on their mathematics, constructing, and validating their own factoring techniques, and also share their work and their thinking during the collective classroom discussion. Technologically speaking, it was expected that the CAS be used as a tool for verifying paper-and-pencil work with factor and expand, and for testing conjectures. We also expected that the surprise brought by the CAS

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( )(x + 1) and − 1). Agree? And for the other one (x − 1)(x 2 + x + 1) the same idea, I

T2: [while writing at the board; see Figure 10.3] When you expand this x − 1

(

add all your terms you get x

2

multiply the –1 throughout, getting −x 2 − x − 1, and that is going to give you x 3 − 1. What do you notice about the middle parts? Ss (several students, all at once): They cancel out. T2: They cancel out, because the x just elevates the degree of everything, and when you bring the −1, all the middle terms will cancel. You are going to have your x3 because you elevated the degree, but you are going to have your −1 at the end as well, and everything in the middle will cancel out. That is why without doing any algebraic manipulations, if I did

(x − 1)(x 3 + x 2 + x + 1), I notice that these (x 3 + x 2 + x + 1) are just a decreasing degree of x, so without doing any distributing, you figure out what the results would be.

Fig. 10.2 Extract from the discussion surrounding the Telescoping Task in T2’s class

through some of the verifications would lead students to strive for deepening their factorization techniques, but that this would require some additional elaboration presented at the blackboard. The Telescoping Task. Let us consider the beginning of the first collective discussion within Activity 6, where T2 conveyed his particular approach to dealing with mathematical issues of a technical and theoretical sort (see Fig. 10.2 and Fig. 10.3). The context was Question 2d: prod explain thefact that the following   How do you ucts (x − 1)(x + 1), (x − 1) x2 + x + 1 , and (x − 1) x3 + x2 + x + 1 result in a binomial? The technique and the theory of the mathematics are being talked about. But notice that T2 is not drawing these aspects from the students, but is rather presenting them himself. If one could say that our general intention about coordination between technique and theory has not been disregarded, our implicit intention with respect to fostering personal mathematical reflection on the part of the students, and on inquiring into their thinking, is clearly set aside by T2’s intervention. This is in contrast with T1’s way of orchestrating a whole class discussion, as is seen with the example of the subsequent Reconciling Task, which is provided in Fig. 10.4. The Reconciling Task. For the factoring of x4 − 1, the CAS had not yielded  for the students what they had expected: not (x − 1) x3 + x2 + x + 1 , but   rather (x − 1)(x + 1) x2 + 1 . In T1’s class, the following discussion ensued (see Fig. 10.5). The extract provided in Fig. 10.5 illustrates the ways in which T1 adapted the researcher-designed resources by filling in some of the unstated gaps in the teacher

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Fig. 10.3 T2’s use of the blackboard during the Telescoping Task

In this activity each line of the table below must be filled in completely (all three cells), one row at a time. Start from the top row (the cells of the three columns) and work your way down. If, for a given row, the results in the left and middle columns differ, reconcile the two by using algebraic manipulations in the right-hand column.

Factorization using paper and pencil

Result produced by the FACTOR command

Calculation to reconcile the two, if necessary

x2 − 1 = x3 − 1 = x4 − 1 = x5 − 1 = x6 − 1 =

Fig. 10.4 The first part of the Reconciling Task

guide. He inquired into students’ thinking and used this as a basis for discussing some of the different approaches to factoring completely x4 − 1. This was done with the stated aim of reconciling the differences between the unexpected result produced by the CAS and the paper-and-pencil result yielded by the general rule.

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T1: What does it turn out is the case? S1: Sometimes they like factor even more. T1: What we did initially is not wrong. It’s just not complete. … So for x 4 − 1, it’s what?

( )( )(

)

( )( )(

)

S1: x − 1 x + 1 x 2 + 1 [teacher writes at the board: x 4 − 1= x − 1 x + 1 x 2 + 1 ] T1: So let’s look at this one. How can we go about getting that without the calculator? S2: Use the rule.

( )(

)

T1: Is that right (as the teacher writes at the board: x − 1 x 3 + x 2 + x + 1 ] Class: Yeah. T1: And what do you do from there? S2: Group it. T1: And how do you group it? S2: [student explains how she would group the second factor, as the teacher writes at the board that which she dictates] T1: that’s one way of doing it. Bob [S3]? S3: [the student Bob then describes how he would factor x 4 − 1 by first breaking the x4 part into two equal halves] T1: What concept have you used? S3: Difference of squares [the student continues his explanation of the technique, which the teacher writes at the board as per S3’s dictation] T1: So both ways reconcile the differences, coming in from different points of view.

Fig. 10.5 Extract from the discussion following the Reconciling Task in T1’s class

T1 also displayed on the blackboard the various factoring approaches offered by the students, which thereby presented a public record of their different techniques. This is in contrast to the manner in which T2 responded to the implicit intentions of the researcher-designers for the same task. As seen in the Fig. 10.6 excerpt, T2 used the blackboard to show only the technique that he wanted to emphasize for the factoring of the x4 − 1 binomial: the difference of squares. It would also appear that student participation was called upon with the sole purpose of providing an opening

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T2: Now for the x 4 − 1, if you use the trick that we were looking at, and we just write it like this

( ) ( )(

)

[teacher writes on the board x 4 − 1 = x − 1 x 3 + x 2 + x + 1 ]; this is factored but not fully factored. When you press Factor on your calculator, what do you get? What did you get, Chris (S1), when you did Factor on your calculator?

( )( )(

)

S1: x − 1 x + 1 x 2 + 1

T2: [teacher wrote this response on the board] Right! Like that. So, how do we reconcile the two? … S2: You could do difference of squares at the start. T2: Yes, you go back to the start, and that is what I said, you can go back to the start, and look

( )

at how you do it paper-and-pencil-wise. If you go back to the start and you’ve got your x 2 − 1 ,

( )

your x 2 + 1 [teacher writes these two factors on the blackboard] – your difference of squares –

( )

and then you have another difference of squares here [teacher points to x 2 − 1 and writes

(x − 1)(x + 1) below it]. … So, in another words, what we are discovering is that our little trick ( )

that we did, that only helps to get the x −1 out. That doesn’t necessarily mean that what is left is not refactorable. …

( )( )(

) ( )(

)

S3: I did the opposite, I mean, [ x − 1 x + 1 x 2 + 1 = x − 1 x 3 + x 2 + x + 1 ] T2: Reconciling the two doesn’t mean just expanding one and showing it is x 4 −1, and I guess that’s what you are saying. S2: Instead, we can just, eh, factor out the other x 2 and make it, for the second factor. T2: So, you grouped two by two. So, that is another way you could have factored this bracket

( )

over here. Because it is four terms, you factor out x 2 here; you get x + 1 , then you factor

( )

( )

( )

your x + 1 out, and you get x + 1 and x 2 + 1 . Ok? All right. [Teacher does not write the grouping method out on the blackboard; he just points to the different terms and states orally S2’s method.]

Fig. 10.6 Extract from the discussion surrounding the Reconciling Task in T2’s class

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for T2’s own preferred approach, the one that he considered more efficient. Other approaches that students offered were dealt with orally. Another instance of T2’s manner of adapting what was implicitly conveyed in the resources concerns the factoring of x10 − 1. Here, the surprise factor of the CAS tended not be taken advantage of, nor allowed to play its intended thought-provoking role. While students were still working on the second part of the Reconciling Task, with the polynomials from x7 −1 to x13 −1, trying to reconcile their paper-and-pencil   factorizations with the results produced by the CAS, T2 rapidly wrote x10 − 1 =    5 x − 1 x5 + 1 on the board. He then stated: ‘The one that may give you some trouble here is the x to the 10th. I will explain why.’ He proceeded to explain at the   board the factorization x5 +1 = (x + 1) x4 − x3 + x2 − x + 1 , with a great deal of ad hoc hand-waving. It appeared here and elsewhere in Activity 6 that T2 assumed ownership of all the main mathematical ideas presented in class, a corollary being that students were not held responsible for thoroughly explaining their own thinking. A rather different situation evolved in T1’s class where the x10 − 1 example led a student to conjecture a new theory involving the factoring of xn + 1 for odd ns – on the basis of the CAS factorization of x5 + 1, supported by the factoring pattern for the sum of cubes, x3 + 1 (for more on the unfolding of this student’s conjecture, see Kieran & Guzmán, 2010). T1 encouraged the student to talk about the way he was thinking and to be as complete as possible in his explanation. The Proving Task. T1 believed that students would need time to get into the last task of the sequence, the Proving Task: Explain why (x + 1) is always a factor of xn − 1 for even values of n ≥ 2. Note that the teacher guide had not included any explicit suggestions in regard to the proving task, simply a possible solution on the basis of the Factor Theorem, i.e., ‘a is a zero of the polynomial p(x) iff (x − a) is a factor of p(x).’ T1 waited patiently until some of the students had ideas to submit to the class. He then asked three of them to go to the board in turn and to write down and explain their proofs. He requested that the class listen carefully to the explanations being offered by these proof-givers: ‘Guys, give him a chance’ and ‘Ok, listen because this is interesting, it’s a completely different way of looking at it.’ After each of their explanations, everyone in the class was encouraged to discuss and try to understand the main approach used in the proof. From time to time, T1 asked for further clarification, offered counter-examples, and pushed students to think more deeply. T1’s way of filling the gaps in the teacher guide for the Proving Task was in sync with the student-related intentions of the researcher-designers, and even enriched them further. In a teacher interview held with T1 at the close of Activity 6, he was asked if he felt that his participation in the project was affecting his teaching practice in any way. This is part of what he had to say (Kieran & Guzmán, 2010, pp. 141–142): I think it’s made me think more, or made me realize that what I like is making them [the students] think a little bit more. And I think I did that anyway, I remember when you came into class last year that there were some things similar happening, but it just made me, just consider a little bit more: Can I let them come through this themselves, let them try this out themselves a little bit more, which I think I always did – but just seeing these activities

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work, it’s made me realize there’s more scope to it than I have done in previous years. There is much more scope to let them really go.

Although he stated that the CAS technology was essential to the changed nature of his students’ mathematical learning, he was quick to point to the role played by the mathematics of the task sequences. The intertwining of novel and substantive mathematical tasks, and technological tools appropriate for these tasks, led to mathematical activity that, according to T1, the students quite enjoyed and from which they learned a great deal. This, in turn, promoted the development of new awarenesses on the part of T1, awarenesses that were reshaping his teaching practice. He realized that he could push his students to think a little bit more about their mathematics and put even more emphasis on having them share their reflections with the rest of the class. We continued to observe T1’s classes during the 2 years that followed. He never stopped using the task-sequences and CAS technology that he mentioned he had found so worthwhile during the present study; at the same time, his practice continued to evolve along all three dimensions of the researcher-designed resources.

10.4.2 Adaptations Observed During the Unfolding of Activity 7 Our analysis continues, this time bearing on the adaptations made to the explicitly stated aspects of the researcher-designed resources. Mathematics-wise, our primary intention in Activity 7 was to make students aware of the possible loss of solutions when they simplify an equation by dividing both sides by some factor. Students were thereby to be directed toward the more reliable solving method of isolating terms on one side and using the zero-product theorem, that is, ‘a product is zero iff either one of the factors is zero.’ Both the teacher guide and the student task-sequence had included explicit mathematical notes in the opening block of Activity 7 (see Fig. 10.7). Student-wise, it was intended, and explicitly asked for, that they describe at a meta-level – perhaps quite loosely – both the patterns they were seeing within the equation and their equation-solving approach, before actually solving the equation. We were expecting that, for the subsequent equation-solving task, most of them would lose a solution. Technology-wise, the CAS was to be used as a follow-up to the paper-and-pencil solving – a verifying device that would yield surprises, such as producing one more solution than they had likely obtained with their paperand-pencil methods. The teacher’s guide suggested a way of handling the class discussion related to lost solutions and their verification with the CAS (see Fig. 10.1, presented earlier in this chapter). In sum, the central explicit components related to the first three tasks of Activity 7 concerned, in this order: (a) a focus on the meta-level aspects of solving a particular equation containing common factors with radicals, (b) the actual solving of a related equation having a similar pattern of common factors (but without radicals) and which could induce a loss of solutions,

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Primary idea: Factoring (taking out a common factor) as a tool for solving equations, particularly when used in conjunction with the ‘zero-product theorem.’ Secondary ideas: •

• •

•

Factoring (taking out a common factor) can be applied not only to constants and variables, but also to algebraic expressions that can be taken as objects to operate upon. Students should be able to bring the methods learned for solving linear and quadratic equations to bear on equations that are neither linear nor quadratic, per se. Simplifying an equation by dividing both sides by some factor may lead to a loss of solutions. In equations in which such simplifications are possible, the strategy of isolating terms on one side of the equation and using the zero-product theorem is generally a more effective solving method; In equations involving variables under the radical sign, verification after solving is not only advisable, but necessary.

Fig. 10.7 Explicit mathematical notes in both the student task-sequence and the teacher guide

and (c) the verification by CAS of the paper-and-pencil solutions which would lead for many students to a required reconciliation of the two sets of solutions. All of the examples given in this section are drawn from T3’s practice, as it was here that we observed the most extensive adaptations to the explicit aspects of the researcher-designed resources. Specifically, his adaptations involved replacing an expression by a letter, inserting a transitional equation, and using the CAS to factor a quadratic. In addition, we note that T3 had an empathetic way of preparing students for possible task difficulties, telling them not to worry and reformulating each question with a phrasing that in his view was better adapted to their level of understanding. A notable sign of his general attitude is the fact that from the outset of the prior Activity 6, in advance of bringing it to class, he told the researchers that he had decided to skip the final Proving Task, it being too difficult in his view. In his defence, recall that T3’s class was the weakest of the three. Replacing An Expression by a Letter. For the first proposed equation in Activity 7, 5

3 √ √ √ x − 4 + 11 x − 4 = (2x + 1) x − 4

(10.1)

a general reflection on how students would proceed to solve this equation was to be elicited. Immediately afterward, they were to be directed toward the simpler equation (y − 2)3 − 10 (y − 2) = y (y − 2)

(10.2)

which was the one to be actually solved. From the start, while reading and rewording the instructions, √ before anything whatsoever had been done by his students, T3 suggested replacing x − 4 by a (see Fig. 10.8).

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T3: Just imagine that this expression, root of x minus 4, is replaced by a. It would give you something much simpler, wouldn’t it? So each time it’s written ‘root of x minus 4’, if we write a instead, we would have quite a simpler equation and then, what would you do? Because the principle would be, say, quite the same. Anyway, some aspects would be the same. So try to see what you could do to solve this equation ... or anyway, what would be your steps. We don’t ask you to solve it, just go with what you think you would do. There are no bad answers, just try.

Fig. 10.8 T3 suggested to students that they replace an expression by a letter

This adaptation interfered with our intention of having students recognize by themselves in what facet Eqs. (10.1) and (10.2) have the same structure, and to what extent the solving steps they were asked to sketch for Eq. (10.1) could be put to the test by actually solving Eq. (10.2). As well, we note that T3 did not follow the explicitly given sequence of holding off on the class discussion until after the students had worked on both equations and had tested the solutions of Eq. (10.2) with the CAS. Following his too early and wordy discourse on Eq. (10.1), T3 had students work on this first equation, but in fact never asked them how they viewed it at a metalevel. Inserting a Transitional Equation. We will now see that T3’s implementation of the activity digressed even further from what was explicitly presented in the researcher-designed resources (see Fig. 10.9). T3’s insertion of a transitional equation, accompanied by replacing the main expression by a letter, was an adaptation that not only further confounded our initial intentions with respect to students’ seeing structural similarities between the two equations, but also presented an added mathematical difficulty for the students: Eq. (10.2) conveying a term in both y and y − 2, the substitution of x for y − 2 gives either a two-variable equation or a term in x and x + 2. The transitional equation introduced by T3 did not involve such a hindrance. Whether T3 proposed it as a transitional stage for the students, or simply did not foresee this snag pertaining to substitution in Eq. (10.2), we do not know. In any case, as the students began working on this second equation, one did complain that the substitution of x for y−2 gave him an xy term, which got him stuck. T3 offered him the following hint: ‘Nothing keeps you from going back to y − 2.’ Still a little later, as T3 was showing at the board a method for handling this equation, he replaced the y−2 by a (while keeping a term in ay), factoring out an a (see Fig. 10.10) and replacing back the a by y−2. (The possibility of substituting a+2 for y was not mentioned.) Finally, the equations displayed   on the blackboard by T3, as a path to solve Eq. (10.2), were: a a2 − 10 − y = 0,   (y − 2) (y − 2)2 − 10 − y = 0, and (y − 2) (y − 6) (y + 1) = 0. The fact that T3 left out many of the intermediate steps of the solving process and used oral commentary to fill the missing steps is quite surprising, in view of his metalevel remarks to the students on how difficult they must be finding this work.

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T3 [reading]: ‘Using paper and pencil, see whether you can first solve the following equation.’ So they are giving you another one that may be less scary, which is y minus 2 to the three ... [He does not finish reading the second equation, but searches for a piece of chalk]. Well ... They are giving you this one so that you can compare, they say that it is somewhat analogous to the ‘monster’ given just before. They say [reading]: ‘Factoring (taking out a common factor) might be useful here.’ So they are giving you a hint. Ok, I have seen that some of you wrote interesting things, in the sense that you have already good ideas about how to solve. Nevertheless I’ll give one to you all, because some of you are facing it without knowing what to do, and that I can understand. I’ll do an example that is completely different.

(

) (

[He writes on the board: 5 a − 3 + 2 a − 3

)2 = 3(a − 3)3.] I'm coming back to what I've said before,

about root of x minus 4, if I remember well, that it could be replaced by a value, say, a. Ok, it may have given some of you a hint, precisely about what could be done with it. So here [he gets a piece of colored

(

)

chalk and circles a − 3 ], if I say here that a minus 3, if all of this parenthesis here would have been, 2

3

say, x. We would be facing [he writes on the board: 5x + 2x = 3x ]. Do you agree? All I have done is that I've been saying to myself: Instead of writing a − 3, to make things easier, I'll replace a − 3 by x. And then, I’m facing this new equation. Is this equation [pointing to the board] less scary?

Fig. 10.9 T3’s use of a transitional equation

Fig. 10.10 An unexpected transitional equation before solving Eq. (10.2)

Using CAS to Factor a Quadratic. Further adaptations by T3 concerned his use of the CAS technology. When discussing the solving of −3x3 + 2x2 + 5x = 0 (derived from the transitional equation 5 (a − 3) + 2 (a − 3)2 = 3 (a − 3)3 by substituting x for a − 3), T3 suggested that, the common factor x having been taken out, students may then use the CAS to factor −3x2 + 2x + 5. Perhaps this suggestion was

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made in the interests of time or to reduce some of the overall complexity of the task. Nevertheless, it was outside the suggested route of solving with paper-and-pencil and, only later, verifying the solutions with the CAS. Moreover, for the third question of this first part of the task-sequence – the one that asked students to check their solutions of Eq. (10.2) with those produced by the CAS – T3 chose to eliminate this question, having introduced Eq. (10.2) with a view-screen display of the three solutions yielded by the CAS and subsequently asking students to find themselves the same three solutions with paper and pencil. Thus, the surprise realization that there might be three solutions, and how it came to be that one of them had been lost through their paper-and-pencil techniques, was never provoked in T3’s class.

10.5 Discussion The discussion that follows touches upon issues related to documentational genesis, as well as the design features of the resources and the influences contributing to their shaping by participating teachers in their practice.

10.5.1 Documentational Genesis The findings of our study suggest that the dialectical process by which resources are considered to both shape and be shaped by teachers in their practice may not be a truly equilibrated process. By this, we mean that over the period of 5 months during which our regular observations occurred, a much great tendency to shape rather than be shaped by was noted. In their attempts to grapple with the complexity of resources that involved novel mathematical material, a technology that they had never before used in class, and specific expectations regarding student participation in the process of learning, the participating teachers seemed more preoccupied with making multiple adaptations to the various aspects of the resources than with being aware of whether or not, and in which ways, the resources might be shaping their practice. In fact, it was really only T1 who spontaneously expressed that both the mathematics and the technology of the resources were pushing his students into going further mathematically and into reflecting more deeply. This in turn was inducing T1 to think a little differently about his practice, a practice that was thereby being coshaped as much by the resources as by his approach to teaching with them. He also stated that the resources encouraged his students to be more active and more involved participants in the process of learning. But here we must add that, during our visits to T1’s class during the year preceding the study, we noted that he was already predisposed to asking his students questions of a reflective nature during whole class discussions. Thus, in a very important sense, T1’s existing practice with respect to student participation allowed his practice to be shaped further in this direction, as well as along the other two dimensions regarding the mathematics and the technology.

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Despite the limitations of our study in terms of number of teachers and time frame, our findings – fragile as they are with respect to the complex interaction between the complementary forces of shaping and being shaped by – suggest that the interaction between these two forces of documentational genesis may at times, and with different individual teachers, be stronger with respect to one force than the other. It seems particularly the case that, during the early stages of using novel resources, the shaping of resources by some, if not many, teachers may be especially strong and may serve as a formidable counterforce to the potential of the same resources to shape teaching practice. Suffice it to say that the relation between the dialectical processes of documentational genesis remains for us an area where further study is warranted.

10.5.2 The Implicit Versus the Explicit in Task-Designers’ Intentions The analysis presented in this chapter indicated that adaptive shaping occurred with respect to all three key features of the researcher-designed resources (the mathematics, the students, and the technology) and to their coordination, whether our intentions with respect to those features were explicitly stated or implicitly suggested. Regarding Activity 6, we observed that our implicit emphasis on fostering personal mathematical reflection on the part of the students, and on inquiring into their thinking, was set aside by most of T2’s interventions. Similarly, he used the blackboard to show only the techniques that he wanted to stress. In a related manner, the surprise factor of the CAS was not taken full advantage of with respect to provoking student thinking. In contrast, our analysis of T1’s adaptive activity indicated a different manner of filling in the unstated gaps in the teacher guide. He thoroughly inquired into students’ thinking and used this as a basis for class discussions. T1 also displayed on the blackboard the various approaches offered by the students, thereby presenting a public trace of their different techniques. Likewise, our analysis of the ways in which T3 adapted the researchers’ explicit intentions with respect to Activity 7 of the researcher-designed resources disclosed significant adaptive activity in much that was expressly documented regarding the mathematics, the students, and the technology. Researchers (e.g., Freeman & Porter, 1989) have argued that, if teachers’ guides were more explicit and less ambiguous, the degree of closeness between teaching practice with these resources and the intentions of the resource designers could be greater. For example, Manouchehri and Goodman (1998) have critiqued certain reform-based curricula for not ‘providing the teachers with detailed methods of how to address the content development’ (p. 36). Our findings are in disagreement with the argument that greater detail will necessarily lead to a closer following of curriculum materials. No matter how explicitly expressed the researcher-designers’ intentions be, adaptation of the resources will take place. Our comparison of the nature of the adaptations that were forged with respect to both the implicitly suggested and explicitly expressed intentions of the researcher-designers showed that,

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in both intentional domains, teachers will adapt the resources that they use. This is not to suggest that researchers and curriculum developers should not attempt to make as explicit as possible their intentions with respect to the content and use of the resources they design. But they should expect that – as will be discussed shortly – the personal beliefs, goals, and habitual classroom practice of the teachers may be at variance with the epistemological and pedagogical assumptions underlying the researcher-designed resources and that this will inevitably lead to adaptation.

10.5.3 The Influences Contributing to the Adaptations That Teachers Made to the Resources Our finding that, whether the intentions of the researcher-designers were explicitly stated or implicitly suggested, teachers adapted the given resources leads naturally to the question as to what it was that underpinned these adaptations, that is, why were the resources adapted in the ways that they were? An additional question concerns the issue of the consistency of these adaptations within individual teachers. The teachers’ mathematical knowledge clearly filtered their interpretation of the mathematical intentions of the researcher-designers. This was seen in T3’s inappropriate choice of example that he used when inserting his substitution technique into the task-sequence. Pedagogical content knowledge also played a role. It underpinned the well-developed ways in which T1 orchestrated the whole-class discussions with students being asked to explain their thinking to the class at large, the differential ways in which T1 and T2 used the blackboard for keeping a written trace of students’ thinking and as a tool for mediating the reconciliation of paperand-pencil and CAS responses, and the varying roles for the CAS technology that were encouraged by each of T1, T2, and T3. Teachers’ beliefs accounted for much of their adaptive activity with respect to their use of the researcher-designed resources, from T1 who believed his students could and should be challenged mathematically and thereby adapted the unfolding of the task-sequences in such a way that students be held responsible for their own mathematical thinking, to T2 who believed that the teacher is the mathematical focal point of the classroom, to T3 who believed his students required a certain social and mathematical security net. In keeping with Schoenfeld (1998), we have attributed beliefs and knowledge to be at the root of the adaptations that the individual teachers carried out in their day-to-day teaching with our resources. However, such attributions do not account entirely for the global picture of each teacher’s approach to adaptation. The enactment of a teacher’s beliefs, which translates into both short- and long-term goals in the classroom, also constitutes a ‘pedagogical contract’ with the students – a certain set of expectations of the teacher for the students and vice versa (note that this pedagogical contract takes in the ‘didactical contract’ of Brousseau, 1997, but also includes more, namely certain attitudes, beliefs, and convictions of the teacher that are not tied specifically to the mathematical content being considered à la Brousseau).

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For example, T3 was extremely sensitive to the perceived needs and abilities of his students, as inferred from his rewording of the task questions and alteration of the content so as to try to make the mathematics more accessible to them. He seemed reluctant to put his students into potentially awkward situations where they might not know how to express themselves; thus he engaged the class in very few collective discussions. He called upon students whom he thought might have the beginnings of an answer and then proceeded to elaborate on their rather sketchy responses. In short, he made few mathematical demands of his students. His interactions with the students always weighed on the side of showing empathy toward them. In contrast, T2 delighted in demonstrating his mathematical prowess to the students. This seemed to be a central part of the identity he had forged for himself in the mathematics classroom. His students, who were very bright, also seemed to appreciate his displays of mathematical competence. He used the beginnings of students’ oral answers as the spark for his own elaborations of the underlying mathematics. He never asked students to respond more fully, but rather attempted to anticipate the direction in which their thinking was headed. There seemed to be an unwritten contract between him and his students that he was the main mathematical resource of the classroom. A rather different pedagogical contract was at play in T1’s class. T1, who was highly respectful of his students, not only encouraged the expression of their mathematical thinking but also asked them for further explanation and justification. During the whole-class discussions, he often assumed what we came to call his discussion posture – sitting on the edge of one of the empty student-desks at a front corner of the classroom, thereby indicating to the class that it was now time for some serious collective thinking and sharing of ideas. He intended that students be pushed mathematically and had confidence that they could rise to the occasion, if encouraged to do so – which they did. The three teachers’ deeply held beliefs, which constituted a manner of interacting with their students, lent a certain consistency to their individual adaptations of our resources. This consistency was also seen when impromptu activity occurred. For example, when faced with the unexpected proofs generated by a few of the students, T1 asked the students to come to the front to explain their thinking to the rest of the class; these exposés were then followed by classroom discussion of the central ideas of the proofs. In contrast, T2 when similarly faced with an unexpected proof idea from a student tried to interpret it on his own and illustrate it himself at the board. T3, as per his intentional goals vis-à-vis his students, decided not to embark at all on the proving task. Thus, even if the unexpected led to different ways of handling the situation, each teacher acted on the spur of the moment in ways that were consistent with his own convictions and ways of interacting with his class. This finding makes contact with Remillard’s (Chapter 6) observation that teachers’ modes of engagement with resources are shaped by their expectations, beliefs, and routines, thereby bestowing a degree of stability on these modes. Sensevy (Chapter 3) makes a similar point with regard to the enactment of the threefold process of teaching practice by which documents, prior intentions, and intentions in action are intimately linked together. Also related to this discussion are the findings reported

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by Drijvers (Chapter 14), who describes the unfolding of teachers’ intentions with respect to their classroom use of computer-based resources in terms of didactical configuration, exploitation mode, and didactical performance. Lagrange and Monaghan (2010) have argued that inconsistency characterizes the practice of teachers in dealing with the complexity of classroom use of technology. While we would agree that the presence of the CAS technology within our researcher-designed resources led to more unplanned and impromptu activity than might otherwise be the case in a mathematics class, we would have to disagree with the substance of their claim. We argue instead that the manner in which individual teachers engaged in this impromptu activity was indeed consistent. An example involves the unexpected complete of x10 − 1 by the CAS with its unan  5 factorization ticipated factorization of the x + 1 factor. This led to on-the-fly decision-making on the part of both T1 and T2: for T2, it was to gain control of the mathematical situation by having himself present to the class the factorization of this ‘new’ class of expressions; for T1, it was to give the student who was provoked into thinking about a new factorization rule for xn + 1 the time to express his new conjecture and the examples that were supporting it. Just as with the proof example above, T1 and T2 each handled differently the impromptu foray occasioned by unexpected results with the technology; nevertheless, their approaches were clearly consistent with their individual deep-seated beliefs and habitual manner of interacting with their students.

10.6 Concluding Remarks In closing, our findings regarding the various ways in which teachers adapted the researcher-designed resources cast light on a particular aspect of the theoretical frame of the documentational approach of didactics, namely the differential role that the same resources can play within the dialectical process of documentational genesis whereby resources occasion the shaping of and are shaped by individual teaching practice. The implicit and explicit aspects of the researcherdesigned resources served as both affordances and constraints that influenced teachers’ activity. Resources are not neutral; they speak to different teachers in different ways – even to teachers using the same resources and sharing the same goal of participating in a research project aimed at developing the technical and theoretical knowledge of algebra students within a CAS-supported environment. The teachers brought into the study their own beliefs, knowledge, and customary ways of interacting with their students. Quite clearly this had an impact for each class on the nature of the mathematical activity engaged in. The different ways in which the same resources were shaped were by no means irrelevant or insignificant in nature; they either promoted or impeded the emergence of different techniques and theoretical-conceptual elements in students. But that is a whole other story. Acknowledgments We express appreciation to A. Boileau, to the participating teachers, and to those who, with C. Kieran and D. Tanguay, collaborated in designing the task-sequences:

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A. Boileau, F. Hitt, J. Guzmán, and L. Saldanha. We also acknowledge the support of the Social Sciences and Humanities Research Council of Canada (Grant #410-2007-1485), the Fonds québécois de recherche sur la société et la culture (FQRSC, Grant #2007-NP-116155), and the PROMEP/103.5/10/5364, México (2010). We thank the editors and reviewers for their helpful feedback on an earlier version of this chapter.

References Adler, J. (2000). Conceptualising resources as a theme for teacher education. Journal of Mathematics Teacher Education, 3, 205–224. Ainley, J., & Pratt, D. (2005). The significance of task design in mathematics education. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of 29th PME Conference (Vol. 1, pp. 93–122). Melbourne, Australia: PME. Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7, 245–274. Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht, The Netherlands: Kluwer. Chavez, O. L. (2003). From the textbook to the enacted curriculum. Unpublished doctoral dissertation, University of Missouri, Columbia, MO. Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathématiques, 19, 221–266. Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75, 213–234. Eisenmann, T., & Even, R. (2011). Enacted types of algebraic activity in different classes taught by the same teacher. International Journal of Science and Mathematics Education, 9, 867–891. Freeman, D. J., & Porter, A. C. (1989). Do textbooks dictate the content of mathematics instruction in elementary schools? American Educational Research Journal, 26, 403–421. Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71, 199–218. Helgesson, G. (2002). What is implicit? Crítica, 34, 33–54. Hershkowitz, R., Dreyfus, T., Ben-Zvi, D., Friedlander, A., Hadas, N., Resnick, T., et al. (2002). Mathematics curriculum development for computerized environments. In L. English (Ed.), Handbook of international research in mathematics education (pp. 657–694). Mahwah, NJ: Erlbaum. Kieran, C., Drijvers, P., with Boileau, A., Hitt, F., Tanguay, D., Saldanha, L., et al. (2006). The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection. International Journal of Computers for Mathematical Learning, 11, 205–263. Kieran, C., & Guzmán, J. (2010). Role of task and technology in provoking teacher change. In R. Leikin & R. Zazkis (Eds.), Learning through teaching mathematics (pp. 127–152). New York: Springer. Lagrange, J.-B. (2003). Learning techniques and concepts using CAS: A practical and theoretical reflection. In J. T. Fey (Ed.), Computer algebra systems in secondary school mathematics education (pp. 269–283). Reston, VA: National Council of Teachers of Mathematics. Lagrange, J.-B., & Monaghan, J. (2010). On the adoption of a model to interpret teachers’ use of technology in mathematics lessons. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the sixth congress of the European society for research in mathematics education, January 28–February 1, 2009, Lyon (Working Group 9, pp. 1605–1614). Lyon, France: Institut National de Recherche Pédagogique. Available on line http://www.inrp.fr/ editions/editions-electroniques/cerme6/ Manouchehri, A., & Goodman, T. (1998). Mathematics curriculum reform and teachers. Journal of Educational Research, 92, 27–41.

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Mounier, G., & Aldon, G. (1996). A problem story: Factorisations of xn − 1. International DERIVE Journal, 3, 51–61. Otte, M. (1986). What is a text? In B. Christiansen, A. G. Howson, & M. Otte (Eds.), Perspectives on mathematics education (pp. 173–203). Dordrecht, The Netherlands: Reidel. Rabardel, P. (1995). Les hommes et les technologies. Paris: Armand Colin. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75, 211–246. Robert, A., & Rogalski, J. (2005). A cross-analysis of the mathematics teacher’s activity. Educational Studies in Mathematics, 59, 269–298. Saxe, G. B. (1991). Culture and cognitive development. Hillsdale, NJ: Erlbaum. Schoenfeld, A. H. (1998). Toward a theory of teaching-in-context. Issues in Education, 4(1), 1–94. Sensevy, G., Schubauer-Leoni, M.-L., Mercier, A., Ligozat, F., & Perrot, G. (2005). An attempt to model the teacher’s action in the mathematics class. Educational Studies in Mathematics, 59, 153–181. Trouche, L. (2004). Managing complexity of human/machine interactions in computerized learning environments. International Journal of Computers for Mathematical Learning, 9, 281–307.

Chapter 11

Classroom Video Data and Resources for Teaching: Some Thoughts on Teacher Education Dominique Forest and Alain Mercier

11.1 Introduction This chapter examines teacher documentation with a focus on the classroom. It has a double objective. Firstly, it aims to show the complexity of the ways in which “test teachers”, involved in a research team, use material and symbolic elements as resources when teaching, and to clarify their use of such resources. Secondly, it investigates the use of videos as resources, for researchers, and finally for teacher education/educators. We analyse the videos with a focus on how teachers can use students’ written work1 as a shared documentation, shaping this as resources for a collective study in classroom, as the instructional sequence proceeds. This use of students’ writings seems to belong to a long tradition if we consider Proust’s analysis of CBS 1215 tablet, in Chapter 9 of this book. Classroom videos are the central kind of data in our study. Large studies (TIMMS, 1999) and related comparative studies (e.g. Andrews, 2009) have provided standard descriptions of teachers’ actions. Our study (or the part of our study corresponding to the first aim described above) complements such works. It focuses indeed, not on ordinary classes, but on experimental classes, where didactical engineering (Artigue, 1989) has been set-up. The teaching we study has thus very specific features. It takes place within an experimental school, the Centre for Observation and Research on Mathematics

1

In the case of this instructional sequence, a first moment (situation d’action) is followed by a moment for communication: students who tried to describe the thickness of sheets of paper use their description for ordering the same paper. The “written work” at stake here in the sequence is composed by the written orders. D. Forest (B) IUFM de Bretagne, Université de Bretagne Occidentale, Rennes Cedex, France e-mail: [email protected]

G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_11,  C Springer Science+Business Media B.V. 2012

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Teaching (CORMT)2 (described in Section 11.2). It has been designed by a team of researchers and teachers working together (Schoenfeld, 1998; Chapter 10). The choices retained for the design of this teaching refer to Brousseau’s theory of didactical situations (Brousseau, 1997). They rest on a thorough didactical analysis of the knowledge at stake; moreover, they correspond to a didactical contract leaving an important responsibility to the students. In the light of more recent studies, it can be considered as inquiry-based teaching (NCTM, 1989). The teaching at CORMT has proven especially beneficial3 in terms of students’ attainments in mathematics (Ratsimba-Rajohn, 1992). School results from 50 students a year were collected from end-of-year standardised test: assessment of scholastic achievement tests of school achievement, and some more: analyses are available from Brousseau (1997, pp. 191–195). In terms of inquiry-based teaching it can be considered as exemplary. This is why we consider it essential to study the actions of CORMT teachers and to analyse their features. It is a preliminary step towards developing a teacher training programme that draws on video data gathered in the classroom. Brousseau, looking at knowledge as a result of pupils’ action within a didactic situation (Brousseau, 1997) did not intend to describe the teacher’s work. But since the whole process was videotaped, it was possible to reuse the data and develop new interpretations of the action of both the teacher and the pupils, according to joint action theoretical framework (Sensevy & Mercier, 2007; Sensevy, SchubauerLeoni, Mercier, Ligozat, & Perrot, 2005; Chapter 3 in this book). Our research proposes a framework for a clinical observation of a classroom activity, grounded in a modelling of didactic systems. The theoretical frame of Joint Didactic Action considers the teacher and student joint actions as a specific social game, a didactic game in which the teacher “wins” if and only if the student “wins” (learns the knowledge at stake). But the success of this game requires that the student produces the expected behaviour by his or her own cognitive movement. This requirement means, among others, that the teacher has to avoid mentioning explicitly the knowledge at stake: we call this behaviour didactical reticence. This requirement also means that any task in the classroom addresses a piece of knowledge. Consequently, from the teacher’s point of view, the game is a secondarised game: The teacher manages the pupil’s game, introducing symbolic elements with which they build some relationship. We use the term of milieu to refer to the set of meanings that stems from the surroundings of action, from a symbolic or a concrete viewpoint.

2 CORMT was the research team in Jules Michelet School, 33400 Talence (France). Brousseau worked in the school for many years (1972–1999) and we got more than 400 video records of CORMT lessons, from 1982 to 1999, as numerised archives in the VISA project (IFE and ENS Lyon). 3 Each year pupils were tested on the basis of standard tests for school achievement evaluation (SAT). Pupils taught by CORMT school teachers scored on average higher (or equal) to those of neighbourhood schools (Brousseau, 1980; Brousseau & Brousseau,1987).

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In this paper, we first analyse a selected part of a lesson by considering language, body and space “fittings” as means of support for pupils’ joint attention. From the video data, we connect non-verbal to linguistic interactions with a specific methodology, on the basis of the proxemics (Hall, 1966), which we detail below. We specify proxemic and linguistic aspects of interactions with content knowledge in use, building up a storyline from photograms and commentaries (see Section 11.2). We examine how such videos, of experimental teaching grounded in didactical engineering, can make didactical phenomena more visible. This point leads us to consider in a third part the use of such video data for pre-service teacher education, and furthermore for in-service professional development.

11.2 Building a Teaching Resource from the Pupils’ Work We analyse a classroom video from the CORMT collection. This Centre for Observation and Research on Mathematics Teaching was created by the French researcher Guy Brousseau in 1972. It was a primary school with facilities to welcome research and observation of classroom situations proposed by the researcher. These situations were designed and constructed, using the theory of didactical situations (Brousseau, 1997). This practice of didactical engineering took into account the necessity of a close collaboration between researchers and school teachers that we can call “test teachers”, who set up the mathematical situations. We chose a lesson that was engineered by Brousseau (1980) for a fifth grade classroom (pupils about 11 years old). It is organised as a 1-year teaching sequence composed by 15 chapters. Four main problems are chronologically revisited in class: (1) to construct the set of rational numbers as measures; (2) to construct the decimal numbers as a subset of fractions and show that they can separate any two rational numbers; (3) to look at problems using these numbers as linear operators from a measure space to another one; and (4) to explore how operating on operators work. The lesson we chose for this study is emblematic because no teacher in ordinary schools would be able to allow pupils to build up the set of rational numbers from a “sense-making situation”. That makes this lesson particularly fruitful for researchers who want to describe teachers’ work. Nevertheless, we do not look at teachers’ and researchers’ use of documentation and documenting teachers’ action, which is didactical engineering in Brousseau’s theorisation. The features of these situations allow students to produce written work that the teachers can use as a document for collective study in classroom as instruction proceeds (see Table 11.1). We analyse creativity of the teacher when using these writings during the process, realizing “prior intentions” in “local intentions” as modelled by Sensevy in Chapter 3. In this lesson, pupils had to describe and distinguish various sheets of paper by their thickness. Their description was construed as valid if it allowed them to

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recognise a pile of paper sheets among five different piles. But the pupils were not able to distinguish the piles at a glance to one sheet, and they could not measure directly the thickness of one sheet with a calliper.4 In this situation, integers were not efficient and pupils needed to find a new code. Some of them proposed “A” the thickest, “E” the thinnest, but this code made it difficult to differentiate more than three thicknesses. Most groups of pupils proposed to measure several sheets together to get a measurable thickness and suggested a code of the following form: (23 sheets; 3 mm). After the pupils worked by themselves to find an answer, and after the teacher asked one of each working groups to present their work, we will see here how a teaching resource was created by the joint action of the teacher and the pupils. These two class sessions led to the production of messages by pupils. For the beginning of the third session, these messages were then communicated to the whole group and organized by the teacher in a table like this: Table 11.1 Messages of pupils are organized in a table which is drawn on the blackboard A Team 1

Twice as thick as C, twice as thick as D

Team 2

1 mm = 9 sheets

Team 3

Team 4 Team 5

20 sheets = 2 mm

B

C

D

E 5 sheets = 1/2 mm 30 sheets = 3 mm 25 sheets = two and a half mm

1 mm = 3 sheets 16 sheets = 1 mm

2 mm = 6 sheets 7 sheets = 1.5 mm 14 sheets = 3 mm

11 sheets = 1/2 mm

10 sheets = 2 mm

27 sheets = two and a half mm, almost 3

To analyse both verbal and non-verbal aspects of the teacher’s action, appropriate theoretical frameworks are needed. We refer here to the framework of proxemics (Hall, 1963, 1966) applied to didactical situations as developed by Forest (2006, 2009). We describe non-verbal phenomena with the notion of distance, in a selfcentred and perceptive way. In addition to the metric distance, we take into account 4 The calliper is a rubber one, the ruler of which is graduated in millimetres. This makes the direct measure of any sub-millimetric thickness impossible.

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eye-contact, mutual gaze, positions and postures, and we include phenomena such as touch, or pitch of the voice. We use this type of distance to describe fittings of teachers, students, and other elements with a principle that can be summarised as follows: what is close (for me) is more important than what is remote. This approach allows us to report and analyse the proxemic comments and allows the identification of teaching/learning phenomena related to the dynamics of the milieu.

T (teacher): I’ve put on the blackboard all the messages that you have written, right?

After completing the table the teacher moves away from the blackboard and stands aside while looking at pupils. This displacement is accompanied by a gesture with both her hands. She moves her hands in front of her as we can see on the picture. From the perspective of a natural semantic of action (Sensevy, 2001), this gesture can be interpreted as: “I haven’t done anything”, or in a weaker version as: “As a teacher, I’ve done my share of the work”. The teacher then turns her back to the blackboard and that way, reinforces the second interpretation when she asserts: “I’ve put on the blackboard all the messages that were written” (implied “by all of you”). The pupils’ agreement she is asking for is purely formal and aims to remind their previous relation to the objects of the milieu. We show this by a photogram (see below). The teacher’s corporal behaviour is described in a four dimensional code: the direction of the shoulders line, the pointing of the hands, the direction in which she looks and the spatial positioning in the classroom. We cut the flow of teacher’s action on the video as one shot for each global proxemic state. Then we give a complete transcription of the flow of language action, and associate each utterance to the related shot. Sequences of photograms Bn are then the data for our analysis of relevant didactical episodes. The choice is guided by both the epistemological importance of didactic phenomena (here, the collective creation of a mathematical notation), and the density of proxemic behaviour of the teacher in the construction of the milieu.

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B1

B2

B3

B4

T: So I, I have put it in

P (pupil): Well, it was 27, T: OK, so actually, it

T: But here we had met

yellow here//come on,

T: Uh, no, we had

was compatible//it was

some problems, and

Akim, come on// have

discussed it, and it was

almost true, we could

Vanessa, you had made

put in yellow what we

Cristobal who came to

accept that message.

a remark here.

talked about on

show that they had

Tuesday.

found 5 sheets, 1/2 mm

The teacher returns to

She quickly shows the

She continues to look

With her hands, she

the blackboard, moves

E5, then returns to E1,

at the table, indicating

maintains the indication

aside and shows the E1

remaining to one side

alternately E5 and E1.

on E5, and looks at the

cell (see Table 11.1

on the left.

above).

pupils, while standing to one side.

When completely moving aside, the teacher leaves the pupils in front of the resource but she does not leave them alone. She organises the discussion, indicates what they should look at in the table (photogram B1) and brings them to focus on a particular issue (photogram B2 then B3). Nevertheless, she does not specify this issue herself: “we could have accepted that message but we had met some problems, and Vanessa you made a remark here” (photogram B4). Then the teacher moves away from the blackboard area and walks between desks, so that she’s close to pupils. After re-emphasizing the differences between the two types of messages, she invites Vanessa to speak:

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“Vanessa: there are 25 sheets and 27 sheets, there are two sheets as deviation. And we had said that it could. . . we had found “25 sheets 2 mm 1/2”, then 27 2 mm 1/2, almost 3. And so it proves that it is. . . we have not finished, need a calliper, uh, a more sophisticated calliper, with more. . . um T: with more. . .? P: strokes. . . T: Yes Peter? More details? But which details would you have needed? P: the half-millimetre T: What? P: half-millimetre T: half-millimetre

By her spatial positioning among the pupils, the teacher indicates that she let them a lot of space in didactical joint action. According to Vanessa, the difficulty here for pupils lies in the precision of the measuring instrument (“a calliper, uh, a more sophisticated calliper”). The teacher is now close to Vanessa (referring to the blackboard). That encourages Vanessa to take more responsibility and allows the emergence of the question of the accuracy of the measuring instrument. That will be resolved by other means: the rising number of sheets. Instead of contradicting Vanessa’s assertion, the teacher accepts it, makes it clearer and asks the pupils to detail information that would be useful to solve the problem: “halfmillimetre” (the graduations of the calliper). This statement sets off controversy about the accuracy of the tool. Pupils can take part in the controversy, with the help of the table on the blackboard. To help pupils to realize that the table could be a possible resource, the teacher moves to the blackboard. B10 T: But here, would half-millimetres be helpful? P: Yes T: To write this message ... P: Because after, there are half-half millimetres, and we can't ... T: We should ... P: That would still make small strokes T: And if it is not enough? P: Still more strokes, and ... T: And then we will never, we'll never cope with it P: We’ll finish with an all black strip T: Well, we'll end up as you say with a black stripe, and we won't be able to read anything (the teacher goes back among the pupils). So we've said we'll have to find another solution.

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During the discussion, the teacher stands near the blackboard while leaving a distance between her and the part of the table she is currently pointing to (27 sheets of paper = two and a half mm almost 3, message previously produced by the pupils). This pointing gesture along with her spatial positioning, her posture and her eyecontact allows both the teacher to be present at the heart of the debate and the pupils to bear the major responsibility. The teacher does not decide whether “graduations” should be used, but instead, she encourages the pupils to debate on how to practically designate each sheet of paper: “but here, would half-millimetres be helpful?”[. . .] “to write this message . . .” Finally, a pupil provides the refutation: “we’ll finish with an all black strip”. Once this proposal has been discussed, the teacher now joins the pupils again. Doing this, she leaves the blackboard area as a free space.

B11

B12

B13

B14

T: Anyway we, on our

T: So therefore, uh, we

T: So, could we not do

T: Have some of you

callipers, we have only

stayed on that problem,

it differently? Some

who have found it

millimetres.

saying half a millimetre,

teams have perhaps

difficult ... did it fall on

met those difficulties…

halves, or almost

with the half.

halves ...

P: I’ve found how to do it. and then another half of P: we’ve found it!

the half, and we’ll never succeed, we must find another solution.

After staying for a while

The teacher indicates

The teacher goes back

The teacher turns her

among the students,

the cell E5 while

among students,

back to the blackboard,

the teacher goes back

looking at the pupils.

looking alternately at

and moves to the back

them and at the table

of the classroom.

to the blackboard.

on the blackboard.

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On the photogram B11, we see the teacher going back to the blackboard. Her movement is accompanied by a statement that refers to the material constraints of the milieu: “Anyway we, on our callipers, we have only millimetres”. The use of the terms “we” and “our” underline the proximity of the teacher and the pupils. We note that the teacher, despite the demands of some students (“we’ve found it!”), takes time to “institutionalise” – to establish as an official reference, shared by the whole class (Brousseau, 1997) – the rejection of sub-graduations, ensuring that the need to find another solution has been accepted by all the students. Moving to the blackboard (B12) is also an opportunity to engage students in seeking further solutions. The photogram B13 shows the teacher’s return to the pupil area. Her movements, along with her successive eye-contacts with the pupils and her gazes on the table on the board, continue the process of creating and maintaining proximity. The teacher does not say: “my” “your” or “their” problem but “our” problem because everybody has worked on it. “Some teams have perhaps met these problems. . . with the half”. The teacher’s spatial positioning close to the students, with her focus on the table (on the board) enable her to remind pupils of the presence of the table where all the notations produced are captured. Even though the teacher makes clear that the table is available and could be useful, the pupils remain responsible for relying on the relevant information. This process is supported by the movement of the teacher to the back of the class (B14), now turning her back to the blackboard. These teacher’s comings and goings are accompanied by many gestures and hand movements that support her activity. The final movement of the teacher leaves room for the pupils to act.

B15

B16

P: I!

P: We are team 2

T: You, yes? So, what team do you belong to?

T: You are team 2. Ah, so how have you solved the problem?

The teacher keeps on fading back of the

She stops at the last rank, next to the pupil who is

classroom.

speaking, while looking ostensibly at the table.

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The teacher implements here a real didactical reticence: she helps the pupils in the process of finding the thickness but the progress of the action remains under their responsibility. Finally the teacher adopts a position of a pupil looking at the table as a collective resource, the main object of attention (B16). At the same time, her proximity with the pupil who is currently speaking allows the teacher to encourage her in producing a proposal (including a slight touch of the student in photogram B15). The support of the pupil’s action is better understood if, like the pupils, we have the table in front of us and in particular the boxes with the messages of the team 2 who has solved the problem (Table 11.1). Supported by the teacher, the pupil explains the approach of her group, which will be commented on later in a collective discussion. Then the teacher will institutionalise the notations invented by the pupils as a mathematical object before starting to fill the empty boxes on the board. From now that point on, these notations are available to all.

11.3 Video Data: A Tool that Makes Didactical Phenomena More Visible In this lesson, we have considered pupils’ actions when measuring the thickness of a paper sheet and coding their results (the action situation according to Brousseau, 1997), in addition to the actions on the various representations pupils have worked with and developed (the formulation-situation). We have linked the movement from the action-situation to the formulation-situation with the teacher’s technique to initiate and manage this movement. We have shown then how the teacher does this by mobilising an artifact5 as a resource for her action (Brousseau, 1997, pp. 195– 202): a two-way table of pupils’ coding results (groups) × (types of paper), on the blackboard (Mercier, Rouchier, & Lemoyne, 2001). She directs pupils’ attention to the table producing in that way a public account of the pupils’ previous actions. This artifact supports a new and unique collective memory about multiple personal actions (Flückiger & Mercier, 2002; Matheron & Salin, 2002). Pupils’ representations, recorded on the blackboard, become mathematical codes and make sense for all: this allows pupils to remember their own action as related to their and others’ previous actions. From now on the teacher can say “What did we do yesterday, and what did you do personally?” and she produces a new situation where mathematisation is the focus. In this lesson the pupils’ actions change, from measurement to proportional reasoning, through the notations they produce. This process supposes a communication 5 We share Tomasello’s position (1999): “The evidence that human beings do indeed have speciesunique modes of cultural transmission is overwhelming. Most importantly, the cultural traditions and artifacts of human beings accumulate modifications over time in a way that those of other animal species do not” (pp. 4–5), including in these artefacts “tool industries, symbolic communication, and social institutions”, this process requiring “faithful social transmission that can work as a ratchet to prevent slippage backward” (p. 5).

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game (in which notations are tested through the information they provide) and a validation game (when notations are verified thanks to theory, here, proportionality) in course of which the pupils come to see their notations as mathematical notations for measures: the first step towards rational numbers. To support this didactical movement, the teacher must be confident that acting in this way is the “right way” of teaching mathematics. Not only because this is an inquiry-based method, but also because this method requests the related didactical movement (Brousseau, 1997). This reflects Brousseau’s epistemological point of view on mathematics: numbers are symbolic systems that give a complete report for measurement operations (Lebesgue, 1935/1975). Every day the “test teacher” in Brousseau’s CORMT asks herself about the pedagogical content knowledge that is at stake for her lesson: “What is the future of this symbolic code that the pupils have proposed as an account for their action?” and “From now, using this code, what could they calculate?”, “What reasoning is then made possible?” She has now to find a way of helping pupils in their work with the codes: testing their usefulness and validating their theoretical consistency. Those two dimensions of the pupils’ action rely on the teacher’s ability to organise and regulate specific situations (Brousseau, ibidem). For those reasons the “teacher’s game” is a very difficult game to play. Playing this game is a subtle action, out of reach for an inexperienced teacher. However, teaching in an inquiry-based way requires such choices. A written documentation is available from CORMT archives but no one could use it for teaching again the CORMT lessons: the documents don’t give enough information about the very decisions a teacher must make (Forget & Schubauer-Leoni, 2008; Schubauer, Leutenegger, Ligozat, Flückiger, & ThevenazChristens, 2010). It leads us to raise the question of the possible use of CORMT videos for the training of novice teachers. We claim that it is much more useful to show to novice teachers some serialised sequences of test teachers’ actions than to give them a lecture upon teaching gestures and their effectiveness (though we will not test this hypothesis here and leave it for future work). In particular, studies of such videos could be organised within collective groups of teachers and researchers (Mercier et al., 2001). Such an approach seems to be fruitful: therefore, we argue for constituting large databases of classroom video, as we began to do it in the frame of the Vidéos de Situations d’Apprentissage (ViSA, videos of learning situations)6 project.

11.4 From Video Data to a Tool for Teachers’ Training Shulman (1992) underlined the relevance of using cases in teacher education. Video has been used extensively to support teacher learning, in a variety of ways (Sherin, 2004), and advances in video technology make it a powerful instructional tool that

6 The provision ViSA is supported by the IFE and ENS Lyon, in the VISA project (http://visa.inrp. fr).

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provides a wealth and variety of cases, and allows users to reflect and analyse teaching and learning process. But we have to take into account that a key component of teaching expertise seems to be the ability to notice and to interpret what happens in one’s classroom (Sherin & Van Es, 2005). Expert teachers make special choices about where to direct their attention. This “noticing” capacity detailed by Sherin is likely to be important if we want to implement inquiry-based lessons like Brousseau’s lessons. Training teachers, with the help of video, to set up such inquiry-based lessons is a challenge for teacher educators, in particular for the following reasons: (1) Analysing teachers’ choices (made visible with the help of video) requires investigating their Pedagogical Content Knowledge (Ma, 1999; Shulman, 1986). This is a delicate matter, which could be approached in training via collective work, as in lesson studies (Chapter 15). (2) Teachers’ proxemic behavior and pragmatic use of language cannot be taught by lectures, because non-verbal phenomena are fundamentally different from verbal phenomena with which they are intertwined. This difference has been theorized by Bateson (1972) through the analogic-digital distinction.7 Therefore we have to use a specific way to make the non-verbal phenomena visible and accessible (Forest, 2009; Wilder, 1998). (3) Teachers’ proxemic behavior and pragmatic use of language are teaching techniques8 but are also individual and personal properties. As Remillard (Chapter 6) recalls, there are many institutional attempts to modify practices by providing teachers with written materials. Such an idea can be grounded in Shannon (1948) or Jakobson (1963) models, which suggest that communication could be viewed as a transmission of information. Video data demonstrates that such models cannot account for what happens in classrooms. A written description of this engineering is not enough for teachers to set up such a lesson. The dynamics of the didactical action cannot be fully reported.

7 Analogic-digital distinction is characterized by Bateson as follows: in verbal language, characterized as “almost (but non-quite) purely digital”, “the signs themselves have no simple connection (e.g. correspondence or magnitude) with that they stand for”. Verbal language is composed of discrete elements and “the name usually has only a purely conventional or arbitrary connection with the class named”: the word “big”, said Bateson, is not bigger than the word “little” (pp. 372–373). Non-verbal communication, however, is said “analogic”: magnitudes that are used correspond to real magnitudes in the subject of discourse. Analogic communication is a continuous process, where what is represented and the “representative” are in a ratio of magnitude, possibly contradictory. “in kinesic and paralinguistic communication”, Bateson said, “the magnitude of the gesture, the loudness of voice, the length of the pause, the tension of the muscle, and so forth, commonly correspond (directly or inversely) to magnitudes in the relationship that is the subject of discourse” (p. 374). 8 We talk about “techniques”, in the meaning introduced by Mauss (1935/1973), including its incorporated and embodied aspects.

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Like mathematics in the CORMT example, all activities which aim at understanding the world, are made possible due to the production of a system of representations which serves as a framework for the work of de-psychologisation and rationalisation of our primitive knowledge (Bachelard, 1965). Studying the teacher’s action requires to build such a distance with the observed reality; it cannot rely only on the natural language. Displaying useful and relevant elements to study teacher’s action (natural and technical language, video shots, photograms, etc.) is a way to build this distance, which is needed to provide access to the phenomena, and progress in the understanding of the teacher’s action. Video shots can show this movement, from the pupils’ action in the world of objects (playing a real game) to their actions in a world of symbolic objects that allow them to understand the world of objects (playing with symbols to master the rules of the real game, if not the game). And videos can show the teacher’s choices in accompanying pupils in this movement related to “umbilical questions” such as the one we choose as emblematic for this issue: “How can we measure objects which are much smaller than a measuring unit?”, for which an universal answer is such an object as “a rational number”. Every didactical exercise constitutes an interpretation of the knowledge to be studied. If this interpretation is worth being studied, the video can provide the researchers with the material, physical and language elements that are part of each joint didactical action. This may produce, thanks to a collective work with teachers, formal systems that can guide the action (Fleck, 1935/1979; Chapter 3). A video documentary can show objects and their functioning, human gestures and enunciations. But the rhythm of the lesson yields an empathetic movement, which hinders the scientific observation. The use of photograms permits to overcome this difficulty. It allows us to consider more systematically what happens in the classroom; this use of photograms is a way of “creating distance”. It seems that the analyses of recordings need to be conceived in a renewed paradigm (Bateson, op. cit.; Winkin, 2000). With the discretisation of data and the resulting photoshots it becomes possible to make particular phenomena visible and to connect semiotic constructions with the underlying didactic intentions (Chapter 3).

11.5 Conclusion: Video as a Resource for Teacher’s Training? Our work illustrates how physical processes and language phenomena participate in the dynamics of the milieu, in which the teacher’s actions rely on a very elaborated epistemology visible through the preparation of the lesson (Brousseau & Brousseau, 1987) and also in the process of teaching. But there is a huge gap between the way mathematics is typically taught in French schools and the teaching on the CORMT videos. If those data provide us with the opportunity to revisit inquiry-based teaching as Brousseau imagined it, we claim that a simple viewing of classroom videos does not allow the teachers to fully and accurately elucidate the teacher’s and pupils’ didactical joint actions. It needs a more sophisticated approach; in the following, we start to explain what such an approach could be.

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Firstly, we need to create distance while watching body gestures and positioning. Furthermore, we need a system of description which allows us “to recognize the hallmarks of a didactical phenomenon” (Leutenegger, 2000, p. 245, our translation), a semiology of didactical facts and acts such as clinicians invented for medicine, during the XIXth century (Foucault, 1973). Moreover, the semiology that we need must take into consideration both the teachers’ knowledge and the pupils’ knowledge, and their evolution. Our multimodal analysis with proxemic, verbal transcript and photograms attempts to build such a system. Secondly, conditions and constraints in ordinary schools are not the same as in the CORMT school. In addition, there is a considerable time gap (30 years). The use of video for teachers’ professional training and development requires an extensive study of didactic systems within the ordinary classrooms. Therefore, we are currently trying to experimentally bring about the phenomena observed in the CORMT school, and to produce video shots for mathematics teachers’ professional training based on these experiments. In various French schools we are following Brousseau’s method and we videotape teachers setting up collaborative interactions with the pupils, and using the representations they produce, constituting them into resources for the learning processes. Thirdly, as Lave and Wenger (1991) argue, “a community of practice is an intrinsic condition for the existence of knowledge” (p. 98). Drawing on “didactical engineered” lessons like those of Brousseau, our videos could be used in collective settings, similar to “lesson studies” (Chapter 15), and with the support of teacher educators and/or researcher (Chapter 17). Practices could be on the basis of the experience of “teacher’s video-club”, as reported by Van Es and Sherin (2009). With this analytical and practical devices, teachers produce, watch and discuss excerpts of videos from their classrooms. In summary, we claim that the production and use of video documents is useful for teachers education, in particular when used in connection with Brousseau’s theoretical framework. This is much more ambitious than producing the photograms of a single episode. However, designing such a resource needs to be thought through carefully. The aim is neither to lead teachers to reproduce what they see on the screen, by mere imitation, nor to introduce theoretical categories. The most promising training device could consist of teachers collectively studying the same situations, which in turn may lead them to re-produce “didactical engineering” of those situations, with the support of researchers. Acknowledgements The authors would like to thank Serge Quilio for providing this video from CORMT and for his suggestions about analysis, Tracy Bloor and Jana Visnovska for helping us to revise the English language in this chapter.

References Andrews, P. (2009). Comparative studies of mathematics teachers’ observable learning objectives: Validating low inference codes. Educational Studies in Mathematics, 71(2), 97–122. Artigue, M. (1989). Ingénierie didactique. Recherches en Didactique des Mathématiques, 9(3), 281–308.

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Bachelard, G. (1965). L’activité rationaliste de la physique contemporaine. Paris: Presses Universitaires de France. Bateson, G. (1972). Steps to an ecology of mind: Collected essays in anthropology, psychiatry, evolution, and epistemology. Chicago: University of Chicago Press. Brousseau, G. (1980). Problèmes de l’enseignement des décimaux » (1ère partie). Recherches en Didactique des Mathématiques, 1(1), 11–58. Brousseau, G. (1997). Theory of didactical situations in Mathematics. Dordrecht, The Netherlands: Kluwer. Brousseau, G., & Brousseau, N. (1987). Rationnels et décimaux dans la scolarité obligatoire. Bordeaux, France: IREM université Bordeaux 1. Fleck, L. (1935/1979). The genesis and development of a scientific fact. Chicago: University of Chicago Press. Flückiger, A., & Mercier, A. (2002). Le rôle d’une mémoire didactique des élèves, sa gestion par le professeur. Revue Française de Pédagogie, 141, 27–37. Forest, D. (2006). Analyse proxémique d’interactions didactiques. Carrefour de l’Education, 21, 73–94. Forest, D. (2009). Agencements didactiques, pour une analyse fonctionnelle du comportement non-verbal du professeur. Revue française de pédagogie, 165, 77–89. Forget, A., & Schubauer-Leoni, M. L. (2008). Inventer un code de désignation d’objets au début de la forme scolaire. Des productions personnelles à la convention collective. In L. Filliettaz & M. L. Schubauer-Leoni (Eds.), Processus interactionnels et situations éducatives (pp. 183–204). Coll. Raisons Educatives. Paris, Bruxelles: De Boeck Université. Foucault, M. (1973). The birth of the clinic: An archeology of medical perception (A. M. SheridanSmith, Trans.). London: Tavistock. Hall, E. T. (1963). A system for a notation of proxemic behavior. American Anthropologist, 65, 1003–1026. Hall, E. T. (1966). The hidden dimension. New York: Doubleday. Jakobson, R. (1963). Essai de linguistique générale. Paris, France: Minuit. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: University of Cambridge Press. Lebesgue, H. (1935/1975). La mesure des grandeurs. Paris, France: Blanchard. Leutenegger, F. (2000). Construction d’une clinique pour le didactique. Une étude des phénomènes temporels de l’enseignement. Recherches en didactique des mathématiques, 20(2), 209–250. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah: Lawrence Erlbaum Associates. Matheron, Y., & Salin, M.-H., (2002). Les pratiques ostensives comme travail de construction d’une mémoire officielle de la classe dans l’action enseignante. Revue française de pédagogie, 141, 57–66. Mauss, M. (1973). Techniques of the body (B. Brewster, Trans.). Economy and Society, 2(1), 70–88. (Original work published 1935) Mercier, A., Rouchier, A., & Lemoyne, G. (2001). Des outils et techniques d’enseignement aux théories didactiques. In A. Mercier, G. Lemoine & A. Rouchier (Eds.), Le génie didactique. Usages et mésusages des théories de l’enseignement (pp. 233–249). Bruxelles, Belgium: De Boeck. NCTM. (1989). The curriculum and evaluation standards for school mathematics. Reston, VA: Author. Ratsimba-Rajohn, H. (1992). Contribution à l’étude de hiérarchie implicative. Application à l’analyse de la gestion didactique des phénomènes d’ostension et de contradiction. Thesis, Université Rennes I, France. Schoenfeld, A. (2008). On modelling teachers’ in-the-moment decision-making. In A. Schoenfeld (Ed.), A study of teaching: Multiple lenses, multiple views (pp. 45–96). Reston, VA: National Council of Teachers of Mathematics.

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Schubauer-Leoni, M.-L., Leutenegger, F., Ligozat, F., Flückiger, A., & Thevenaz-Christens, Th. (2010). Producing lists of objects to be remembered and communicated. The « treasure game » with 4 and 5 year old children. Fapse Genève University, Translated from French by N. Letzelter & F. Ligozat. Sensevy, G. (2001). Théories de l’action et action du professeur. Raisons Educatives, théories de l’action et éducation (Vol. 4, pp. 203–224). Bruxelles, Belgium: De Boeck Université. Sensevy, G., & Mercier, A. (dir.). (2007). Agir ensemble, l’action didactique conjointe du professeur et des élèves. Rennes, France: PUR. Sensevy, G., Schubauer-Leoni, M.-L., Mercier, A., Ligozat, F., & Perrot, G. (2005). An attempt to model the teacher’s action in the Mathematics class. Educational Studies in Mathematics, 59, 153–181. Shannon, C. E. (1948). A mathematical theory of communication. The Bell System Technical Journal, 27, 379–423. Retrieved on October 20, 2008, from http://cm.bell-labs.com/cm/ms/ what/shannonday/shannon1948.pdf Sherin, M. G. (2004). New perspectives on the role of video in teacher education. In J. Brophy (Ed.), Using video in teacher education (pp. 1–27), NY: Elsevier Science. Sherin, M. G., & van Es, E. A. (2005). Using video to support teachers’ ability to notice classroom interactions. Journal of Technology and Teacher Education, 13(3), 475–491. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. Shulman, L. S. (1992). Toward a pedagogy of cases. In J. H. Shulman (Ed.), Case methods in teacher education (pp. 1–30). New York: Teachers College Press. TIMSS. (1999). Video studies. Retrieved on October 2010, from http://www.lessonlab.com/ TIMMS/index.htm Tomasello, M. (1999). The cultural origins of human cognition. Cambridge, MA: Harvard University Press. Van Es, E. A. & Sherin, M. G. (2009). The influence of video clubs on teachers’ thinking and practice. Journal of Mathematics Teacher Education, 13, 155–176. Wilder, C. (1998). Being analog. In A. Berger (Ed.), The postmodern presence (pp. 239–251). London: Sage. Winkin, Y. (2000). La nouvelle communication. Paris, France: Seuil.

Chapter 12

Interactions of Teachers’ and Students’ Use of Mathematics Textbooks Sebastian Rezat

12.1 Introduction Many studies tackling the question “What kinds of curriculum materials do teachers select and use, and how?” that has been raised in the introduction of this volume point to the importance of mathematics textbooks. The mathematics textbook was and still is considered to be one of the most important resources for teaching and learning mathematics (Valverde, Bianchi, Wolfe, Schmidt, & Houang, 2002). Studies on teachers’ preparation work, and on teachers’ use of textbooks – even though carried out in different countries and focusing on different grade-levels – draw a very coherent picture of how and how often teachers use their mathematics textbooks: In planning activities teachers rely heavily on textbooks (Bromme & Hömberg, 1981; Chávez, 2003; Hopf, 1980) and the mathematical content of the classroom is heavily influenced by the text (Johansson, 2006; Schmidt, Porter, Floden, Freeman, & Schwille, 1987). Mathematics textbooks are used by teachers in two dominant ways, namely as a source for tasks and problems (Pepin & Haggarty, 2001, p. 168), and as a guide for instruction. The latter relates to decisions about what to teach, which instructional approach to follow, and how to present content (Valverde et al., 2002, p. 53). But, where is the student? Although some authors regard students as the main readers of textbooks (Kang & Kilpatrick, 1992; Love & Pimm, 1996), studies on the use of textbooks by students are rare. That mathematics textbooks are directed at learners is already apparent in the mode of address (see Chapter 6) of most textbooks: The voice of mathematics textbook is directed to students, e.g. students are invited to do tasks and activities, mathematical concepts are explained in a way that is appropriate for students (Remillard, 2000; Valverde et al., 2002). Nevertheless, students are usually only considered when textbooks are analysed in terms of opportunities to learn. How students take advantage of these opportunities has only been studied at a rudimentary level. S. Rezat (B) Institut für Didaktik der Mathematik, Justus-Liebig-University Giessen, 35394 Giessen, Germany e-mail: [email protected]

G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_12,  C Springer Science+Business Media B.V. 2012

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Furthermore, the mathematics textbook is a resource that is shared by teachers and students in many countries; and both, teachers and students, actively shape the enacted curriculum or – as Sensevy (Chapter 3) puts it – both teachers and students are active players of the didactic game. In the mid-1950s Cronbach already pointed to the fact that the “text-in-use is a complex social process wherein a book, an institution, and a number of human beings are interlaced beyond the possibility of separation” (Cronbach, 1955, p. 188). Nevertheless, studies on the use of textbooks tend to either put a focus on the teacher or on the student. According to one main aim of this volume – to deepen our understanding of collective aspects of the use of resources – it seems likely to ask for interactions of teachers’ and students’ uses of textbooks. Although this issue has hardly been tackled in empirical research before, the notions of the teacher as a mediator of the text and the use of textbooks pervade the relevant literature (Johansson, 2006; Pepin & Haggarty, 2001). Teachers are regarded as mediators of the text in the way that teachers assist students learning from the textbook by providing expositions of the text and explaining the contents of the text (Love & Pimm, 1996, p. 398). This meaning of teacher mediation implies that the teacher might even act as a mediator of the text when the textbook is not apparent in the classroom. In the analysis of three Swedish teachers organization of their mathematics lessons, Johansson (2006) found that the mathematical content in the classroom is influenced by the textbook to a large extend even when the textbook is not apparent for two of the three teachers. Stodolsky (1989) concludes from her investigation of six fifth grade teachers’ use of mathematics textbooks that “math textbook content tends to place something like a cap on content coverage in classrooms” (p. 176). Attention is also drawn to the teacher as mediator of textbook use: Teachers decide which textbooks to use; when and where the textbook is to be used; which sections of the textbook to use; the sequencing of topics in the textbook; the ways in which pupils engage with the text; the level and type of teacher intervention between pupil and text; and so on. (Pepin & Haggarty, 2001, p. 165)

But, Pepin and Haggarty (2001) also point out that “the ways in which the teacher mediates the text are largely unknown” (p. 166). The notion of interactions of teachers’ and students’ use of textbooks has different facets. In a narrow sense it relates to the question of how teachers’ and students’ uses of the textbook interact with one another. More broadly “interactions” also refers to impacts of the teachers’ use of the textbook on students learning, for example motivational aspects, and to how students’ use of textbooks affects the teacher in a more general sense, for example the planned succession of the lesson. This chapter will elaborate on teachers’ mediation of textbook use and on impacts of students’ use of textbooks on teachers’ plans. The findings presented here build on a study that was initially carried out with a focus on students’ use of textbooks (Rezat, 2009). In the next section the methodology of the study will be introduced. The major findings of the study concerning impacts of teachers’ use of textbooks on students and vice versa will be presented in Sections 12.3 and 12.4. These findings will lead to the proposal of a more comprehensive

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framework for the investigation of the use of textbooks and other resources that are shared by students and teachers in section 12.5.

12.2 Methodology Data on teacher-mediation and interrelating uses of textbooks of teachers and students were gathered in the context of a study with a primary focus on students’ use of mathematics textbooks (Rezat, 2009). The main aims of the study were to identify activities in which the mathematics textbook is incorporated in as an instrument for learning, and to analyse the ways students utilise their mathematics textbooks to learn mathematics. Since teacher-mediation of texts and teacher’s use of textbooks were regarded as major influential factors on students’ use of mathematics textbooks data on teacher-mediation of textbook use were collected. Furthermore, the data on students’ uses of textbooks provided an account for teacher-mediation of textbook use. The data collection method is characterised by a triangulation of questioning, observation and interviews. Firstly, data on teacher mediation of textbook use in the classroom were collected by classroom observation. Field notes captured every use of the textbook in class. Both, the uses of the textbook by students and by the teacher were taken into account. Secondly, the students were asked to highlight every part they used in the textbook. Additionally, they were asked to explain the reason why they used the part they highlighted in a small booklet by completing the sentence “I used the part I highlighted in the book, because . . .”. This method was developed to get the most precise information about what the students actually use and why they use it. Furthermore, this method facilitated data collection at different locations, for example at school and at home. Thirdly, stimulated recall interviews were conducted with selected students. The method was used in the way that students were confronted with their own markings and comments in the book and were asked to explain their way of proceeding. These data on students’ use of textbooks gathered in the booklets and in the interviews has proven particularly illuminating in revealing information about teacher-mediation of textbook use. Data were collected for a period of 3 weeks from 4 teachers and 74 students in 6th and 12th grade from two German secondary schools (Gymnasium). Within the German tri-partite school system, these schools are considered to be for highachieving students. They are located in two German small towns. Most of the students are from medium and high socioeconomic status backgrounds. As outlined earlier, the phenomenon of interactions of teacher’s and students’ use of mathematics textbook has different facets. In a narrow sense it refers to direct interactions of the respective uses. In a broader meaning it refers to how one user’s use of the textbook effects the other participants of the enacted curriculum more generally. Therefore, the analysis of interactions between teacher’s and students’ use of textbooks requires the utilisation of different but related theoretical frameworks to draw a comprehensive picture of the phenomenon under study.

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First of all, the data were coded according to the procedures of Grounded Theory (Strauss & Corbin, 1990). Open coding of teachers’ explicit references to the textbook in the classroom was used to identify and develop dimensions of teacher-mediation of textbook use. To understand the teachers’ mediatory role on students’ use of mathematics textbooks, students’ utilisations of the book were reconstructed using the theoretical lens of Rabardel’s (1995, 2002) theory of the instrument. According to Rabardel an instrument is a psychological entity that consists of an artefact component and a scheme component. In using the artefact with specific intention the subject develops utilisation schemes which are shaped by both, the artefact and the subject. Furthermore, the effects of students’ use of mathematics textbooks on teachers’ plans were analysed using the documentational approach (Gueudet & Trouche, 2009; see also Chapter 2) as a theoretical lens. The documentational approach is based on Rabardels’ notions of instrumentalisation and instrumentation of artefacts. According to Gueudet and Trouche (2009; Chapter 2), teachers’ documentation work encompasses teachers’ interactions with resources, their selection and teachers’ work on them (adapting, revising, reorganising . . .). A document consists of a set of resources and utilisation schemes linked to these resources and to specific situations. A pivotal aspect of both theoretical lenses used in this analysis is the notion of utilization scheme. According to Vergnaud (1996) a scheme is an invariant organisation of behaviour for a certain class of situations. It is characterised by operational invariants, inference possibilities, rules of action and goals. Vergnaud stresses the importance of operational invariants. They represent knowledge that is implemented in a scheme and therefore determine the particular structure of the scheme. Therefore, the reconstruction of operational invariants can contribute substantially to the understanding of teachers’ and students’ use of resources and related interactions. While different theoretical lenses are used to grasp different aspects of interactions related to teachers’ and students’ uses of textbooks these frameworks are complementary because both the instrumental approach and the documentational approach, are related to Rabardel’s theory of the instrument. By referring to analogous theoretical concepts, such as instrumentalisation and instrumentation, they are capable of drawing a comprehensive picture of the phenomenon under study by approaching it from different perspectives. How to integrate these frameworks into a more comprehensive framework will be discussed at the end of this chapter.

12.3 Impacts of Teachers on Students’ Use of Mathematics Textbooks All four teachers in the study referred explicitly to the book in their lessons but the ways in which they refer to the book varies considerably. The analysis of the teachers’ explicit references to the textbook in the classroom that were recorded in

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the field notes enabled the identification of different dimensions in which teachers mediate textbook use. Teacher 1 (6th grade) uses tasks from the book only once in a while but he draws attention to the book in a general way almost every lesson. Whenever he asks the student to do a task or problem he points to the book as a helpful means, for example “If you don’t know how it is done you can look it up in your textbook”1 or “in order to do the following task there is a resource you might use: your book”. In addition, he arranges teaching scenarios where students have to use their books to find assistance. He let the students work individually on a worksheet with questions about symmetry and congruence mappings. After they finished the worksheet he asked them to look up their answers in the book without referring to specific sections. Accordingly, the students had to find the relevant sections in the book on their own. Furthermore, teacher 1 addresses textbook use on a meta-level, for example “What could you do if you don’t find anything helpful skimming through your book? You know, it can happen that you overlook something”. Teacher 2 (12th grade) uses tasks and problems from the book mostly for assigning homework, for example “Homework, due Monday: Page 184, number 5 b, e, h”. After introducing the integration by parts rule as a new subject she points to the book in a general manner: “What we have done today you can find on page 182”. Teacher 3 (6th grade) and teacher 4 (12th grade) use the mathematics textbook predominantly in a way that is regarded as typical in the relevant literature, namely as a collection of tasks and problems. The structure of the lessons of teacher 4 can be best described as a sequence of tasks and problems from the book with short instructional interludes. Typically, he refers to the book in the following way: “And now we want to look inside our books on page 22, number 7, please”; “Let’s look inside the book on page 33, number 2, please”; “For Wednesday, please work on numbers 3, 4, and 5 on page 33”. These four teachers’ explicit references to the textbook can be characterised according to three dimensions. The first dimension relates to the way in which the students’ use of the textbook is affected by the teacher. Students’ use of textbooks might be influenced directly or indirectly by the teacher. All references to the textbook quoted above are explicit: The teacher is talking about the book in some way. Thus, the students’ attention is drawn directly to the book. However, it will be shown later that students’ utilisations of mathematics textbooks are also influenced indirectly by the mere use of the textbook by the teacher in the classroom. In these cases, the use of the textbook is not caused by an explicit reference to the book by the teacher, but the use of the book in the classroom is a prerequisite for the students’ self-directed use. Indirect mediation of textbook use is usually not planned by the teacher and often eludes his attention. The second dimension of mediation of textbook use relates to the specificity in which the teacher refers to the book. Teacher 4 typically refers to a specific

1 All citations from teachers and students are originally in German und were translated by the author.

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section in the book, for example page 22, task number 7. On the contrary, teacher 1’s references to the book are general. He does not refer to specific sections, but draws attention to the book in a general way. In the case of teacher 1, the students have to decide themselves which section on which page they are using. Specific and general mediation of textbook use can only appear in combination with direct mediation, because the characteristic of indirect mediation is that the teacher does not refer to the book at all, but the students are influenced by its use in class. The third dimension relates to the binding character of the mediation. Teachers’ mediation of textbook use might be voluntary or obligatory. Teacher 1 mediates the use of the textbook as a voluntary task. He reminds the students that they can use the book in order to get assistance. But, the students do not have to use the book if they do not need assistance. In contrast to teacher 1, the mediation of textbook use of teacher 4 is obligatory. The students do not have a choice to use the book or not. They are supposed to work on the assigned tasks and problems. The matrix in Fig. 12.1 summarises the conceptualisation of these different ways teachers mediate textbook use and draws attention to the fact that all three dimensions are intertwined in a concrete mediation of textbook use. Fig. 12.1 Conceptualisation of ways teachers mediate textbook use

Mediation

Obligatory

Voluntary

Specific Direct General Indirect

Five of these six combinations of the different ways teachers mediate textbook use were actually observed in the study. The five ways of mediation will be explained in the following sections.

12.3.1 Direct, Specific, Obligatory Teacher Mediation Direct, specific, obligatory teacher-mediation always occurs when the teacher explicitly refers to the textbook and asks students to do a specific task or to read a specific section from the book in class or for homework. Teacher 4’s explicit references to the book are prototypical for this kind of mediation.

12.3.2 Direct, Specific, Voluntary Teacher Mediation It was also observed that teachers sometimes explicitly refer to a specific section in the textbook in a non-obligatory sense. Teacher 2’s reference to the section in the book where students can find the subject of the mathematics class is a typical case of direct, specific and voluntary mediation. The teacher refers explicitly to a specific page of the book and thus mediates textbook use directly and specifically, however,

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students are not obliged to read the page as the prompt is just a hint to students for further study. From the interview data on students’ use of textbooks it becomes apparent that students sometimes also ask the teacher for such recommendations. For example, Lilli – a sixth grade student – explained in the interview: “I usually ask the teacher for recommendations which parts of the book I can use for further practicing”.

12.3.3 Direct, General, Obligatory and Voluntary Teacher Mediation In the study, teacher 1 did not always refer to specific sections in the textbook. In the case of the teaching scenario with the worksheet, he explicitly asked the students to look up their answers in the book without telling them a specific section. Other times, he advised his students to use the book when they needed assistance with tasks and problems. In these cases, the use of the book was voluntary. But again, he did not tell them specifically where the relevant information can be found.

12.3.4 Indirect Teacher Mediation of Textbook Use Indirect teacher-mediation refers to a mediation of textbook use that is related to the teacher’s own use of the textbook in the classroom. Whereas direct teachermediation is recognised without difficulty because the teacher explicitly refers to the book, indirect teacher-mediation easily remains unnoticed because it can only be inferred from students’ utilisations of the book. The detailed reconstruction of students’ utilisation schemes of textbooks reveals that the teacher has a major influence on the selection of sections from the book even when he does not refer to the book explicitly. Regarding interactions between teachers’ and students’ utilisations of the textbook, indirect teacher-mediation is the most interesting because it elicits how the teacher’s own use of the textbook affects student’s use of it. Indirect teachermediation only occurs in combination with voluntary uses of the book. How students’ and teachers’ utilisations of the book interact from the perspective of indirect teacher-mediation, will be exemplified by the cases of Emma and Merle. Emma (6th grade) uses tasks from her mathematics textbook voluntarily for consolidation. First of all, she repeats tasks that were done in the mathematics class. Additionally, she picks tasks that are adjacent to these tasks from the mathematics class in the book. She explains her way of proceeding as follows: “If we did no. 4 in the mathematics class then I will do no. 5, because it is similar.” Emma’s choice of tasks from the book is dependent on the use of tasks from the book by the teacher. The teacher-mediated tasks guide Emma’s own choice of tasks from the book. Furthermore, Emma’s argument elicits that she infers characteristics of the task from its position: She believes that adjacent tasks in the book are similar. Emma’s selection of tasks is prototypical for many students’ scheme of selecting tasks for consolidation. This scheme only works, if the teacher uses tasks from the

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book in the mathematics class. Therefore, the teacher’s use of the textbook in class is a perquisite for Emma’s voluntary utilisation of the book. Students’ like Emma are most affected by implicit teacher mediation. This became evident in the mathematics class of teacher 1. Teacher 1 did not use tasks from the book for a period of 2 weeks. Some students did not use their books for consolidation during this time. But, after the teacher let them work on some tasks from the book they used their book heavily for consolidation in the same way as Emma. Whereas in Emma’s case the teacher’s use of the textbook is a prerequisite for her use of the book the case of Merle illustrates that a close connection between the sequencing of textbook contents and instruction can foster student’s voluntary learning. One of Merle’s (6th grade) utilisation schemes of the textbook is her use of kernels and excerpts from the expository section of the textbook lesson that follows the textbook lesson corresponding to the latest topic of her mathematics class. She infers the relevance of the section from its position in the book. Underlying this inference is the assumption that the order of the textbook lessons corresponds to the succession of the topics in the mathematics class. Her utilisation-scheme also interacts strongly with the use of the textbook by the teacher. But, the interaction is of a different kind than in the case of Emma. Emma’s use of the textbook was dependent on the teacher’s use of the textbooks. If the teacher does not use tasks from the book, Emma will not utilise her textbook. In contrast to Emma the teacher’s use of the textbook is not a prerequisite for Merle’s utilisation. She can find out the lesson in the textbook that corresponds with the latest topic in the mathematics class herself. However, if the teacher does not follow the sequence of the book her scheme is not useful. Merle’s case sheds a different light on teachers’ close adherence to the book. Whereas an instruction that closely follows the book is sometimes connoted negatively (Ewing, 2004), it provides the foundation for an effective utilisation of the book by Merle. The two cases exemplify how the teacher’s use of the textbook can affect students’ use of textbooks. Both students used the textbook of their own initiative, but in both cases the use of the textbook by the teacher was related to a successful utilisation. Whereas in the first case the teacher’s use of textbooks in class is a prerequisite for Emmas’ self-directed selection from the book, the second case shows that a close adherence of teacher’s instruction to topics and sequencing in the textbook might afford students’ self-regulated learning. Some students are highly dependent on teachers’ implicit mediation of the textbook.

12.4 Impacts of Students’ Uses of Mathematics Textbooks on Teachers The investigation of the influence of teachers on students’ use of mathematics textbooks is only one aspect of interactions between students’ and teachers’ use of resources. The question remains if and how students’ uses of textbooks affect

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teachers. In the study, situations where the students’ use of the textbooks interfered with the teachers’ plans were documented during classroom observation. One exemplary situation observed in sixth grade will be analysed in detail to substantiate this assertion: The subject of the lesson is the rule for multiplying decimals. The aim of the teacher is that the students discover the rule themselves. The resources he uses are four tasks from different textbooks and group work. The textbooks from which the tasks are chosen are not used by the students. These tasks are supposed to guide students’ discovery of the rule for multiplying decimals in different ways: one by estimation in an everyday context, a second one by transforming decimals into ordinary fractions, a third one by using the calculator and the fourth one by calculating as if there was no decimal point and setting the decimal point afterwards on the basis of estimation. The situation is characterised by a tension between the requirements of the tasks and the students’ knowledge. While working on the task in small groups two students use the box with the rule from the textbook-lesson “multiplying decimals”. The reason Denise gives for using her book is: “I want to know how to multiply decimals”. Mia argues that she uses her book because she “want(s) to know it in advance.” Following the group work the teacher summarises the findings from the work in small groups in whole class discussion. The resources he uses are students’ verbalisations of the rule. While writing the rule for multiplying decimals onto the black board, one student complains: “But, in the textbooks it says that you have to determine the algebraic sign first.” This statement reveals that the student compared the rule on the black board with the rule in the book. He wants to know why both rules are not identical. The teacher is obviously not prepared for this kind of intervention and answers: “I don’t care what is written in the book.” The analysis of this situation in terms of the documentational approach reveals that these students’ use of the textbook as an instrument affected the teacher’s plan. The document of the teacher comprises the four tasks, group work, students’ verbalisations of the discovered rule and the black board as resources. From his utilisation of the resources the following operational invariants might be inferred: “To guide students’ discovery of the rule for multiplying decimals is a good way of introducing the new rule”. “Using students’ own verbalisations of a new rule is a good way to formulate a new rule”. In this situation the document is constructed in use. The statements of the two students using their textbook during group work reveal that both students use the book in order to acquire knowledge that is required to solve the problem. Consequently, both of them avoid the discovery of the rule for multiplying decimals and skip directly to the essential result by using their textbooks. Thus, the instrumentalisations of the textbook by the two students as a means to acquire knowledge interferes with the teacher’s instrumentalisation of the four tasks, that is with a central feature of the teacher’s document. The incident, occurring during the verbalisation of the new rule in whole class discussion, is also characterised by a conflict of the students’ behaviour with the operational invariant of the teacher’s utilisation scheme. While the student seems to

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be interested in a correct and consistent formulation of the rule and therefore relies on the authority of the textbook the teacher wants to connect to students’ creative mathematical activity by using the resource “students’ verbalisation”. In both the incidents the teacher is confronted with unexpected student interventions. As a consequence the document cannot be implemented in the way it was planned. An interesting question here is how the document changes because of the students’ behaviour. To answer this question information about the teacher’s repeated instrumentalisation of tasks to guide students’ discovery of mathematical rules and his instrumentalisation of students’ verbalisations are required. The data used here do not contain any more information about similar situations. But, since students’ use of textbooks in a group work situation easily eludes the teacher’s attention, it is likely that the document will not be affected. This might be different for the incident occurring in whole class discussion. Nevertheless, this observation points to the importance of knowledge about students’ use of resources as part of teachers’ professional knowledge. If the teacher is familiar with students’ instrumentalisations of textbooks this knowledge is likely to influence his documentational genesis. These observations point to the important fact, that students’ use of resources can play a crucial role in teachers’ documentational work, especially if the evolution of documents in use is considered. Therefore, the role of students in the documentational process should not be reduced to resources that are used by the teacher, but the active part of the student in shaping the enacted curriculum should also be considered. As Cohen, Raudenbush, & Ball (2003) write “Learning depends on students and teachers making bits of lessons develop and connect” (p. 126). Therefore, Cohen et al. argue for a model in which the key causal agents are situated in instruction. In the following section, a model fulfilling this need will be outlined.

12.5 Towards a Comprehensive Theoretical Framework In the introduction of the chapter it was pointed out that research on the use of textbooks either focuses solely on teachers or on students. The same seems to apply for research on the use of ICT (see Chapter 3). Consequently, theoretical frameworks conceptualising the use of these resources emphasise either the role of teachers or students. The other user of the resources and its role as an active designer of the enacted curriculum tends to be marginalised, respectively. In the previous section it was argued that the documentational approach conceptualising teachers’ interactions with resources proposed by Gueudet and Trouche (2009) does not comprise students’ use of resources as an influential factor affecting teachers’ design process of the curriculum. As long as the focus is on the design area (Remillard, 2005), that is teachers’ activities outside the class, this model suits the situation as the teachers’ activities are only affected by his beliefs about students’ use of resources. Gueudet’s and Trouche’s framework (Chapter 2) considers teachers’ beliefs as an important aspect affecting planning decisions and therefore is an appropriate model for the design area. Although the documentational approach focuses on teachers’ planning activities outside the class (Gueudet & Trouche, 2009,

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p. 201). Gueudet and Trouche point out that the design of documents continues in use (p. 207). But, as soon as the document in use is considered insufficient attention is paid to the active role of the student in the documentational approach. This was exemplified by the situation analysed in the previous section. The active part that students’ use of resources plays in the design of documents is not accessible via teachers’ beliefs in the documentational work. Therefore, if the framework is supposed to comprise the design of documents in use as suggested by Gueudet and Trouche (2009), the use of resources by students influencing this process has to be considered. A comparison of the documentational approach with the framework conceptualising teachers’ interactions with curriculum materials2 proposed by Remillard (2005) reveals that Remillard pays more attention to the students’ active role. In her framework the student is included as an influential factor on the enacted curriculum and therefore affects the teachers’ participatory relationship with curriculum materials. But still, the framework explicitly conceptualizes the teacher–curriculum relationship and marginalises the students’ role. Students are regarded as only one influential factor, among others, on the enacted curriculum. His role as an active user of the same material is not taken into account. The previous analysis of both the effects of students’ use of resources on the implementation of teachers’ documents and the impact of teachers’ use of resources in the classroom on students’ use of them indicates that a comprehensive investigation of the use of resources must consider both users of the resources: the teacher and the student. Only focusing on one of these aspects will lead to an incomplete picture because the interactions between the use of resources by teachers and by students are neglected. A more comprehensive model for the study of the use of resources is provided by the didactical tetrahedron in Fig. 12.2.

Resources

Student

Fig. 12.2 The didactical tetrahedron – a comprehensive model for the study of the use of resources

Teacher

Mathematics

2 Curriculum materials are only one aspect of resources in the wide meaning underlying the documentation approach. Therefore, Remillards’ framework is more limited in scope than the documentation approach because of the focus on curriculum materials.

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In this model resources are situated in relation to the didactical system in the narrow sense as described by Chevallard (1985). In line with Cohen et al. (2003), the teacher and the students are regarded as active agents of instruction and as active users of resources. Therefore, the model allows for the analysis of interactions between teachers and students over content including their independent, but interrelated use of resources. Regarding the broad conceptualisation of “resources” that is put forward in this book (Adler, 2000; Chapter 1), it might seem surprising that mathematics and resources appear at two different vertices in the didactical tetrahedron. Adler (2000; Chapter 1) argues for a conceptualisation of knowledge as a resource, and thus mathematics, understood as mathematical knowledge, should not be separated from the resources in the didactical tetrahedron. But, Adler herself separates mathematics and resources when she refers to resources as being “both visible (seen/available and so possible to use) and invisible (seen through to the mathematical object intended in a particular material or verbal representation) if their use is to enable access to mathematics” (see Chapter 1). This also becomes apparent in the definition of what she calls an “evaluative event”, that is “an interactional sequence in a mathematics classroom aimed at a particular mathematical concept or skill” (see Chapter 1). Additionally, it is suggested in the activity theory origins of the instrumental approach (Rabardel & Bourmaud, 2003; Chapter 2) to distinguish between the object of the activity and the mediating artefacts (Engeström, 1987; Vygotsky, 1978). In line with the previous considerations the didactical tetrahedron puts forward the view that the object of the activity of teaching and learning mathematics is mathematics. Furthermore, the distinction between resources and mathematics affords the inclusion of other aspects, such as that of semiotic mediation (Sträßer, 2009, p. (1)75). Thus, it seems to be capable of further expansion and integration of other important aspects in the field of the use of resources, for example, the Theory of Semiotic Mediation (see Chapter 3). In the following discussion. it will be substantiated that this model is capable of integrating different theoretical perspectives on the basis of the instrumental approach and relates these perspectives to one another. The documentational approach conceptualises teachers’ interactions with resources (Chapter 2). The outcome of this interaction is a document consisting of a set of resources and utilisation schemes which comprise teachers’ beliefs as operational invariants. The scope of a document is linked to a class of professional situations which are defined in mathematical terms. Altogether, the documentational approach conceptualises activities of teachers with resources related to specific mathematical content. In the didactical tetrahedron this activity is represented by the triangle with the vertices teacher, resources and mathematics. As outlined above, a shortcoming of the documentational approach is that the active role of students in the design of documents in use is not included. This aspect is added in the didactical tetrahedron. The triangle with the vertices student, resources and mathematics represents the student as an active user of resources in order to learn mathematics. In this triangle, the use of resources by the student is conceptualised by the instrumental approach (Rabardel, 1995, 2002; Trouche, 2005). The teacher’s task is an intentional and systematic organisation of resources available in a mathematical

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task situation to guide students’ instrumental genesis. This activity is captured by the metaphor of orchestration (Trouche, 2004; see also Chapter 14) which is introduced in the context of the instrumental approach. Thus, the teacher’s perspective on the triangle “student–resources–mathematics” is a perspective of orchestration. Finally, the effects that the instrumental geneses3 of the teacher and of the students have on one another are represented by the triangle with the vertices teacher, students and resources. Both, teachers and students are active users of resources with their own individual instrumental genesis. In the triangle “teacher–students–resources”, these instrumental geneses interact with one another. On the one hand, the student is capable of affecting the instrumental genesis of different resources within the documentational genesis of the teacher as outlined in section 12.4 by being an active user of resources himself or herself. On the other hand, teachers mediate the use of resources by their own use of them. By incorporating the two human players in the mathematics classroom, the tetrahedron model allows for integrating the different theoretical perspectives related to the instrumental approach. Therefore, it provides a more comprehensive model for the investigation of the use of resources. It is not only suitable for the analysis of issues related to the two players’ instrumental geneses of different resources, but draws attention to interrelations of these instrumental geneses. Finally, the classroom situation referred to in this article draws attention to a particular, but nevertheless important aspect of teachers’ professional knowledge: knowledge about students’ use of resources. The widely accepted and influential conceptualisation of teachers’ professional knowledge by Shulman (1986) only relates to content knowledge and therefore does not comprise knowledge about students. Accordingly, many conceptualisations of teachers’ professional knowledge developed from Shulmans work (e.g. Bromme, 1992) do not comprise knowledge of students’ use of resources as an important aspect. Fortunately, this situation has changed. The Teacher Education And Development Study in Mathematics (TEDS-M) conducted by the International Association for the Evaluation of Educational Achievement (IEA) – one of the latest international surveys on teachers’ professional knowledge – incorporates knowledge of students in terms of knowledge of learning theories, predicting typical students’ responses, including misconceptions, analysing or evaluating students’ mathematical solutions, arguments and questions (Tatto, Schwille, Senk, Ingvarson, Peck, & Rowley, 2007). But still, this conceptualisation does not comprise knowledge about students’ actual ways of

3 The notion of instrumental genesis refers to the use of artefacts, usually focusing on only one artefact. If a set of resources is considered, it would be appropriate to speak of a teacher’s documentational genesis. However, the distinction between artefacts and resources would need further elaboration. Furthermore, there has not been introduced an analogous notion for student’s use of multiple resources yet. Therefore, it is only referred to instrumental genesis for both teachers and students because it would be inconsistent to refer to the use of a set of resources for the teacher (documentational genesis) and to the use of one artefact for the student (instrumental genesis). Since documentational genesis encompasses the instrumental geneses of multiple artefacts this framework also relates to documentational genesis of teachers.

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learning and their use of resources. If learning is conceptualised from a sociocultural perspective as “learning is always learning to do something with cultural tools” (Säljö, 1999, p. 147), insight into students’ use of resources provides a better understanding of students’ ways of learning. Therefore, knowledge about students’ use of resources is an important aspect of teachers’ professional knowledge.

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Rabardel, P., & Bourmaud, G. (2003). From computer to instrument system: A developmental perspective. Interacting with Computers, 15(5), 665–691. Remillard, J. T. (2000). Can curriculum materials support teachers’ learning? Two fourth-grade teachers’ use of a new mathematics text. Elementary School Journal, 100(4), 331–350. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246. Rezat, S. (2009). Das Mathematikbuch als Instrument des Schülers. Eine Studie zur Schulbuchnutzung in den Sekundarstufen. Wiesbaden: Vieweg+Teubner. Säljö, R. (1999). Learning as the use of tools. A sociocultural perspective on the human-technology link. In P. Light & K. Littleton (Eds.), Learning with computers: Analysing productive interactions (pp. 144–161). New York: Routledge. Schmidt, W. H., Porter, A. C., Floden, R. E., Freeman, D. J., & Schwille, J. R. (1987). Four patterns of teacher content decision-making. Journal of Curriculum Studies, 19(5), 439–455. Shulman, L. S. (1986). Those who unterstand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. Stodolsky, S. S. (1989). Is Teaching Really by the Book? In P. W. Jackson, & Haroutunian-Gordon (Eds.), From Socrates to Software: The Teacher as Text and the Text as Teacher (Vol. 1, pp. 159–184, Yearbook of the National Society for the Study of Education). Chicago: University of Chicago Press. Sträßer, R. (2009). Instruments für learning and teaching mathematics. An attempt to theorise about the role of textbooks, computers and other artefacts to teach and learn mathematics. In M. Tzekaki, M. Kaldrimidou, & C. Sakonidis (Eds.), Proceedings of the 33rd conference of the international group for the psychology of mathematics education (Vol. 1, pp. 67–81). Thessaloniki: PME. Strauss, A., & Corbin, J. (1990). Basics of Qualitative Research: Grounded Theory Procedures and Techniques. Newbury Park: Sage. Tatto, M. T., Schwille, J., Senk, S., Ingvarson, L., Peck, R., & Rowley, G. (2007). IEA Teacher Education and Development Study in Mathematics (TEDS-M). Conceptual Framework. Field Trial. Retrieved Janurary 7, 2010, from http://tedsm.hu-berlin.de/publik/Downloads/ framework_juli07.pdf Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. [10.1007/s10758-004-3468-5]. International Journal of Computers for Mathematical Learning, 9(3), 281–307. Trouche, L. (2005). An instrumental approach to mathematics learning in symbolic calculators environments. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators. Turning a computational device into a mathematical instrument (pp. 137–162). New York: Springer. Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). According to the book – Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Dordrecht: Kluwer. Vergnaud, G. (1996). The theory of conceptual fields. In L. P. Steffe, P. Nesher, C. Paul, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 219–239). Mahwah, NJ: Lawrence Erlbaum. Vygotsky, L. (1978). Mind in society: The development of higher psychological process. Cambridge: Harvard University Press.

Chapter 13

Teachers Teaching Mathematics with Enciclomedia: A Study of Documentational Genesis Maria Trigueros and Maria-Dolores Lozano

13.1 Introduction The integration of new technologies to the classroom and its relationship with students’ learning and appreciation of mathematics has received attention in a wealth of studies (e.g. Artigue, 2004; Hegedus & Moreno-Armella, 2009; Hoyles & Lagrange, 2010; Lagrange, Artigue, Laborde, & Trouche, 2003; Lozano, Sandoval, & Trigueros, 2006; Mariotti, 2002; Trigueros & Lozano, 2007; Sandoval, 2009; Swan, Schenker & Kratcoski, 2008). Teachers’ practices have also received some attention (Assude, 2008; Kynigos & Argyris, 2004; Roschelle, Shechtman, Tatar, Hegedus, Hopkins, Empson, et al., in press; Trigueros & Sacristán, 2008). In México, in the last decades, there have been two national projects designed to integrate technology to teaching. One of them, “Enciclomedia”, was designed to support the teaching and learning of all subjects in grades 5 and 6 of primary school by working with one computer and an electronic whiteboard in the classrooms, together with existing teaching mandatory materials and curriculum. Enciclomedia provides teachers and students with digital resources – interactive programs, animations, activities for dynamic geometry software and spreadsheets – which are linked to different parts of the curriculum and the official textbooks. Evaluations of the Enciclomedia project by several institutions show positive results in terms of resources’ usability and interactivity, a high potential for promoting meaningful and high order operations learning, as well as high motivation of students (Díaz de, Guevara, Latapí, Ramón, & Ramón, 2006; Holland, Honan, Garduño, & Flores, 2006; Trigueros, Lozano, & Lage, 2007). Issues related to infrastructure and teacher training were, however, found problematic. One challenge our educational system still faces is how to help teachers to integrate Enciclomedia’s resources into their teaching of mathematics so that it contributes to student learning. Some studies (e.g. Díaz de et al., 2006; Sagástegui, 2007) have reported that these

M. Trigueros (B) Instituto Tecnológico Autónomo de México, CP 1000 México City, Mexico e-mail: [email protected]

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resources can change the way technology is used in the classroom and help teachers develop a learning environment where students can participate actively and profit from their use. However, there are few studies concerned with the ways teachers use Enciclomedia resources while participating in professional development courses designed to help them use these resources effectively (Chávez, 2007). In this chapter we examine teachers’ professional development through the analysis of their appropriation and transformation of Enciclomedia resources. To do this we use the documentational approach of didactics (Chapter 2) to analyse information about teachers’ interactions with Enciclomedia obtained from different sources. In particular we intend to answer the following research questions: how do teachers develop documents, which comprise several resources, including digital programs from Enciclomedia, throughout time? What are the components of those documents? Is it possible to describe patterns in the way teachers use these resources? After a brief discussion of the theoretical framework, we will describe the methodology used in this study. We then focus on the results found on how three different teachers develop and use resource systems (see Chapter 5) in their classrooms, and how these systems evolve in time. We will finally discuss those results focusing on answering our research questions.

13.2 Theoretical Framework Recently, Gueudet and Trouche (2009; Chapter 2) introduced a new theoretical approach to extend instrumentation theory (Rabardel, 1999, 2005) by considering the whole resource system teachers use in their lessons and how that system evolves in terms of teacher professional development through the analysis of their appropriation and transformation of resources in their everyday practice. In this approach documentation work and professional development are interrelated; and thus both must be studied at the same time. This is why a study must take the material aspects of documents and the evolution of usages into account. Gueudet and Trouche hypothesise that teachers develop a structured documentation system, and that this system and their professional development evolve together. For this reason, the analysis of the evolution of a documentation system, in different time scales, gives valuable information about the changes introduced by the development of particular documents, and about the development of the teacher.

13.3 Methodology The research project described in this chapter was carried out by two researches who were involved in the development of Enciclomedia resources, teacher training, and research. Data came from different sources (lesson observation, written materials, interviews, and requested letters). Firstly, we reviewed all the information

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we had collected for 6 years from our direct involvement with 50 teachers working with Enciclomedia. We organised professional development workshops through which we had intermittent contact with 10 teachers who stayed with us during 4 years. Selected workshop sessions were video recorded, and teachers’ written materials were collected. In addition, during these 4 years we carried out classroom observations of selected teachers to investigate the ways the resources were used. From the analysis of the collected data, we chose, for this study, to focus on three teachers we considered representing the characteristics of different groups of teachers using Enciclomedia. Two of them showed professional development pathways that were representative for most of the 50 teachers we had closely worked with. The first (T1) of these two was selected from a group of school teachers who were finishing a 1-year professional development course where teachers not only learnt more about how to use Enciclomedia resources, but also shared with each other their experiences and practices in their classrooms. By the time he started this course, he had already been using Enciclomedia for 2 years in 6th grade. The second teacher (T2) teaches 6th grade students and has been using Enciclomedia for 6 years. She was trained by the team who developed Enciclomedia resources and had the opportunity to observe how they taught lessons using the programs. She has also taken two other workshops on Enciclomedia, which were offered by the local ministry of education. Teacher (T3) is a teacher-researcher who had been teaching for 5 years in a primary school before she included digital resources in her lessons. She has been involved in mathematics education for several years, is interested in reading the mathematics education literature and is eager to learn new things. She got involved in “Enciclomedia’s” workshops from its early stages, and has been using it for 4 years. She always has the latest version of the program installed in her computer at home. We selected her because she represented a small group of teachers who were particularly successful in integrating resources in their practice. After selecting the above-mentioned teachers, we carefully revised all the collected data related to them, and decided to carry out new classroom observations using a semi-structured observation guide and to interview each of them. Additionally we designed, specifically for this research study, an instrument which consisted of a series of four letters, written by the teachers one every 2 months and addressed to a fellow teacher in which they shared their experience while working on a particular mathematical topic with Enciclomedia’s resources and where they included elements of success and difficulties encountered with students. These letters were used as an indirect means of obtaining information that otherwise could not have been obtained, and with the purpose of helping teachers reflect on their practice (this is similar to the reflexive investigation approach described in Chapter 2). To make our results reliable, each researcher analysed all the data, using the same general analysis guide, looking for teachers’ particular ways of using Enciclomedia and other resources, together with their approach to teaching specific lessons. We then came together and discussed the results obtained.

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13.4 Results Obtained The analysis of the information revealed differences in the ways teachers use teaching resources, in particular, in their approach to the use of technology in the classroom, lesson preparation and actual work with students. In this section we discuss documentational genesis of the three selected teachers. We focus our attention to the regularities found in each documentation process and on the possible factors which influenced the genesis of each of them. At the same time we relate this process with teachers’ professional development. Our intention is not to compare teachers or diagnose their professional needs. The use of the documentation theoretical framework is used as a means to understand the processes of genesis itself.

13.4.1 Influence of a Professional Development Seminar Teacher 1 is an experienced teacher who started using Enciclomedia in his lessons 5 years ago. He attended the first two Enciclomedia workshops where he learnt “how to use it, but only the technical aspects of the use of the electronic board and the computer. . .”. He does not have access to Enciclomedia outside the classroom and he currently teaches 6th grade at two schools. We describe T1’s documentational genesis through his teaching volumes of prisms in the last 4 years. Before his professional development course, T1 taught mathematics in a “traditional” way, focusing on mathematical content. His planning consisted of a list of specific things to do, on the basis of the use of the textbook and a teachers’ book that all teachers have access to (see Remillard’s Chapter 6 about relationships that teachers develop with curriculum resources as they use them). When asked why the use of technology did not appear in his lesson plan, he nevertheless responded “I introduce it where I consider it to be necessary . . . I always use it”. He regularly introduced Cubícula (Fig. 13.1), an Enciclomedia program that can be used to construct different solids by using unit cubes, and where solids can be rotated to see them from different perspectives and can be decomposed in sections to explore their volume and superficial area. T1 used it mainly to explain the meaning of the symbols in the formulae for the volume of the cube and of rectangular prisms. During the interviews he commented “I consider they need to know how to use the formulae. . .”. We observed that T1’s material components of the sets of resources employed were: lesson plan, textbook, teaching guide and Enciclomedia. The mathematical component consisted of the use of formulae for volumes and measurement units. From the whole set of data we were able to infer rules of action: “give students formulae first”; “emphasize the importance of units” and “use technology to help students understand the formulae”. The didactical component of the resources he used is based on what he knew and in sequencing it so that all converged in the possible operational invariant “understanding the meaning of the symbols in volume formulas is needed to calculate volumes”. His use of some resources can be

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Fig. 13.1 Cubícula

considered stable, but his use of technology was not as he utilised it differently in different occasions. While some of the rules of action can be considered usages, we observed that Cubícula was utilised in a haphazard manner. During the professional development course, T1 had opportunities to reconsider his beliefs. He reviewed his lesson plan on volumes, and taught it again. During interviews T1 said that he “did not feel completely at ease during the lessons”; we observed that he continuously had to modify his plan. He included the use of paper cubes before using “Cubícula”, the combination of the program with activities from the textbook, and demonstrations by rotating the solids or decomposing them in slices. His students participated more but he still gave most definitions and explanations and devoted many activities to the understanding of the meaning of symbols in formulae. During discussions in the professional development course T1 said he was “still not sure about how to guide students’ explorations and conclusions”, and “how to link exploration with important mathematical ideas”. During the interview he commented he didn’t feel confident about “how to handle students’ participation”. We observed the influence of the activities from the professional development course on T1’s document genesis and structure. The document’s material component was complemented by concrete paper cubes, the blackboard and a worksheet with questions about volumes of specific solids. Its mathematical component stayed the same. The didactical component, however, changed. It involved exploration by

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students and demonstrations by the teacher with Cubícula. New rules of action appeared “it is important to start by knowing what students think about the topic”, “start the lesson by using concrete materials”; “making these operations with concrete materials is not easy, so ‘Cubícula’ is needed.”; “immediate feedback from the computer is important to check what students did with the program”, “use of the program helps students visualize solids from different perspectives, it makes counting of the cubes easier”, “use decomposition of figures by levels of cubes to calculate their volumes”. We consider T1’s document evolved by the introduction of concrete materials and new rules of action. The use of Cubícula is probably being integrated in usages of resources. We can only infer that “formulae are important” is still an operational invariant for T1, since this belief seems to guide T1’s planning and teaching actions. As a final requirement from his training course, T1 had to plan and teach a lesson, and discuss some data with the instructor and colleagues. He chose the same lesson. He added to his lesson plan: “make sure that students explore, work in teams, more active participation”. T1 asked a colleague to take notes during his lesson to be used for discussion in his course. He had opportunities to use the computers to prepare his class. He told the instructor “I will let students work with concrete material and try to make them rotate it and examine its different layers. Students need to practice and use all their senses. When they face difficulties manipulating the materials, I will suggest to use Cubicula”. During the lesson he had students construct regular and irregular solids in small groups and asked them to calculate their volume. Then he said “Let’s work with ‘Cubícula’; you will be able to see things that are not easy to see with the solids you constructed”. He let students participate and verify results by going to and from Cubícula’s activities and textbook problems. Students also worked with a worksheet he designed before doing the most difficult textbook exercises and were given another worksheet as homework. During discussion at the professional development course, he stressed some ideas that are related to his changes as teacher: “mathematical discussions are important”, “Cubícula helps developing students’ imagination and to find out ways to calculate volume of solids”, “they also had opportunities to understand the formulae and use them”. After the course T1’s resources system was enriched in each of its components. The material component was enriched by the integration of the two files he introduced; the mathematical component by his making an explicit distinction between capacity and volume, and the didactical component changed considerably as he motivated his students to participate, used small group work, work with the computer, work with worksheets, and he made sure to close the session. We found new rules of action for T1: “compare operations with concrete materials to those done with Cubícula”; “use decomposition of figures by levels of cubes to calculate volume of solids” “avoid counting the cubes without reflection”, and “students need reflection to relate work with ’Cubícula’ with formulae to calculate volumes”. After observation of another class the following year, we realised some of those rules had become usages and that Cubícula was clearly integrated into the document. We were able to infer some operational invariants: “construction of

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solids with concrete material and with ’Cubícula’ helps students understand volume formulae” and “formulae are important, students need to know how to use them”. In summary, even when T1’s emphasis on the use of mathematical formulae remained, their role changed from being the central focus of the didactical approach to be the result of joint exploration with concrete materials and the program, which became part of the document. However, we consider that in T1’s document the resources system and the associated rules of action had not stabilised yet.

13.4.2 Teaching Difficult Mathematical Topics Teacher 2 has been a teacher for 10 years. After reviewing our data, we decided to exemplify her documentational genesis by analysing her lesson on combinatory. At first, T2 didn’t feel confident with lessons on these subjects. The first time T2 used Enciclomedia’s resources to teach combinatory, we observed her lessons and interviewed her after the class. Beforehand, she had looked at the program “Tree diagrams”, which allows users to construct tree diagrams level by level and in which users can highlight different paths in the diagram using different colours, to facilitate counting processes. Even when T2 liked the program, she still felt hesitant about her mathematical understanding: “. . . During the weekend I worked with ‘Tree diagrams’, . . . but also reviewed Encarta (the encyclopaedia, which is included in Enciclomedia) to look at some definitions about. . . combinations or permutations. . . I still don’t understand them well. . . for example the differences when there are repeated elements. . . using the program has helped me . . .”. During the lesson, we observed that she relied completely on the programs. She brought her notes to the classroom and repeated what she had read. She had prepared a PowerPoint slide showing a tree diagram she had constructed and explained it to the students. Then she used “Tree diagrams” together with activities from the textbook. During the interview she commented: “Students had many difficulties, . . . I did not feel comfortable”. T2’s lesson plan, the PowerPoint slide, the textbook and “Enciclomedia” constitute the material components of the first document she constructed. The mathematical component was limited to what she had read before the lesson. The didactical component consisted in explanations of the importance of considering all the possibilities, and the way tree diagrams can help organize the information and count. The rules of action we inferred are: “students need explanations in order to work with ‘Tree diagrams’, this topic is too difficult”, “consider both types of problem situations: different and repeating elements”, “present advantages of tree diagrams as a representation tool”, “explain the relationship with products to count the different possibilities”. Some operational invariants we detected are: “there is no understanding without explanation”, “’Tree diagrams’ is needed for students to understand explanations”, “it is important to differentiate between different combinatory situations”. After she underwent training on the use of “Enciclomedia”, T2 started feeling more comfortable with this topic. When we observed her during the third

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year, we could appreciate how her planning was based more on “Enciclomedia’s” teaching guides. She mentioned: “It is important to work with this activity a lot in order to learn this topic, . . . to help students in the classroom”. She started the lesson by posing the questions suggested in the teaching guides. She then asked students to work with the program while she worked on the same problems on the blackboard and discussed, with the whole class, the usefulness of tree diagrams and products. During the interview she said “I worked hard with the program. . ., together with the teaching guides I was able to reflect and learn many things that I hadn’t understood before. . . ” The material component of T2’s second document consisted of her new lesson plan, the teaching guides from “Enciclomedia”, the blackboard, the digital program and the textbook. The mathematical component was enriched by her reflection on combinations and permutations. Its didactical component changed. After working with the program, T2 gave students more opportunities to work and think by themselves, asked important questions to help them reflect on the need of organizing information to count, and on the advantages of the use of tree diagrams. Some of her rules of action changed: “asking questions to students helps them reflect on their actions”, “exploration with ‘Tree diagrams’ favours understanding”. We found in her work, a new operational invariant: “‘Tree diagrams’ promotes reflection and understanding”. In the interview she said she had started using the paperboard to record students’ strategies and discussions so she could review them and make future decisions. T2 follows now, 5 years after her first training, a specific plan for the class of situations where she teaches combinatory: before the lesson she gives students a homework task with the intention of reviewing students’ knowledge. She reviews her plan for the lesson and works with “Tree diagrams” (Fig. 13.2) at home. She uses students’ strategies from the homework task to discuss advantages and limitations

Fig. 13.2 Tree diagrams

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of different methods and stresses the usefulness of tree diagrams. Students draw a tree diagram for the homework task on their notebooks. T2 then uses the program to review their work, asks students interesting questions about counting, and relates them to tree diagrams. Later, T2 introduces other tasks for students to work with “Tree diagrams”, first with different elements then with repeated elements and organises whole class discussion about different counting strategies with different types of sets. T2 added the use of paperboards and notes on her lesson plan relating activities with competencies students need to develop and specific difficulties she has observed when teaching combinatory to the material component of her document. The mathematical component did not change and the didactical component was enriched by adding a previous activity and by interweaving small group work, whole class discussion and institutionalisation of the results obtained. We found in T2’s work specific usages demonstrated by new rules of action that were confirmed by her interview’s responses: “using a homework problem before the lesson provides a means to recover students’ ideas and strategies”, “start by combinations of different elements reviewing what students did before”, “let students work in groups and discuss their strategies before using ‘Tree diagrams’”, “introduce products as a way to count”. We also inferred from her work new operational invariants: “students need to think on problems by themselves before they work with others or the whole group”, “the program helps students organize their thinking”, “counting has to be related with the use of products”.

13.4.3 Integrating Resources into a Document T3 became familiar with the first version of Enciclomedia. She reported a particular interest in a program called “The Balance” (Fig. 13.3), which shows a problem situation where scales need to be balanced by using fractions. The program provides the users with automatic feedback that helps them in identifying which parts of the mobile toy are balanced and which are not. Before the lesson, T3 prepared, with the help of a colleague, some preliminary activities to promote reflection on fractions and decimal numbers to be carried out with “The Balance”. On these activities, students were asked to compare pairs of numbers and find equivalent expressions of a given one. Later, she worked with textbook problems using “The Balance”. These resources constitute the material components of her first document. Other resources include the conversations she had with her colleagues and “Enciclomedia’s” teaching guides which T3 read beforehand. From the first year of her involvement with Enciclomedia and after each lesson, T3 wrote down notes on her logbook about the development of her classroom when she used these digital resources (see Chapter 2). She used notes to refine the activities and the teaching sequences for the following year. After the first experiences using “The Balance”, she noted that, without a proper sequencing of activities and the introduction of whole class discussions at appropriate times, students tended to

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Fig. 13.3 The balance

move too fast and did not reflect while working with the program. As a result, some of them got stuck and could not solve the problems in the textbook, with or without the program. T3 then prepared a worksheet with specific problems for students and in interviews she said she spent more time discussing the mathematics with the whole group: T3 – During the first year I just used a couple of examples to find equivalent expressions with “The Balance”, asking them to balance the scales before getting to the textbook problems . . . I realized that I moved too soon, and I let them explore the most difficult problems on their own . . .. So for the second year I gave them a worksheet to work with . . . and we had long discussions on how equivalent expressions could be found and on why they were equivalent.

The worksheet and the new activities involved more mathematical concepts than those included in the textbook. The mathematical component expanded as T3 refined her activities. The didactical component related to this new document also included activities which were not present before the introduction of “Enciclomedia”. T3 mentioned having to change her teaching strategies to get students’ attention since they got absorbed by the computer program. Additionally, introducing a particular sequence of activities before starting the work on the textbook also meant a didactical development. In the end, after having tried this for two different school years, T3 said she thought the experience had been useful both for students and for her as “more students were able to solve the really difficult exercises on the book”. In this initial 2-year period, possible rules of action related to the class of situations “working with equivalent fractions and with operations with fractions” include “developing very specific sequences of activities to be used both

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Fig. 13.4 The number line

with ‘The Balance’ and with paper and pencil and later using the program to work with problems in the textbook” and “allowing students to work on their own and having whole class discussions regarding the mathematics involved”. As time went by, T3 became familiar with other programs such as “The number line” (Fig. 13.4), which also includes work with fractions. She mentioned having used it in class and having observed that students became excited when playing with it. She mentioned that she used this program “not linked to a particular chapter in the textbook, but whenever I wanted them to work with fractions and have fun”. At a mathematics education seminar, T3 heard colleagues talking about the different uses of fractions, she read about it and decided to analyse what the different uses both in “The Balance” and in “The number line” were. She commented: “there are different things I can do with each program, even when they are both related to fractions . . . I wanted to find a way to use them in a more productive way . . . like . . . when to use which, and how”. She developed a teaching sequence including the use of both programs, and developed specific activities based on the previous ones. This exemplifies a third phase of documentation for T3, where she drew from a variety of resources in order to create a teaching sequence that addressed the learning of fractions from a wider perspective: “I was aware that fractions are used in different ways, and that students must be supported in learning all the different uses, so I thought of strategies using the ‘The Balance’, ‘The number line’ and specific activities that could enhance this learning.” Even when the mathematical components still included the same concepts, the didactical aspect was widely modified by the introduction of the different uses of

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fractions into the sequence: “I think that, having to find a number between consecutive integers might help them think of a fraction as number in itself, not as an operation or as a result of an operation”. Finally, at a later stage, when the number of Enciclomedia resources related to fractions increased significantly, T3 decided to reconsider her teaching sequence, to include the use of more digital programs, keeping her focus on the teaching of the different uses of fractions. She analysed if the new programs could contribute to this and how, and decided to include them in her teaching sequence. She developed a long sequence including all the programs to address different uses of fractions in different ways. T3 decided to include problems from different textbooks, and developed new worksheets for students. The mathematical component stayed the same. Didactically, the sequence was longer, and different: “I am taking more time, I am using more resources in order to go deeper into each one of the uses of rational numbers. I am making them reflect more on their work with the programs by posing more difficult problems I am getting from other books, but also I am including operations”. In the end, for a class of situations linked to “designing activities for the teaching and learning of fractions”, T3 developed a complex set of action rules including: “posing particular problems and exercises for the different uses of fraction”, “using the programs in ‘Enciclomedia’ related to fractions in a specific order and for a long time, so that students become familiar with the different uses of fraction in different contexts and deepen their knowledge”, and “combining digital resources with worksheets and a variety of word problems in a precise way”. The operational invariants in this latest document might comprise “working with the different uses of fractions in different ways, both using digital an non-digital resources enhances students’ learning”, “students have to be able to solve a variety of problems related to fractions in order to deepen their knowledge of this concept and these include both complex activities and drill exercises” and “an effective teaching sequence that allows students to learn fractions is necessarily long, students need time to develop their knowledge”. T3 has been teaching this sequence for a few years now. Her document has evolved from specific activities to a well structured didactical sequence. We believe she has developed a scheme of utilisation, since patterns in the way she uses the digital resources from Enciclomedia together with other resources can be observed.

13.5 Discussion and Conclusions We now explore the findings described above, in terms of the ways in which the introduction of Enciclomedia has shaped the development, and organization, of documents of the three teachers. In the development of documents, both instrumentalisation and instrumentation (see Chapter 2) occur “naturally”. Using these ideas to analyse the differences found in teachers can be illuminating. Similarly

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to Drijvers (Chapter 14) we observed that the use of technology together with the documentational process was paralleled by a process of professional development with a greater focus on the mathematics involved than that showed when the process started. The teachers described in this chapter changed their teaching practice as a result of using Enciclomedia. Changes, however, have not been the same for all three. Each developed different kinds of documents in the process of incorporating the digital resources into their activities, and in this process they transformed the material resources from Enciclomedia by using them in particular ways. T1’s last document consists of material resources which included elements that were not present before such as concrete materials and the program Cubicula together with usages described as rules of action in the paragraphs above, which are linked through operational invariants related to the contribution of concrete materials and to the understanding and importance of volume formulae. Enciclomedia enabled T1 to include different teaching strategies and actions in order to achieve this desired outcome. Contact with the program did change T1’s actions in the classroom, together with the documents he produced, in a way he thought students learned the mathematical content. T1 used Cubícula as a “means to an end”, showing the program’s features such as rotation of solids as a way for justifying and give meaning to the algebraic formulae employed: “I use it so that I can show them where the formulae come from”. This, of course, is not the only way that the program can be used, but it is the one T1 found useful to achieve his goals. We have shown how T2 developed, through her years of experience with Enciclomedia, documents for a mathematical topic she was originally afraid to teach and had previously avoided altogether during her lessons. Throughout the process of instrumentation, “Enciclomedia” shaped not only her teaching strategies and group dynamics: “I use it a lot for whole class discussions, . . .students really become engaged, . . . they talk a lot about counting and how many possibilities there are, so it is not just me asking questions and them answering, . . .”, but also the mathematical content she included and, moreover, her own mathematical understanding became enriched. “I really couldn’t understand a thing about combinations and permutations, but the teaching guides and the program help, because I can do it several times . . . Now I can teach these chapters in the textbook”. She used the program not only to justify mathematical procedures but as spaces for exploration, in which students were allowed to build their own tree diagrams. Finally T3 also showed ways of working different from her acting as a teacher before she started using Enciclomedia. She developed a documentation system that comprised a number of resources that included work with several “Enciclomedia” programs and with the different uses of fraction, both through complex word problems and practice exercises. The variety of resources T3 found in Enciclomedia related to fractions allowed her to develop a long teaching sequence in which different kinds of activities for the different uses of fractions were involved. It became clear that the integration of resources into the initial document followed a specific purpose. In the latest form of the document this purpose was reflected in the already described operational invariants.

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This constituted new kinds of actions for T3, since previously she had followed the textbook approach which did not include long sequences devoted to specific mathematical content and which presents mainly complex problems for students to solve without much practice or algorithms. “Through this sequence I included also more drill and practice exercises and I stayed with the same topic for a longer period of time”. Her understanding of the didactical component and her understanding of mathematics learning were also deepened “I am more aware of certain things now, because when using the programs I ask children about their strategies and some things emerge, which I did not consider before. They really can use many interpretations and fraction representations; they use several when solving one problem or when explaining. I was not aware of this, maybe because the programs give them a lot of freedom, they are so enthusiastic and they employ all the resources available to them, so they can show me, in a way, what they are thinking”. T3 used the programs both for exploration and for solving problems and exercises, which enriched her documentational genesis. The instrumentalisation process also involved using the programs for “working with the different uses of fractions in different contexts.” In the end T3 produced a complex document which included material resources, usages and operational invariants which had not been present before. The changes we have described, including the different documentational geneses, cannot be conceived as simply the result of introducing Enciclomedia. The teachers’ personal history, as well as institutional affordances and constraints and external circumstances have also had an important influence. Opportunities for reflection on their own teaching practice have been, from our perspective, crucial, as was also found by Drijvers. T1, for example, had to discuss his teaching plans with colleagues from the professional development course. T2, was able to discuss some of her ideas with the “Enciclomedia’s” mathematics programs developers, and through these discussions she was able to try out the materials and develop lesson plans for difficult mathematical topics. We also consider that the opportunities she had of working with Enciclomedia at home and her own persistence and hard work were very important factors in her documentational genesis. Possible factors which might be involved in the documentational genesis in the case of T3 include the integration of new material resources, both digital programs from “Enciclomedia” and problems T3 found in different textbooks. Her participation in a mathematics education seminar, discussions with colleagues and her reading of mathematics education literature also played an important role in her development as a teacher and in her creation of more complex and rich documents for her teaching. We can see, therefore, that the introduction of the program Enciclomedia, can influence teaching practices and documentational genesis in powerful and different ways, especially when it is accompanied by reflection and discussion with fellow teachers and researchers. At the same time, each teacher used the programs in a unique way. They can be used, for example, for exploration of mathematical ideas, for justification and for solving exercises. It would be useful to include discussions on the different uses that can be afforded by the programs during training workshops, together with opportunities for extensive reflection on teaching practices.

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As we have found that all three teachers greatly benefitted from discussions both with peers and other expert colleagues, we also contend that creating working groups inside and outside the schools constitute an essential element in professional development programs. As seen through this chapter, the process of documentational genesis, that includes resources from Enciclomedia, takes place over long periods of time, and cannot be observed or studied without the consideration of the many aspects that are part of it. For several years we have been involved in research on the use of Enciclomedia mathematics’ resources by teachers and students. We have studied students’ learning with some specific resources and some aspects of the resources themselves. Throughout this time we have observed many lessons where “Enciclomedia” was used together with other resources, and have been involved in training programs for different groups of teachers. All these activities have involved the collection of large amounts of data. Although we have gained knowledge of and experience about the use of these resources, reviewing our data from the point of view of documentation framework, and locating interesting examples of documentational genesis, enabled us to highlight some aspects related to the evolution of teachers’ work that we had not previously considered. We believe that these new aspects that were brought to our attention are important in understanding how teachers’ knowledge grows not only through development programs but also through their own work and interests. This is also relevant and important for future research. The use of the new theoretical framework of documentational genesis, in a context different from the one it had been originally created, may be seen as a contribution of this work to research about teachers, and to provide evidence of its pertinence and usefulness. Many questions remain unanswered. For example, in future research it would be interesting to explore patterns in the resources, their usages and operational invariants that might be found when investigating documentational genesis with a greater number of teachers. Additionally, it would be important to deepen our understanding of the processes of documentation when using different resources from “Enciclomedia” and how the students’ use of Enciclomedia and the textbook influence teachers’ documentation process as Rezat (Chapter 12) explored in his study. Acknowledgements This project was partially supported by Asociación Mexicana de Cultura A.C. and the Instituto Tecnológico Autónomo de México.

References Artigue, M. (2004). The integration of computer technologies in secondary mathematics education. In J. Wang & B. Xu (Eds.), Trends and challenges in mathematics education (pp. 209–222). Shanghai: East Normal University Press. Assude, T. (2008). Teachers’ practices and degree of ICT integration. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the fifth congress of the European society for research in mathematics education (pp. 1339–1349). Larnaca: CERME 5. Retrieved 16, September, 2009, from http://ermeweb.free.fr/CERME5b/

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Chávez, Y. (2007). Enciclomedia en la clase de matemáticas. Tesis de Maestría en Desarrollo Educativo en la Línea de Especialización en Educación Matemática. Retrieved 20, August, 2009, from http://biblioteca.ajusco.upn.mx/pdf/24746.pdf Díaz de, C. R., Guevara, N., Latapí, S., Ramón, B., & Ramón, C. (2006). Enciclomedia en la práctica. Observaciones en veinte aulas 2005–2006. Centro de investigación educativa y actualización de profesores A.C. México. Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for teachers? Educational Studies in Mathematics, 71, 199–218. Hegedus, S., & Moreno-Armella, L. (2009). The transformative nature of “dynamic” educational technology. ZDM, The International Journal on Mathematics Education, 41(4), 397–398. Holland, I., Honan, J., Garduño, E., & Flores, M. (2006). Informe de evaluación de Enciclomedia en: F. Reimers (Coord.) Aprender más y mejor. Políticas, programas y oportunidades de aprendizaje en educación básica en México. México: FCE/ILCE/SEP/Harvard Graduate School of Education. Hoyles, C., & Lagrange, J. B. (Eds.). (2010). Mathematics education and technology-rethinking the terrain. The 17th ICMI study (Vol. 13). New ICMI Study Series. Dordrecht: Springer. Kynigos, C., & Argyris, M., (2004). Teacher beliefs and practices formed during an innovation with computer-based exploratory mathematics in the classroom. Teachers and Teaching: Theory and Practice, 10(3), 247–273. Lagrange, J. B., Artigue, M., Laborde, C., & Trouche, L. (2003). Technology and mathematics education: A multidimensional study of the evolution of research and innovation. In A. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second international handbook of mathematics education (pp. 239–271). Dordrecht: Kluwer. Lozano, D., Sandoval, I., & Trigueros, M. (2006). Investigating mathematics learning with the use of computer programmes in primary schools. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 89–96). Prague: PME. Mariotti, M. A. (2002). Influence of technologies advances in students’ math learning. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 757–786). Mahwah, NJ: Lawrence Erlbaum Associates Publishers.. Rabardel, P. (1999). Eléments pour une approche instrumentale en didactique des mathématiques. In M. Bailleul (Ed.), Actes de l’école d’été de didactique des mathématiques (pp. 203–213). IUFM de Caen. Rabardel, P. (2005). Instrument subjectif et développement du pouvoir d’agir (subjective instrument and development of action might). In P. Rabardel & P. Pastré (Eds.), Modèles du sujet pour la conception. Dialectiques activités développement (pp. 11–29). Toulouse, France: Octarès. Roschelle, J., Shechtman, N., Tatar, D., Hegedus, S., Hopkins, B., Empson, S., et al. (2010). Integration of technology, curriculum, and professional development for advancing middle school mathematics: Three large-scale studies. American Educational Research Journal, 47, 833–878. Sagástegui, D. (2007). Usos y apropiaciones del programa Enciclomedia en las escuelas primarias de Jalisco. Memorias del Congreso del Consejo Mexicano de Investigación Educativa. Retrieved 16, September, 2009, from http://www.comie.org.mx/congreso/memoria/ v9/ponencia/ato7/PRE118953481.pdf. Sandoval, I. (2009). La geometría dinámica como una herramienta de mediación entre el conocimiento perceptivo y el geométrico. Educación Matemática, 1(27), 5–27. Swan, K., Schenker, J., & Kratcoski, A. (2008). The effects of the use of interactive whiteboards on student achievement. In J. Luca & E. Weippl (Eds.), Proceedings of world conference on educational multimedia, hypermedia and telecommunications (pp. 3290–3297). Chesapeake, VA: AACE. Trigueros, M., & Lozano, M. D. (2007). Developing resources for teaching and learning mathematics with digital technologies: An enactivist approach. For the Learning of Mathematics, 27(2), 45–51.

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Trigueros, M., Lozano, M. D., & Lage, A. (2007). Development and use of a computer based interactive resource for teaching and learning probability. Practice of primary classrooms, in International Journal for Technology in Mathematics Education, Special ICTMT7 issue, 4(13), 205–211. Trigueros, M., & Sacristán, A. I. (2008). Teachers’ practice and students’ learning in the Mexican programme for teaching mathematics with technology. International Journal and Continuing Engineering Education and Life Long Learning, 15, 5–6.

Chapter 14

Teachers Transforming Resources into Orchestrations Paul Drijvers

14.1 Introduction Several chapters of this book address the fact that teachers nowadays have access to a myriad of both material and electronic knowledge resources for mathematics teaching (e.g. Chapter 1). These resources can be accessed through technological means and are available on the internet (Bueno-Ravel & Gueudet, 2007). However, resources do not transform teaching practices in a straightforward way. Documentational work, as part of the teacher’s process of documentational genesis, is needed (Chapter 2). Several studies show that teachers may perceive difficulties in orchestrating mathematical situations which make use of technological tools and resources, and in adapting their teaching techniques to situations in which technology plays a role (Doerr and Zangor, 2000; Lagrange and Degleodu, 2009; Lagrange and Ozdemir Erdogan, 2009; Monaghan, 2004; Sensevy, SchubauerLeoni, Mercier, Ligozat, & Perrot, 2005). Also, different teachers may adapt the same set of resources into quite different teaching arrangements (Chapter 9). As Robert and Rogalski (2005) point out, teachers’ practices are both complex and stable. Building on this, Lagrange and Monaghan (2010) argue that the availability of technological resources amplifies the complexity of teaching practices and, as a consequence, challenges their stability. It is not self-evident that techniques and orchestrations which are used in “traditional” settings can be applied successfully in a technological-rich learning environment. A new repertoire of orchestrations, instrumented by the available tools, has to emerge. This involves professional development of the teacher, in which both professional activity and professional knowledge may change. This process of transforming sets of technological and other resources into orchestrations is the topic of this chapter, which focuses on the question of how teachers orchestrate the use of digital resources in teaching practice and how these orchestrations change over time.

P. Drijvers (B) Freudenthal Institute, Utrecht University, PO Box 85170, 3508 AD Utrecht, The Netherlands e-mail: [email protected]

G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_14,  C Springer Science+Business Media B.V. 2012

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14.2 Theoretical Framework For describing and investigating teacher practice and professional development, several models are available (e.g. see Chapter 5; Ruthven & Hennessy, 2002). The main theoretical perspective used in the present chapter is the notion of instrumental orchestration. It is widely acknowledged that student learning needs to be guided by the teacher through the orchestration (McKenzie, 2001) of mathematical situations (Mariotti, 2002). For example Kendal and colleagues (Kendal & Stacey, 2002; Kendal, Stacey, & Pierce, 2004) showed that teachers privilege certain techniques for using technological tools over others and, in this way, guide the students’ acquisition of tool mastery and their learning processes. To describe the teacher’s role, Trouche (2004) introduced the metaphor of instrumental orchestration. An instrumental orchestration is defined as the teacher’s intentional and systematic organization and use of the various artefacts available in a – in this case computerized – learning environment in a given mathematical task situation, to guide students’ instrumental genesis (Trouche, 2004). Within an instrumental orchestration we distinguish three elements: a didactic configuration, an exploitation mode and a didactical performance. A didactical configuration is an arrangement of artefacts in the environment, or, in other words, a configuration of the teaching setting and the artefacts involved in it. In the musical metaphor of orchestration, setting up the didactical configuration can be compared with choosing musical instruments to be included in the band, and arranging them in space so that the different sounds result in a polyphone music, which in the mathematics classroom might come down to a sound and converging mathematical discourse. An exploitation mode is the way the teacher decides to exploit a didactical configuration for the benefit of his or her didactical intentions. This includes decisions on the way a task is introduced and worked through, on the possible roles of the artefacts to be played, and on the schemes and techniques to be developed and established by the students. In terms of the metaphor of orchestration, setting up the exploitation mode can be compared with determining the partition for each of the musical instruments involved, bearing in mind the anticipated harmonies to emerge. A didactical performance involves the ad hoc decisions taken while teaching on how to actually perform in the chosen didactic configuration and exploitation mode: what question to pose now, how to do justice to (or to set aside) any particular student input, how to deal with an unexpected aspect of the mathematical task or the technological tool, or other emerging goals. In the metaphor of orchestration, the didactical performance can be compared to a musical performance, in which the actual interplay between conductor and musicians reveals the feasibility of the intentions and the success of their realization. Didactical configurations and exploitation modes were introduced by Trouche (2004). As an instrumental orchestration is partially prepared beforehand and partially created “on the spot” while teaching, we felt the need to add the actual didactical performance as a third component (Drijvers et al., 2010). Establishing the didactical configuration has a strong preparatory aspect: often, didactical configurations need to be thought of before the lesson and cannot easily be changed

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during it. Exploitation modes may be more flexible, while didactical performances have a strong ad hoc aspect. Like every metaphor, the metaphor of instrumental orchestration has its limitations. If one think of a teacher as a conductor of a symphony orchestra consisting of highly skilled musicians who enters the concert hall with a clear idea on how to make the musicians play Beethoven the way he himself reads the century-old partition, one may feel uneasy with it. However, if one thinks of the class as a jazz band (Trouche & Drijvers, 2010) consisting of both novice and more advanced musicians, and the teacher being the band leader who prepared a global partition but is open for improvization and interpretation by the students, and for doing justice to input at different levels, the metaphor becomes more appealing. It is in the latter way that we suggest to understand it. We also point out that the metaphor of orchestration in fact includes multiple roles for the teacher, who may act as a composer, as an orchestrator, as a director, and as a conductor. Earlier research focused on the identification of orchestrations within wholeclass technology-rich teaching. Drijvers et al. (2010) identified six types of such orchestrations, termed Technical-demo, Explain-the-screen, Link-screen-board, Discuss-the-screen, Spot-and-show and Sherpa-at-work, with the following global descriptions. 1. The Technical-demo orchestration concerns the demonstration of tool techniques by the teacher. It is recognized as an important aspect of technology-rich teaching (Monaghan, 2001, 2004). A didactical configuration for this orchestration includes access to the technology, facilities for projecting the computer screen, and a classroom arrangement that allows the students to follow the demonstration. As exploitation modes, teachers can demonstrate a technique in a new situation or task, or use student work to show new techniques in anticipation of what will follow. 2. The Explain-the-screen orchestration concerns whole-class explanation by the teacher, guided by what happens on the computer screen. The explanation goes beyond techniques, and involves mathematical content. Didactical configurations can be similar to the Technical-demo ones. As exploitation modes, teachers may take student work as a point of departure for the explanation, or start with their own solution to a task. 3. In the Link-screen-board orchestration, the teacher stresses the relationship between what happens in the technological environment and how this is represented in conventional mathematics of paper, book and blackboard. In addition to access to the technology and projection facilities, a didactical configuration includes a blackboard and a classroom setting such that both screen and board are visible. Similarly to the previously mentioned orchestration types, teachers’ exploitation modes may take student work as a point of departure or start with a task or problem situation they set themselves. 4. The Discuss-the-screen orchestration concerns a whole-class discussion about what happens on the computer screen. The goal is to enhance collective instrumental genesis. A didactical configuration once more includes access to the technology and projecting facilities, preferably access to student work and a

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classroom setting favourable for discussion. As exploitation modes, student work, a task or a problem or approach set by the teacher can serve as the point of departure for student reactions. 5. In the Spot-and-show orchestration, student reasoning is brought to the fore through the identification of interesting student work during preparation of the lesson, and its deliberate use in a classroom discussion. Besides previously mentioned features a didactical configuration includes access to the students’ work in the technological environment during lesson preparation. As exploitation modes, teachers may have the students whose work is shown explain their reasoning, and ask other students for reactions, or may provide feedback on the student work. 6. In the Sherpa-at-work orchestration, a so-called Sherpa student (Trouche, 2004) uses the technology to present his or her work, or to carry out actions the teacher requests. Didactical configurations are similar to the Discuss-the-screen orchestration type. The classroom setting should be such that the Sherpa student can be in control of using the technology, with all students able to follow the actions of both Sherpa student and teacher easily. As exploitation modes, teachers may have work presented or explained by the Sherpa student, or may pose questions to the Sherpa student and ask him/her to carry out specific actions in the technological environment. The above categorisation, with three more teacher-centred and three more student-centred orchestrations, resulted from a study on the use of applets for the exploration of the function concept in grade 8, and emerged from observation of three teachers in a relatively guided situation (Drijvers et al., 2010). Of course, from these limited data from a specific context, we cannot claim completeness. Rather, we wonder how specific this categorisation is with respect to the type of technology, the mathematical topic, the whole class teaching format, the level and age of the students, and the amount of guidance teachers were provided with. Therefore, the goal of the study presented in this chapter is to investigate in another teaching context in which types of orchestrations teachers transform the available technological resources and how these results relate to the above categorization. Also, we are interested in the professional development that takes place while teachers include technological resources into their teaching. This professional development is a process of change, which involves both the teachers’ own instrumental genesis (Drijvers and Trouche, 2008) as well as documentational genesis (Gueudet and Trouche, 2009; Chapter 2). Therefore, we want the present study to shed light on the change processes that occur when teachers engage in an experimental setting.

14.3 Research Setting The research was carried out in the context of a pilot initiated by the publisher of the main Dutch textbook series for secondary mathematics education. The publisher, seeking for ways to improve their product and to integrate technology, decided to

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offer to their customers’ schools an online, interactive version of a chapter on algebraic skills for grade 12, the final year of pre-university secondary education. The skills addressed include solving equations and recognising specific equation types and corresponding solving strategies. For this online module, the Freudenthal Institute’s Digital Mathematics Environment (DME)1 was used. DME is a web-based environment which integrates a content management system, an authoring tool and a student registration system, and which already contains content in the form of an impressive amount of applets and modules (Bokhove and Drijvers, 2010). The new module for this pilot was designed by the authors of the textbook series, with support of the Freudenthal Institute DME experts. The module includes tasks as well as video clips with elaborated examples. The tasks provide feedback to students’ answers, with decreasing feedback levels as the module advances. A pdf file of the original textbook chapter was also made available online, with embedded links to the new online activities.2 Figure 14.1 shows a part of the book file on the left, and one student’s work in the digital environment on the right. The book text includes a reference to the online module and the task to solve two equations. In the right screen, the student makes a mistake in the last line, and gets feedback saying “This step contains both correct and incorrect parts. Remove or replace the incorrect parts”. A message from the publisher generated reactions by 69 teachers, who volunteered to join this pilot teaching sequence. These teachers were provided with online

Fig. 14.1 Screen shots from book (left) and digital environment (right)

1 2

See www.fi.uu.nl/dwo/en/. The module (in Dutch) is available through http://www.fi.uu.nl/dwo/gr-pilot/.

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guidelines for the use of the module and with support to create login accounts for their students, and so the pilot begun. For the students, the set of resources in this pilot includes the regular textbook, the online book chapter, digital modules including feedback and video clips, and the traditional resources such as paper and pencil and calculator. As the work is stored on a central server, students can access, revise and continue their work at any time and from any place with internet access. For the teacher, the set of resources is similar, but provides the additional option of access to student work. Overviews of whole class results as well as individual student work can be monitored by the teacher through the internet and can be used in whole-class teaching settings, which seem appropriate for discussing interesting or erroneous strategies that students applied in their homework.

14.4 Methods The research methods include a case study focusing on one teacher, a survey among all 69 participating teachers, and interviews with five teachers. The case study was carried out in two classes of one of the pilot schools, a school in a small, prosperous town in the Netherlands with mainly “white” student intake. Both classes, with 30 and 14 students, respectively, were taught by the same, experienced teacher, who was close to his retirement. This teacher initially volunteered for the pilot, but later intended to step back, because, on the one hand, computer facilities in school were insufficient and, on the other hand, his students objected to the idea of practicing algebraic skills with the computer, whereas they would need to master them with paper and pencil in the national central examination. Concerning the first issue, we were able to offer him a loan set of 30 netbook computers for the period of the teaching sequence. On the students’ concerns, we convinced the teacher that practicing skills with computer tools was expected to directly transfer into better by-hand skills. He spoke again with his students, and they accepted to participate in the pilot. During the period of the pilot, this teacher had a heavy teaching load, with twenty-six 50-min lessons a week to teach, and an additional remedial teaching practice at home. A technical assistant was available in school to set-up the classes with the netbook computers, and to make other practical arrangements such as charging the batteries, etc.. Most of the lessons (23 out of 36 during an 8-week period) were observed and videotaped. The video registration was done by a mobile camera person, who followed the teacher very closely during individual teacher--student interactions, so as to capture all speech and screens. Video data were completed with field notes from observations. A final interview with the teacher took place after the teaching sequence. Data analysis took place with software for qualitative data analysis3 and focused on the identification of orchestrational aspects of the teaching. 3

We used Atlas ti, www.atlasti.com

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The case study was set up to enable us to address the research question in a qualitative and in-depth manner. To complement the very specific data from the case study with a more general view on the orchestrations used in this teaching sequence, a survey among all participating teachers was set up. It consisted of two online questionnaires, one before and one after the teaching sequence. The response was 49 out of 69 for the pre-questionnaire, and 41 out of those 49 for the post-questionnaire. Non-response was caused by the fact that not all teachers who originally volunteered for the pilot really started their participation, and that some of the teachers who filled in the pre-questionnaire did not start either, or stopped the pilot before bringing it to an end. Some of them sent messages by e-mail, indicating reasons such as time constraints, lesson cancellation because of illness or other unforeseen circumstances. To bridge the gap between the detailed case-study data and the global survey data, interviews were held after the teaching sequence with five teachers, including the one engaged in the case study. These interviews had a semi-structured character, the post-questionnaire providing the backbone of the interview. We will only use interview results to illustration the findings.

14.5 Results 14.5.1 Results from the Case Study The results of the case study show that one particular orchestration, which we call Work-and-walk-by, was highly dominant. The didactical configuration and the corresponding resources basically consisted of the students sitting in front of their netbook computers, with wireless access to the online module and their previous work as well as to the textbook chapter in pdf format. In addition to this, a blackboard or whiteboard allowed the teacher to write down additional explanations. A data projector showing the online environment was available in most lessons. As exploitation mode, the students individually worked through the online module on their netbook computers, and the teacher walked by and sat down with students to answer questions and eventually monitor the students’ proceedings (see Fig. 14.2). As a reaction to student questions, the teacher in some cases went to the blackboard to write down an algebraic explanation or technique, but still speaking to the individual student who had raised the issue. The data projector was hardly used. Concerning the didactical performance, the initiative for teacher--student interaction was taken by the student in almost all cases. If an interaction with a student led to a new insight for the teacher, such as an understanding of a technical issue, he sometimes went back to students whom he had previously spoken to on a similar issue, as to disseminate the news. An interesting aspect of this Work-and-walk-by orchestration concerns the determination of students’ difficulties. If a student has a question while the teacher walks by, the latter is faced with the issue of where the heart of the problem lies: is it

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Fig. 14.2 The teacher (left) helping a student (right) individually

a lack of the student’s algebraic understanding or skill? Is it a technical problem caused by the student, for example a mistake in entering an expression? Or is it a limitation of the online module, which in some cases gave inappropriate feedback, or was very strict in expecting a specific answer, such as 3.5 instead of 7/2? As the teacher’s resource knowledge was limited, he was not aware of the peculiarities of the online module. Therefore, determination in some cases was difficult and took time. Mismatches between student problems and teacher reactions could be observed, but became less frequent as the pilot advanced. As an example of such a mismatch, one of the students was stuck when she had rewritten an equation to e log(x) = −5. The teacher understood this as a mathematical issue and walked to the whiteboard. He wrote down 2 log 8. Teacher: 2 log 8, what is that? Student: 3. Teacher: Why? Student: Because 2 to the power 3 is 8. Then the teacher continued with e log(x) = 5, which was solved by the student as well. Walking back to her it turned out that her problem was not a mathematical one, but rather how to enter e, the base of the natural logarithm, into the digital environment. The teacher at the end solved this, after consulting another student. Instead of focusing on the meaning of e log(x), he might have considered the technical issue at once, which shows that determination difficulties can lead to interactions that are longer and less efficient than needed. The previous episode shows that determination in some cases was hindered by the technical issues the teacher encountered. Some technical problems, such as students who forgot their login code or the web address of the online module, or

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netbooks which lack battery power or fail to connect to the wireless internet, were dealt with by the technical assistant, who attended the classes most of the times, and always in the first part of the teaching sequence. Technical problems within the online module, however, often appeared in the individual student–teacher interactions during the Work-and-walk-by orchestration. As the teacher himself was not familiar with the module, he often was unable to solve students’ problems, which led to uncertainty about whether it was a mathematical mistake or a technical problem that caused the technology to report an error. Compared with the Technical-demo orchestration described earlier, there was little technical guidance or attention to students’ instrumental genesis, even if he used to “spread the news” in individual interactions, as soon as one of the students solved a technical issue or found a convenient technique. The analysis also shows that such technical complications interfering with the mathematical content of the student--teacher interactions became less frequent as the teaching sequence advanced. This Work-and-walk-by orchestration took at least 90% of the lesson time in the lessons we observed, and remained dominant throughout the pilot teaching sequence as a whole without much variation; still, some changes over time in its didactical performance, and in the type of teacher–student interactions in particular, could be noticed. First, as the teaching sequence advanced and he found out how it worked, the teacher used the data projector to show the overall advancements of the students, so that each individual student could monitor if he or she was more or less on schedule (see Fig. 14.3). Second, as both teacher and students during the teaching sequence got more familiar with the online module, its technical demands and its feedback, the student questions and the student--teacher interactions gradually focused more on algebra and less on technical issues. As a consequence, the character of these interactions changed from technical discussions into “Explainthe-screen” or “Discuss-the-screen” interactions. Also, the teacher went to the board

Fig. 14.3 Overview of student results generated by the DME

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less frequently, but instead used the online module more often as an environment to check algebraic claims or techniques. He encouraged students to type something in to see if it is correct, and used this as a way to explain the algebra. As an example of this, one of the students walked to the teacher carrying his notebook computer. The task on the screen was to simplify a radical expression, and the student ended up with 43

1 6 a.

Student: And it [the learning environment] says it is good, but it wants it to be easier. Teacher: Yes that’s right, because there is a fraction [points at the 1/6 at the screen] in the, . . . eh, under the nominator [points at the square root sign, and that is actually what he means]. It is that, that he does not want, I think. [. . .] Teacher: The 1/6, you can also see that as 6/36 . . . Then the teacher explained what to do with the 6/36 as to further simplify the expressions under the square root sign and asked the student to type this in. Teacher: You have to remove as much as possible under the square root sign, and no fraction. In this interaction, the teacher focused on the mathematical issues, and used the online module as an environment to have things found out by the student. The teacher ended with some more general guidelines. If we relate the findings presented in this section to the six whole-class teaching orchestrations types identified above, we already noticed some Explainthe-screen and Discuss-the-screen elements within the didactical performance of the Work-and-walk-by orchestration. The same holds, to a lesser extent, for the Technical-demo orchestration: technical issues regularly emerged in the individual student--teacher interactions, even if the teacher was in many cases not able to solve them. Elements of the Link-screen-board orchestration could also be observed, as the teacher regularly walked to the whiteboard to explain the algebra, or used paper and pencil to do so. The Spot-and-show opportunities that the didactical configuration offers were not exploited. The same holds for the Sherpa-at-work, even if the teacher by the end of the teaching sequence invited students to carry out a specific technique in the digital environment, which can be seen as an individual “Sherpa-at-work light”. All together, the case study reveals a teaching practice which heavily relies on one single orchestration type, the Work-and-walk-by orchestration. Little variation was found, and the available resources were exploited to a limited extent. To understand these observations, we reflected on this teaching practice in the interview with the teacher after the pilot, and we observed two regular lessons taught by this teacher in different classes. In the final interview, the teacher admitted that he had not had

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the time to prepare his lessons or to familiarise himself with the online module and its technical peculiarities: ‘Well, I don’t know much about it [the technology] myself. I did not invest time in preparation’. In addition to this, he explained his attitude of leaving much initiative to the students and of giving limited attention to whole-class teaching: “I refrained from explaining a chapter. The kids are just listening passively, and at the end of the lesson I learned a lot, and they just said ‘yes’. I prefer the kids act, and raise questions based on their actions.” He admitted that he had to explain some things several times to different students, as he was moving to the students one by one. Through the use of the board for individual explanations, he hoped to make these explanations also accessible to other students. To compare the case study teacher’s pilot lessons with his regular teaching, two ‘normal’ lessons in different classes were observed. Even if his teaching in this pilot was similar to his regular teaching, the analysis of these lessons suggests that the teacher was more central in his orchestrations in the regular lessons. For example some whole-class explanations could be observed, and the teacher seemed more confident, also in guiding the use of technology, in this case graphing calculators. As a final remark on the case study, it is worth while noticing that the students’ original objection against using computers to practice by-hand algebraic skills gradually disappeared. More and more, they used the netbooks, and textbooks and notebooks were hardly seen by the end of the teaching sequence.

14.5.2 Findings from the Questionnaires Even if the word “orchestration” was not mentioned in the questionnaires, some of the responses provide insight in the orchestrational choices made by the participating teachers. One question on both the pre- and the post-questionnaire was: which ICT-means were used? In the pre-questionnaire this concerned the use of technology in the teacher’s lessons preceding the pilot; in the post-pilot questionnaire, this concerned tool use during the pilot. Participants could click on more than one answer. Table 14.1 summarizes the findings. Data shows that the technological

Table 14.1 ICT means used during the pilot ICT-means used (more answers possible)

Pre-pilot (N = 47), frequency (%)

Post-pilot (N = 41), frequency (%)

Data projector Teacher’s computer Interactive whiteboard Computer lab Student computers in classroom Students’ home computers

57 57 55 0 0 0

46 32 37 83 29 83

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P. Drijvers Table 14.2 Expected and effectuated working formats used during the pilot Expected (pre-pilot, % of N = 47)

Effectuated (post-pilot, % of N = 40)

Working formats

Not

Sometimes

Often

Not

Whole-class explanation Whole-class demonstration Whole-class homework discussion Whole-class presentation Individual work Work in pairs Group work Homework

0 19 4 57 6 9 53 23

36 62 47 38 26 30 38 28

64 19 49 2 66 60 4 47

32 38 40 100 2 28 93 7

Sometimes

Often

48 47 47 0 2 25 5 53

20 15 13 0 96 47 2 40

devices which are most frequently used during the pilot are the computer lab and students’ computers at home, which contrasts to the more teacher-driven “regular” use of ICT before the pilot. Teachers seem to have changed the didactical configurations for the case of the pilot. Another question on the pre-pilot questionnaire concerned the working formats the teachers were expecting to use during the pilot, and a similar one on the postpilot questionnaire asking which working formats they used indeed. Table 14.2 summarizes the findings. It shows that individual work, work in pairs and homework are the most frequently used working formats, whereas whole-class explanations and whole-class homework discussion occurred less than expected beforehand, in spite of the opportunities the didactical configuration offers for it. A follow-up question in the post-pilot questionnaire was whether technology was used in the mentioned working formats. The results shown in Table 14.3 confirm the impression from Table 14.2, namely that technology during the pilot was mainly used for individual work, work in pairs and homework and not so much in wholeclass orchestrations.

Table 14.3 ICT used in working formats Post-pilot (% of N = 41) ICT in working formats

Not

Sometimes

Often

Whole-class explanation Whole-class demonstration Whole-class homework discussion Whole-class presentation Individual work Work in pairs Group work Homework

58 41 61 98 15 51 93 24

32 24 34 2 5 17 5 32

10 20 5 0 80 32 2 44

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The most interesting outcome in Tables 14.2 and 14.3 is that the option to show students’ homework by means of a data projector or an interactive whiteboard, and to use it as a catalyst for whole-class discussion, was hardly used, whereas the teachers usually used such technology in whole-class teaching settings according to the pre-pilot questionnaire results. Even if the teachers beforehand expected some more individual work or work in pairs, this seems to have happened to a larger extent, and opportunities for using ICT in the way they were most familiar with, remained unexploited. To summarize the findings from the questionnaires, we conclude that before the pilot, teachers indicated that they used technology mainly in whole-class teaching settings, probably with the teacher operating the technology. In spite of this preference and experience, during the pilot they privileged individual work and work in pairs, which turn out to be the dominant orchestrations, and thereby neglected options for whole-class teaching offered by the technology. Even if the orchestrational variety among all teachers seems to be greater than was observed in the case study, the results point into the same direction by suggesting that student-centred orchestrations, for example in computer lab and home settings, got more frequent at the cost of whole-class orchestrations using tools such as a data projector or an interactive whiteboard. Compared to the teachers’ previous experiences with technology in their teaching and their expectations, this is a shift. It is not clear if the six identified whole-class orchestration types also appear in the context of this pilot. The questionnaires do not offer enough information. The focus on individual work and work in pairs is clear, but we do not know what happened besides that. Spot-and-show orchestrations and Sherpa-at-work orchestrations, however, do seem to be very rare, even if some teachers in the interviews reported incidentally using these orchestration types. While interpreting these findings we should notice that most of the teachers engaged in this pilot were not experienced, at least not in using the specific technology, and were left over to themselves with little support. We also observed an expert teacher, who was the main designer of the online module, in one of his lessons. As a result of his own instrumental genesis, he was aware that entering formulas in the digital environment can be laborious, and that shortcut keystrokes and copy--paste options can help a lot. As an experienced teacher, he knew that students initially complain saying that writing down formulas with paper and pencil is much faster than entering them in a digital environment. Combining the results of his own instrumental genesis with his pedagogical experience, he set up a Technicaldemo orchestration in which he demonstrated the main editing techniques and highlighted their importance. He also included this as a suggestion in the teacher guide that came with the instructional material, but probably many teachers did not read it, which can be interpreted as a limitation of the preparatory documentational work.

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14.6 Conclusion and Discussion 14.6.1 Conclusion What answers to the initial questions do the findings suggest? A first question was to investigate in which types of orchestrations teachers transform the available technological resources. The findings from both the case study and the questionnaires – albeit the first to a greater extent than the second – suggest that individual, student-centred orchestrations are dominant when teachers use the resources that were developed in the frame of this pilot. Teachers tended to privilege students working individually or in pairs on the online module tasks, and devoted little time to whole-class explanation or homework discussion, whereas their expectation before the pilot were different. The case study resulted in the identification of a Work-andwalk-by orchestration, which in itself is not very surprising one. However, we were surprised by its dominance and by the fact that other orchestrational opportunities of the available technology were not exploited, whereas more variation could be observed in this teacher’s regular lessons. Several factors may explain these phenomena. The subject, practicing algebraic skills, probably is more suitable for individual work or work in pairs than for whole-class teaching. Also, the computer labs, in which many lessons apparently took place, may be less suitable for whole-class teaching. Individual orchestration types are probably the easiest thing to do for a teacher, who is not feeling confident about his or her own technical skills. It may be the technology itself that invites student work rather than whole-class teaching. The case study results suggest a clear relationship between the teacher’s orchestrational choices and his pedagogical intentions (Chapter 4). The interviews with teachers suggest that all these factors play a role; however, data is insufficient to decide on the impact of each of them. A second point of interest is how these results relate to the categorisation of orchestration types described in Section 14.2. The latter typology emerged from whole-class teaching episodes, whereas in this pilot mainly individual orchestrations were found. Still, from the case study observations we conclude that the six whole-class teaching orchestration types identified earlier have their counterparts, or at least similar aspects, in the context of the present study. Even if many teachers seem to prefer individual interactions to whole-class teaching in this case, at the level of the didactical performance we see elements that are more explicitly part of the didactical configurations of the typology found earlier. The overall conclusion, therefore, is that the six whole-class orchestration types of course are not exhaustive, but do contain elements that can be observed in other orchestrations as well. As a new orchestration type, the Work-and-walk-by orchestration was identified. We expect the list of possible orchestrations to be extended in future, not as to strive for a complete list, but as to provide teachers with a diverse repertoire of possible orchestrations as source of inspiration to their professional activity.

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A third and final point of interest concerns the change processes that occur when teachers engage in an experimental setting. The conclusion here is twofold. First, the case study provides insight in the change process during the pilot teaching sequence. The findings suggest a stable and not so dynamic orchestration, in which there is not much change, at least not at the superficial level. Meanwhile, at the level of didactical performance a process of professional development was observed, showing for example an increased focus on the algebra and on what we might call “Explain-the-screen”, at the cost of attention to technological issues. Second, the findings of the questionnaires shed light on the change that takes place when teachers engage in such a pilot, compared with their regular teaching practices before the pilot. The data suggest that many teachers, who were used to integrating technology in a teacher-centred way – the teacher using a computer connected to a projector, or using an interactive whiteboard – in the frame of this pilot switched to studentcentred orchestrations. It seems that most of them during the pilot sequence did not extend their teaching technique repertoire with, for example a Spot-and-show orchestration type, even if the technology supports the monitoring of student work by the teacher anytime and anyplace.

14.6.2 Discussion The study that we report on here has some important limitations. First, the danger of presenting one single case study is that the results are too much influenced by the particular situation and at the particular teacher involved. Second, the additional data has the weakness of providing just global information on teachers’ use of resources and the resulting orchestrations and teaching practices. Even if the latter issue is partially solved by additional interviews, we should be careful with interpretations from these results. And finally, comparing whole-class orchestrations from Drijvers et al. (2010) with the more individual orchestration types found here is not a straightforward thing to do. In fact, we had not expected such a big shift in orchestration types for this pilot; to observe this happening is one of the most interesting aspects of this study, and matches with the observations made by Lagrange and Degleodu (2009), who claim that teachers do not articulate the use of technology as a working environment for students and as a teacher resource. These limitations being noticed, the findings further evidence the difficulties that teacher may encounter when integrating technological resources into their teaching practices. Re-sourcing as mentioned by Adler (Chapter 1) does not always seem to appear, and documentational geneses (Chapter 2) take time indeed. In terms of semiotic mediation (Bartolini Bussi and Mariotti, 2008; Chapter 3) is not easy for a teacher to exploit the semiotic potential of resources. Resources invite the professional development of a repertoire of appropriate orchestrations. The genesis of such a repertoire seems to be related to the teachers’ own processes of instrumental genesis and documentational genesis (Chapter 2). To engage in such a process, a sense of ownership for the teaching is needed: if teachers are used to just following the text book, and don’t have the time or don’t see an interest in designing their

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teaching, creating multiple repositories of resources cannot be expected to influence teaching practice very much. Furthermore, support for teachers is a precondition. Such support might be organised in professional developments activities, in which co-design and networks for collaboration might be expected to be productive. Acknowledgements Particular thanks are due to the teachers and their students for their involvement in this study, as well as to Nora Niekus, who was involved as a research assistant.

References Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: artefacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education, second revised edition (pp. 746–805). Mahwah, NJ: Lawrence Erlbaum. Bokhove, C., & Drijvers, P. (2010). Assessing assessment tools for algebra: Design and application of an instrument for evaluating tools for digital assessment of algebraic skills. International Journal of Computers for Mathematical Learning, 15(1), 45–62. Bueno-Ravel, L., & Gueudet, G. (2007). Online resources in mathematics: Teachers’ genesis of use? In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the V congress of the European society for research in mathematics education CERME5 (pp. 1369–1378). Cyprus: Larnaca. Doerr, H. M., & Zangor, R. (2000). Creating meaning for and with the graphing calculator. Educational Studies in Mathematics, 41, 143–163. Drijvers, P., Boon, P., Doorman, M., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool: Whole-class teaching behavior in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75(2), 213–214. Drijvers, P., & Trouche, L. (2008). From artefacts to instruments: A theoretical framework behind the orchestra metaphor. In G. W. Blume & M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics: Vol. 2. Cases and perspectives (pp. 363–392). Charlotte, NC: Information Age. Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71(3), 199–218. Kendal, M., & Stacey, K. (2002). Teachers in transition: Moving towards CAS-supported classrooms. ZDM, The International Journal on Mathematics Education, 34(5), 196–203. Kendal, M., Stacey, K., & Pierce, R. (2004). The influence of a computer algebra environment on teachers’ practice. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators: Turning an computational device into a mathematical instrument (pp. 83–112). Dordrecht: Kluwer. Lagrange, J. B., & C.-Degleodu, N. (2009). Usages de la technologie dans des conditions ordinaires : le cas de la géométrie dynamique au collège. Recherches en Didactique des Mathématiques, 29(2), 189–226. Lagrange, J.-B., & Monaghan, J. (2010). On the adoption of a model to interpret teachers’ use of technology in mathematics lessons. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (pp. 1605–1614). Lyon: INRP. Retrieved July 6, 2010, from http://www.inrp.fr/ editions/editions-electroniques/cerme6/ Lagrange, J.-B., & Ozdemir Erdogan, E. (2009). Teachers’ emergent goals in spreadsheetbased lessons: Analyzing the complexity of technology integration. Educational Studies in Mathematics, 71(1), 65–84. Mariotti, M. A., (2002). Influence of technologies advances in students’ math learning. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 757–786). Mahwah, NJ: Lawrence Erlbaum.

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McKenzie, J. (2001). How teachers learn technology best. The Educational Technology Journal, 10(6). Retrieved October 20, 2009, from http://www.fno.org/mar01/howlearn.html Monaghan, J. (2001). Teachers’ classroom interactions in ICT-based mathematics lessons. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 383–390). Utrecht: Freudenthal Institute. Monaghan, J. (2004). Teachers’ activities in technology-based mathematics lessons. International Journal of Computers for Mathematical Learning, 9, 327–357. Robert, A., & Rogalski, J. (2005). A cross-analysis of the mathematics teacher’s activity. An example in a French 10th-grade class. Educational Studies in Mathematics, 59, 269–298. Ruthven, K., & Hennessy, S. (2002). A practitioner model of the use of computer-based tools and resources to support mathematics teaching and learning. Educational Studies in Mathematics, 49(1), 47–88. Sensevy, G., Schubauer-Leoni, M. L., Mercier, A., Ligozat, F., & Perrot, G. (2005). An attempt to model the teacher’s action in the mathematics class. Educational Studies in Mathematics, 59(1–3), 153–181. Trouche, L. (2004). Managing complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9, 281–307. Trouche, L., & Drijvers, P. (2010). Handheld technology: Flashback into the future. ZDM, The International Journal on Mathematics Education, 42(7), 667–681.

Reaction to Part III On the Cognitive, Epistemic, and Ontological Roles of Artifacts Luis Radford

1 Introduction Galileo opens his Discourses and Mathematical Demonstrations Relating to Two New Sciences with a remark about the famous 16th century Venetian arsenal, which he praises for its impressive amount of instruments and machines; this arsenal, he says, offers an opportunity to wonder and think. With their unprecedented variety of tools and artifacts, contemporary classrooms may have looked like the Venetian arsenal to Galileo. True, some of the artifacts that are part of our educational settings have been there for a long time now – for example, textbooks. Others, however, made their appearance with the digital technological progress during the 20th century. And, like the instruments and machines of the Venetian arsenal, they offer new possibilities for thinking and learning. Now, for these possibilities to be materialized in the classroom, the conditions surrounding the use of artifacts in processes of teaching and learning need to be clearly understood. Indeed, since artifacts are artificial devices, neither the understanding of their use nor the best exploitation of their epistemic possibilities is self-evident. This is why investigating the proper conditions of artifact use in educational settings constitutes an important research problem. The various chapters in this part of the book tackle this problem and offer interesting theoretical and methodological contributions to current debates in the field. Thus, seeing the chapters from a general viewpoint, the various authors inquire about the manner in which teachers adapt and use specific resources in their own practice – for example, CAS (Kieran, Tanguay, and Solares), Enciclomedia (Trigueros and Lozano), a digital-based algebra environment (Drijvers), material objects and symbolic artifacts (Forest and Mercier), and textbooks (Rezat).1 Naturally, the authors tackle the

1 I use the term artifact in its most general sense: as “an object made by a human being, typically an item of cultural or historical interest,” as defined by The New Oxford American Dictionary. The category of artifact (or its synonymous term tool) includes the one of didactic resources.

L. Radford (B) École des sciences de l’éducation, Laurentian University, Sudbury, ON, Canada P3E 2C6 e-mail: [email protected]

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general research problem from different perspectives and ask questions of different kinds. Kieran et al. inquire about the adaptations of researcher-designed resources by teachers. Trigueros and Lozano move along similar lines and try to detect what they call the ‘operational invariants’ in the teachers’ use of resources. Drijvers attempts to elicit the kind of ‘instrumental orchestrations’ to which the teachers resort in their classrooms, while Rezat explores the forms of textbook use undergone by both teachers and students. Taken together, the various case studies presented in these chapters show some of the difficulties that teachers face in the integration of resources in the classroom. They pinpoint, to various degrees of explicitness, some aspects of a more general nature that are imbricated in the educational use of artifacts. One of them relates to changes in our conceptions of classroom practices that result from the use of digital technologies. Indeed, traditional conceptions of what a good classroom practice is need to be revisited in light of the teachers’ and students’ use of artifacts. Thus, in Kieran and coworkers’ study, one of the teachers fails to use CAS to promote a deep mathematical understanding. The teacher does use the digital artifact, yet the artifact use seems to remain within the confines of traditional forms of teaching centered on direct content presentation. The teacher, it seems, fails to notice that the use of artifacts in the classroom introduces a new division of labor and that, in this new digital context, his or her role is thereby modified. To be properly exploited, the cognitive potential that an artifact brings with it requires not only a suitable understanding of the artifact itself but also of how it modifies the roles of the teacher and the students. The manner in which we understand the division of labor that artifacts induce in the classroom depends on our own theoretical views about cognition. In fact, the possible roles that we attribute to artifacts or resources derive from the manner in which we conceive of cognition in the first place. It is only within a specific view of cognition that artifacts are endowed with particular cognitive, epistemic and ontological roles. Let me briefly dwell on these roles in the following sections.

2 The Cognitive Role of Artifacts There seems to be a consensus around the idea that artifacts are mediators of activity. But what do we mean by the mediating nature of artifacts? There are several ways in which this question can be answered. One way is to understand the artifact as something that allows us to do something. It is from this perspective that artifacts are seen as a possible extension of the individual. Artifacts are considered here as something like prostheses or amplifiers: they are aids to accomplish actions. They help us without changing our cognitive landscape. What they do is to make accessible to us realms of reality that remain hidden because of our human sensorial limitations. The microscope and the telescope are good examples. By allowing the students to visualize and decompose three-dimensional figures, the Cubícula software mentioned in Trigueros and Lozano’s chapter could be seen in this way. But Cubícula and other artifacts could also be seen as playing a deeper cognitive role. In

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this view, artifacts are not only facilitators of knowledge acquisition. They become part of the way in which we come to think and know. The first meaning of mediation has been put forward in cognitive psychology (Cole & Griffin, 1980). The second meaning of mediation is at the heart of Vygotsky’s view of cognition, where tools are seen as psychological. Within this conception of cognition, artifacts are considered cultural devices that modify our cognitive functioning. As Vygotsky put it in one of the foundational texts of the historical–cultural school of psychology: By being included in the process of behavior, the psychological tool alters the entire flow and structure of mental functions. It does this by determining the structure of a new instrumental act just as a technical tool alters the process of a natural adaptation by determining the form of labor operations (Vygotsky, 1981, p. 137).

Within the historical–cultural Vygotskian conception of cognition, an artifact is considered to be a bearer of historical intelligence (Pea, 1993). It is a bearer of historical voices that need to find an interactional space in the classroom to enter into a dialogue with the teacher’s and the students’ voices. Now, how to promote dialogical spaces susceptible to including the artifact’s sedimented voices and intelligence is a problem in its own right. It entails a reflection on both the epistemological and ontological roles that we attribute to the artifacts.

3 The Epistemological Role of Artifacts The aforementioned Vygotskian artifactual mediated view of cognition has epistemological implications that we still have to explore, for it changes the traditional view of what we mean by learning and knowing. To make this point clearer, let me go back to the discussions that originated from the introduction of calculators in primary schools a few decades ago. Calculators were seen as an object of interference and even an inhibitor to the development of students’ arithmetic thinking. Students were supposed to be able to carry out calculations without the help of the calculator. Once, one of my students told me that in his Grade 1 class he was even forbidden to count with his fingers or to make any gestures. Within this epistemology, knowing was understood as something purely mental. If we consider artifacts as more than aids, their epistemic status changes. Knowing becomes knowing-with-tools as opposed to knowing via the tools. Artifacts become imbricated in the way we think and come to know. The epistemic status of artifacts can be summarized as follows: As artifacts change, so do our modes of knowing. However, this view of artifacts needs further development. Otherwise, it risks remaining anchored in the traditional knowing subject – object of knowledge epistemic schema. Indeed, the only modification to this schema is the insertion of the artifact. The schema becomes: subject + artifact – object of knowledge. In this case, the Piagetian research question about schema formation is barely modified: instead of the subject’s purely mental schemas it becomes the subject’s schemas of artifact

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use. This account fails to make explicit (at the theoretical and methodological levels) the fact that artifacts embody particular forms of cognition and communication, and that thinking emerges not out of patterns of actions with artifacts but in joint activity, out of actions with artifacts carrying social and historical meanings. What is missing in this account is the fact that knowing is a social and cultural practice. More specifically, knowing is a historical collective act. As a result, knowing is accomplished not only through invariant patterned actions with signs and artifacts but also in interaction with other individuals against the background of historical and cultural modes of thinking and communicating (Radford, 2010). The question is not, hence, how artifacts become appropriated or mastered, but how they mediate joint activity. Naturally, in the case studies presented in the various chapters, this question emerges either implicitly or explicitly. It appears in particular when the authors focus on the way the teachers mediate or orchestrate for the students the historical intelligence deposited in the artifacts. Cubícula, for example, conveys ideas of decomposing figures to think mathematically about their measurements. These are historical ideas that have been refined through centuries of human cognitive activity, from sand sketches in ancient Greece to 21st century digital representations.

4 Mathematics and the Ontological Role of Artifacts In his chapter, Drijvers distinguishes three elements of didactic orchestration: a didactical configuration, an exploitation mode of the didactical configuration and a didactical performance. The latter corresponds to the actual classroom activity. It is in the last part that artifacts come to be used and that teachers, through the use of certain techniques, have the opportunity to guide the students in their processes of learning. Drijvers invites us to see the teacher’s actions as a form of didactical performance, involving expected, and unexpected aspects that take into account the students’ inputs. I would like to argue that the didactical performance is part of a more general activity – the activity of mathematics making. Let me explain. In the previous section I suggested that thinking and knowing are social practices. In this section, I want to extend the idea to mathematics. My argument is not that mathematics is governed by social and cultural norms. This, of course, is true. But what I have in mind is something of an ontological nature, something about what mathematics is. Let us start by noticing that, ontologically speaking, mathematics is not really different from music. Both are cultural forms of expression, action, and interpretation. Naturally, there are obvious differences. But there are some important similarities as well. The most important similarity is this: musical and mathematical ‘objects’ share the same ontological nature. Thus, in the same way as music does not reside in musical scores, mathematics does not reside in written theorems. Mathematical objects do not coincide with mathematical written texts. Texts and other artifact are embodiments of the existence of their objects. As Lektorsky (1995, p. 193) put it, ‘man-made objects are in [a] certain sense modes of the embodiment and existence of knowledge’. Thus, I want to argue, in the same way that music resides in its performance, mathematics resides in the activity of

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its enactment. In this sense, mathematics is always new and different, in the sense that each event is always unique and singular. But, at another level, it is similar to other contemporary and past events, without which we would not distinguish an activity about geometry from one about algebra. This similarity of events does not, however, preclude mathematics from living – in an ontological sense – in the event of its execution. Considering mathematics from this viewpoint has some implications on classroom practice and on the ontological role of artifacts and resources. Artifacts can no longer be considered as a means to access mathematical objects and mathematical forms of reasoning, as these are not conceived of as transcendental entities. Artifacts, rather, are considered part of mathematics as material practice. Within this context, mathematics appears as a collective activity, spatially situated, which unfolds in a certain span of time, where the historical voices embedded in artifacts and the voices of students and teachers merge. Let us note, en passant, that in this perspective, the discussions about mathematical proofs assisted by computers (Devlin, 1992) take a different turn. The computer is not helping the mathematician carry out some calculations. Both become part of one chorus singing a polyphonic song. This conception of mathematics as enactment or performance does not mean, however, that all performances are equally good. Each will be more or less successful depending on the historical–cultural understanding of mathematics. But because mathematics is something that is in the making, performances will also be considered to be more or less good depending on how teachers and students understand and coordinate their coemerging and evolving sense of involvement in the collective endeavor in which all of them participate. It is against this polyphonic context that the question of the artifacts and the division of labor that they induce reappear. If thinking mathematically is an artifactual mediated collective endeavor where each participant learns to critically situate herself within cultural and historical constituted modes of thinking (Radford, 2008), the question of responsibility and orchestration must then be seen in a new light. It appears as a voix à trois: the teacher’s, the students’, and the artifacts’. To end this short commentary, I come back to Rezat’s interesting chapter. Rezat’s chapter shows the tensions that are caused in some classrooms by the presence of a textbook, particularly when the textbook brings a perspective that is different from the teacher’s. If the teacher considers her voice as the official one, the artifact has little room to operate. If, in contrast, the teacher considers her view as one of various possible views on a same problem, she can take advantage of the textbook to add its differing and subverting voice to hers and invite the students to reflect on the differences and nuances so that they can end up with a more polyphonic understanding of the matter under scrutiny. The making of mathematics would consist precisely in the understanding of differences and similarities that are brought to the fore by the students’ understandings as they are interwoven with the voices of the teacher and the historical intelligence deposited in artifacts.

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References Cole, M., & Griffin, P. (1980). Cultural amplifiers reconsidered. In D. R. Olson (Ed.), The social foundations of language and thought, essays in honor of Jerome S. Bruner (pp. 343–364). New York/London: W. W. Norton & Company. Devlin, K. (1992). Computers and mathematics. Notices of the American Mathematical Society, 39(9), 1065–1069. Lektorsky, V. A. (1995). Knowledge and cultural objects. In l. Kuçuradi & R. S. Cohen (Eds.), The concept of knowledge. The Anakara Seminar (pp. 191–196). Dordrecht: Kluwer. Pea, R. D. (1993). Practices of distributed intelligence and designs for education. In G. Salomon (Ed.), Distributed cognitions (pp. 47–87). Cambridge: Cambridge University Press. Radford, L. (2008). The ethics of being and knowing: Towards a cultural theory of learning. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom, and culture (pp. 215–234). Rotterdam: Sense Publishers. Radford, L. (2010). The anthropological turn in mathematics education and its implication on the meaning of mathematical activity and classroom practice. Acta Didactica Universitatis Comenianae. Mathematics, 10, 103–120. Vygotsky, L. S. (1981). The instrumental method in psychology. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 135–143). Armonk, NY: Sharpe.

Part IV

Collaborative Use

Chapter 15

A Comparative Perspective on Teacher Collaboration: The Cases of Lesson Study in Japan and of Multidisciplinary Teaching in Denmark Carl Winsløw

Documentational genesis, as introduced in Chapter 2, is clearly an enterprise which goes much beyond the individual teacher’s domain of action and responsibility. Even if we restrict our attention to individual teachers’ work, it is obvious that they draw on resources developed by other teachers. More generally, teachers’ knowledge about teaching is developed and circulated among teachers in many ways, and it is a general hypothesis of the present section of this volume that those processes are crucial to the identity and functioning of the teaching profession. In this chapter, we pursue this contention from a comparative and institutional perspective. In Western countries, the prototype of the teacher is a person with individual responsibility to teach a number of classes. It has been noticed in widely known studies (e.g. Ma, 1999; Stigler & Hiebert, 1999) that teachers in some East Asian countries seem to have a more collective organisation of their work and also a more developed “professional language” about specific pedagogic and didactic phenomena which are important to discuss and organise their teaching. Moreover, these studies (and subsequent developments) suggest that the collective organisations of teacher work are important factors in explaining the consistently impressive results of East Asian education in international evaluations. All this leads to a strongly motivated research interest in how teachers organise their work, and in particular to develop a properly comparative viewpoint on this. Because teacher practices are always conditioned and constrained by a number of invisible factors (cultural, institutional, intellectual . . .), this requires that we develop very precise and explicit models of what we want to study and compare. In this chapter, we shall present and compare two quite different organisations of teachers’ collaborative work, each conditioned by school level infrastructures in a sense which is close to that mentioned by Chevallard and Cirade (2010) and further elaborated in the first section of this chapter. Then, using our theoretical model, we present two contexts and infrastructures for teachers’ collaborative work: that of lesson study as a means for professional development of mathematics teachers C. Winsløw (B) Department of Science Education, University of Copenhagen, 1350 København K, Denmark e-mail: [email protected]

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in Japan, and that of Danish high school teachers’ collaboration in the setting of multidisciplinary modules. The two cases have been chosen both for their intrinsic interest but also because the comparison of them illustrates the importance of infrastructure (while it is almost always ignored in studies of single or similar contexts for teacher work).

15.1 Preliminaries on Paradidactic Infrastructures The basic unit of teaching and learning in school institutions is that of a didactic system (DS) (Brousseau, 1986; Chevallard, 1991). It is constituted by a group G of people studying some object or organisation O of knowledge (in a broad sense to be further specified) while making use of a set A of artefacts (non-human objects like texts, signs, media and so on – cf. also Chapter 3). To say that the system (G, O, A) is a DS usually requires more structure on G, to identify among one or more members of G an intention for other members of G to study (or learn) O. This, in normal school settings, corresponds to the partition of G into teachers and students. Of course, more precise models for G as well as for the other elements are also possible and usual. In particular, O may be modelled as praxeological organisation in the sense of Chevallard (1999), and to study the kinds and uses of artefacts one may want, for instance to identify semiotic systems, such as languages and codes, which could support our interpretation of observations of how G makes use of A. While teachers work to prepare, regulate and evaluate the work of students in a DS, they assume different positions with respect to the DS: inside (as participants in the DS) and outside (as constructors and observers of the DS). These positions and roles are defined by their relation to the DS. We propose the following simple model of them: – teachers in pre-didactic systems (PrS) who prepare the work in a DS (whether or not they will directly participate in it); – teachers in didactic observation systems (DoS) who observe and reflect on the work in a simultaneously unfolding DS (in which they do not participate); – teachers in post-didactic systems (PoS) who evaluate and otherwise reflect on the work in a completed DS (whether or not they directly participated in it). We subsume these three kinds of systems under the name paradidactic system (as all have teachers as the human agents, G), abbreviated PS. A paradidactic system is thus a triple (G, O, A) where G is a group of teachers, O their shared practice and knowledge about one or more DS and A the set of artefacts they use to develop and mediate O. A PS can be clearly situated relatively to a DS by the temporal position, as shown in Fig. 15.1. This temporality of the different parts of PS, relatively to DS, is central to our model. The group G of teachers may be the same in PrS and PoS, while the organisation O is likely to develop in time; the sets of artefacts could be different but are

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Fig. 15.1 Graphic representation of the main components of our theoretical framework, showing the distribution in time of the systems in which teachers (may) work: pre-didactic (PrS), observation (DoS), didactic (DS) and post-didactic (PoS)

still likely to contain some common elements (like a text book). Researchers may find that to study paradidactic systems presents a number of practical and methodological challenges, which are not necessarily smaller in the “simplest” case of an individual teacher who prepares, delivers and reflects upon her own teaching (cf. Chapter 2). Now we simply define the paradidactic infrastructure (in a school system, or in a single school) as everything which conditions and constraints the PS in its different phases and in the interplay between the phases. For instance the absence of work facilities for teachers in a school (desks, computer access and so on) may be an important element of the pedagogic infrastructure. The actually available physical elements of the school environment could also be important artefacts in all phases of the PS (and, of course, in DS) and they could therefore also be significant elements of the paradidactic infrastructure in a particular school. Traditions, habits and policies pertaining to the organisation of teachers’ work (in PS) are more ephemeral elements of the paradidactic infrastructure, some of which may be less local (e.g. be valid for an entire school system, within a discipline or across disciplines). This fact makes them particularly interesting to identify and analyse, and makes a comparative perspective particularly pertinent, as they would have a tendency to appear as “natural” within a given school system. As the finality of PS is to produce the DS, it is clear that paradidactic infrastructures must be studied along with the DS they lead to produce. This is another reason for the pertinence of a comparative perspective: considering two very different paradidactic infrastructures and the resulting didactic systems we can say more about the implication of the first on the latter than if we considered just minor variations within a single teaching system.

15.2 First Case: Collaborative Teacher Work in Japan The forms of interaction between PS and DS which will be presented and analysed in this section have been objects of international research since about 1990. They concern collaborative practices which involve practically all Japanese primary and

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lower secondary school teachers and which have existed – and, of course, developed – for at least a century (cf. e.g. Isoda, Stephens, Ohara, & Miyakawa, 2007; Chapter 2). Using our model, we first analyse the general form of these practices, and we then illustrate this analysis by observations of a concrete paradidactic system (from a lesson study).

15.2.1 Study Inside the School The Japanese practice of konaikenshu, which means literally “study inside the school” is well documented in the international research literature (e.g. Fernandez & Yoshida, 2004). This practice affects all of the primary and lower schools in Japan: to participate in konaikenshu is, indeed, considered as a part of the work of a teacher along with teaching. Concretely, konaikenshu is carried out by a group of teachers which, in Japanese, is called the study group (notice here that the Japanese word translated here as “study” can also be translated “research”). The work of the group involves common meetings and individual study of texts – much as in an academic seminar. The work is driven by one or more explicit study themes set by the group itself, with the aim of developing common understandings of the theme. The study theme could be, for instance a new topic or objective in the national curriculum at a certain grade level, and to relate this to more local objectives: in Japan, every school fixes its own local objectives which, while themselves very general, are to be pursued in every aspect of the school’s life, including every discipline taught. A typical study theme, which includes both local and national objectives, and which could be considered by mathematics teachers across grades, is: “develop, among students, generosity and a strong sense of motivation, by guiding them to acknowledge their individuality; develop lessons which encourage students to learn from each other”. This example comes from the study of Fernandez and Yoshida (2004, pp. 12–13), who describe the study themes pursued by study groups in 35 Hiroshima schools over several years. In particular, they estimated that on average a study theme is retained by groups for 4 years. According to our definitions, the affordances of konaikenshu study groups constitute a PS related to a more or less wide class of DS (for which it may be a significant part of all three phases). It is particularly important to promote a vertical coherence from overall objectives to actual teaching. Through the work, the group members establish concrete relations between general objectives (from the national curriculum, or for the school) and actual lessons taught in specific disciplines; these relations are formulated in reports and also often in presentations and texts for teachers from outside the group. This format of work enables teachers act collectively in the practical interpretation of educational objectives, and it has for that reason been proposed as a major explanation of the fact that major reforms of the educational system in Japan are integrated more smoothly and efficiently “on the ground”, and that teaching practice (DS) in Japan present a relatively high degree of homogeneity (Lewis & Tsuchida, 1997).

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15.2.2 Lesson Study The most important concrete form of konaikenshu, for our problematique, is what is called in Japanese jugyokenkyu and known in English as lesson study. Lesson study typically occurs as one element within a wider “study inside the school”. A lesson study is characterised by a special kind of theme: to design and study a particular lesson (e.g. a first lesson on similarity of polygons in grade 6) while paying close attention of the alignment of the lesson with the objectives of the programme discipline and the school. During the lesson study, the lesson will be taught a number of times by different members of the lesson study group, while the others observe the lesson. In between, the group of teachers will meet to plan or revise the lesson. Several authors, like Stigler and Hiebert (1999, p. 112–114), have presented lesson study as a cycle of phases, which run roughly as follows: 1. Study and planning (PrS). Meetings in group and individual study, with the aim of: –

choosing a lesson to develop, called the study lesson, and to identify and formulate the general and specific objectives of this lesson; – study of a collection of artefacts (a kind of resources, in the sense of Chapter 2): texts books and their accompanying material for the teacher, lesson plans (cf. below), national programme, etc.; – elaborate a plan for the study lesson, including both instructions about the teaching and about the observation (by the teachers not teaching), related to explicit hypotheses of the group (the resulting document is the lesson plan, to be further described below). 2. Test and observation (DS and DoS). A member of the lesson study group teaches the study lesson in her class, according to the lesson plan (DS). The other members of the group observe the lesson, but usually do not intervene. There are two main forms of teaching which correspond to two different behaviours of the observing teachers: – during whole class teaching, the observing teachers remain in the back or to the sides of the classroom, where they take notes either freely or according to a scheme prepared beforehand (included in the lesson plan); – as the students work individually or in group (and the teacher circulates among the tables to answer questions and observe the students), the other members of the group also circulates in the class to observe the work of individual students. 3. Evaluation and revision (PoS). The group meets soon after the test and observation session (normally on the same day) to discuss: – Experience and observations: normally, the teacher who taught the lesson first presents his impressions and reflections about the lesson, taking into

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account his personal acquaintance; then the other group members present their observations and reflections; Appropriate revisions of the lesson are also discussed.

Notice that, with the indications given above for the phases, Fig. 15.1 represents rather well the lesson study cycle, with the realised study lesson as the DS on which the whole PS centres. Whether or not the three phases are repeated in a cycle (this is not always the case!) the lesson plan remains an artefact in all subsystems of the PS, as indeed is a shared and emerging document in the sense of Chapter 2. We now describe it in more detail.

15.2.3 The Lesson Plan and an Example of a Lesson Study Miyakawa and Winsløw (2009) analysed, from the point of view of two paradigms of design research (French “didactical engineering”, Japanese “open approach method”), a case of lesson study from an ordinary school in Tsukuba, Japan, involving a group of 13 teachers. The study lesson took place in grade 6 and forms part of a lesson sequence on ratio and proportion. In the preparatory work, teachers study and discuss different approaches to proportionality, and an outline of their reflections is included in the first section of the lesson plan. The teachers distinguish “basic approaches” to proportions of two magnitudes A and B. In the most basic approach, taught from the first years of elementary school: A is measured in terms of B (e.g. A is three times B). Here B plays the role of a unit. In the second approach, some number or expression (like 1:3, 1/3, 0.33) is used to indicate the proportion. The teachers have conducted a small test in the class to gauge the status of students’ knowledge of the first approach. They have also examined a number of concrete phenomena previously used to introduce students to expressions for proportionality, such as the taste of salad dressings (made of vinegar and oil in different proportions). Because of various problems experienced with these, the teachers decide to use a simple geometry task: find out whether two given rectangles are of the same form, and justify the answer. In this case, proportionality may of course occur in several ways, involving proportions of the lengths of bases and heights. In a preparatory meeting, which was videotaped and analysed in detail, the teachers discuss the study lesson on the basis of this idea. One important variable in the design of the task is the dimensions of the two rectangles, to enable a rich discussion. To confront at the same time the erroneous idea of “magnification by adding the same to each side”, they decide to introduce first one rectangle, then another larger one in which the base and height are increased by the same length relatively to the smaller rectangle. Two candidates for the pair of rectangles are discussed: (1) 3 cm × 6 cm and 5 cm × 8 cm, (2) 3 cm × 5 cm and 5 cm × 7 cm. While (2) could lead to some confusion due to the equality of base in one and height in the other (both 5 cm), this could also enrich the discussion. And (1) is

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discarded because the fact of being a “double square” (3 cm × 6 cm) could be used as an easy explanation why the two rectangles are not of the same form, without taking into account the lengths. Once the basic “problem” (hatsumon, in Japanese) to be posed in the lesson is decided upon, the discussion focuses on the dynamics between hypotheses on student strategies and details of plans for the teacher’s action during the lesson. On the basis of their experience and imagination, the teachers propose possible answers of the students – both justifications of “same form” and of “different form”. To promote the presence of the answer “same form because the height and base both differ by two centimetres” they decide to introduce a “warm up problem” with squares (3 cm × 3 cm, 5 cm × 5 cm): asked if they are of the same form, students will surely answer “yes”, and some will generalise the reasons to rectangles. The material form of presenting the “warm up problem” and the “main problem” is also discussed. The idea of providing the figures on Xeroxed handouts is discarded as it might lead to students superposing and folding the figures rather than reflecting on the proportions of their lengths. To maximise the students’ attention to the figures and their dimensions, they should be asked to draw consecutively, in their notebooks, the small and large square, before being asked the question about same form. It is anticipated that all students will agree the two squares have the same form, and so this question is treated through whole class interaction. Next, the students are asked to draw, below, the small and the large rectangle, and then to write their immediate impression as to whether the rectangles are of the same form. During a 5-min period of personal work, each student must then write a justification of his answer in the notebook. The main part of the lesson is a discussion of students’ different solutions, beginning with the explanation of those students who have answered “same form”. Here, the teacher should maximise the variety of explanations put forward and to do so, he will circulate during the 5-min personal work to identify students with different solutions. This method (kikanshido in Japanese, cf. Clarke, 2004) is very common in Japanese lessons and it is not even mentioned in the lesson plan. The lesson plan consists in this case, and also more generally (Isoda et al., 2007, p. 87), of the following elements: – Short description of the teaching unit (here, seven lessons): its overall goals and the theme of each lesson. – Reflections on the challenges of the lesson: past experiences and designs, students difficulties. – Goals of the lesson. – Detailed “script” for the teaching process, shown in a table (cf. Shimizu, 1999, p. 113), the hypothetical action by the students (strategies for solving the problems proposed), and important points for evaluating the students’ work. Notice that the “script” included in the lesson plan is far from being complete in the sense of specifying everything the teacher should do or say. Its main focus is on the goals (ultimately for student learning) and the ways in which they could be reached.

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The lesson plan is of course an important artefact in the predidactic system that involves all teachers of the team. It is crucial also for the other parts of the paradidactic system (cf. Fig. 15.1): to guide and focus observations of the teachers during the lesson (an artefact of the DoS) as well as the evaluation and revision of the lesson during post-lesson discussions and study (PoS). While the group splits between DS and DoS during the “lesson phase” of the lesson cycle, the lesson plan ensures a common focus of the activity of the group – before, during and after the lesson. More precisely, the lesson plan helps the group develop a common and explicit organisation of their knowledge about the lesson (contents and goals), about the various strategies for teaching it, about knowledge and strategies of the students and so on. This organisation of knowledge develops through all three main phases of the lesson study. In the lesson study we consider here, important details of the lesson (including anticipating students’ action) were identified in the planning meeting referred to above. As the lesson unfolds, the teacher focuses very strongly on making the students’ express their ideas and understanding those of others. The main part of the lesson (about 25 min) consists of this whole-class exchange of ideas, orchestrated by the teacher who calls upon individual students to give their answer and justification, without explicitly assessing them. This strategy is a general one (related to the so-called “open approach” method of teaching, cf. Nohda, 1991; Tsubota, 1977), and it maximises the possibility for observing teachers to identify strategies among the students and hence to support the post-didactic evaluation and revision of the lesson plan. Lesson plans do not contain anything like theorems about teaching. Nevertheless, their public character – that is the possibility of sharing them with colleagues – show that lesson plans constitute a potential for documentation processes (in the sense of Chapter 2) that goes beyond the group and institution in which they were produced.

15.2.4 Lesson Study as a Format of Pre-service Teacher Training It is well known that lesson study plays a significant role in the in-service induction of new teachers in Japan (Howe, 2005; Padilla & Riley, 2003; Shimizu, 1999). In fact, teachers can experience this form of work already in the practice part of their pre-service education. Winsløw (2004) present a study of this aspect of lesson study. The experiences of novice teachers with lesson study and other forms of konaikenshuu is an important explanation why these are so widely established parts of the paradidactic infrastructure in Japanese schools. At present, we know of a lot of experiences with transplanting these infrastructures – particularly lesson study – to other countries (e.g. Fernandez, 2002). Among the obstacles found to such “transplants” we find conditions which are relatively easy to change (like teachers’ schedule or lack of habit to observe and be observed as teachers) but also more deeply rooted constraints in the paradidactic infrastructure, such as the lack of a precise, shared language about didactic phenomena.

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15.3 Second Case: “Multidisciplinary Teacher Teams” (Denmark) We now move to a completely different context of teaching where the PS corresponding to one or more DS require in principle, and in fact in this case by official prescription, the involvement of a larger group of teachers. Following a major reform from 2005, in Denmark at secondary school the scientific and literary streams were suppressed to give way to “study lines” with a freer combination of major disciplines, and with new formats devised to make these disciplines interact. We consider here two of these formats and the ways in which they include mathematics teachers: – The so-called “modules of general study preparation”, which occupy 10% of the total student time after the reform, and where the students work on broad themes constructed to draw on at least two of the three principal “faculties” of the upper secondary school (natural, human and social sciences). – The final project to be done individually by each student in the third year of upper secondary school (corresponding to 12th grade), and in which two of the majors of the student’s study line should be combined (e.g. mathematics and history, or mathematics and physics). Each of these represent in themselves particular forms of DS characterised, in the first place, by requiring multidisciplinary organisations of practice and knowledge. The need for paradidactic systems involving more teachers comes from the combinations of disciplinary knowledge required.

15.3.1 General Study Preparation The modules of general study preparation occupy about 90 class hours in each of the 3 years of upper secondary school. They are organised in clusters of between 5 and 30 h (10–15 h on average) unified by a theme and by a set of (upper secondary school) disciplines which should contribute, in various ways, to the students’ work with the theme. Here are some examples of themes: 1. What is music? (mathematics, physics, music; 1st year), mathematical component: the model f (t) = A sin (ωt + ϕ), mathematical meaning of the parameters A, ϕ and ω. 2. Truth (mathematics, Danish, philosophy, physics, 2nd year), mathematical component: Euclidean proofs (sum of angles, theorem of Pythagoras, . . .). 3. Growth and climate change (mathematics, sociology, geosciences, 3rd year), mathematical component: construction of models (on the basis of empirical data and different forms of instrumented regression: linear, exponential, logistic) and evaluation of their fit.

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To realise the DS required by these parts of the curriculum, teams of teachers are established for each class and module or project, representing the cluster of disciplines involved. The task of coordinating the schedule for these teams with other activities is left to the individual school, and a certain variation exists (EVA, 2006, p. 18). A very common model is that the management of the school appoints a committee G∗ of teachers (often with representatives of the direction) in charge of coordinating the modules for a whole year, and in particular to define the themes of each class in collaboration with the teams of teachers Gi,j made available to each cluster of disciplines i and class j (about 3–5 clusters per class per year). At the level of the school, there is an ongoing debate among teachers about “good themes”, and many schools organise “pedagogical days” including this topic. So, to some extent the whole group of teachers of the school participates in a PS occupied with overall and theme level aspects of these modules; it even extends nationally, as one of the most lively sections of the Danish Ministry of Education’s web platform for teachers (www.emu.dk) is the one devoted to the exchange of themes and materials for general study preparation modules. However, for practical reasons, the overall organisation of the modules is done by G∗ at each school. The teams Gi,j are primarily constituted by a certain combination of disciplines. They can, to some extent, include the students in the choice and phrasing of the details of the theme. After quite a lot of confusion during the first years of teaching these modules, there seems to be strong convergence towards a model where the teams Gi,j distribute the allotted hours among themselves and teach them individually, treating their disciplines’ perspective on the overall theme. In many cases, we thus have a hierarchal organisation of the teachers work: G∗ appointing the teams Gi,j , who in turn distribute the teaching among the members, while the theme sometimes becomes a relatively loose unifying “heading” with only a few substantial bridges between disciplinary DS constituting the module.

15.3.2 Bi-disciplinary Final Projects In many respects, the final “study line projects” represent a challenge to teachers which is similar to the general study preparation modules. The common necessity of two teachers is again dictated by the combination of disciplines, which in this case can be from the same faculty. But in this case the DS to be organised contains only one student, and the study process is not one of class teaching but of individual and relatively autonomous work on a set of questions given by the teachers. After receiving these questions, the student has two weeks entirely reserved for the work on his project, resulting in a 15–20 page report which is graded with the participation of external examiners. The teachers can only give very limited direction to the work of the student during the two weeks. The DS itself is therefore only partially “observable” to researchers, as is (also in other contexts!) the PS. However, a meaningful didactic analysis of the interaction can be made on the basis of the formulations of questions together with the report of the students, and it has led to interesting results. For instance, Hansen (2009) studied

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Fig. 15.2 Graphic representation of the typical organization of parallel disciplinary teaching with no DoS, a collaborative PrS0 and PoS0 (each with at least two teachers) and mono-disciplinary DSs prepared and evaluated by one teacher (and with the same students in both DS). The knowledge organisations of the collaborative and individual paradidactic systems have very limited overlap

a large numbers of question formulations and reports within the combination of mathematics and history. Not surprisingly, most of these turn out to contain – at the level of questions and at the level of the report – two rather separate parts, sometimes of very unequal quality as regards the requirements in mathematics and history, respectively. A few examples of projects where the two disciplines interacted more closely were also noted. It turns out that teachers of the two disciplines have difficulties to relate to – and in particular to assess – students’ work in the other discipline. This means that not only the DS splits up in to mono-disciplinary parts (with sub questions “assuring” a basic work is done in each) but that in fact the PS around it remains only loosely connected by a common heading. Just as in the case of general study preparation, there is a document which stipulates the overall agenda, involving two or more disciplines, and which is elaborated by a group of teachers in a PrD; but their collaboration also, more or less officially, lead to a separation of other parts of the work (including the construction and realization of a DS which then often splits into two mono-disciplinary ones, cf. Fig. 15.2). The student(s) remain at the intersection of the resulting mono-disciplinary DSs, but in some cases, there is not much more – just as in the case of the usual mono-disciplinary DS of school institutions.

15.3.3 Teachers’ Reactions to the Reform The evaluation of the reform by the national evaluation institute EVA (2009) shows that teachers have experienced a lot of difficulties with the two formats described above. The necessity of teacher teams appears as one of the main difficulties. Only 37% of the teachers think that the time used in these teams is “well spent” when compared with time spent on ordinary (individual) teacher work, and 60% of the teachers estimate that the work in teacher teams is “mostly about administrative issues”. In 2008, about 1/3 of the teachers signed a petition to the government, pointing out a number of problems with the reform and not least with the general study preparation format. Since 2004, the teachers’ magazine Gymnasieskolen has printed

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hundreds of critical comments from teachers about the reform. While it would be difficult to summarise all of the criticism in a few lines, the following comment can be taken as relatively typical: The reform now (. . .) enforces a collectivization of the teaching. Team work and cross disciplinarity are the new fashions, echoing the seventies. (. . .) we, the members of the new teacher teams, did not choose a teaching career to become consultants or teamworkers, but because we wanted to teach. Yes, let’s say it the way it is: we want to be masters in our own classrooms – because we believe that’s what we are good at. (Dahl, 2004, our translation)

15.4 Comparative Perspectives It is interesting to compare the two cases above in terms of paradidactic infrastructures. In both cases, teachers collaborate on the preparation and evaluation of one or more DSs and their work can be studied through artefacts – in particular common documents – produced in the process of organising and developing an organisation of knowledge and practice for teachers (that of the paradidactic systems) and for students (that of the DSs). However, the differences are also striking, as Figs. 15.1 and 15.2 suggest in a schematic way (as far as paradidactic systems are concerned). The case of lesson study produces one complete paradidactic system with a group of teachers present in all phases, and it is clear that the underlying paradidactic infrastructure is firmly established both in the school institution and through the formation and induction of teachers into its professional practices. We notice that while experimental settings often involve DoS (e.g. Chapter 7) and collaborative PrS (e.g. Chapter 16), their presence in an “ordinary” PS is a distinct feature of lesson study. On the other hand, the most common way to realise multidisciplinary formats in the Danish upper secondary school appears to be more complicated (cf. Fig. 15.2). The requirements for a common theme of the DS to be set up implies that a group of teachers engages in a paradidactic system PS0 to plan and evaluate the work in a DS with organisations of practice and knowledge coming from the teachers’ disciplines. However, for a variety of reasons – including the teachers’ disciplinary backgrounds – there is a tendency for this work to be reduced to organising a distribution of the work. This results in setting up two or more paradidactic sub-systems PS1 , PS2 , etc., each with just one teacher, and each assuming the responsibility for a corresponding mono-disciplinary DS. The impression that PS0 is mainly of a bureaucratic nature is then not surprising, as its tasks refer only to the official regulations of the DSs (including the standards for assessment) but not directly to the development of the systems, for example in terms of the artefacts and the organisations of practice and knowledge which they confer to students. One may then speculate that a primary difference between the two cases is to be found in the paradidactic infrastructure. Teachers with a long experience from lesson study might approach the challenges of creating multidisciplinary DS in a less bureaucratic way. A key to developing the necessary shared organisations of practice and knowledge would no doubt be to establish the necessary conditions for

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DoS. Opportunities to observe the teaching of peers (not to be confounded with DS having more than one teacher) are rare to non-existent in most Danish schools. The participation of Japanese teachers in implementing educational reforms (cf. Lewis & Tsuchida, 1997) and the difficulties encountered with recent reforms in Denmark (cf. above) are clearly conditioned by the differences in paradidactic infrastructures for in-school teacher collaboration. Therefore, comparative analysis of such differences may be of a quite practical value when it comes to identify conditions and constraints for developing the organisations of teachers’ practice and knowledge needed to implement ambitious reforms.

References Brousseau, G. (1986). Fondations et méthodes de la didactique des mathématiques. Recherches en didactique des mathématiques, 7(2), 33–115. Chevallard, Y. (1991). La transposition didactique: du savoir savant au savoir enseigné (2ème édition). Grenoble: La pensée sauvage. Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en didactique des mathématiques, 19(2), 221–266. Chevallard, Y., & Cirade, G. (2010). Les ressources manquantes comme problème professionnel – (Missing resources as a professional problem). In G. Gueudet & L. Trouche (Eds.), Ressources vives: Le travail documentaire des professeurs en mathématiques (pp. 111–128). Rennes: Presses Universitaires de Rennes et INRP. Clarke, D. (2004). Patterns of participation in the mathematics classroom. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th conference of the international group for the psychology of mathematics education (vol. 2, pp. 231–238). Bergen: Bergen University College. Dahl, E. (2004). Den politisk ukorrekte privatpraktiserende. Letter to the editor, Gymnasieskolen 4–04. EVA (Danish Evaluation Institute). (2006). Almen studieforberedelse og studieområdet. Copenhagen: Danmarks Evalueringsinstitut. EVA (Danish Evaluation Institute). (2009). Gymnasiereformen på HHX, HTX og STX. Copenhagen: Danmarks Evalueringsinstitut. Fernandez, C. (2002). Learning from Japanese approaches to professional development. The case of lesson study. Journal of Teacher Education, 53(5), 393–405. Fernandez, C., & Yoshida, M. (2004). Lesson study: A Japanese approach to improving mathematics learning and teaching. Mahwah, NJ: Lawrence Erlbaum. Hansen, B. (2009). Didaktik på tværs af matematik og historie. Master Thesis, University of Copenhague, May 2009. Retrieved from http://www.ind.ku.dk/publikationer/studenterserien/ studenterserie10/ Howe, E. (2005). Japan’s teacher acculturation: Critical analysis through comparative ethnographic narrative. Journal of Education for Teaching, 31(2), 121–131. Isoda, M., Stephens, M., Ohara, Y., & Miyakawa, T. (2007). Japanese lesson study in mathematics. Its impact, diversity and potential for educational improvement. Singapore: World Scientific. Lewis, C., & Tsuchida, I. (1997). Planned educational change in Japan: The case of elementary science instruction. Journal of Educational Policy, 12(5), 313–331. Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah: Lawrence Erlbaum. Miyakawa, T., & Winsløw, C. (2009). Didactical designs for students’ proportional reasoning: An “open approach” lesson and a “fundamental situation”. Educational Studies in Mathematics, 72(2), 199–218.

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Nohda, N. (1991). Paradigm of the “open-approach” method in mathematics teaching: Focus on mathematical problem solving. Zentralblatt für Didaktik der Mathematik (now: ZDM – International Journal on Mathematics Education), 23(2), 32–37. Padilla, M., & Riley, J. (2003). Guiding the new teacher: Induction of first-year teachers in Japan. In E. Britton, L. Paine, D. Pimm, & S. Raizen (Eds.), Comprehensive teacher induction: Systems for early career learning (pp. 261–295). Dordrecht: Kluwer. Shimizu, Y. (1999). Aspects of mathematics teacher education in Japan: Focusing on teachers’ roles. Journal of Mathematics Teacher Education, 2, 107–116. Stigler, J., & Hiebert, J. (1999). The teaching gap. Best ideas from the world’s teachers for improving education in the classroom. New York: The Free Press. Tsubota, K. (1977). Opunendo no mondai wo toushite suugakutekina kangaekata wo nobasu [On developing mathematical way of thinking through open end approach problems]. Journal of Japan Society of Mathematical Education, 59(2), 2–5. Winsløw, C. (2004). Quadratics in Japanese. Nordic Studies in Mathematics Education, 9(1), 51–74.

Chapter 16

Communities, Documents and Professional Geneses: Interrelated Stories Ghislaine Gueudet and Luc Trouche

16.1 Introduction This chapter can be considered as following Chapter 2, which has presented the foundations of the documentational approach of mathematics didactics.

16.1.1 A Collective Point of View on Documentation Work We try in this chapter to deepen this theoretical approach by emphasizing the importance of social aspects of teachers’ documentation work. Human work always takes place in an institution (Douglas, 1986), which encompasses a cultural, historical and social reality (Engeström, 1987). A teacher’s documentation work is both supported and constrained by curriculum resources (Chapter 6) and more generally by a resource system (Chapters 2 and 5). Thus, the French Dictionary of Pedagogy1 claims that “Teaching is collaborating”. In some cases, for example the case of Japanese lesson studies (Chapter 15), collective aspects of teachers’ work are readily identified. In other cases, collective aspects are less visible, but we argue that they are always present: each teacher necessarily has relationships with her colleagues, and further, teachers are related through their documentation work. We chose the word “collective” to represent this complex and diverse social reality. At some points in this chapter, we use it as an adjective to qualify something done by several people together. We also use it as a noun to name the most general social form: a group of persons doing something together. Note that we take the notion of “a collective” as not necessarily implying cohesion or involvement in a 1

The “Nouveau Dictionnaire de Pédagogie”, coordinated by Ferdinand Buisson, has been published in 1910. It is now online, http://www.inrp.fr/edition-electronique/lodel/dictionnaireferdinand-buisson/. The given quotation can be found at the entry “Conseil des maîtres”. G. Gueudet (B) CREAD, Université de Bretagne Occidentale, IUFM Bretagne site de Rennes, 35043 Rennes Cedex, France e-mail: [email protected]

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common project. Each teacher takes part in a variety of collectives. Some of these are institutional collectives that are compulsory (such as a school team) or chosen (e.g. a training session). Others are associations, which can be large (e.g. a national association, open to every mathematics teacher), or more restricted (e.g. Sésamath in France, see Section 16.3). Moreover, some collectives correspond to experimental contexts associating teachers and researchers, as described in Chapter 17.

16.1.2 A Focus on Digital Resources As with Chapter 2, this chapter focuses on digital resources, particularly online resources, which both modify some collective aspects of documentation and illuminate existing phenomena. Digitalization establishes various forms of collectives (e.g. email lists) as pointed out by Pédauque (2006, p. 12): “That is actually what digitalization changes: it makes virtual, flowing, unlimited, elusive, communities”2 . In this chapter, we look at the conditions promoting emergence of collectives, at the links between collectives and documentation work, and at individual and collective documentation work. Our main point (following Chapter 2) is that documentation relates both to resources and teacher knowledge. Looking at collective aspects of teachers’ documentation, we will be sensitive to this last aspect, keeping in mind what are “the participants in mathematics teacher education: individuals, teams, communities and networks” (Krainer & Wood, 2008). Therefore, we regard teacher knowledge as situated within and distributed among members of communities, rather than a characteristic of individuals. We first explain our choices: theoretical, methodological and definitional. We then consider the case of a teachers’ association in France, Sésamath, and of one of its members, Pierre. Finally, we draw conclusions and indicate opportunities for further research.

16.2 Collectives and Documentation Work Our approach is rooted in activity theory, as introduced by Vygotski (1978) and Leont’ev (1979). This theory has been supplemented by Engeström (1987), adding specific implications for communities, which were further developed by Lave and Wenger (1991). We focus, here, on collective aspects of teachers’ documentation work. This means that we distinguish among a variety of collectives in which a teacher can be involved, and we take into account different sets of collective resources that are linked to these collectives. This study presents several levels of complexity: complexity of the boundaries of each resource set (for a given collective, teachers’ resources are more or less shared); complexity of overlapping of collectives (a given teacher always participates in different groups – within 2

Our translation.

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her classes, within the grade-level team, within the mathematics department, etc.); complexity of time (a collective of teachers working together is subject to strong schedule constraints). Describing (as far as possible) this complexity necessitates making conceptual and methodological choices.

16.2.1 Communities of Practice: A Theoretical Framework We have chosen, among some possible theoretical frameworks (Gueudet & Trouche, 2008; Chapter 3), the concept of communities of practice (CoP), introduced by Lave and Wenger (1991) to designate a group of people sharing an interest, a craft or a profession. Our choice is coherent with a flow of current research on mathematics teacher professional growth (Jaworski, 2008; Lerman & Zehetmeier, 2008). Each CoP is a community of learning: learning of the rules (Engeström, 1987) allowing the community to develop, sharing information and experiences within the group and learning from the activity itself. The community members have thus an opportunity to develop themselves personally as well as professionally. Wenger (1998) defined CoPs according to three essential conditions: mutual engagement (members establishing norms and building collaborative relationships), joint enterprise (members creating a shared understanding of what are the common objectives) and shared repertoire (members producing resources – material or symbolic – which are recognized as their own by the group and its members). Wenger (ibidem, p. 55) also emphasizes two key processes, participation and reification, that are also crucial to our research. Participation “refers to a process of taking part and also to the relations with others that reflect this process”; it supposes a personal contribution to the shared project, coming with a negotiation, within the community, of what is to be done, how and why. Reification refers to “the process of giving form to our experience by producing objects that congeal this experience into thingness” (ibid., p. 58); it supplies the shared repertoire of the community, result of the engagement and the participation of each member. Participation and reification are interrelated: reification results from participation, and the shared repertoire supports each member’s participation to the shared objective.

16.2.2 Communities Geneses, Member Trajectories and Congealed Experience A CoP is not a fixed entity, it emerges and develops naturally because of the dynamic of the joint enterprise. Wenger, McDermott, & Snyder (2002, p. 68–69) distinguish five possible steps for what we will name a community genesis: potential, coalescing, maturing, stewardship and transformation (we elaborate on these notions in Section 16.3). This conceptualization fits our objective of studying the documentation work of teachers in a collective, in describing a variety of teachers’ collectives at various steps of development. It does not mean that each collective of teachers is

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fully characterized as a CoP, even at a first step of potential: for example the set of mathematics teachers working in a same school does not, generally, gather the three features of mutual engagement, joint enterprise and shared repertoire. Our theoretical choice thus lies on two hypotheses: (1) a teachers’ CoP constitutes a good case study for understanding collective aspects of teacher documentation and (2) what we learn from this case study sheds light on what happens in other types of collectives. The evolution of the community goes with the evolution of its members’ identities: the identity is defined by Wenger (2002, p. 149) as “the profound issue of how to be a human being”, and thus is a process of becoming. And so, Wenger indicates that “Identity in practice arises out of an interplay of participation and reification” (p. 153). The identities “form trajectories, both within and across communities of practice” (p. 154). Wenger distinguishes four types of trajectories: peripheral trajectories (never leading to full participation), inbound trajectories (joining the community to become full participants), insider trajectories (full membership and continuous evolution of practice) and boundary trajectories (spanning boundaries and linking communities of practice). A CoP is not an isolated entity, either; Wenger (1998, p. 117) states that “CoP can connect with the rest of the world by providing peripherical experiences [. . .] to people who are not on a trajectory to become full member. This kind of peripherality can include observation, but it can also go beyond mere observation and involve actual forms of engagement”. According to his/her trajectory, each member may have a particular role, integrated in an explicit or implicit division of labour (Engeström, 1987) within the community. We will illustrate these notions further.

16.2.3 Towards an Extended Model We modelled an individual documentational genesis as an interplay between a teacher and a set of resources, leading to a document, mixed entity composed of resources and a scheme of utilisation (Chapter 2, Fig. 2.1). We propose here

Fig. 16.1 A representation of a community documentational genesis

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(Fig. 16.1) an extended model drawing on the conceptualisation of CoP: we consider the interplay between a teacher’s CoP and sets of resources, mobilized for achieving common goals. Instead of reification, we coin the expression community documentational genesis for describing the process of gathering, creating and sharing resources to achieve the teaching goals of the community. The result of this process, the community documentation, is composed of the shared repertoire of resources and shared associated knowledge (what teachers learn from conceiving, implementing, discussing resources). Furthermore, these resources and this knowledge evolve together over time. We will thus consider the duality between participation and documentation: on the one hand, documentation is an outcome of participation; on the other hand, the shared repertoire and associated knowledge supports each member’s participation in the shared objective. Interpreting these processes in terms of geneses, we can point here to the duality between two geneses: the community genesis (the emergence of mutual engagement and joint enterprise), and the community documentational genesis (creation of shared repertoire and building of shared knowledge). There are complex relationships between individual and community documentational geneses. Firstly, the shared repertoire is a component of each member’s resource system: for example we will describe in Section 16.3, how Pierre considers the resources of his association as an essential part of his resource system. Secondly, there is a strong interplay between each member’s knowledge and the shared knowledge embedded in her community documentation: each member learns from her community and the shared knowledge is built by the community documentation genesis. The community documentation cannot be defined by the gathering of the documentation of its members. For example the work in a teachers association produces objects (mailings, webpages, workplace tools) that a teacher alone could not create. Similarly, a teacher’s documentation exceeds what it could give to a community she is part of: for example what teacher learns from her direct contacts with her students work is not totally sharable with her community. We will demonstrate the interest of this theoretical approach in the next section. We develop before some methodological aspects linked with this approach, complementing what we have presented in Chapter 2.

16.2.4 Methodological Choices We have already presented (Chapter 2) a particular methodology of reflective investigation, aiming at analyzing a teacher’s individual documentation work. This methodology needs to be adapted to the case of a community. We begin to make this adaptation, here, looking at both the community documentation work and the individual documentation work. To analyze the community documentation work, we have used classical tools: questionnaires; analysis of the resources collaboratively designed; following of critical meeting of the members (seminaries, training sessions both remote and face to face, emails), where participation and negotiation occur. It is certainly necessary to conceive of other tools to understand the

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interrelated community and individual documentation geneses, including: schematic representation of the community resources (in the thread of SRRS, see Fig. 16.4), schematic representation of the community structure, quantitative and qualitative analysis of emails lists. These tools are still in progress. We have chosen for our purpose a case – a teachers online association – where a single CoP plays a distinctive role in the documentation process. We are aware that a same teacher could be involved in several CoPs, influencing her documentation. Nevertheless we assume that, if a given CoP has a preponderant influence for a given teacher, we could more easily study the interactions between this teacher’s individual documentation work, and the documentation of this community of which she is part. The case study, described in the next section, is not “representative” of an “average” teacher’s CoP. It could be said that this is an advanced case, because of the elaborated design and usage of online resources. It could also be said that this case could be considered a laboratory for the future, or a good representative of future possibilities of teachers’ communities of practice. From the same perspective (exploring advanced cases) we have investigated some of the most active members within this association.

16.3 Sésamath, Individual and Collective Resources Systems In this section, we study community documentation geneses, focusing on a French mathematics teachers online association, known as Sésamath, aiming to collaboratively design free teaching resources. We start with the presentation of what appears as common features of online teachers’ associations in France. We consider then Sésamath itself, the largest teachers’ online association in France. Finally, we focus on one of its members, trying to understand the interactions between his documentation work in the association, in his school, and in his classroom.

16.3.1 The Emergence of Teachers’ Online Associations The rapid growth of teachers’ online associations, designing and sharing resources, appears as a result of digitalization and of a larger accessibility of Internet. We studied three of teachers’ associations,3 asking their leaders and a sample of members to explain how their associations worked (their history, the problems they met, the solutions they have built and with what results, etc.). The responses allow us (Gueudet & Trouche, 2009) to see these associations as instantiations of CoPs, as defined earlier. This opens a view on the community geneses, which seem to have common features: the more active members of these 3 Sésamath (http://www.sesamath.net/) regarding Mathematics teaching, Weblettres ((http://www. weblettres.net/) regarding French teaching and Clionautes (http://www.clionautes.org/) regarding Geography teaching.

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associations describe their geneses as a rolling stone (Fig. 16.3), with three main stages of development. These stages could be related to the first Wenger’s steps: – First step (a potential one, according to Wenger, corresponding at a gathering of persons sharing a common interest), a group of pioneers meets for sharing resources and constitutes a first kernel. – Second step (a coalescing one, according to Wenger, corresponding at a common decision on “what we want to do together, and how”), this initial group deepens the documentation work in cooperating4 (that is the beginning of a shared repertoire). In this dynamic, this group draws around it a periphery of teachers interested only in using the resources of this shared repertoire, and perhaps give back some personal resources in it. – Third step (a maturing one, according to Wenger, corresponding to the efforts for incorporating new members in the community) the “cooperation kernel” passes to a stage of collaboration, reflecting together about what must be done, which means a step of blooming of the initial CoP. This dynamic draws the sharing periphery to a stage of cooperation and gathers a new periphery of “sharing teachers.” The periphery of a CoP appears thus as successive crowns wrapped around the kernel from the rolling of the CoP-stone. There are thus different roles in the community, not static, but evolving in the dynamic of the association. This dynamic is fostered by a permanent reflection of the kernel about the organization and the ways to cultivate (Wenger et al., 2002) the CoP. Propositions are permanently sent to the members of the successive peripheries (Fig. 16.2), to support their progression towards the centre, favouring inbound trajectories. The kernel is neither closed, nor invariant: it is always renewed (“old” members quitting, new members arriving), resulting from different members’ trajectories. The development of an association is probably linked to its openness.5

Fig. 16.2 The genesis of a teachers online association, seen as a rolling stone (Gueudet & Trouche, 2009) 4

Dillenbourg (1999) distinguishes cooperative and collaborative work: “In cooperation, partners split the work, solve sub-tasks individually and then assemble the partial results into the final output. In collaboration, partners do the work ‘together’”. 5 The name of the association itself, Sésamath, is certainly revealing, as a wink to “Open sesame”, the famous phrase from the Arabian Nights.

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The shared knowledge of these communities appears as a set of beliefs (sharing resources is a necessity, for the teachers professional growth and for the quality of the resources), values (promoting free resources and free software) and professional knowledge (on the teaching domain involved, and the way it has to be taught). Strong interactions between participation to the association and documentation, supporting each other, clearly appear. To deepen this analysis, we need to focus on one of these associations, Sésamath; this association is the focus of the next section.

16.3.2 Sésamath, a Teachers Association Designing and Sharing Teaching Resources Sésamath was created in 2001. Its growth, since, has been rapid. Today, Sésamath gathers 100 subscribers (the kernel, see Fig. 16.2), 5,000 teachers participating in various documentation projects (the cooperation crown), and sends a letter each month to 30,000 teachers (the sharing crown). One reason for this growth could be the existence of the French network of IREM,6 which has, in some sense, paved the way since 1970. Sésamath (http://www.sesamath.net/) essentially gathers in-service mathematics teachers, aiming to “freely distribute resources for mathematics teaching”. Its website front page claims “mathematics for everybody”, “working together, supporting one another, communicating!”. Its shared repertoire consists in resources for teaching: online exercises (Mathenpoche), digital textbooks (also with printed copies available at half of the price of other books), a dynamic geometry system (TracenPoche, TeP), simulated geometry instruments (InstrumenPoche, IeP),7 etc. All these materials are free. The audience of Sésamath is very large: about one million visits, each month, to its website. The community documentation exceeds this shared repertoire: Sabra (2009), by way of a questionnaire proposed to 30 of the most active members of the association,8 evidences that knowledge is produced by the community documentation work (knowledge on mathematics, on mathematics teaching and on teaching). Let us have a closer look at the so productive documentation work in this community. What allows this productivity, and fosters the Sésamath genesis, is the development of tools favouring collaborative work. The main tool consists in a platform for collaborative work (Sésaprof), which gathers thousands of teachers, for achieving a given project (e.g. the design of a textbook). Each project concerns about 50 teachers, a reasonable number for a real collaboration. We hypothesize that a CoP emerges as the project takes form, and we have found evidence for this hypothesis in the groups we have studied. The development of Sésamath appears thus as strongly linked to digitalization (Gueudet & Trouche, 2009): rapid technological evolution 6

Institute for Research on Mathematics Teaching. Which means “Trace-in-the-Pocket” and “Instrument-in-the-Pocket”. 8 Those who have accepted to follow a training session organized by researchers at the National Institute for Pedagogical Research. 7

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brings needs and means for creating new resources, and digital environment allows the organization of a collaborative work at a large scale. Borba and Gadanidis (2008) consider “virtual environments and tools both as factors mediating teacher collaboration and as co-actors in the collaborative process” (p. 182). The use of digital tools permits collective documentation work; it also shapes this work.

16.3.3 Pierre’s Documentational Genesis, Involvement in Sésamath and in His School We were looking for a Sésamath member with a given profile: a teacher between 30 and 40 years old (thus, in the middle of her career, with a past and a future as a teacher), member of the association for at least 5 years and actively involved in one of its projects. Pierre (already presented in Chapter 2) met these requirements, and agreed to participate in research for 2 years. He is 35 years old and teaches in a middle school (grade 6–9). His father had a passion for sciences; his mother and her parents were teachers (his grandfather had written textbooks). He started by studying physics, and has kept a vision of mathematics as “a tool necessary for designing scientific models”.9 He has finally chosen mathematics due to its double aspect: “a world both formal and dream-like”. Solving problems constitutes for him the heart of mathematics teaching. He describes a mathematics teacher as a real oneman band: “artist, actor, human resource manager, psychologist for individuals and groups, mathematician, cultural reference. . .”. He evinces a strong collective involvement both in his school and in Sesamath: he is “teacher in charge of technology”,10 treasurer of the school cooperative, responsible of the school’s chess club. These activities are not all dedicated to mathematics. In Sésamath, as of 2008, he was a member of the board for 5 years. This meant that he spent approximately 1 h a day reading emails and participating in forums “that engage the association life”. He was also a member of a project developing a grade 6 textbook, which is still in progress at this time. He was, finally, the pilot of a new Sésamath project entitled “mathematics files for primary schools”. Documentation work takes place within each of these collective involvements and each of them is part of Pierre’s work, as he said: “Consuming time in collective activities is a component of my teaching activity”. He particularly emphasizes the importance of the primary school project (“it gives a better understanding of what my pupils know when arriving at secondary school”), the Sésamath board (“it makes me aware of the questions asked to the profession as a whole”) and the “grade

9

Pierre’s quotations, in Sections 16.3.3 and 16.3.4, are extracted from a questionnaire (October, 2008) and an interview (November, 2008). 10 Which means responsible for computer and software equipment, for giving his colleagues advice for use (website, software. . .). Hard work, his logbook shows that it is time consuming (6 h in 3 weeks), for a limited financial reward of the institution (1 h paid each month).

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At the top of this representation, appear Sésamath resources: textbooks, exercises (Mathenpoche) and software designed by the association. Different arrows appear, allowing to distinguish different types of activity: thick for preparing lessons, medium for preparing exercises, thin for preparing activities. Most of the arrows concern activities (i.e. problem solving, open ended questions, which constitute the heart of Pierre’ teaching). Pierre does not renew his personal resources: actually, he essentially contributes to feed Sésamath repertoires. Apparently no interactions with his own colleagues.

Fig. 16.3 Pierre’s SRRS (February 2009), handmade, our translation

6 textbook”. It is actually this last project, which appeared as fostering Pierre’s documentation. For all the duration of the project (2 years), Pierre decided to have only grade 6 classes (three classes, for 6 h teaching in it), to “align” his documentation work with the community documentation. Thus, the documentation work that Pierre accomplished in 2008–2009 for the grade-6 level concentrated his main efforts, and connected individual and community documentation. It is possible to analyze Pierre’s trajectory, related to Sésamath, from two points of view. To the kernel of the association, it was clearly, an insider trajectory during our follow-up period, and it was both fed and guided by his Sésamath documentation work. It appears also as a boundary trajectory, spanning several collectives (collectives in Pierre’s school, Sésamath board, grade 6 textbook). For example when proposing to his colleagues to choose the Sésamath textbook for their own classes, Pierre appears as a go-between for the two collectives. The interplay between Pierre’s and community documentation appears also through the schematic representation of his resource system (SRRS, Chapter 2), as depicted in Fig. 16.3. For Pierre, Sésamath’s repertoire (textbooks, exercise books, software), constitutes the main reference of his system. What is described as “personal resources” (down right) are archives and seem to be congealed (no arrow comes to renew them); what seems to be evolving (“lived”, with the meaning conveyed by the title of this book) are resources of the shared Sésamath repertoire. Pierre’s explanations on “how it works” help to understand his documentation work:

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• In the first direction (from Sésamath resources to his own resources) he “digs up” what he needs, and “customizes” it (“generally, I pick up an exercise, I keep its main idea, and I rephrase its questions”); • In the second direction, Pierre acts as a “brocanteur”11 : he “bargain-hunts” resources (on the web as well as in old books found in libraries or bookstores), captures them in his computer. The space dedicated to his documentation work on his computer has an important role. Pierre gave it a name, Piwosh, standing for “Pierre’s workshop”.12 Piwosh looks like an incubator of resources. Pierre jots down his ideas on Piwosh as they come (Pierre sometimes has difficulties finding them again). He develops them when the need occurs, and tests them with his students. The resources thus follow a path, from test phases to revision phases, until they are good enough (according to Pierre’s judgment) to be added in the shared Sésamath repertoire. There is not only an interplay between Pierre’s and Sésamath resource systems: it is a more complex interplay, where other members of the textbook project act as active partners: Pierre proposes his ideas for discussion on Sésamath discussion lists, and he also discusses the resources proposed by the members of the project group, which we regard as an emerging community of practice. The Sésamath resource system therefore appears as both a result of Pierre’s documentation work, and as one of its essential sources. This situation constitutes a culmination of the collaborative process within Sésamath. This is not only the resource content that is shared (“sharing the same exercises”), this is not only the type of material resources that is shared (“sharing the same type of textbook”), it is, physically, the same resources which are shared, on the same remote host, and which are available, from anywhere, for each member of the project. To this collaborative documentation corresponds a collaborative form of teaching, which we portray in the following section.

16.3.4 Documentation Work Going on in Pierre’s Class Using online resources is an important feature of Pierre’s documentation work, within or without his students (for preparing his teaching or collaborating in Sésamath projects). Within his classroom, a connected computer, a projector and an interactive whiteboard (IWB) are used to work with online resources. For example at the beginning of each lesson, the teacher opens Pronote (Chapter 2), an application allowing displaying the students list, to note the absentees, to memorize what has been done, and what is still to do . . . Another example of this continuous Internet 11

French word standing for “secondhand goods dealer”. The English expression is interesting, evidencing that a resource is never a firsthand one, but always inherits from some older ones. 12 Our translation from French TafPi, literally “Taf de Pierre”; Taf is a slang French word, meaning “work”.

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use: the teacher exploits Google to do any arithmetic operation exceeding students’ capacities of mental computation (it was amazing to observe that handheld calculators remain in students’ schoolbags!). For continuing to interact with his students outside of the classroom, he developed a collaborative website on which he regularly uploads mathematics problems (that he calls enigma). Students try to solve them and write their solutions on a forum. Sésamath resources are widely used during each lesson, and they contribute to Pierre’s instrumentation processes. For example the simulated geometrical instruments (Sésamath IeP) allow students to visualize geometrical constructions. It clearly contributes to the development by Pierre of a scheme for “teaching how to draw a geometrical construction corresponding to a mathematical text.” Firstly, he expects his students to work without any help; then, when the construction is almost complete, he shows the construction process on the IWB. Finally, he shows the construction as if it were a film, playing in a loop, which helps the students who have not succeeded in completing their figure. Pierre explains why this method is important: – “students have to establish a direct relationship with the construction” (actually, when looking at the film, they receive no scaffolding from the teacher), – “geometry goes on as a film, not as a picture”, – “repetition, in pedagogy, is essential” and – “students have to follow their own rhythm” (which is the case with the film: if students are lost, they can always wait for the next passage of the film, thanks to the loop), etc. We consider these declarations as indicating operational invariants, components of a scheme, fostered by the resource (IeP) involved in this situation. Pierre has developed a strong professional knowledge within Sésamath by demonstrating that “learning is collaborating”.13 Pierre’s online collaboration has evident consequences for the orchestrations (Chapter 14) of mathematical situations. Standing in front of the students is not the teacher, but rather the blackboard and the IWB (Fig. 16.4). Pierre combines these two boards, mostly for the purpose of using Sésamath resources, which express his pedagogical theory that “Comparing different resources is the way to make his/her own idea.” Pierre explains that what he names the “chevron” configuration (Fig. 16.4, Pierre’s classroom) of the students’ tables fosters debate in the classroom, students facing the two boards and their peers to discuss a given problem. Problem solving is, for Pierre, the heart of learning mathematics. He privileges phases of joint construction (geometrical figures, conjectures), more than phases of discussion of the correctness of a solution (validation). For him, learning can be viewed as a process of using, adapting and sharing resources. Thus, this process is dynamic, collective, and cannot be fully planed by the teacher. Joint design of 13

Which echoes our introductory quotation “teaching is collaborating”.

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Fig. 16.4 Pierre’s classroom configuration (drawn by Pierre), and Pierre in his classroom, showing something on the IWB

resources in Sésamath and joint construction of knowledge (Chapters 3 and 11) in the classroom seem associated. We notice that Pierre and Myriam (Chapter 2), although they have a common interest for Sésamath resources and for problem solving, seem to develop very different documentation systems. For Myriam, the official texts are very important resources, intervening in many documents she develops, providing problems worksheets. Pierre naturally also refers to the official texts, but does not look for ideas of exercises in them. Myriam uses the Sésamath textbooks and Sésamath exercise sheets to prepare her own sheets; but she does not use, for example TeP (geometry software) and IeP (simulated instruments). For her, Sésamath is only one resource, among several others. She sometimes video-projects online exercises during her classes, but she does not often use the Internet during her lessons. We hypothesize than this difference is linked with the difference of position towards the association. There is, in Pierre’s case (and not in Myriam’s), a strong interaction between community and individual documentation system, between the documentation work for his association and for his own classes. We describe this situation as a symbiosis between two documentation systems. This can be linked with two points: firstly, Pierre is member of the Sésamath kernel; secondly, his documentation work for the association (making a grade 6 textbook) perfectly meets his documentation work for himself (making the teaching of his own grade 6 classes). What will happen at the end of these unusual conditions – which means the end of textbook elaboration? To answer this, we describe the second year of Pierre’s teaching that we observed.

16.3.5 Association vs. School, Two Faces of a Same Medal In 2009–2010, Pierre’s situation within Sésamath has deeply evolved. The work on the grade 6 textbook is finished, and Pierre is no longer a member of the Sésamath board. He explains: “After a strong investment, it is necessary to take a step back”.14 14

Pierre’s quotations, in Section 16.3.5, are extracted from his interview in November, 2009

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This step back is certainly an effect of the completion of the textbook (“the work has been done”), but also a consequence of a personal event (Pierre has had a new child). Pierre is still a Sésamath member, but involved in a single project (mathematics files for primary school), which is not as time consuming as the previous one (instead of 10 h a week for the association the previous year, he now spends about 1 h a week). This enlightens the possible variations of trajectories (Wenger, 1998) inside the association’s rolling stone (Fig. 16.2). Even in the association’s kernel, complex trajectories take place due to both community documentation geneses and personal stories. After having been an insider one, Pierre’s trajectory became a peripheral one. This evolution goes with a greater care by Pierre about what could be collectively done within the school: for example the website that Pierre developed for communicating with his students migrated from a private host to the school common website, for sharing with colleagues. The new SRRS (Fig. 16.5) that Pierre draws evidences this phenomenon. When discussing with Pierre about the data collected 1 year before, he notices also the interest of the classroom arrangement (Fig. 16.4) in relation with this collaboration with his colleagues: he sometimes exchanges his classroom with his neighbour’s one, for organizing small groups work, which fits well in this room. This exchange yielded an introduction of this neighbour (who teaches French) to the interest of IWB, then to some ideas about how to use it, etc. This re-evaluating of the existing collaboration within the school is certainly a consequence of Pierre’s refocusing on his school, but also an indirect effect of our methodology of reflective investigation (Chapter 2) itself: Working with researchers (Fig. 16.5: “after our

Fig. 16.5 New SRRS made by Pierre in February 2010 (our translation)

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reflection, I thought . . .”) on his own documentation work makes Pierre more aware of his colleagues as part of “the sources of his resources.” Pierre also distances himself from Sésamath resources (Fig. 16.5 is to be compared with Fig. 16.3 from this point of view). Sésamath is not only Pierre’s horizon, but also emerges as an important external resource through his new SRRS (films, readings, . . .). Once the textbook design was finished, Pierre takes up a more critical stance: the whole Sésamath resources are no more directly applied from the association’s website. When they seem to be not as relevant, they are modified, and saved in Pierre’s personal repository for future usages. There is a sort of balance between Pierre’s investment in his association and in his school, not only as communicating vases (less in the association, more in his school and vice versa), but also the various types of community documentation work that feed each other. We could say, extending a formula we met twice in this chapter: documenting is collaborating. The case of Pierre evidences also the interplay between documentation geneses and professional geneses. Pierre’s drawings (Fig. 16.6), representing the evolution of his classroom configurations, are very interesting from this point of view: – First configuration: the beginning (4 years ago), when the IWB entered the classroom, he “put it in a corner, on its feet”, and the students “in front of the boards, as looking at a film”. – Second configuration: this new tool, and the discussions in Sésamath about the resources to be designed for this purpose (Pierre wrote, with a colleague, a paper on this theme, for the journal of the association15 ) led him to a new configuration. The IWB is now installed on a wall (“it is now part of the classroom”), the students’ desks faced both the blackboard and the IWB (necessary to compare the information “without privileging one of them”). Implementing new resources

Fig. 16.6 Evolution of equipment and classrooms configurations (drawings from Pierre)

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creates new needs for writing: a white board appears on the side of the blackboard, to keep the memory of what appears on the IWB without changing its current display. – Third and last configuration: the “chevrons” (already seen Fig. 16.4) appearing for encouraging debates around problem solving. We conceptualize this professional growth as a professional genesis, encompassing several documentational geneses. The collectives, under various forms, foster these processes. This occurs in the context of associations, and in the context of “natural life” of schools. Is it possible that the schooling institution takes profit also of this dynamic of collectives for teacher professional development programs? We examine this question in the next section.

16.4 Discussion We have argued in this chapter that the collective is everywhere in teachers’ documentation work and that it takes very different forms. The notion of communities of practice is useful to grasp the dynamics of teachers collectives sharing a project of documentation. It has often been used in the context of teacher training (Krainer & Wood, 2008), with cultivated communities of teachers. But each community is both spontaneous and cultivated (Wenger, 1998), we observed it here for the teacher association on line – Sésamath – that we have studied in this chapter. Teachers freely join this association, and the board of Sésamath takes care of its development. Each community is a tumultuous aggregation of members – tumultuous in several different senses: some teachers enter the community while other ones get out; teachers’ roles inside the CoP permanently change, sometimes suddenly; as a rolling stone, a community gathers in successive crowns various groups attracted in some way by the practice of the community and its shared repertoire. Paraphrasing Lave and Wenger (op. cit.), saying that each community of practice is a community of learning, we could say that each teachers’ community of practice is a community of documentation, which means that community geneses and documentation geneses act in concert. The documentation work leads to the production of temporary objects, as “lived” resources, always engaged in new evolutions. We assume that these phenomena are not specific to local situations, but concern, at different levels, each teacher, involved in various collectives. No collective is an isolated one. Pierre, for example, is member of Sésamath, member of various collectives within his school. These collectives are acting as co-stimulating agents. Following the work of a teacher means following interrelated stories: stories of the collectives she is part of, stories of their documents, and stories of her professional growth. Instead of story, we have used, both in Chapters 2 and 16, the term genesis to underline the idea of development boosted by itself, fed by an environment, directed towards a higher level of organization. Our study proceeded by successive levels of investigation. We described (Chapter 2) how studying a teacher’s activity requires encompassing documentational geneses, considering activities outside of and inside school, and sets of

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resources intervening in her documentation work. This “resources” point of view led us to situate the teacher within a set of collectives where these resources are living (this chapter). It would certainly be necessary, this time by successive closer looks, to more precisely study what is at stake inside these collectives over time. We are aware that the understanding of community documentational geneses requires a refinement of methodology and new tools, allowing us to collect and to analyse new types of data (extracts of online forum, emails, verbatim of communities meeting – online or face to face – annotations of resources . . .). We are now developing these new tools, towards a methodology of community reflective investigation. Acknowledgments We deeply thank Janine Remillard and Joshua Taton for their help in reviewing the English language in the last version of this chapter.

References Borba, M. C., & Gadanidis, G. (2008). Virtual communities and networks of practising mathematics teachers. In K. Krainer & T. Wood (Eds.), Participants in mathematics teachers education: Individuals, teams, communities and networks (Vol. 3, pp. 181–206). Rotterdam/Taipei: Sense Publishers. Dillenbourg, P. (1999). What do you mean by collaborative learning? In P. Dillenbourg (Ed.), Collaborative-learning: Cognitive and computational approaches (pp. 1–19). Oxford: Elsevier. Retrieved on May, 2011, from http://tecfa.unige.ch/tecfa/teaching/aei/papiers/Dillenbourg.pdf Douglas, M. (1986). How institutions think? Syracuse: Syracuse University Press. Engeström, Y. (1987). Learning by expanding. An activity-theoretical approach to developmental research. Helsinki: Orienta-Konsultit. Gueudet, G., Soury-Lavergne, S., & Trouche, L. (2009). Soutenir l’intégration des TICE: quels assistants méthodologiques pour le développement de la documentation collective des professeurs? Exemples du SFoDEM et du dispositif Pairform@nce. In C. Ouvrier-Buffet & M.-J. Perrin-Glorian (Eds.), Approches plurielles en didactique des mathématiques (pp. 161–173). Paris: Laboratoire de didactique André Revuz, Université Paris Diderot. Gueudet, G., & Trouche, L. (2008). Du travail documentaire des enseignants: genèses, collectifs, communautés. Le cas des mathématiques. Education et didactique, 2(3), 7–33. Gueudet, G., & Trouche, L. (2009). Conception et usages de ressources pour et par les professeurs: développement associatif et développement professionnel, Dossiers de l’Ingénierie Educative, 65, 78–82. Retrieved, on May, 2011, from http://www.cndp.fr/archivage/valid/139699/13969918418-23865.pdf Jaworski, B. (2008). Building and sustaining inquiry communities in mathematics teaching development. In K. Krainer & T. Woods (Eds.), Participants in mathematics teachers education (pp. 309–330). Rotterdam/Taipei: Sense Publishers. Krainer, K., & Wood, T. (Eds.). (2008). Participants in mathematics teachers education: Individuals, teams, communities and networks (Vol. 3). Rotterdam/Taipei: Sense Publishers. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press. Leont’ev, A. N. (1979). The problem of activity in psychology. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 37–71). New York: M.E. Sharpe. Lerman, S., & Zehetmeier, S. (2008). Face-to-face communities and networks of practicing mathematics teachers. In K. Krainer & T. Woods (Eds.), Participants in mathematics teachers education (pp. 133–153). Rotterdam/Taipei: Sense Publishers. Pédauque, R. T. (Coll.) (2006). Le document à la lumière du numérique. Caen: C & F éditions.

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Sabra, H. (2009). Entre monde du professeur et monde du collectif: réflexion sur la dynamique de l’association Sésamath. Petit x, 81, 55–78. The Design-Based Research Collective (Baumgartner, E., Bell, P., Brophy, S., Hoadley, C., Hsi, S., Joseph, D., Orrill, C., Puntambekar, S., Sandoval, W., & Tabak, I.). (2002). Design-based research: An emerging paradigm for educational inquiry. Educational Researcher, 32(1), 5–8. Vygotski, L. (1978). Mind in society. Cambridge, MA: Harvard University Press. Wenger, E. (1998). Communities of practice. Learning, meaning, identity. New York: Cambridge University Press. Wenger, E., McDermott, R. A., & Snyder, W. (2002). Cultivating communities of practice: A guide to managing knowledge. Boston: Harvard Business School Press.

Chapter 17

Mathematics Teachers as Instructional Designers: What Does It Take? Jana Visnovska, Paul Cobb, and Chrystal Dean

17.1 Introduction Early perspectives on instructional improvement in mathematics conceptualized curriculum materials as the primary means of bringing about instructional changes (Bruner, 1960; Dow, 1991). This perspective is apparent in the development of ‘teacher-proof’ curricula in the 1950s and 1960s that failed to produce the desired instructional improvements (Ball & Cohen, 1996; Remillard, 1999). As Ball and Cohen (1996) argue, this failure is often explained as failure to take account of teachers’ current knowledge and practices, and the approach has been critiqued for attempting to ‘de-skill’ teaching (Apple, 1990). In contrast, contemporary research on curriculum design and implementation distinguishes between the designed and the enacted curricula, thereby highlighting the key role of teachers in using textbooks and associated materials to support students’ learning (Ball & Cohen, 1996). Following de Certeau (1984), it is in fact reasonable to view implementation as a second act of creation (Cobb, Zhao, & Visnovska, 2008). Although designed curricula and textbooks are important instructional resources, teachers are the designers of the curricula that are actually enacted in their classrooms (Doyle, 1992; Remillard, 1999). This shift in perspective has brought to the fore the need to understand the work of teaching and what is involved in supporting teachers’ development of effective instructional practices. The approach proposed by Gueudet and Trouche (2009) emphasizes teachers’ central role in mediating the reform efforts of curriculum designers, policy makers, and school leaders, and focuses attention on teachers’ documentation work. According to Gueudet and Trouche, teachers’ documentation work includes looking for resources (e.g., instructional materials, tools, but also time for planning, colleagues with whom to discuss instructional issues, and workshops dedicated to specific issues), and making sense of and using them (e.g., planning individual tasks and sequences of instructional tasks, aligning instruction with the objectives and J. Visnovska (B) School of Education, The University of Queensland, St Lucia, QLD 4072, Australia e-mail: [email protected]

G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8_17,  C Springer Science+Business Media B.V. 2012

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standards to which the teachers are held accountable). The products of this work at a given point in time are characterized as documents (e.g., records of the mathematical ideas that are the goals of an instructional unit; a sequence of tasks along with a justification of their selection). These documents can in turn become resources in teachers’ subsequent documentation work. The process of documentational genesis therefore foregrounds interactions of teachers and resources, and highlights how both are transformed in the course of these interactions (see also Chapters 2 and 16 of this book). Although teachers necessarily make changes in their documentation work whenever they use new instructional materials for the first time, the challenge of designing resources to proactively support specific changes (e.g., toward practices that the research on student learning indicates are effective) remains nontrivial (cf. Chapters 10 and 6). This is in part due to the complexity of resources involved in a productive instructional design. Our purpose in this chapter is to both acknowledge this complexity and to present a case in which a group of mathematics teachers’ documentation work was guided in productive directions over an extended period of time. We outline some of the resources that the teacher group used routinely in the last year of a 5-year study, and draw contrasts with ways that these resources were used in the group’s documentation work several years earlier. In doing so, we draw on the documentation genesis framework described by Gueudet and Trouche (2009) to explain how the use of the same material resources (e.g., an instructional sequence in statistics) came to have significantly different meanings in the group activities over time. We thus illustrate and substantiate the argument that teachers themselves, in working with and reworking different resources, play a central role in developing the sophisticated documents that are needed to facilitate their instructional improvement efforts (cf. Rabardel & Bourmaud, 2003; see also Chapters 7 and 5). We also stress that the productive use of social resources (e.g., the coparticipating teachers) followed a similar pattern in that these resources, central to the teachers’ effective documentation work, were not readily available from the outset but were instead developed in the course of sustained professional development (Dean, 2005).

17.2 Background to the Professional Development Design Experiment We draw on a professional development design study (Brown, 1992; Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003) that we1 conducted with a group of middleschool mathematics teachers in a diverse urban school district with a high-stakes accountability program (Dean, 2005; Visnovska, 2009). We began working in the district to provide teacher development in statistical data analysis at the invitation of 1 Presented study was a part of a larger research project. The research team included the authors, Kay McClain, Teruni Lamberg, Qing Zhao, Melissa Gresalfi, Lori Tyler, and Jose Cortina.

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the district’s mathematics coordinator. We conducted a 2-day summer institute and three 1-day work-sessions during the first year of the study, a 3-day summer institute and six 1-day sessions during each of the subsequent 4 years, and a concluding 3-day summer institute. Our goals in working with the teachers were to support them in (a) deepening their understanding of central statistical ideas, (b) making sense of individual students’ statistical interpretations and solutions, and (c) adapting instructional sequences developed in prior classroom design experiments to their needs and to the constraints of their instructional situations (Cobb & McClain, 2001). The research question that we addressed concerned the process of supporting teachers’ development of instructional practices in which they place students’ reasoning at the center of their instructional decision-making.

17.2.1 Statistics Instructional Sequences One of the major issues we examine in this chapter is how the statistics instructional sequences introduced by the research team (Cobb, McClain, & Gravemeijer, 2003; McClain & Cobb, 2001) became a key resource in the teachers’ documentation work in year 5 of the collaboration. The intent of instructional activities included in the two sequences2 was that students would conduct genuine data analyses to address problems that they considered significant. The tasks typically involved comparing two data sets to make a decision or judgment (e.g., analyze the T-cell counts of AIDS patients who had enrolled in two different treatment protocols to determine which treatment was more effective). Three computer tools provided the students with a variety of options for organizing data sets (for descriptions and analyses of these tools, see for example Bakker & Gravemeijer, 2003; Cobb, 2002). The students were usually required to write a short report for an audience that would make a policy decision on the basis of their analyses (e.g., advise hospital administrators about which treatment they should use and why). In the classroom design experiments in which the instructional sequences had been developed, students compared their recommendations in classroom discussions and justified them by explaining how they had analyzed data (Cobb, 1999; Cobb et al., 2003). The rationale for the instructional sequences consisted of a documented trajectory for students’ statistical learning together with specific means of supporting that learning that had been substantiated during the classroom design experiments. The means of support included not only the instructional tasks and tools, but also the organization of classroom activities and the nature of the classroom discourse (McClain, Cobb, & Gravemeijer, 2000). Our goal, as we worked with the teachers, was that they would examine issues of teaching and learning statistics as they adapted, tested, and modified the sequences in their classrooms. We did not focus

2 The first of the instructional sequences focused on supporting students to reason about univariate distributions and the second about bivariate distributions.

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on specific teacher moves, but instead pressed the teachers to justify the moves and actions that they chose by considering opportunities for student learning. We conjectured that professional development activities in which instructional decisions came to be justified explicitly in terms of student learning opportunities would constitute an effective means of supporting the learning of the teacher group.

17.2.2 Organization of the Professional Development Sessions We engaged the teachers in resource-rich activities designed to support their reconstruction of the rationale for the statistics instructional sequences. During the initial 4 years of the collaboration, the central professional development activities frequently followed the pattern of teachers (a) solving a statistics task in a work session, (b) using the same task with their students, and (c) bringing students’ written work to the following work session for analysis and group discussion. In addition, during years 3 and 4, we frequently videorecorded two teachers coteaching a lesson using a statistics task, and used the recording as a focus for group discussion in the following session (see Fig. 17.1 for a timeline example). Our overarching goal in these

Fig. 17.1 The timeline of the professional development activities in year 3. Each whole-day session is depicted by a vertical rectangle. The rectangle is subdivided into four sections that represent the foci of the main activities on that day. The 3 by 4 array depicts the 3-day summer workshop. The single square in August depicts an informal session organized by the continuing teachers for the newcomers to the group

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sessions was to support the teachers’ collective construction of a long-term learning trajectory for the development of students’ statistical reasoning and the means of supporting it. The activities in which the teachers engaged during year 5 were designed as a performance assessment of the group’s documentation work and required the teachers to develop statistics instructional units that could be used in their school district. The teachers first reviewed and critiqued two sets of instructional units in statistics. They then selected tasks from these units, modified them for their needs, and organized them into an instructional sequence. One set of materials was an inquiryoriented textbook series that the district had adopted. The second set of materials comprised three units that had been designed by a group of teachers at a second research site in a different US state. The units included sequences of tasks accompanied by teacher notes and were based on the teachers’ work with our statistics instructional sequences. We continued to support the teachers throughout year 5, but we did not press them to develop instructional units that aligned with our view of effective design.

17.3 Data Sources and Method of Analysis The data that we analyzed consist of videorecordings of all professional development sessions together with a set of field notes, copies of all the teachers’ individual and collective work, and nine classroom videorecordings of their statistics instruction that were produced for use in professional development sessions during years 3 and 4. In addition, we collected modified teaching sets (Simon & Tzur, 1999) each year that entailed videotaping a lesson in each teacher’s classroom and then conducting follow-up audiorecorded teacher interview that focused on issues that emerged in the course of the lesson. This chapter builds on two studies of our collaboration with the teachers. Together, these studies document the actual learning trajectory of the teacher group over the 5-year period. Dean (2005) studied the first 2 years of the collaboration and documented the transition of the group into a genuine professional teaching community. Visnovska (2009) studied the remaining 3 years of the collaboration and focused on the mathematical and pedagogical learning of the community. We identified shifts in the teachers’ reasoning as they participated in professional development activities by analyzing patterns and regularities in their ongoing interactions. In making claims about teachers’ views and perspectives, we do not rely on teachers’ self-reports but instead base our claims on the ways in which they participated in professional development activities and on their classroom instruction. The specific approach we used have been described by Cobb and Whitenack (1996). This method is an adaption of Glaser and Strauss’ (1967) constant comparative method and was developed to analyze longitudinal data sets of the type generated during classroom design experiments. Tentative conjectures are continually tested and revised while working through the data chronologically. As new episodes are

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analyzed, they are constantly compared with currently conjectured themes or categories, eventually resulting in the formulation of empirically grounded claims or assertions that span the entire data set.

17.4 Teachers’ Documentation Work We focus on the teachers’ documentation work in relation to two key resources (cf. Chapter 1) that the group had developed: the professional teaching community (a social resource) and the statistics instructional sequences (a material resource). We first outline the end points of the teachers’ learning as revealed by their documentation work during year 5 of the collaboration. We then discuss the community genesis and community documentational genesis (Chapter 16), illustrating that the manner in which the teachers used the resources in successful instructional design was an achievement that required guidance over an extended period of time.

17.4.1 Teachers’ Documentation Work in Year 5 The basis upon which the teachers came to make instructional judgments became explicit when they reviewed, critiqued, and proposed adaptations to statistics tasks in the later half of year 5 of our collaboration. To illustrate the documentation work of the group at the time, we first discuss one of the task adaptations that the teachers proposed and then outline how they reasoned about the ‘big ideas’ of the statistics sequences. The first episode comes from the group discussion of the instructional task in which the students were to analyze the T-cell counts of AIDS patients who had enrolled in two different treatment protocols (186 patients in traditional and 46 patients in experimental treatment) to advise hospital personnel about which treatment was more effective (Fig. 17.2).

Fig. 17.2 Data for the AIDS task: T-cell counts per mm3 of blood. The data are represented in second of the three computer tools designed to support students’ comparisons of univariate data

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During the discussion, the teachers proposed that the distribution of the ‘traditional treatment’ data should look more like a hill – the term that students often use when they first focus on shape of data distributions. The teachers also pointed out that once the distribution of the data had been smoothed out, it would be important that a break point in the data that splits each data set into two groups should be selected so that students who used the additive argument would arrive at a different conclusion than students who reasoned about the data proportionally (see Fig. 17.3). During the classroom design experiment in which this task was developed, the researchers had purposefully constructed data sets with significantly different numbers of data points so that the contrast between absolute and relative frequency might become explicit. The teachers’ comments indicated that they were aware of (a) the kinds of arguments that their students might make as they engage in the AIDS activity, (b) how characteristics of data sets influenced students’ arguments, and (c) the value of comparing different arguments during classroom discussions to support all students in coming to interpret data sets in proportional rather than additive terms. For the teachers, reasoning proportionally about data was both a sophisticated way of identifying patterns in data and a big mathematical idea that the AIDS task could help them pursue. The second episode comes from the summer institute at the end of year 5, when we asked the teachers to outline the main instructional goals in each phase of teaching univariate data analysis. The purpose for this activity was to reach consensus about major goals and then use these goals to orient the design of an instructional

Fig. 17.3 ‘Smoother’ shape of AIDS data sets proposed by the teachers. The datasets were designed to facilitate the contrast between additive and proportional comparison of subgroups with T-cell count above 550 (i.e., although a greater number of patients in the traditional treatment have T-cell counts above 550 than in the experimental treatment, greater proportion of patients in the experimental treatment improved when compared to the traditional treatment)

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sequence that they could use in their classrooms. The teachers first spent about 20 minutes making notes individually while reviewing tasks. In the subsequent discussions, each of the teachers stated one goal and a researcher typed-up their proposals. The following excerpt is representative of the entire discussion and concerns the comparison of data sets with unequal number of data points. Muriel: [One of the goals for the class is] seeing the shape without the numbers, going from the absolute to the relative [comparisons]. Researcher: Not without the numbers, without the data, like when you’ve hidden the dots (see Fig. 17.4.) Muriel: Yes, the middle 50%. Researcher: Are you thinking about four equal groups? Muriel: [describes a specific graph, using AIDS task as an example] Researcher: Being able to see what the graph looks like. Lisa: Comparing the different partitions in the data, not just the middle 50%, but each quartile. Researcher: If we compare the entire data set, how the entire thing is distributed. Lisa: Yeah, but you are comparing different pieces. ... Erin: We were always talking about using percentages, using proportions [not just counts of absolute frequencies]. Bruno: Unequal groups leads to concepts like histograms, [relative] frequency. Researcher: We talked about shapes before, now that we have unequal data sets, we need to have ways to make comparisons explicit so

Fig. 17.4 AIDS data organized into four equal groups with the data hidden

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that thinking in terms of percentages becomes a useful tool. . . Bruno mentioned equal interval widths which lead naturally to histograms. The whole idea of how many of a number. [Rephrases] What proportion of the total. . . . And again, the whole idea of where the data comes from, data generation. Looking at general trends instead of actual numbers. Shape, trend of the distribution – how the data are distributed.

(Summer at the end of year 5, Day 1, Jun 2005) Importantly, the teachers did not simply recite a list of the ‘big ideas’ for the statistics sequence but instead grounded them in their experiences of using specific instructional tasks in their classrooms and of analyzing videorecordings of others’ classrooms. For instance, they referred to specific task scenarios to elaborate the relatively cryptic initial descriptors of the goals (e.g., AIDS datasets as an example of “natural breakpoint” in data). They articulated the major shifts in students’ statistical reasoning (i.e., from starting to genuinely analyze data, through beginning to look at patterns in data, to reading how data were distributed from graphs) along with the means of supporting these shifts. In addition to the types of tasks, the teachers made numerous references to the three computer tools and how they anticipated that students would use them (e.g., initial partitioning or grouping of data, using equal intervals to describe shape, work with data hidden). They also referred to the nature of classroom discourse and the types of normative practices that they intended to support in their classrooms (e.g., discussions in which students justify partitioning with regard to the problem context). In both activities, the teachers built on shared repertoire of ‘lived resources’ that the group had generated both in professional development sessions and when the teachers had experimented with the instructional activities in their classrooms over the past 5 years. Even though the group generated a table of ‘big ideas’ (a physical artifact), it was the collective reconstruction of the lived resources that was the goal as well as the product of this activity (cf. Chapter 5) and could be conceived as a document in the sense proposed by Gueudet and Trouche (2009). When the teachers subsequently used (and edited) the table and associated notes as they made decisions about which activities to include and which to omit from their instructional unit, they spoke about and worked with the meanings that these notes came to have in this group through the collective reconstruction process. The teachers’ participation in professional development sessions in year 5 indicated that the group had developed instructional planning practices that attended to both students’ reasoning and to the major ideas that were the goal of instruction. A number of studies have documented that instructional practices of this kind can be effective in supporting all students learning of significant mathematical ideas (e.g., Hiebert & Grouws, 2007). We cannot speak to the actual effectiveness of subsequent instruction in the teachers’ classrooms, nor can we make claims about the teachers’ planning practices in settings other than the professional development sessions. However, we can demonstrate that the teachers had developed

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sophisticated mathematical (or more specifically, statistical) knowledge for teaching (Ball & Bass, 2003) as well as other complex resources on which they drew routinely in their documentation work.

17.4.2 Community Genesis The ways in which the teachers used each other as resources3 in their documentation work changed substantially in the course of our collaboration with them. Dean (2005) analyzed the development of the teacher group during the first 2 years of the collaboration and reported that, initially, the participation structure within the group could be characterized as turn taking with the teachers directing comments to the researchers rather than each other. The group was a pseudocommunity (Grossman, Wineburg, & Woolworth, 2001) in that the teachers treated challenges and conflicts as violations of established norms. In addition, they initially kept their classroom practices private: they did not allow other teachers to observe their instruction and did not share their instructional challenges during professional development sessions. These types of interactions were not conducive to substantial pedagogical learning because the rationale for specific instructional decisions remained implicit, private, and uncontested. They were also not conducive to collaborative documentation work. The ongoing and retrospective analyses revealed that the ways in which the teachers initially participated in the professional development sessions, and in particular how they interacted with their colleagues, were influenced to a significant extent by the school settings in which they worked. The school administrators monitored their instruction on content coverage and student behavior, and assistance to improve instruction was limited (Cobb, McClain, Lamberg, Dean, 2003). As a result, the teachers worked in almost complete isolation. Given that school settings of this type are relatively typical in the United States and in a number of other countries, it is unreasonable to assume that mathematics teachers who work in the same school, or who engage in a professional development program together will necessarily serve as a useful resource for one another. One of our major goals during the first two years of the collaboration was therefore to support the development of more effective forms of participation in the professional development sessions. As Dean (2005) documented, it was not until 19 months into the collaboration that the group finally became a genuine professional teaching community (cf. community of practice, Wenger, 1998; Chapter 16). The joint enterprise of the community centered on ensuring that students came to understand central mathematical ideas while simultaneously performing more than adequately on high-stakes student assessments. The norms of mutual engagement that were key to the teachers’ documentation work included building on others’ 3

We specifically attend to the relationships and methods of communication among the group of teachers engaged in joint activities that serve as a social resource (cf. Carpenter et al., 2004; Cobb, McClain, Lamberg, & Dean, 2003).

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contributions to group discussions, asking clarifying questions, challenging others’ assertions, as well as openly sharing problems or challenges experienced during instruction. Additional norms were specific to mathematics teaching and included the standards to which the members of the community held each other accountable when they justified pedagogical decisions and judgments. In supporting these group developments, especially in helping teachers to open up their teaching practices to their colleagues, addressing teachers’ perceptions of their school settings was of key importance (Dean, 2005; Visnovska & Zhao, 2011).4 It was in the context of expressing the frustrations about institutional pressures that the teachers first shared their teaching experiences and asked each other for advice. It was not until the link between administrators’ evaluative style of leadership and the teachers’ desire to keep their instructional practices private became explicit that the teachers started to share their concerns and began to view each other as resources for their learning. In years 2 through 5, the nature of teachers’ participation in the sessions allowed for genuine discussions of the pedagogical issues. At least one but sometimes as many as seven teachers spontaneously shared problems that they encountered in their teaching. The following excerpt in which Muriel explained that her students partitioned data into three rather than four equal groups when using a computer tool (Figs. 17.4 and 17.5) in year 4 is representative in this regard. Muriel was concerned that this way of organizing data would not provide grounding for box and whisker plots:

Fig. 17.5 An example of data organized into three equal groups

4 The activities designed to support these changes included conversations about the supports and constraints available to teachers in their schools, principals’ understanding of effective mathematics teaching, and how the group could support principals in developing more productive views and in valuing teachers’ professional judgment.

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Muriel: My class, when we went to the computer lab, they were dividing it [data] like into thirds. And we did that. Didn’t we do that at first? Thirds? [Wesley nodding] Muriel: But they, none of them divided into the 4 equal groups. Researcher: No, that’s fine. Muriel: Which did not lead to box and whisker plots at all [laughs]. Researcher: No, that’s fine. That’s actually kind of what we expect, by the way. Muriel: Ok. Researcher: Ok? Let’s be clear about this. . . . None of your kids are gonna just invent box and whiskers plots. . . . That’s not what we are aiming for. What we are aiming for is, I think it’s just fine to introduce stuff to kids. Not that they just create it out of nothing. But it’s at a point when they see a need for it. And it’s gonna make sense when they can see it as a useful tool. Wesley: And the “industry standard” is to divide it into 4ths. Researcher: Yes. Wesley: And if they’re dividing into the thirds, they’ve got the idea in their head, to divide. And then you’d say: “Well, you know, most people talk about it in terms of 4ths” and they’re not gonna have a hard time making that leap to there. (Year 4, Session 2, Nov 2003) With support from a researcher, the teachers responded to Muriel’s account of an instructional problem by proposing explanations and suggesting possible courses of action that built on students’ solutions (e.g., Wesley’s suggestion). By the fourth year of the collaboration, problems that the teachers shared were routinely constituted as cases in which to talk through broader instructional issues (e.g., inventing versus telling). Moreover, the teachers viewed instructional improvement as the collective responsibility of the group, and valued collaboration as a means of understanding and improving instruction. Once the group had established productive norms of mutual engagement, other teachers proved to be an invaluable resource for effective documentation work. However, the collective development of productive norms was a nontrivial accomplishment. Although structures that facilitated collaboration were necessary (e.g., time, space, group leaders), it was as the teachers worked together with considerable guidance that effective social resources developed in the group.

17.4.3 Documentational Genesis It would be a mistake to assume that the ways in which the teachers used instructional sequences in year 5 were either ‘natural’ for this group or in some way necessitated by the sequences. Indeed, both the use of sequences, and the documents

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(or collections of meanings) that the teachers produced in the process, changed substantially in the course of our collaboration. Resource for statistical learning. Prior to the professional development collaboration, the teachers had limited experience in conducting statistical analyses and thus in dealing with variability and distribution. There were clear indications that they recognized the sequences as resources for their statistical learning relatively early in the collaboration (Dean, 2005). As the teachers engaged in activities from the instructional sequences as learners, they became increasingly competent in analyzing data, developing data-based arguments, and providing justifications for their solutions. By the end of year 2, all the teachers reasoned about distributions multiplicatively (i.e., in terms of relative rather than absolute frequency), developed increasingly sophisticated strategies for comparing how data sets were distributed, and could infer the shape of these distributions from a variety of statistical representations. However, the ways in which the teachers initially used instructional activities from the sequences with their students indicated that the teachers assimilated the sequences to their current instructional practices. In other words, the teachers’ use of the sequences did not lead them to reorganize their instructional practices: the sequences were not, by themselves, effective resources for the teachers’ pedagogical learning. Source of benchmarks and prescriptions. During years 1–3 of the collaboration, the teachers repeatedly requested explicit prescriptions for how they should enact statistics activities in their classrooms. Despite our repeated attempts to reorient discussions toward the bases for making informed instructional decisions, the teachers typically focused on what they were supposed to do when they used the instructional activities. It appeared that, from the teachers’ point of view, it ought to be possible to script what they should do irrespective of the ways in which their students engaged in the activities. As an illustration, in the first session of year 2 the researchers attempted to support the teachers in identifying the ‘big ideas’ of the statistics sequence in an activity similar to the one that we reported from the summer session at the end of year 5. In contrast to their responses at the end of year 5, the teachers created a list of benchmarks that they should ensure students ‘got’ as they went through the sequence. They also requested that the benchmarks be worded as objectives similar to those in state mathematics standards for student achievement, and asked whether the particular objectives (e.g., ‘developing data-based arguments’) should be written on the board prior to lessons or told to the students after the lesson (Dean, 2005). The meanings of the instructional sequences that were collectively reconstructed in this session (i.e., documents produced by the group) differed significantly from those established in later years. As the discussion of Muriel’s concern about three equal groups illustrated, by year 4 the teachers focused on how they could proactively support the emergence of specific forms of student reasoning (e.g., partitioning data into four equal groups, reasoning about comparisons proportionally). In contrast, in year 2, the teachers generated benchmarks to be used retrospectively

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to assess student learning. The activities in which the teachers engaged in the intervening 2 years were explicitly designed to support the teachers’ development of more productive views of classroom instruction that went beyond benchmarking students’ solution methods. Resource for supporting students’ statistical interests. Throughout year 3, we pressed the teachers to adopt a student’s point of view when they examined classroom situations and attempted to make sense of students’ statistical analyses and explanations. In designing the three-day summer institute conducted at the end of year 3, we planned to continue supporting the teachers in shifting their focus from the teacher’s performance to ways in which students might be making sense of classroom activities. However, because our prior attempts to support this shift by focusing directly on students’ reasoning had been repeatedly unsuccessful,5 we instead chose an issue that was already instructionally important to the teachers, student motivation. An analysis of the teaching sets collected before the summer institute of year 3 to document the teachers’ classroom practices revealed that all the teachers considered student motivation to be a major determinant of both students’ engagement in classroom activities and their mathematical learning (Zhao, Visnovska, Cobb, & McClain, 2006). However, the process by which teaching resulted in students’ learning was largely a black box for the teachers. Whether students learned or not depended to a great extent on their motivation, which the teachers attributed to societal and economical factors beyond their control. Student motivation and engagement were thus highly problematic issues for the teachers. These analyses oriented the design of a series of professional development activities in which we supported the teachers in reconceptualizing students’ motivation in terms of cultivating students’ disciplinary interests (Dewey, 1913/1975). We have documented elsewhere (Visnovska, 2009; Visnovska & Zhao, 2010) that the teachers came to view students’ motivation as being within their control to influence, and found it meaningful to investigate how they might support the development of students’ interest in analyzing data. Commencing with the summer institute at the end of year 3, the teachers began to use the statistics instructional sequences as a means of supporting students’ development of statistical interests (Visnovska, 2009). They initially did so by focusing on task scenarios and the opening phase of statistics lessons in which the task was introduced. It was as they attempted to envision which types of task scenarios were likely to be of interest to their students that the teachers began to adopt a student’s perspective. Resource for supporting students’ statistical reasoning. Once it became normative in the group to adopt a student’s perspective when considering students’ interests (session 4, year 4), most of the teachers came to view the whole class discussions of different student solutions as a source of students’ continued interest. With support from the researchers, the teachers started to attend to the

5 Students’ reasoning appeared to be irrelevant to high quality mathematics instruction as it was defined within the district, in terms of content coverage and classroom management.

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listening students when they analyzed classroom videorecordings and gradually realized that if these students could not understand their classmates’ explanations it would be difficult for them to remain engaged. The teachers therefore became committed to developing increasingly effective ways of supporting their students’ understanding of others’ explanations. This in turn required that they could anticipate the range of solutions that their students might produce at different points in statistics sequences. The sequences and their underlying rationale then became a resource for understanding both how students’ statistical reasoning might develop and how these developments could be proactively supported. As we have illustrated, this orientation was apparent during the performance assessment activities in year 5 when the teachers drew on the statistics sequences while constructing an instructional unit.

17.5 Discussion and Conclusions To summarize, the same colleagues and the same statistics instructional sequences functioned as very different resources in the teachers’ work at different points during the professional development collaboration. The teachers’ initial interactions in the professional development group reflected their daily experiences in school environments characterized by monitoring and control but little assistance. The group became a professional teaching community in which genuine conversations about problems of instructional practice were possible only after the teachers developed insights into the institutional context of their work and how it influenced both their practices and their relations with each other. These insights were a precursor to the development of new and more productive ways of working together. Even though the teachers readily recognized the statistics sequences as a resource for their statistical learning, they did not initially reorganize their instructional practices when they used the sequences in their classrooms. We illustrated that the reorganization of planning practices involved a series of shifts in the ways that the group used the statistics sequences when addressing the problems that the teachers viewed as relevant to their teaching. Importantly, the teachers did not simply choose to use the sequences in new ways or for new purposes. Rather, the evolution of normative ways of interpreting and using the sequences was closely related to the teachers’ development of increasingly sophisticated forms of mathematical and pedagogical reasoning. Conversely, the teachers’ guided participation with the statistics instructional sequences supported the development of their mathematical and pedagogical reasoning. In this process, the teachers collectively developed shared repertoire of resources that enabled them to (re)construct a rationale for the instructional sequences and to adapt the sequences to their classrooms in ways that were consistent with underlying design principles. We conclude this chapter by expanding on the two aspects of teachers’ collective documentation work on which we have focused in the illustrative case: the development of social resources and of material resources. Our first observation relates to

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the frequent calls that have been made to establish collaborative teacher communities as a resource for supporting teachers in improving their instructional practices (Ball & Cohen, 1996). The illustrative case indicates that professional teaching communities cannot be legislated into being merely by providing time, space, and instructional leadership (see also Chapter 15). Instead, the teachers became social resources for each other as a result of engaging in professional development activities with considerable guidance. As the case illustrates, the ease or difficulty with which teachers can become social resources for each other depends to a considerable degree on the school contexts in which they work. In absence of proactive guidance, teacher groups might merely perpetuate the patterns of interaction that are typical in their schools (i.e., engage in pseudo-agreement to protect themselves from being negatively evaluated) and fail to become effective social resources. As a related observation, the ways in which teachers initially use new instructional materials are likely to involve assimilation to current instructional practices. Although use of the new materials might result in some teachers reorganizing their practices significantly, the extent to which they do so is influenced by whether and how teachers’ use of the new materials is supported as well as by the school contexts in which they work. In school contexts in which teachers’ instructional performance is monitored and where they receive little assistance to improve their teaching, the introduction of new instructional materials is unlikely to support substantial improvements in teachers’ instructional practices. As our case illustrates, sustained proactive guidance as the teachers’ engaged with the instructional sequences was crucial in supporting the development of new instructional planning practices (e.g., anticipating students’ solutions to particular tasks). Two characteristics of the professional development activities proved to be important in this regard: (a) the teachers had to view the new ways of engaging with the instructional sequences as relevant to their classroom instruction, and (b) the activities had to challenge some of the teachers’ existing assumptions about classroom instruction, thereby giving rise to opportunities for them to develop new instructional insights. It, therefore, seems unlikely that teachers will reorganize their instructional practices when they are required to meet for common planning while using new instructional materials, but are left to their own devices to determine how this planning work should proceed unless some of the teachers have already developed relatively sophisticated practices. In the framework proposed by Gueudet and Trouche (Chapters 2 and 16), the resources include both the social resources and the proactive role of facilitators in supporting substantive learning of teacher communities. This is adequate for the purpose for which the framework was developed – to capture documentation genesis as a phenomenon. However, it is important to add that the types of resources on which we have focused cannot necessarily be provided externally, but instead have to be developed locally with ongoing support and guidance. We raise this issue because frameworks developed by educational researchers are often appropriated by others and translated to policy recommendations, resulting in recommendations and requirements that are frequently counterproductive to efforts

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to support the improvement of mathematics teaching and learning. The rhetoric of teachers as instructional designers serves to illustrate this detrimental process. The common assumption that groups of teachers are capable of designing coherent instructional sequences from provided materials with little if any ongoing support is a dangerous misinterpretation of both the potential of teacher collaboration and the fact that implementation is necessarily an act of design. This latter observation acknowledges that teachers design their instruction in what de Certeau (1984) termed a second act of creation that builds on the prior creative acts of the developers who produced the materials that the teachers adapt and use. It is important to emphasize that this observation is descriptive rather than prescriptive: it does not make any claims about the quality and effectiveness of the resulting instruction. The observation, therefore, orients how we might view teachers’ work, but it does not offer a prescription for improving instruction that involves leaving the teachers to their own devices to design and adapt instructional materials. The analysis we have presented in this chapter illustrates that this prescription is inadequate. We therefore conclude by suggesting that attention be given to forms of ongoing support that might facilitate teachers’ development of increasingly effective documentation practices. Acknowledgments The preparation of this chapter was supported in part by the US National Science Foundation under grant No. ESI 0554535 and by The University of Queensland under NSRSU grant No. 2009002594. The findings and opinions expressed here are those of the authors and do not necessarily reflect the views of the funding agencies.

References Apple, M. (1990). Is there a curriculum voice to reclaim? Phi Delta Kappan, 71(7), 526–531. Bakker, A., & Gravemeijer, K. (2003). Planning for teaching statistics through problem solving. In R. Charles & H. L. Schoen (Eds.), Teaching mathematics through problem solving: Grades 6–12 (pp. 105–117). Reston, VA: National Council of Teachers of Mathematics. Ball, D. L., & Bass, H. (2003). Toward a practice-based theory of mathematical knowledge for teaching. In B. Davis & E. Simmt (Eds.), Proceedings of the 2002 annual meeting of the Canadian mathematics education study group (pp. 3–14). Edmonton, AB: CMESG/GCEDM. Ball, D. L., & Cohen, D. (1996). Reform by the book: What is – or might be – the role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25(9), 6–8, 14. Brown, A. L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. Journal of the Learning Sciences, 2, 141–178. Bruner, J. S. (1960). The process of education. Cambridge, MA: Harvard University. Carpenter, T. P., Blanton, M. L., Cobb, P., Franke, M., Kaput, J. J., & McClain, K. (2004). Scaling up innovative practices in mathematics and science. Retrieved 2006, from http://www.wcer. wisc.edu/NCISLA/publications/reports/NCISLAReport1.pdf Cobb, P. (1999). Individual and collective mathematical development: The case of statistical data analysis. Mathematical Thinking and Learning, 1(1), 5–43. Cobb, P. (2002). Reasoning with tools and inscriptions. Journal of the Learning Sciences, 11(2&3), 187–215. Cobb, P., Confrey, J., diSessa, A. A., Lehrer, R., & Schauble, L. (2003). Design experiments in education research. Educational Researcher, 32(1), 9–13.

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Cobb, P., & McClain, K. (2001). An approach for supporting teachers’ learning in social context. In F. L. Lin & T. Cooney (Eds.), Making sense of mathematics teacher education (pp. 207–232). Dordrecht: Kluwer. Cobb, P., McClain, K., & Gravemeijer, K. (2003). Learning about statistical covariation. Cognition and Instruction, 21(1), 1–78. Cobb, P., McClain, K., Lamberg, T., & Dean, C. (2003). Situating teachers’ instructional practices in the institutional setting of the school and school district. Educational Researcher, 32 (6), 13–24. Cobb, P., & Whitenack, J. W. (1996). A method for conducting longitudinal analyses of classroom videorecordings and transcript. Educational Studies in Mathematics, 30, 213–228. Cobb, P., Zhao, Q., & Visnovska, J. (2008). Learning from and adapting the theory of Realistic Mathematics Education. Éducation et Didactique, 2(1), 105–124. de Certeau, M. (1984). The practice of everyday life. Berkeley, CA: University of California Press. Dean, C. (2005). An analysis of the emergence and concurrent learning of a professional teaching community. Unpublished Dissertation, Vanderbilt University, Nashville, TN. Dewey, J. (1913/1975). Interest and effort in education. Carbondale, IL: Southern Illinois University. Dow, P. (1991). Schoolhouse politics. Cambridge, MA: Harvard University. Doyle, W. (1992). Constructing curriculum in the classroom. In F. K. Oser, A. Dick & J. Patry (Eds.), Effective and responsible teaching (pp. 66–79). San Francisco: Jossey-Bass. Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. New York: Aldine. Grossman, P., Wineburg, S., & Woolworth, S. (2001). Toward a theory of teacher community. Teachers College Record, 103(6), 942–1012. Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71, 199–218. Hiebert, J., & Grouws, D. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Reston, VA: NCTM. McClain, K., & Cobb, P. (2001). Supporting students’ ability to reason about data. Educational Studies in Mathematics, 45, 103–129. McClain, K., Cobb, P., & Gravemeijer, K. (2000). Supporting students’ ways of reasoning about data. In M. Burke (Ed.), Learning mathematics for a new century (2001 Yearbook of the National Council of Teachers of Mathematics) (pp. 174–187). Reston, VA: National Council of Teachers of Mathematics. Rabardel, P., & Bourmaud, G. (2003). From computer to instrument system: A developmental perspective. Special Issue “From computer artifact to mediated activity”, Part 1: Organisational issues, Interacting With Computers, 15(5), 665–691. Remillard, J. T. (1999). Curriculum materials in mathematics education reform: A framework for examining teachers’ curriculum development. Curriculum Inquiry, 29(3), 315–342. Simon, M. A., & Tzur, R. (1999). Explicating the teacher’s perspective from the researchers’ perspective: Generating accounts of mathematics teachers’ practice. Journal for Research in Mathematics Education, 30, 252–264. Visnovska, J. (2009). Supporting mathematics teachers’ learning: Building on current instructional practices to achieve a professional development agenda. Unpublished Dissertation, Vanderbilt University, Nashville, TN. Visnovska, J., & Zhao, Q. (2010, May). Focusing on interest in professional development of mathematics teachers. Paper presented at the annual meeting of the American Educational Research Association Conference, Denver, CO. Visnovska, J., & Zhao, Q. (2011). Learning from a professional development design experiment: Institutional context of teaching. In J. Clark, B. Kissane, J. Mousley, T. Spencer & S. Thornton (Eds.), Proceedings of the 34th annual conference of the Mathematics Education Research Group of Australasia. Alice Springs, NT: MERGA.

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Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge: Cambridge University. Zhao, Q., Visnovska, J., Cobb, P., & McClain, K. (2006, April). Supporting the mathematics learning of a professional teaching community: Focusing on teachers’ instructional reality. Paper presented at the annual meeting of the American Educational Research Association Conference, San Francisco, CA.

Reaction to Part IV Teacher Agency: Bringing Personhood and Identity to Teaching Development Barbara Jaworski

The title of this book, “Mathematics Curriculum Material and Teacher Development: From Text to “Lived” Resources” is fittingly brought to a conclusion in this final part which focuses on the collaborative aspects of teacher documentation. The three chapters in this part offer a range of theoretical perspectives as well as specific practical insights to issues in developing mathematics teaching for the effective learning of students. Each of the chapters addresses a tension/dilemma for teachers: that is, the engagement of self within the collective of institutionalized practice and an exciting panorama of resources and their associated challenges. In his seminal discussion of “self”, Harré (1998) adapts the terminology of Apter (1989, p. 75) to speak of “personhood” as having characteristics as follows: In displays of personhood, of our singularity as psychological beings, we express “a sense of personal distinctness, a sense of personal continuity, and a sense of personal autonomy” (p. 6). In what we read in this part we gain a sense of how teachers’ personhood, in terms of distinctness, continuity and autonomy, relates to the panorama of resources within which they make sense of their teaching role, in which they become the teacher they are. All three chapters build on Gueudet and Trouche’s Chapter 2 (Part I) to make reference to “documentational genesis”, in which genesis means becoming: becoming a mathematics teacher; becoming a professional user of resources; becoming a knowledgeable professional. In his book Communities of Practice, Wenger (1998) talks of learning as “a process of becoming” (p. 215). On the one hand, this, he claims, is “an experience of identity” (p. 215), where identity “serves as a pivot between the social and the individual, so that each can be talked about in terms of the other” (p. 145). Harré, on the other hand, sees a person’s identity “not their singularity as a unique person, but the group, class or type to which they belong” (p. 6). He sees this as being the opposite of the characteristics of singularity, distinctness, continuity and autonomy. In his terms, identity and personhood are opposites.

B. Jaworski (B) Mathematics Education Centre, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK e-mail: [email protected]

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My brief references here to the writings of Harré and Wenger draw attention to a philosophical grounding to these chapters, which juxtaposes ideas of self, identity and agency. Harré speaks of agentive power as “power [of the individual] to initiate action” (p. 116). Wenger juxtaposes agency with knowledgeability, and suggests a dichotomy between theories of social structure that deny agency to individual actors, and theories of situated experience that emphasise agency and intentions, and address “the interactive relations of people with their environment” (pp. 12–13). Wenger suggests a middle ground, that of “learning as participation” which “takes place through our engagement in actions and interactions” and “embeds this engagement in culture and history” (p. 13). Documentational genesis, a term which captures the process of the mathematics teacher becoming a professional user of resources and, concomitantly, a knowledgeable professional, navigates the ground between the personhood of the teacher and the teacher’s belonging (Wenger, 1998) to social structures and communities in which resources take meaning. Gueudet and Trouche suggest an associated community genesis including five steps potential, coalescing, maturing, stewardship and transformation as distinguished by Wenger, McDermott and Snyder (2002); they write: “This conceptualization fits our objective of studying the documentation work of teachers in a collective, in describing a variety of teachers’ collectives at various steps of development” (p. 307). Winsløw (Chapter 15) uses the concept of documentational genesis to introduce his theory of paradidactic systems; for him this concept is “clearly an enterprise that goes much beyond the individual teacher’s domain of action and responsibility”. He emphasizes that even in a context where teachers work mostly alone, peer learning and team work can be crucial factors for teachers’ development. Gueudet and Trouche (Chapter 16) write of teachers’ documentational geneses and professional geneses with particular reference to community and collaboration, drawing on Wenger’s theory of community of practice. For them the idea of teacher-in-community seems central to their conceptualization. Visnovska, Cobb and Dean (Chapter 17) discuss teachers’ documentation work with a helpful rephrasing of the concept of documentational genesis: they write, with reference to Gueudet and Trouche, that “teachers’ documentation work includes looking for resources (e.g., instructional materials, tools, but also time for planning, colleagues with whom to discuss instructional issues, and workshops dedicated to specific themes) and making sense and use of them (e.g., planning instructional tasks and sequences, aligning instruction with the objectives and standards to which the teachers are held accountable). The products of this work at a given point in time are characterized as documents (e.g., records of the big mathematical ideas that are the overall goals of an instructional unit; a sequence of tasks along with a justification of their selection). These documents can in turn become resources in teachers’ subsequent documentation work. The process of documentational genesis therefore foregrounds interactions of teachers and resources, and highlights how both are transformed in the course of these interactions” (pp. 323–324).

I, therefore, consider documentational genesis, and the associated professional genesis for teachers, in relation to their involvement with and use of resources, to be fundamentally related to teachers’ agentive power and development of teaching identity.

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Winsløw (Chapter 15) develops the idea of a didactic system, the basic unit of teaching and learning in school institutions (drawing on theories from Brousseau and Chevallard), to offer a threefold collaborative process which he calls a paradidactic system. The components are the predidactic system (PrD) involving design and planning, the observation system (DoS) in which classroom teaching is observed and documented and the postdidactic system (PoS) in which the didactic system is evaluated and its design may be revised. He goes on to apply this model to examples of practice: firstly an example of Japanese lesson study and secondly an implementation of interdisciplinary modules in Danish schools. While Japanese lesson study is historically and culturally rooted, the Danish project was imposed onto the existing culture and systems and proved problematic for teachers to accept and implement. The challenge posed by the specific requirement for teachers to work in teams cut across what teachers saw as their motivation for becoming teachers. Gueudet and Trouche expand on the ideas of Wenger and illustrate community documentational genesis in practice through the case of a teacher Pierre and his activity within the digital network Sésamath. Their case study shows how the many facets of Pierre’s documental work coalesce, mature and transform to contribute to the teacher that Pierre has become. They contrast the activity of Pierre with that of Myriam, detailed in Chapter 2. The two teachers navigate differently between the resources offered in Sésamath and their own use of these resources. We might say that their patterns of instrumentation/intrumentalisation are different, and hence also their personal agency in designing teaching. Visnovska et al. (Chapter 17) discuss documentational and professional genesis in a project involving teachers as instructional designers – a 5-year developmental programme with mathematics teachers in which teachers developed knowledge of statistical concepts and associated pedagogical knowledge to grow into more principled modes of practice with their students. The authors emphasise the complexity of resources, including social resources, and point to key shifts in teachers’ participation in the project relate to their co-participating teachers and the pre-designed instructional sequences. In some cases it was clear that school norms influenced teachers more than project goals. It was pointed out that the teachers would not have developed the desired ways of working central to the innovation if left to their own initiative; the innovation was of central importance to teachers’ mediation of reform effort. In all these cases, relative to the particularities of the case, we see three key dimensions, in one case, four. As Gueudet and Trouche point out, each teacher takes part in a variety of collectives, sometimes institutional compulsory and sometimes chosen by themselves. They are a part of an institution which imposes norms and expectations into which the personal activity of the teacher must fit or “align” (Wenger, 1998). They use a variety of resources of different kinds: curricular, collegial, text and Internet, classroom interaction, for example. So we see a teacher’s agency in relation to these collectives: 1. teacher as person, with social and cultural identity; 2. teacher as member of an institution, with a complexity of demands and interrelationships;

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3. teacher as operational designer drawing on a web of resources. In the Visnovska et al. study, we see also a fourth dimension: 4. teacher as participant in an innovation driven by external designers. These dimensions are of course deeply inter-related, but we can see different emphases in the activity portrayed in these chapters. For example, we might see Japanese lesson study as emphasizing points 1 and 2, the Danish reform as emphasising 2; Myriam and Pierre as emphasizing points 1, 2 and 3, and the teachers in the Visnovska et al. study as emphasizing points 1, 2 and 4. In making these observations and thinking about theory and practice as portrayed here, I have unsurprisingly been challenged to draw my own recent developmental research with teachers into this complexity of teacher agency. In the project Learning Communities in Mathematics in Norway, didacticians from the university formed communities with teachers in schools from lower primary to upper secondary to develop inquiry-based activity with students in classrooms and inquire into the teaching design process that this involved (Jaworski, 2008). The project sought to create communities of inquiry between didacticians and teachers to encourage teacher agency in developing inquiry in schools (in collaboration with colleagues) and in mathematics in classrooms with students. Didactician agency was also a central focus of research. We analysed relationships between teachers and didacticians, recognizing the knowledge and experience brought by each group as a resource for the other. We extended Wenger’s notion of alignment, which he characterizes (along with engagement and imagination) as one of the key elements of belonging to a community of practice, to one of “critical alignment” as being central to a community of inquiry. Essentially, critical alignment through inquiry allows questioning of established practices, their norms and expectations, while aligning institutionally with them. Unsurprisingly, there were many issues arising for teachers and didacticians in this collaboration, in some cases leading to tensions and potential conflict. We found activity theory, rooted in Vygotsky and Leonte’v, as detailed also by Gueudet and Trouche, valuable to analyse situations and make sense of the tensions in relation to the full sociocultural complexity of institutions, project and relationships. Gueudet and Trouche say little about how they have used activity theory and I do not have the space either to do so here. However, it seems to me to be well worth further consideration as to how activity theory can throw light onto the complexities inherent in these projects. Such consideration can illuminate teachers’ professional activity and make sense of what we see and experience in classrooms against the panorama of practical and theoretical possibilities on which this book throws light. I end with a return to notions of teacher agency and its relation to concepts of personhood and identity: that is, bringing personhood and identity to teaching development. I have been struck in these chapters by the different examples of how teachers’ “personal distinctness, a sense of personal continuity, and a sense of personal autonomy” (Apter, 1989, cited in Harré, 1998) sit alongside teachers’

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navigation of resources within social and cultural settings in which they develop identity. What we see in classrooms has to be interpreted in this full sense. I see the concept of critical alignment as offering teachers, as well as the didacticians who work with them, a way of dealing themselves and with their colleagues knowingly with the issues and tensions involved.

References Apter, M. (1989). Negativism and the sense of identity. In G. Breakwell (Ed.), Threatened identities (75). London: Wiley. Harré, R. (1998). The singular self. London: Sage. Jaworski, B. (2008). Building and sustaining inquiry communities in mathematics teaching development: Teachers and didacticians in collaboration. In K. Krainer & T. Wood (Eds.), Participants in mathematics teacher education: Individuals, teams, communities and networks. Volume 3 of the International Handbook of Mathematics Teacher Education (pp. 335–361). Dordrecht: Sense Publishers. Wenger, E. (1998). Communities of practice: Learning, meaning and identity. Cambridge: Cambridge University Press. Wenger, E., McDermott, R. A., & Snyder, W. (2002). Cultivating communities of practice: A guide to managing knowledge. Boston: Harvard Business School Press.

Afterword: Using and Designing Resources for Practice Deborah Loewenberg Ball

Closing Reaction The chapter authors in this volume examine and theorize about the nature, role, and use of resources in instruction. The range of “resources” they investigate is vast— from commercial to teacher-made curriculum, to videos, to technology-based tools and environments, to artifacts—and generated by a range of creators, from teachers to professional designers and researchers. Certainly textbooks remain a mainstay of mathematics instruction. Research on modal mathematics classrooms suggests that much teaching is “text-driven.” Reformers often turn their attention to the design of text materials as a means of leveraging teaching and thus improving learning. Others praise teachers who do not “follow” curriculum materials, but who invent their own curriculum, lessons, and examples. At the same time, the range of resource material used for instruction is expanding to many other forms. That said, understanding resources-in-use, or “lived” resources, matters across these forms. Both the concepts of resources and use are fundamental to the inquiry. The improvement of learning depends on many factors, but clearly, the resources used by learners and their teachers form a vital medium of instruction. At their best, resources are both attentive to learners’ ideas and responsible to mathematical learning goals. Not all curriculum resources are created by outsiders: In highly responsive and interactive teaching, teacher-invented materials constitute the pertinent resources. The authors of this volume consider what counts as a “curriculum resource” for mathematics instruction and examine how design and use interact in real-time teaching and learning. As the chapter authors make visible, “using a textbook”––or any curriculum resource––is a process that is both interpretive and dynamic. Teachers read and make sense of curriculum developers’ ideas, adapting them to their own ideas and contexts. Learners, too, interpret and use textbooks, not necessarily as writers intended

D.L. Ball (B) School of Education, University of Michigan, Ann Arbor, MI 48109-1259, USA e-mail: [email protected] G. Gueudet et al. (eds.), From Text to ‘Lived’ Resources, Mathematics Teacher Education 7, DOI 10.1007/978-94-007-1966-8,  C Springer Science+Business Media B.V. 2012

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or envisioned. The difference between the text as written and the text as enacted is significant. The chapters in this book support an important direction in the field, toward curriculum resources that are designed for reasoned and responsive use, not for control or for loose and unspecified improvisation. This is the heart of the notion of curriculum as “lived resources.” Three questions stand out to which the authors of this book make vital contributions: 1. What counts as a “curriculum resource” for teaching mathematics? 2. How are resources used in instruction? 3. How can resources be designed for use, and for learning, in and from practice? First, what counts as a “curriculum resource”? Nominal categories such as textbooks, programs, problem sets, learning goals, artifacts, and tasks insufficiently specify the notion of a “resource.” Helping learners develop understanding and skill with mathematics depends on the development of pathways, spaces, and tools in and with which to work on key ideas and processes. This translation of disciplinary content into forms that are accessible and manipulable by those learning the subject is an old and core problem of instruction. On the one hand, such translations between learners’ thinking and the mature ideas of the field involve managing between what John Dewey (1902/1956) referred to as the “psychological and logical aspects” of the subject. Dewey wrote with sympathy about the challenges inherent in managing these two different aspects of the subject. Young children thinking about integers do not have the real line as a mental object, yet developing their understanding requires sensitivity both to their current ways of thinking (numbers refer to “real” counts of objects or measures of actual lengths) and to an eye on their mathematical horizons, as well as a focus on the integrity of the subject matter itself (Ball, 1993). Curriculum resources, of many different forms, comprise a wide range of translational materials and objects, but all designed to support this fundamental challenge of building bridges between learners and the discipline. Designing such resources is no small task and requires substantial knowledge and skill, and deep understanding of the nature of instruction, or teaching and learning. Instruction consists of interactions among teachers and learners, around content, in environments (Fig. 1, from Cohen, Raudenbush, & Ball, 2003). Teachers interpret learners, learners interpret their teachers, and both bring past experience and understandings of the material. As teachers and learners work with curriculum resources, whether texts, or problems, or software environments, these mutual interpretations, in context, shape what these resources become in practice. Two teachers, teaching with the same text, will make different decisions, some deliberate and some as a product of their beliefs and assumptions, about everything from what to emphasize and what to omit, to how to modify and where to stick close to the material, to how to “speak” the tools and how to engage learners with them. These interpretations profoundly affect the resource as “lived” out in class; further, learners also interpret and shape resources-in-use. Two different groups of learners will work differently with the same lesson as offered by the same teacher, as a function

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Fig. 1 The instructional triangle: instruction as interaction in environments

of individual and collective and mutual interpretation. As such, as the authors of this volume make clear, curriculum tools are converted in real-time use from potential to actual resources. Consequently, design produces material with potential for use; it cannot determine actual use. What then does this conception of curriculum resources, and this understanding of their use in practice, suggest about their design? The chapters in this book reveal many subtleties inherent in the crucial work of design. First, for curriculum resources to support and guide deliberate use as envisioned for learning, designers must have a sensitivity for practice and its demands. For example, in real-time teaching, teachers cannot read detailed instructions as they listen to and interpret learners and manage the trajectory of content through the discourse and activity of the class. How then can their decision-making be supported? Further, how can additional examples, questions, and guidance be designed in ways that are usable in practice? Another concern regards learners: they say and do many things that are predictable and patterned; they also produce unexpected and novel ideas and conceptions. Designers can seek to provide forecasts and guidance for the predictable and open teachers’ readiness for the unanticipated. Thoughtfulness here can increase the support provided through design. And other issue pertains to the content: the mathematics itself is often also complex, and support for teachers’ learning often weak. Designers can develop usable opportunities for teachers’ own learning, but how can this be done well? Building curriculum resources with an eye toward their potential to support teachers’ development requires a multifocal approach, with an eye on the mathematics, on learners, on teachers, and on their learning and interactions. Such demands point to the importance of educationally oriented design, based on backward mapping from an understanding of practice and of resourcesin-use, or “lived resources,” in order to support resource use for improved learning. Figure 2 proposes an expansion of the instructional dynamic represented in Fig. 1, which affords a view of the dynamics of supporting instruction and of teachers’

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Fig. 2 The instructional triangle of teachers’ learning in and from practice

professional learning of and from practice. This learning occurs through interaction with resources and with other professionals, as well as in and from practice itself. At the heart of improving learning is to understand instruction as the complex weave of interactions and interpretations. It is on this foundation that this volume provides rich analyses and examples for the development of curriculum resources designed to be used, or “lived,” and from which both teachers and their pupils can learn. Bringing together such sophisticated design with a detailed understanding of practice can contribute to both better research on curriculum and its use, as well as better resources for use, and better outcomes for learners. This volume is itself a wonderful resource for this important agenda.

References Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal, 93(4), 373–397. Cohen, D., Raudenbush, S., & Ball, D. (2003). Resources, instruction, and research. Educational Evaluation and Policy Analysis, 25(2), 1–24. Dewey, J. (1902/1956). The child and the curriculum. Chicago: University of Chicago Press.

Conclusions Ghislaine Gueudet, Birgit Pepin, and Luc Trouche

Dynamic and Nested Documentation Systems in/for Teaching and Teacher Learning: Re-conceptualising Mathematics Curriculum Resources and Their Use “An organised being is then not a mere machine, for that has merely moving power, but it possesses in itself formative power of a self-propagating kind which it communicates to its materials though they have it not of themselves; it organises them, in fact, and this cannot be explained by mere mechanical faculty of motion.” Immanuel Kant (in “Critique of judgment”)

Reading through the 17 chapters and reactions in the book, we are impressed by the rich and varied perspectives provided by the authors and reactants. This underlines the fruitfulness of the position proposed at the beginning of this book: viewing teachers as designers and creative users of their own resources, considering the implications of teacher ‘interactions’ with resources for teacher professional development and hence the deepening of our understanding of ‘teacher documentation’. The authors have considered a great variety of resources, encompassing and reconceptualising artefacts and tools: from clay tablets, to textbooks and websites, including student work, and language; to name but a few. They have explored these resources in a creative and encompassing way, and their findings evidence the richness that lies in seeing resources as ‘lived resources’, when teachers work with them in their resource systems, and how these processes become part of teacher professional development. In this respect the use of digital resources raises particular questions. For example, some software is difficult to integrate into a teacher’s resource system, whilst other online resources are widely used and contribute to create new networks and

G. Gueudet (B) CREAD, Université de Bretagne Occidentale, IUFM Bretagne site de Rennes, 35043 Rennes Cedex, France e-mail: [email protected]

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communities. Studying interactions between teachers and (digital) resources, and various types of collectives, helps to deepen our understandings of such phenomena. This book has also provided evidence for the contextual nature of the processes involved in teacher–resource interactions. Interestingly, whilst being influenced by the context, there is evidence that these interactions are not restricted to particular countries considered to be ‘developed’. Furthermore, in any context, teacher documentation work, teacher agency and their interaction with teacher knowledge and professional development are evident. The aim of this conclusion chapter is to draw together the book’s chapters and synthesise the main results, and hence develop a deeper understanding of ‘teacher documentation’ as a construct and with respect to teacher learning. We propose four key issues that permeate the four sections of the book: 1. 2. 3. 4.

The intentions and affordances of a resource in terms of its use; The adaptation, appropriation and work with resources; From resources to orchestration and collaborative use of resources; Interrelations between documentation process and teacher knowledge.

In the following, we will attend to the four themes in turn, giving and relating to examples from the book’s chapters, before providing the conclusive remarks.

Theme 1: The Characteristics, Intentions and Affordances of a Resource in Terms of Its ‘Use’ We regard the affordances of a resource as the attributes and characteristics of the resource which provide potential for its use with peers/colleagues and students/pupils in the course of teachers’ work. This means that by virtue of their support for particular actions in a setting, the affordances may foster particular actions, and inhibit other actions which are less desirable. However, affordances of resources must also be considered in relation to the intentions of the participants in the activity they support. Thus, affordances, in this view, are ‘potentials’ or pre-conditions for activity. A particular resource provides an affordance for some activity; this does not imply that the activity will occur, although it may contribute to the likelihood of that activity. Additional conditions include ‘characteristics’ of an ‘agent’, that is beliefs and principles of practice of the teacher with respect to the resource (and its affordances). An example is Ruthven’s (Chapter 5) description of the ‘resource system’ and its affordances, shaping the integration of technology, or indeed, in Pepin’s chapter (Chapter 7), the (mathematical task analysis) tool’s affordances in terms of reflection and feedback. Schmidt (Chapter 8) investigates the affordances of school mathematics textbooks in terms of opportunities to learn demanding and engaging mathematics, and hence students having different experiences in school mathematics courses. The most striking example is probably provided by Proust (Chapter 9), who examines ancient Mesopotamian resources.

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Here the resources, the master and the tablet (including the text), may have different affordances and intentions (as defined in Chapter 4), depending on who has written the tablet and for what purpose (e.g. mathematics teaching, providing cultural background). Interestingly, Remillard (Chapter 6) uses the notion of ‘positioning’ to analyse affordances of curriculum materials. She contends that curriculum materials have particular ‘modes of address’, ways of ‘talking’ to teachers, and that these prescribe particular roles for teachers. This links to Sensevy’s contention (Chapter 3) that documents have particular ‘pedagogic intentions’. The question remains where the agency of the teacher lies.

Theme 2: The Adaptation, Appropriation and Work with the Resource – Its ‘Use’ This theme is at the heart of the documentation process and runs through most of the book’s chapters. It relates to the instrumental approach introduced by Verillon and Rabardel (1995) where the subject (in our case, the teacher) plays a crucial role in creating, modifying and using tools as instruments. Verillon and Rabardel claim that instruments are created when they are used and integrated into the subject’s activities – this process, the instrumental genesis, is linked to the tool’s characteristics and affordances (or constraints) and to the subject/teacher’s knowledge and principles of practice. According to this approach, there is an inter-relationship between the tool and the subject/teacher: the subject/teacher uses the tool and in the process evolves and develops, and in turn the instrument evolves. Two processes are crucial here: instrumentation, that is the implicit modes of actions and knowledge, and instrumentalisation, that is how the subject/teacher shapes the tool. In Chapter 2, Gueudet and Trouche develop these ideas in their documentational genesis approach where teachers interact with resources, select and work with/on them. The work in Chapter 4 (by Mariotti & Maracci) is sensitive to the semiotic aspects and potential of an artefact, and the authors explore how such an artefact (e.g. ICT tool) can be a resource for the teacher. In Chapter 13 Trigueros and Lozano describe a case of documentational genesis when working with teachers in ‘Enciclomedia’: teachers analysed and transformed texts in particular ways due to the resources affordances. In Chapter 7 by Pepin teachers’ work with the tool changed the tool, to become a ‘catalytic tool’, and in the process it changes its character, from tool as artefact to ‘epistemic object’ at the interface between task design and enactment. Kieran et al. (Chapter 10) theorise how teachers adapt ‘researcher-designed’ resources considering teachers’ own beliefs, knowledge and principles of practice.

Theme 3: From Resources to Orchestration and Collaborative Use of Resources Documentation can be considered (Chapters 2 and 16) as a continuous process, the work in class being only one of its components. However, in most contexts, most of the interactions between teachers and students appear to happen in class. This in turn

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confers a particular status to the classroom and leads us to focus on the orchestration of resources as a central part of the documentation process. Originally introduced by Trouche (2004), orchestration can here be at different levels: at the level of documents, or sets of documents, or at the level of the participant (e.g. teacher) working with, and relating to, the documents/sets of documents. In Chapter 14 Drijvers reports on teachers privileging orchestrations where students work individually or in pairs, and he contends that teacher beliefs and agency play an important role in the development and enactment of the processes involved in transforming resources into orchestrations. The collaborative use of resources relates to collaborative work of teachers in terms of resources and in the larger frame of scaling-up of the process of documentation and use of resources. In terms of teacher learning, collaborative use of resources is illustrated when groups of teachers work together on documents (likely to be important for their teaching) to analyse, search for understanding and meaning, and to create a common resource of their learning. Sensevy (Chapter 3) develops an understanding of collective thought (influenced by the institutional thought style) by identifying ‘patterns of didactic intentions’ which in fact are said to lie in the documents (used by teachers) and the positioning of teachers towards these documents. Linking this to ICT communication, collaborative learning networks can develop, via electronic dialogue, and where participants share a common purpose of/for documentation. In Chapter 16 (Gueudet & Trouche) the common ‘purpose’ is Sésamath, both an individual and a collective resource. The processes involved in collective documentation are exemplified by Gueudet and Trouche, when ‘sharing’ turns into ‘cooperation & sharing’, into ‘collaboration & cooperation & sharing’ before another cycle develops. The scaling-up collaborative process is evident in Chapter 17 (Visnovska, Cobb, & Dean), where the authors drew on a five-year-long interventionist professional development study where teachers collectively (e.g. in a professional development group) designed resources for teaching of a statistics unit and at the same time made meaning of the objectives prescribed by the State. Interestingly, Winsløw (Chapter 15) compared two very different genres of teacher collaborative work (using the frame of paradidactic infrastructure): the Japanese lesson study and the Danish teacher collaboration in ‘multidisciplinary modules’. He concludes that collaborative work forms, also for documentation work, are influenced by the cultural and educational traditions of the country concerned and that particular practices would be ‘unthinkable’ in certain environments, whereas in others they are common practice – hence the importance and influence of the context in which the documentation process is taking place.

Theme 4: Interrelations Between Documentation Process and Teacher Knowledge Teachers working with resources, we have presumed, is an interactive and dialectic process: teachers shape the resources, and the documentation processes involved influence teachers in turn. Teachers, it is argued, develop deeper understandings with

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respect to particular resources, and they may adopt new roles in their interactions with the resources initiating or constructing new processes in terms of learning situations, or indeed they may communicate and interact in particular collaborative ways with their colleagues – all acts of teacher learning that are connected to the documentation process. In Chapter 1, Adler argues for ‘professional knowledge in use’, and in her study illuminates ‘knowledge resources in use’ in two different pedagogic practices. Pepin (Chapter 7) claims that the task analysis ‘tool’ provided feedback to teachers, at four different levels, and in turn helped them to develop deeper understandings. Interestingly, Forest and Mercier (Chapter 11) provide evidence for using video as a tool for professional development, in particular considering the teacher’s attitudes and gestures as resources and connecting them with the use of language in the mathematics classroom. Whilst these four themes capture most of the authors’ work, there is a permeating strand that runs through all of the chapters: the pupils’ influence and involvement in the documentation process. As an example, Rezat’s work (Chapter 12) considers the orchestration of resources in and outside the classroom when exploring pupil/student use of the textbook as resource, which in turn is said to have an influence on teacher use. Interestingly, Sensevy (Chapter 3), as well as Forest and Mercier (Chapter 11), conceptualise the teaching processes as the joint didactic action of the teacher and the students. Schmidt (Chapter 8) develops a way to quantify student curricular experiences in different courses, their exposure to particular curriculum materials, which in turn is likely to have an influence on their opportunities to work with and learn from mathematics resources.

Concluding Remarks and Looking Ahead Considering the issues raised by authors and reactants, one wonders what makes a ‘documentation system’, and how does such a system evolve? It seems that the key factors that can be argued to explain the ‘workability’ of a documentation system are the nature of the system, its constituents and the feedback ‘loops’ that characterise and shape such a system. In each study an important step to develop a documentation system appeared to have been when reflective capacity was built, such as between teachers and resources, and/or amongst peers, and/or between teachers and academics. With this reflective capacity, the participants of the system had information about the nature of the resources and their potential dynamics (also with participants). However, it is not evident that this reflective capacity develops as a matter of course. As Visnovska et al. (Chapter 17) point out, teachers need support to design and implement ‘coherent instructional sequences’. Moreover, the participants of such a ‘workable’ documentation system need a shared purpose (see Kieran et al.’s Chapter 10), and it appears to develop more ‘easily’ in collectives (see Chapters 3 and 16). It can be argued that the documentation system needs a ‘minding of the system’ (Vickers, 1995) in order to be workable.

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Systems such as documentation systems have to be acknowledged to be inherently turbulent, and also inherently unique in the way that they adapt to external intervention (such as inclusion of web books, web-based learning groups, etc.). The different sub-systems can be regarded as inter-dependent, or inter-related – the concept often used is that of nested systems, with each system nested within a larger one. This book constitutes one step in an ongoing work, and the key issues and results outlined above need to be further investigated. Considering the new perspectives crucial questions emerge and these need further investigation:

(1) Many resources are available for mathematics teachers, but which resources do they crucially need for their work? Are there resources that could be regarded ‘universal’, as ‘resources of the (mathematics teaching) profession’? What are the national and cultural differences among resources, what are the individual differences? In which ways could such resources be designed, and differences catered for? How could they be made available to all teachers (e.g. ‘broadcasted’)? (2) Considering Shulman’s (1986) major categories of teacher knowledge in connection with Ball, Thames, & Phelps’s (2008) categories, one wonders where the ‘documentation process knowledge’ is situated. In particular its dynamic and creative nature, in addition to its ‘position’ at the interface between design and enactment, does not make it ‘fit in easily’. We contend that an additional teacher knowledge category (perhaps ‘hors categorie’) may be necessary, which we call documentation knowledge and which would include knowledge about resources/materials in use, individually or collectively, and their interaction with the teaching/learning process of both teachers and learners (including the teacher as learner). (3) All the book’s chapters focus on the teaching of mathematics. In mathematics the documentation work of teacher educators, or of mathematicians (Chapter 9), may be similar, or different, to teachers’ documentation work. Turning to other subject areas, similar (or different) phenomena may be evident for teacher documentation work in other domains. Investigating these is likely to deepen our understandings of the documentation process.

In conclusion, closing the cycle and linking to the book’s title, we have developed deeper understandings about mathematics curriculum materials as ‘lived’ resources – which points to their use in the past. We now suggest viewing them as ‘living resources’ emphasising their present and continuous use in teachers’ work. Teacher documentation, we have learnt, is a creative and dynamic process where participants work in a collaborative system and with the aim of teacher learning – this provides challenges, and at the same time a positive outlook both for teachers and reformers.

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References Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching – What makes it special? Journal of Teacher Education, 59(5), 389–407. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. Trouche, L. (2004). Managing complexity of human/machine interactions in computerized learning environments: Guiding student’s command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9(3), 281–307. Verillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrument activity. European Journal of Psychology in Education, 9(3), 77–101. Vickers, G. (1995). The Art of Judgement: The study of policy making. London: Sage centenary Edition.

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A Activity theory, 23–24, 78, 137, 242, 306, 346 Adaptation, xi, 28, 30, 83–84, 90, 97–99, 185, 189–212, 284–285, 309, 328, 354–355 Algebraic skills, 269–270, 275, 278 Algebra teaching, 145 Applets, 89, 268–269 Artefact, v–vii, x–xi, 24–25, 46, 59–63, 66–67, 73–74, 78–79, 124, 137–138, 224, 234, 242–243, 266, 292–293, 295 C CAS, xi, 190–199, 202–211, 283–284 “Catalytic tool,” 355 Classroom teaching practice, 189–212 Collectives, xii, 29, 35, 37, 306–308, 314, 320, 321, 344–345, 354, 357 Communities documentation, xii, 308–310, 312, 314, 318–319, 321, 328, 345 geneses, 310, 320 of practice, vi, xii, 307–308, 310, 320 Craft knowledge, 87 Curriculum, passim Curriculum reform, 114 D Decimal numbers, 217, 255 Didactical cycle, 60–63 Didactical situations, 49, 59, 216–218 Didactics, v–vi, x, xii, 5, 23, 25, 38, 43, 190, 211, 248, 305 Digital resources, ix–x, 3, 23–25, 32, 35, 37–38, 98, 138, 184, 247, 249, 255, 258–259, 265, 306, 353–354 technology, 87, 114, 283–284 Document, passim

Documentational approach, x, xii, 23–27, 37–39, 59, 190, 211, 234, 239–242, 248, 305 Documentational genesis, 23, 25–26, 38, 43, 49, 78–79, 136, 190–191, 197, 207–208, 211, 240, 243, 247–261, 265, 268, 279, 308–309, 313–315, 324, 328, 334–337, 343–345, 355 Documentation work, x, xii, xiii, 3, 23–30, 33, 35, 37–39, 43–44, 46, 55, 124, 161, 234, 248, 305–307, 309–317, 319–321, 323–325, 327–332, 334, 337, 344, 354, 356, 358 E Education, 123–124, 215–228 Enciclomedia, xii, 247–261, 283, 355 Epistemic object, xi, 138–139, 355, 358 F Feedback & teacher learning, 123–140, 184, 225, 353–354, 356–358 Forms of address, 106–117 Forms of engagement, 115–117, 308 G Geneses, x, xii, 23–39, 60, 243, 260, 279, 305–321, 344 H Historic source, 64 I Implementation, 211 Institution, 30 Instructional design, vi, xii, 47, 55, 323–339, 345

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362 Instructional materials, 277, 323–324, 338–339, 344 Instructional practices, 323, 325 Instrumental genesis, vi, 243, 266–268, 273, 277, 279, 355 Instrumentalisation, 25, 33–35, 136, 234, 239–240, 258, 260, 355 Instrumental orchestration, 59, 99, 191, 266–267, 284 Instrumentation, 25, 33–35, 91, 97, 234, 248, 258–259, 316, 345, 355 Instruments, 27, 88, 137, 266, 283, 312, 316–317, 355 Integration, x–xi, 20, 29, 37–38, 83, 89, 99–100, 235, 242, 247, 252, 259–260, 284, 354 Intention, x, xii, 5, 18, 20, 43–56, 74, 77–79, 84, 114, 126, 135, 181, 192–198, 200, 203, 205, 208–211, 217, 227, 234, 242, 250, 254, 266, 278, 292, 344, 354–356 J Joint action, x, 25, 43, 50, 55–56, 216, 218, 221, 227 theory in didactics, 43 K Knowledge resources, x, 3–21, 265, 357 L Lesson study, 291–303, 345–346, 356 M Materials, 83–100, 123–140 Mathematics education, v–vii, ix, 5, 9, 15, 79, 88, 113, 123–124, 181, 189, 249, 257, 260, 268, 343 task analysis, 123–140, 354 teaching, 10, 83, 107, 116, 119, 125, 128, 131, 191, 225, 247–261, 350 textbooks, xii, 10, 105, 109, 117, 144, 146, 149, 153, 231–244, 354 Measures of curriculum coverage, 145, 157–158 Mediation, x, 24, 39, 59–74, 77–80, 96, 100, 137, 185, 232–238, 242, 279, 285, 345 Modes of address, 106–108, 120, 355 of engagement, xi, 105–121, 210

Index N Non-verbal communication, 226 O Operational invariants, 25–27, 33–34, 234, 239, 242, 250, 252–255, 258–261, 284, 316 Opportunities to learn, 125, 231, 354 Orchestration, xii, 34, 55, 59, 99–100, 191, 243, 265–280, 284, 286–287, 316, 354–357 P Photograms, 217, 219, 227–228 Practitioner thinking, 92–97 Primary artifact, 74 Professional adaptation, 97 Professional development, vi, ix–x, xii–xiii, 20–21, 38–39, 78, 120, 125–128, 135–136, 181, 184, 217, 248–253, 259–261, 265–266, 268, 279–280, 320, 324–327, 331–332, 335–338, 353–354, 356–357 Professional geneses, x, 23–40, 305–321, 344 Professional teaching community, 327–328, 332, 337 Proxemics, 217–218 Q Quantitative Index of Curriculum Exposure, 155–156 R Reflection, 7, 20, 36, 45, 63, 96, 99, 126–128, 135, 145, 149, 152, 154, 161, 182, 194–196, 198, 203–204, 208, 211, 252, 254–255, 260, 285, 295–297, 319, 354 Researcher-designed resources, xi, 189–212, 284, 355 Resources artifacts, 287 Role of curriculum materials, 106, 125 S Scheme, 20, 44, 48, 50, 55, 95, 116, 129, 132, 134–135, 138, 182, 254–257, 259–260, 269–270, 297–298, 335 School mathematics, 3–21, 83, 85, 88, 100, 109, 125, 132, 157–158, 166–173, 189, 192, 324, 354 Scribal schools, xi, 161–164, 172–175, 182 Secondary artifact, 74

Index Semiotic mediation, x, 59–74, 78, 242, 279 Semiotic potential, x, 60–64, 68, 74, 279 Standards-based curriculum materials, 105, 114 Strategic rules, 44, 50, 53, 55 Students’ use of textbooks, xii, 232–233, 235, 237–238, 240 T Teacher action, x, 78 education, v, xii, 4–7, 10, 20–21, 74, 78, 215–228, 243, 306 learning, xi, 112, 123–140, 184, 225, 353–354, 356–358 professional development, x, xii–xiii, 127, 248, 320, 353 Teacher-curriculum interactions, 106 Teachers’ activity, 23, 38, 211 associations, 306, 310 beliefs and goals, 209 task design, 74 professional growth, 23–24, 26, 37 resource, 35, 37–38, 272, 353 shaping of resources, 208 use of textbooks, 231–232 Teaching with CAS tools, 193 practice, vi–vii, 105, 118, 173–174, 177, 189–212, 259–260, 265, 270, 274, 279–280, 294, 333

363 being shaped by, 190, 197, 208 resources, xiii, 80, 250, 310, 312–313 Technology, xi, 25, 37–38, 78–80, 84, 88–94, 100, 184, 190–197, 203, 206–209, 211, 225, 247–248, 250–251, 259, 265, 267–268, 273, 275–279, 313, 349, 354 Textbooks as implemented curriculum, 145–146, 148, 182 Theories of cognition, 284, 286 Thinking, mathematical, 6, 9, 19, 48, 51, 86, 92–97, 99, 117, 124, 132, 134–136, 138–139, 194–195, 197–199, 202, 208–211, 255, 260, 283, 285–287, 330–331, 346, 350 Thought style, 44, 51, 55–56, 356 Tracking in US Schools, 144, 155 Transaction, 106, 114 U Usages, 248, 251–252, 255, 259–261, 319 V Video analysis, 226 Vygotsky, L. S., 60, 114, 137, 242, 285, 306, 346 W Written texts, 62, 64, 66–69, 72, 74, 78, 173, 286, 313