Making Connections: Comparing Mathematics Classrooms Around the World

Making Connections

Making Connections Comparing Mathematics Classrooms Around the World Edited by

David Clarke University of Melbourne Australia

Jonas Emanuelsson Göteborgs Universitet Sweden

Eva Jablonka Freie Universität Berlin Germany

Ida Ah Chee Mok University of Hong Kong Hong Kong

SENSE PUBLISHERS ROTTERDAM / TAIPEI

A C.I.P. record for this book is available from the Library of Congress.

ISBN 90-77874-79-8 ISBN 90-77874-90-9

Published by: Sense Publishers, P.O. Box 21858, 3001 AW Rotterdam, The Netherlands http://www.sensepublishers.com Printed on acid-free paper

Cover design: Cameron Mitchell, ICCR, Melbourne, Australia Cover Photo: Simulated data generation at Flinders Peak Secondary College, Corio, Victoria, Australia. Used with permission

All Rights Reserved © 2006 Sense Publishers 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.

SERIES PREFACE

The Learner’s Perspective Study provides a vehicle for the work of an international community of classroom researchers. The work of this community will be reported in a series of books of which this is the second. The documentation of the practices of classrooms in other countries causes us to question and revise our assumptions about our own practice and the theories on which that practice is based. International comparative and cross-cultural research has the capacity to inform practice, shape policy and develop theory at a level commensurate with regional, national or global priorities. International comparative research offers us more than insights into the novel, interesting and adaptable practices employed in other school systems. It also offers us insights into the strange, invisible, and unquestioned routines and rituals of our own school system and our own classrooms. In addition, a cross-cultural perspective on classrooms can help us identify common values and shared assumptions, encouraging the adaptation of practices from one classroom for use in a different cultural setting. As these findings become more widely available, they will be increasingly utilised in the professional development of teachers and in the development of new theory. David Clarke Series Editor

TABLE OF CONTENTS

Acknowledgements 1

2

The Learner’s Perspective Study and International Comparisons of Classroom Practice David Clarke, Jonas Emanuelsson, Eva Jablonka and Ida Ah Chee Mok Addressing the Challenge of Legitimate International Comparisons: Lesson Structure in the USA, Germany and Japan David Clarke, Carmel Mesiti, Eva Jablonka and Yoshinori Shimizu

ix

1

23

3

Beginning the Lesson: The First Ten Minutes Carmel Mesiti and David Clarke

47

4

Kikan-Shido: Between Desks Instruction Catherine O’Keefe, Li Hua Xu and David Clarke

73

5

Student(s) at the Front: Forms and Functions in Six Classrooms from Germany, Hong Kong and the United States Eva Jablonka

6

How Do You Conclude Today’s Lesson? The Form and Functions of ‘Matome’ in Mathematics Lessons Yoshinori Shimizu

7

‘Learning Task’ Lesson Events Ida Ah Chee Mok and Berinderjeet Kaur

8

Interaction, Organisation, Tasks and Possibilities for Learning about Mathematical Relationships: A Swedish Classroom Compared with a US Classroom Johan Liljestrand and Ulla Runesson

9

The Introduction of New Content: What is Possible to Learn? Johan Häggström

107

127

147

165

185

vii

10

11

A Study of Mathematics Teachers’ Constraints in Changing Practices: Some Lessons from Countries Participating in the Learner’s Perspective Study Herbert Bheki Khuzwayo Deconstructing Dichotomies: Arguing for a more Inclusive Approach David Clarke

201

215

Appendix A: The LPS Research Design David Clarke

237

Author Index

253

Subject Index

259

viii

ACKNOWLEDGEMENTS

The Editors would like to express their gratitude to Mary Barnes for her meticulous work in formatting the majority of the chapters in this book, her work in establishing many of the conventions employed in the text and her contribution to developing the Subject Index to this book. Carmel Mesiti’s careful work in formatting the remaining chapters and in constructing both the Subject Index and Author Index is also acknowledged with gratitude. The research reported in this book benefited substantially from funding awarded by the following agencies, centres and universities: The Australian Research Council Bank of Sweden Tercentenary Foundation Centre for Research in Pedagogy and Practice, National Institute of Education, Nanyang Technological University (Singapore) The Collier Charitable Trust (Australia) Committee for Research and Conference Grants, University of Hong Kong, (Hong Kong SAR, China) Council for International Exchange of Scholars Fulbright Research Scholar Award (USA) Japan Society for the Promotion of Science Mathematics Association of Victoria (Australia) Ministry of Education, Science, Sports and Culture (Japan) National Research Foundation (South Africa) The Potter Foundation (Australia) Research Commission, Freie Universität Berlin (Germany) Research Grants Council (Hong Kong SAR, China) The Spencer Foundation (USA) Swedish Research Council The University of Macau, Academic Community (China) The University of Melbourne (Australia) University of Zululand (South Africa) All editors and authors would like to thank the teachers and students, whose cooperation and generous participation made this international study possible. Publication of this work was assisted by a publication grant from the University of Melbourne.

ix

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK

CHAPTER ONE The Learner’s Perspective Study and International Comparisons of Classroom Practice

INTRODUCTION

The practices of classrooms are the most evident institutionalised means by which the policies of a nation’s educational system are put into effect. The curriculum can be viewed as the embodiment of the aspirations of the school system in which the classroom is situated. To a significant extent, the teacher is the agent of the system by whose actions the curriculum is put into effect. Teachers, however, interpret the curriculum in idiosyncratic fashion, within the constraints and affordances of both system and culture. Research in the Learner’s Perspective Study (LPS) has made clear just how culturally-situated are the practices of classrooms around the world, and the extent to which students are collaborators with the teacher, complicit in the development and enactment of patterns of participation that reflect individual, societal and cultural priorities and associated value systems (this book and Clarke, Keitel, & Shimizu, 2006). International comparative and cross-cultural research has the capacity to inform practice, shape policy and develop theory at a level commensurate with regional, national or global priorities. This book, its companion volume, and the series of publications of which these are a part, provide evidence of various possibilities of combining analyses that address macro and micro level concerns in mathematics education and suggest possible directions for practical advances. The community at large has the right to expect any advocacy of practice to be evidence-based – which shifts the debate from the potential value of research to consideration of what constitutes evidence sufficient to support the advocacy of any particular practice in any particular classroom. Within any educational system, the possibilities for experimentation and innovation are limited by more than just methodological and ethical considerations: they are limited by our capacity to conceive possible alternatives. They are also limited by our assumptions regarding acceptable practice. These assumptions are the result of a long local history of educational practice, in which every development was a response to emergent local need and reflective of changing local values. Well-entrenched practices sublimate this history of development. In the school system(s) of any country, the resultant amalgam of tradition and recent D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 1–22. © 2006 Sense Publishers. All rights reserved.

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK

innovation is deeply reflective of assumptions that do more than mirror the encompassing culture: they embody and constitute it. International comparative research offers us more than insights into the novel, interesting and adaptable practices employed in other school systems. It also offers us insights into the strange, invisible, and unquestioned routines and rituals of our own school system and, in the case of this book, our own mathematics classrooms. International comparative research has an additional power: the capacity to reveal similarity within difference, structure within extreme diversity. AndersonLevitt (2002) noted the “significant national differences in teacher gender, degree of specialization in math, amount of planning time, and duties outside class” (p. 19). But these differences co-exist with similarities in school organization, classroom organization, and curriculum content. Anderson-Levitt (2002, p. 20) juxtaposed the statement by LeTendre, Baker, Akiba, Goesling, and Wiseman (2001) that “Japanese, German and U.S. teachers all appear to be working from a very similar ‘cultural script’” (p. 9) with the conclusions of Stigler and Hiebert (1999) that U.S. and Japanese teachers use different cultural scripts for running lessons. The apparent conflict is usefully (if partially) resolved by noting with Anderson, Ryan and Shapiro (1989) that both U.S. and Japanese teachers draw on the same small repertoire of “whole-class, lecture-recitation and seatwork lessons conducted by one teacher with a group of children isolated in a classroom” (Anderson-Levitt, 2002, p. 21), but they utilise their options within this repertoire differently. Given the cultural dissimilarities and the separate development over time of traditions of practice in schools as geographically distant as Sweden and China or Germany and Japan, any similarities should startle us just as much as any differences. If, in fact, educational policy and practice represent the enactment of both societal and cultural values, the classroom, as a profoundly social setting, seems a sensible place to look for explanations and consequences of the differences and similarities identified in international comparative studies of curriculum, teaching practice, and student achievement (see Clarke, 2003; Lerman, 1994). Within the specific focus of classroom practice, the central problem of international comparative research (Emanuelsson & Clarke, 2004) translates into: How best might the practices of classrooms be compared internationally if our purpose is to inform those practices? Both the curriculum and the teacher have been the focus of recent international comparative studies. Among the studies of curriculum and teaching practice, we can lose sight of the student. Thorsten makes this point beautifully. What is absent from nearly all the rhetoric and variables of TIMSS pointing to the future needs of the global economy is indeed this human side: the notion that students themselves are agents. TIMSS makes students from 41 countries into passive objects of 41 bureaucratic gazes, all linked to the seduction of one global economic curriculum (Thorsten, 2000, p. 71). Educational research has increasingly drawn our attention to the importance of the social processes whereby competence is constructed and in which competence is 2

LEARNER’S PERSPECTIVE STUDY

constituted (for both teaching and learning). In particular, the agency of the student, the nature of learner practice, and the cultural specificity of that agency and that practice must be accommodated within our research designs. The Learner’s Perspective Study (LPS) has effected that accommodation. International Comparative Studies: What can we hope to learn? International comparative research in mathematics education is a growing field. There is a need for enlightened discussion as to how the results of international studies relate to the research methods and techniques used, to the theoretical and analytical perspectives enacted in the research, and to the political aims of the stakeholders who promote them. It is essential that the research community consider the different comparative approaches available and the consequent differences in project outcomes. The abiding challenge for classroom research is the realization of structure in diversity. The structure in this case takes the form of patterns of participation: regularities in the social practices of mathematics classrooms. Only through the identification of patterns of classroom participation and their connection with learning, can our research inform individual and collective ways of understanding learning and teaching in mathematics, and inform the development of classroom practices likely to support the effective participation of all members of the classroom community. The expansion of our field of view to include international rather than just local classrooms increases the diversity and heightens the challenge of the search for structure, while increasing the significance of any structures, once found. Educational research has tried on a variety of methodological attire. Processproduct studies came and went. Local constraints on what can and cannot be manipulated in school-like settings place serious constraints on experimentation. International research offers opportunities to study settings and characteristics untenable in the researcher’s local situation. Contextual elements such as class size, use of technology, combinations of social background, and the multi-cultural or mono-cultural composition of the class are available for study and comparison across the data set of an international comparative research project. Importantly, international comparative studies can reveal possibilities for practice that would go unrecognized within the established norms of educational practice of one country or one culture. Our capacity to conceive of alternatives to our current practice is constrained by deep-rooted assumptions, reflecting cultural and societal values that we lack the perspective to question. The comparisons made possible by international research facilitate our identification and interrogation of these assumptions. Such interrogation opens up possibilities for innovation that might not otherwise be identified; possibilities that might then become the focus of design-based research in specific cultural settings (cf. The Design-Based Research Collective, 2003). Such comparisons need not be competitive evaluations. It is possible to benefit from the identification of the similarities and differences in locally-identified good practice without identifying one set of practices as “better than” another set of 3

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK

practices, except perhaps in so far as one set of practices are better aligned with the needs and aspirations of a particular setting. Of course, our capacity to make any such evaluative judgments, even locally situated judgments, requires evidence of outcome; and the confident attribution of learning outcomes to specific instructional practices poses significant methodological challenges. The postlesson video-stimulated interviews employed in the LPS provide the basis for such attributions. International comparisons of student achievement such as the Program for International Student Assessment (PISA) or the Third International Mathematics and Science Study (TIMSS) are not only producing data for comparative purposes, they also produce conceptions of what is important and of value in mathematics education. They are not only comparing, they are participating in the social construction of curricula in mathematics education. This thought is well developed in the work of Ian Hacking (1999). From this point of view, international comparisons are about homogenization of mathematics education. An analogous point was made powerfully by Keitel and Kilpatrick (1999). A pseudo-consensus has been imposed (primarily by the English-speaking world) across systems so that curriculum can be taken as a constant rather than a variable, and so that the operation of other variables can be examined (Keitel & Kilpatrick, 1999, p. 253). Keitel and Kilpatrick (1999) problematised the assumptions on which international comparative studies of school mathematics had been predicated. In particular, they questioned the treatment of the mathematics curriculum as unproblematic and the associated assumption that a single test could give comparable measures of curriculum effects across countries. They further suggested that the spectre of an “idealized international curriculum” lay behind even the most sophisticated research designs, including text and document analyses and the use of video to study classroom practice. The analogue of this idealized curriculum is the idealized international classroom with idealized participants. Such a conception ignores fundamental differences in affluence and aspiration, in values and in norms of social exchange. The power of international comparative research is to expand the range of the possible, not to constrain it to a culturally-neutral prescription of practice. Our aim in this project and in this book is to increase cultural and contextual sensitivity in comparative and international educational research, and to enrich both theory and practice, by reporting the diversity of practice in the classrooms of competent teachers around the world and by identifying the patterns of participation that we find there. What is compared? The question of what to compare is at the heart of international comparative research and has been addressed very differently in various studies. In most international comparisons of mathematics education, it is achievement in terms of test results that is compared. From such outcome comparisons we can conclude 4

LEARNER’S PERSPECTIVE STUDY

that students in some countries are doing better than students in other countries according to the achievement constructs operationalised in the tests. Why this is the case is impossible to tell without further information. Recent studies also generated data about the possible prerequisites for learning mathematics. For example, the PISA 2003 context questionnaires included questions on student characteristics, student family background, student perceptions, school characteristics and school principals’ perceptions. Some of these data were used to compute an index on economic, social and cultural status of the students. A variety of correlations with achievement data have been reported. In the IEA’s Repeat of the Third International Mathematics and Science Study (TIMSS 1999), the impressive list includes amongst others: an index of home educational resources, the frequency with which students speak the language of the test at home, the students’ expectations for finishing school, an index of out-of-school study time, an index of students’ self-concept in mathematics, an index of teachers’ emphasis on mathematics homework, schools’ expectations of parental involvement or the frequency and seriousness of student behaviour threatening an orderly school environment (cf. Mullis et al., 2000). Achievement data also have been correlated to class size, education expenditure as percentages of GDP as absolute amounts per student in relation to distinct levels of education, formal teacher qualification and other indicators of resources. If any positive correlation of achievement with one of these variables was identified, this often gave rise to simplistic interpretations of causality, assuming one direction of causation. However, any such correlation evidence is both rare and contentious, and difficult to interpret. We need to know what is happening in the teaching process in order to understand the outcomes of this process. And this was exactly what the TIMSS-99 video study (Hiebert et al., 2003) did in the most advanced attempt to produce useful descriptions of mathematics teaching in different high-achieving countries. One hundred eighth grade classes were selected by random sampling in seven countries. In each class, one lesson was video-recorded. This project supported the comparison of the mathematics classrooms in the different countries with regard to, for instance: length of lesson, time devoted to mathematical work, time devoted to problem segments, percentage of time devoted to independent problems, time per independent problem, time devoted to practising new content, time devoted to public interaction, number of problems assigned as homework, number of outside interruptions, number of problems of moderate complexity, number of problems that included proofs, number of problems using real life connections, number of problems requiring the students to make connections, time devoted to repeating procedures, number of words said by teacher (publicly), number of words said by students (publicly), number of lessons during which the chalkboard was used, and number of lessons during which computational calculators were used. Even if the classrooms of each country could be identified with distinctive combinations of attributes, the connection between any particular classroom attributes and national mathematics achievement remained problematic. For example, the national average scores for middle school students in Japan and Hong Kong on the TIMSS 1999 mathematics assessment were very similar, yet the level of procedural complexity 5

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK

of the problems used in the mathematics classrooms in the two countries was very different (see Figure 2, in Hiebert et al., 2003, p. 6). The average proportion of time spent in mathematics classrooms in Hong Kong and Japan in reviewing previous content was identical, but there were significant differences in the amount of time spent introducing (rather than practicing) new content (see Figure 1, in Hiebert et al., 2003, p. 5). These difficulties in drawing conclusions that might support the advocacy of any particular instructional strategy derive in part from the need (inherent in a nationally representative study) to characterize classrooms by dissociated, quantifiable characteristics, whose connection to a single national outcome (average student mathematics achievement) can only be conjectured. Underlying any such international comparison is the assumed legitimacy of characterizing ‘typical instructional practice’ in each country, and the concomitant assumption that such typical practice might be causally connected to national achievement measures. In order to consider alternative answers to the question “What should we compare?” it is necessary to consider alternatives to either competitive comparisons or correlational studies that can offer opportunities for multidisciplinary endeavors and collaboration between researchers and practitioners in order to reach better performance within our different educational systems for the benefit of all students. It seems reasonable to clarify the goals comparative studies try to achieve and the levels and units of analysis that can be accomplished by the types of data produced. Comparisons, such as “Lesson Studies” (Fernandez & Yoshida, 2004), can be driven by the desire to find good examples of teaching or classroom organisation in order to adapt these to local needs and conditions and to implement them in our own country. Other countries’ practices can also serve as mirrors in the quest to understand the practice of our own country. On the other hand, reference to an international data set can be motivated by the search for similarities, which can be analysed with the help of theories that focus on structural features of the school setting. These approaches are not mutually exclusive, but each involves a different level of analysis and consequently produces results with a different character informed by a different frame of reference. The data set produced in the LPS is sufficiently complex to allow for different levels and units of analysis, such as lesson structure, forms and functions of particular lesson elements (Lesson Events), structure of tasks and forms of classroom interaction, forms of the evolvement of a distinct mathematical topic, teachers’ intentions and students’ rationales. As was the case with the earlier work of Clarke and his colleagues (Clarke, 2001), the LPS research team has produced multi-faceted analyses of a commonly held database, undertaken from different educational and theoretical positions. This approach offers a variety of mutually informing perspectives intended, in their combination, to provide a much richer portrayal of classroom practice than would be possible from any single analysis; richer also than a similar number of disconnected analyses of different databases.

6

LEARNER’S PERSPECTIVE STUDY

Similarity and Difference in International Comparative Research Schmidt, McKnight, Valverde, Houang and Wiley (1997) investigated the mathematics curricula of the “almost 50” countries participating in the Third International Mathematics and Science Study (TIMSS). The documented differences in curricular organisation were extensive. Even within a single country differentiated curricular catered to communities perceived as having different needs. Countries differed in the extent of such differentiation, in the complexity or uniformity of their school systems, and in the distribution of educational decisionmaking responsibility within those school systems. Given such diversity, the identification of any curricular similarity with regard to mathematics should be seen as significant. And there were significant similarities. There were similarities of topic, if not of curricular location; broad correspondences of grade level and content that, on closer inspection, revealed differences in the detail of content and sequence; differences in the range of content addressed at a particular grade level, but which repeated particular developmental sequences where common content was addressed over several grade levels. In another international study of mathematics curricula, the OECD study of thirteen countries’ innovative programs in mathematics, science and technology found that, “Virtually everywhere, the curriculum is becoming more practical” (Atkin & Black, 1997, p. 24). Yet, despite this common trend, the same study found significant differences in the reasons that prompted the new curricula (Atkin & Black, 1996). These interwoven similarities and differences are the signature of international comparative research in mathematics education (Clarke, 2003). Schmidt, McKnight, Valverde, Houang, and Wiley (1997) reported that differences in the characterization of mathematical activity were extreme at the Middle School level; from ‘representing’ situations mathematically, ‘generalizing’ and ‘justifying’ to ‘recalling mathematical objects and properties’ and ‘performing routine procedures.’ Despite the apparent diversity, it was the latter two expectations that were emphasised in the curricula studied. Given the documented diversity, it is the occurrence of similarity that requires explanation. Some curricular similarities may be the heritage of a colonial past. Others may be the result of more recent cultural imperialism or simply good international marketing. LeTendre, Baker, Akiba, Goesling and Wiseman (2001) claimed that “Policy debates in the U.S. are increasingly informed by use of internationally generated, comparative data” (p. 3). LeTendre and his colleagues went on to argue that criticisms of international comparative research on the basis of “culture clash” ignored international isomorphisms at the level of institutions (particularly schools). LeTendre et al. reported yet another interweaving of similarity and difference. We find some differences in how teachers’ work is organised, but similarities in teachers’ belief patterns. We find that core teaching practices and teacher beliefs show little national variation, but that other aspects of teachers’ work (e.g., non-instructional duties) do show variation (LeTendre, Baker, Akiba, Goesling & Wiseman, 2001, p. 3) 7

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK

These differences and the similarities are interconnected and interdependent and it is likely that policy and practice are best informed by research that examines the nature of the interconnection of specific similarities and differences, rather than simply the frequency of their occurrence. One of the dangers of identifying any facet of classroom or school practice for independent comparison is that the object of analysis is disconnected from the local (national) context that provides its rationale. Hence we find a growth in utilitarian curricula motivated differently from country to country. Classrooms are inherently social places, where teacher and students improvise their interactions within the constraints and affordances of cultural, societal and institutional norms. Eugene Ionescu is reputed to have said, “Only the ephemeral is of lasting value.” Social interactions are nothing if not ephemeral and, since it is through social interaction that we experience the world, the understanding of social interactions must underlie any attempts to improve the human condition – in this case, the effectiveness of our mathematics classrooms. Our difficulties in characterizing social interactions for the purpose of theory building are compounded by the fluid and transient nature of the phenomena we seek to describe. While social practice may become progressively crystallized as culture, cultures evolve and reinvent themselves in response to local situations. Attempts to categorise social behaviour run the risk of sacrificing the dynamism, contextualdependence and variation that constitute their essential attributes. This poses a challenge both for methodology and for theory. The ephemeral nature of social interactions is something that must be honoured in the methodology but transcended in the analysis. Data and Approaches to Analysis in the Learner’s Perspective Study The Learner’s Perspective Study documented sequences of at least ten lessons, using three video cameras, supplemented by the reconstructive accounts of classroom participants obtained in post-lesson video-stimulated interviews, and by test and questionnaire data, and copies of student written material (Clarke, 1998, 2001, 2003). In each classroom, formal data generation was preceded by a oneweek familiarization period in which the research team undertook preliminary classroom videotaping and post-lesson interviewing until such time as the teacher and students were accustomed to the classroom presence of the researchers and familiar with the research process. In each participating country, the focus of data generation was the classrooms of three teachers, identified by the local mathematics education community as competent, and situated in demographically different school communities within the one major city. For each school system (country), this design generated a data set of at least 30 ‘well-taught’ lessons (three sequences of at least ten lessons), involving 120 video records, 60 student interviews, 12 teacher interviews, plus researcher field notes, test and questionnaire data, and scanned student written material. Well-taught, in the context of this study, meant that the teachers in each country were recruited according to local criteria for competence: visibility as presenters at conferences for other teachers, 8

LEARNER’S PERSPECTIVE STUDY

leadership roles in professional organizations, and, acclamation by colleagues and students. It is not surprising, therefore, that the classroom of a competent teacher in Uppsala might look a little different from the classroom of a competent teacher in Shanghai or San Diego. The local construction and enactment of competence was one of the most appealing aspects of this study. Greater detail on data generation procedures is provided in the appendix to this book. The generation of data has been completed in Australia, China (Hong Kong, Shanghai and Macau), the Czech Republic, Germany, Israel, Japan, Korea, the Philippines, Singapore, South Africa, Sweden, and the USA. The teacher and student interviews offer insight into both the teacher’s and the students’ participation in (and reconstruction of) particular lesson events and the significance and meaning that the students associated with their actions and those of the teacher and their classmates. Erickson (2006) distinguishes three approaches to the analysis of video: – Whole-to-part, inductive approaches, which he associates with context analysis, ethnographic/sociolinguistic discourse analysis and conversational analysis; – Part-to-whole, deductive approaches, related to speech-act analysis, in which instances of research interest are identified within an interactional event and the distribution of these instances displayed for that event; and, – Manifest content approaches, derived from subject matter/pedagogical knowledge, and emphasizing the manifestation of subject matter knowledge by whatever means. To these we want to add a fourth point, "Whole-to-part, abductive approaches," which is similar to Ericksons first point but the analysis proceeds by an iterative process shifting between inductive and deductive steps. All four approaches are in evidence in this book, sometimes in the same chapter. Erickson draws attention to the danger that the latter two of his approaches may “fit so closely with conventional wisdom about manifest curriculum and subject matter pedagogy that they suffer from tunnel vision and from a literalist approach to the way in which meanings are communicated in social interaction” (Erikson, 2006, p. 187). Inevitably, any analysis of classroom video data reflects the interests and theoretical and cultural orientations of the researcher conducting the analysis; cultural orientation offering another aspect of the “tunnel vision” suggested by Erickson. The interrogation of a shared data set by researchers that are differently situated with respect to educational system, theoretical orientation and cultural affiliation provides an excellent safeguard against any researcher’s projection of a pre-existing agenda or value system onto those data going uncontested. Shepard (1991) argued that psychometricians’ test construction and consequent analyses represented the enactment of a specific theory of learning that remained largely unacknowledged and, therefore, uncontested. The combination of perspectives present within the international LPS research team created a form of methodological dialectic that acted to reveal any such implicit frames of reference. The most interesting current work combines serious attention to subject matter and learning with close attention to the behavioural organization of the

9

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK

social interaction, verbal and nonverbal, within which teaching and learning take place, as well as to the ways in which spoken and written discourse in classrooms relate to social and cultural processes in operation across wide spans of time and social space, beyond the walls of the classroom as well as within it (Erickson, 2006, p. 187). While the data analysed and reported here remain situated within the confines of the classroom, we would like to think that the other attributes listed by Erickson constitute a fair description of the work reported in this book considered as a totality, but with the significant omission of the additional dimension offered by the capacity for comparison between classrooms situated in culturally disparate communities. The analysis of video is inevitably enacted as a series of selective choices concerning what to attend to, what distinctions to draw, what patterns to privilege with the status of categories, and what relationships between categories or data types to explore. It is a virtue of video that it permits revisiting, not just for the purposes of secondary analysis, but as part of the inevitable reconsideration, after progressive cycles of code refinement, of data previously coded – revisited both to confirm the resilience of the elaborated code and to encompass the full data set in the most elaborated classification scheme. The cyclic iteration of coding and code revision is undertaken while holding the research purpose firmly in mind. Issues such as inter-rater reliability take on different meanings according to whether or not the intention is to make generalizations about nations or cultures or to construct empirically-grounded explanations of learning processes or instructional practice. In the first TIMSS Video Study, national representative sampling was a priority (Stigler & Hiebert, 1999). An immediate pragmatic consequence was the documentation of single lessons only. The power to make generalizations about national patterns of lesson structure was bought at the cost of explanatory power related to the antecedent and consequent conditions by which the motivations and consequences of teachers’ actions might be understood. Similarly, the researchers’ interest in teaching practice led to an exclusive focus on ‘public talk’ (Kawanaka & Stigler, 1999) at the expense of documenting student ‘private’ collaborative work. In the case of the Learner’s Perspective Study, the pragmatism functioned differently. Because the documentation of “the learner’s perspective” was a priority, a three camera approach was necessary to record student as well as teacher practices, student-student ‘private’ talk had to be documented, and post-lesson interviews utilised to unpack the multiple subjectivities in play in any classroom interaction. Similarly, the decision to document sequences of lessons for a particular class traded representative sampling for the power to study patterns of practice over several lessons and to situate a given teacher or learner action in terms of the events that led to or arose from that action. Recent classroom research (Alton-Lee, Nuthall & Patrick, 1993; Clarke, 2001; Sahlström & Lindblad, 1998), backed by more sophisticated ways of generating and analysing data, has shown that some of the findings of the classroom research classics such as Bellack et al. (1966), Sinclair and Coulthard (1975) and Mehan (1979) are seriously skewed because of technological issues in data generation. In particular, this has concerned the ability to simultaneously record both student and 10

LEARNER’S PERSPECTIVE STUDY

teacher interaction, and the ability to facilitate ways of working with these data. The Learner’s Perspective Study is highly dependent on the recounting of various texts: classroom dialogue (‘public’ and ‘private’); teacher and student written material; and teacher and student interviews. These texts provide the basis from which to consider how the individuals in the classroom are positioned by the discourses in which they participate. It is important, however, to note that the discourse of educational research also acts to position participants in ways that afford and restrict certain interpretations. For example, analyses that attribute characteristics such as interest, motivation or values to individuals require a theory of psychology of the individual, albeit a socially-situated individual, that recognises personal histories and perceptions. Analyses intended to identify patterns of social interaction characteristic of social groups or settings require a theory of social situations in which social events and social structures are the constituent elements, and in which the generation of data on individual subjectivities is subordinated to group behaviours. The theoretical eclecticism that enriched the Learner’s Perceptive Study can be seen in the chapters in this book, which report different analyses constructed upon quite different theoretical foundations. To reiterate the principles on which the methodology is founded: A study of learning in classroom settings is too restricted without the simultaneous documentation of the social and cultural practices in which the learner participated, the instructional materials, physical configuration of the classroom, and other contextual features with which the learner interacted, the teacher actions that preceded and followed the learning under investigation, and the extent to which the practices of others were reflexively related to the learner’s activities and the personal consequences of those activities. Such research requires a methodology that accords value and voice to all participants in the classroom. Such a methodology must document both the practices in which individuals participate and the meanings that individuals associate with those practices. One participates in social practice as a member of a social group, but this membership is a matter of interpretive affiliation by the participating individual. It is an oversimplification to discuss classroom practice as though it were constituted the same for each individual. The nature of an individual’s participation can in itself be seen as an interpretive act. To draw the distinction between social and cognitive processes is not to preclude the influence of one upon the other (in either direction). Cobb (1994) framed the relationship as one of reciprocal contextuality, where the reflexivity between social and cognitive processes can be located in the implicit presence of each perspective in the other. Learning as acculturation via guided participation implicitly assumes an actively constructing child . . . Learning as cognitive self-organization implicitly assumes that the child is participating in cultural practices (Cobb, 1994, p.17). Erickson’s characterization of the classroom was strongly reflexive, “The 11

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK

researcher seeks to understand the ways in which teachers and students, in their actions together, constitute environments for one another” (Erickson, 1986, p. 128). This is highly compatible with the LPS conception of classroom practice as being constituted as a single conjoined constructive process engaged in by teacher and students, experienced and co-constructed by the participants. The publication Scientific Research in Education (National Research Council, 2002) triggered widespread debate concerning the tenets of rigorous, scholarly research in education. Among the principles espoused in that report, were the following: – – – – – –

Pose significant questions that can be investigated empirically; Link research to relevant theory; Use methods that permit direct investigation of the questions; Provide a coherent and explicit chain of reasoning; Yield findings that replicate and generalize across studies; and Disclose research data and methods to enable and encourage professional scrutiny and critique (Feuer, Towne, & Shavelson, 2002).

We would argue that the Learner’s Perspective Study, as reported in this book, conforms to five of these six criteria. In the Learner’s Perspective Study, the study design sought to juxtapose the observable practices of the classroom (documented through videotape and written product) and the meanings attributed to those practices by individual participants (documented through video-stimulated postlesson interviews and questionnaires). The analysis of such data takes the form of inter-textual analysis identifying linkages and tensions between forms of classroom text. The analyses reported in this book examine the similarities and differences in practice across many lessons documented in well-taught classrooms in a variety of countries and a variety of cultures. The fifth point above, relating to replicability and generalisability, is somewhat problematic in the context of a multi-cultural study of the scale reported here. However, with a more contemporary perspective on generalisability, where the locus of generalization is determined by the reader of the accounts rather then the author of the accounts (e.g. Eisner, 1991), we argue that the network of interrelated data and complementary accounts accumulated in relation to any one lesson or even any one lesson event accords the data a degree of trustworthiness that transcends mere replicability. Furthermore, providing the community of teachers internationally with multi-faceted portrayals of some of the ways in which their competent colleagues elsewhere construct their practice has the potential to expand the repertoire of mathematics teachers internationally. Educational consequences of this type, coupled with the capacity to advance theory in both learning and instruction, offer a form of fruitful inquiry that transforms older ideas of generalisability into more useful conceptions of consequence, utility, impact, adaptability, and the reconstruction of knowledge. COMPLEMENTARITY AS ESSENTIAL

The distinguishing characteristic of the research design for the Learner’s 12

LEARNER’S PERSPECTIVE STUDY

Perspective Study is the inclusion of four levels of complementary accounts: (a) At the level of data, the accounts of the various classroom participants are juxtaposed; (b) At the level of primary interpretation, complementary interpretations are developed by the research team from the various data sources related to particular incidents, settings, or individuals; (c) At the level of theoretical framework, complementary analyses are generated from a common data set through the application by different members of the research team of distinct analytical frameworks; and (d) At the level of culture, complementary characterizations of practice and meaning are constructed for the classrooms in each culture (and by the researchers from each culture) and these characterizations can then be compared and any similarities or differences identified for further analysis, particularly from the perspective of potential cross-cultural transfer. Complementarity Between Participant Accounts: Establishing the Co-Construction of Classroom Practice Like Wenger (1998), Clarke’s (2004) analysis of patterns of participation in classroom settings stressed the multiplicity and overlapping character of communities of practice and the role of the individual in contributing to the practice of a community (the class). Clarke (2001) has discussed the acts of interpretive affiliation, whereby the learners align themselves with various communities of practice and construct their participation and ultimately their practice through a customizing process in which their inclinations and capabilities are expressed within the constraints and affordances of the social situation and the overlapping communities that compete for the learner’s allegiance and participation. By examining sequences of ten lessons, the Learner’s Perspective Study provides data on the teacher’s and learners’ participation in the coconstruction of the possible forms of participation through which classroom practice is constituted (cf. Brousseau, 1986). An example of utilizing the complementarity of teacher and student accounts can be found in several studies drawing on LPS-data (e.g. Emanuelsson & Sahlström, in press; Clarke, 2004). Clarke (2004) examined the legitimacy of the characterisation of kikan-shido (Between-Desks-Instruction) as a whole class pattern of participation, and situated the actions of teacher and learners in relation to this pattern of participation. By drawing on classroom video evidence and juxtaposing teacher and student interview data, it is possible to demonstrate that while engaging in kikan-shido, the teacher and the students participate in actions that are mutually constraining and affording, and that the resultant pattern of participation can only be understood through consideration of the actions of all participants. A key characteristic of kikan-shido, as it was practiced in the Australian LPS classrooms, was the implicit devolution of the responsibility for knowledge generation from the teacher to the student, while still institutionalizing the teacher’s obligation to scaffold the process of knowledge generation being enacted by the students. Comparison with the enactment of kikan-shido in other classrooms (Hong Kong, Shanghai, and San Diego, for example) provides significant insight into the pedagogical principles 13

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK

underlying the practices of different classrooms internationally. This example is discussed in much greater detail in chapter four of this book. Complementarity Between LPS Researcher Accounts: A More Comprehensive Portrayal of Classroom Practice Classrooms are complex social settings, and research that seeks to understand the learning that occurs in such settings must reflect and accommodate that complexity. This accommodation can occur if the data construction process generates a sufficiently rich data set. Such a data set can be adequately exploited only to the extent that the research design employs analytical techniques sensitive to the multifaceted and multiply-connected nature of the data. We need to acknowledge the multiple potential meanings of the situations we are studying by deliberately giving voice to many of these meanings through accounts both from participants and from a variety of “readers” of those situations. The implementation of this approach requires the rejection of consensus and convergence as options for the synthesis of these accounts, and instead accords the accounts “complementary” status, subject to the requirement that they be consistent with the data from which they are derived, but not necessarily consistent with each other, since no object or situation, when viewed from different perspectives, necessarily appears the same (Clarke, 2001, p. 1). In the LPS project, multiple, simultaneous analyses are being undertaken of the accumulated international data set from a variety of analytical perspectives. For example, while Ference Marton and his colleagues in Sweden and Hong Kong analyse the practices of classrooms in Shanghai informed by the perspective of Marton’s Theory of Variation, Clarke and his co-workers in Melbourne are undertaking analysis of the same lessons in relation to the Distribution of the Responsibility for Knowledge Generation. These two analytical approaches do not appeal to the same theoretical premises, but nor are they necessarily in conflict. They represent complementary analyses of a common body of data, aspiring to advance different theoretical perspectives and to inform practice in different ways. Complementarity Between Project Accounts: Approaches to Studying Lesson Structure Lesson structure can be interpreted in at least three ways (see Chapter 2): • At the level of the whole lesson - regularity in the presence and sequence of instructional units of which lessons are composed; • At the level of the topic – regularity in the occurrence of lesson elements at points in the instructional sequence associated with a curriculum topic, typically lasting several lessons; • At the level of the constituent lesson events – regularity in the form and function of types of lesson events from which lessons are constituted. 14

LEARNER’S PERSPECTIVE STUDY

A research design predicated on a nationally representative sampling of individual lessons, as in the TIMSS Video Studies (1995 and 1999), inevitably reports a statistically-based characterization of the representative lesson (the first of the alternatives listed above). The characterisation of the practices of a nation’s or a culture’s mathematics classrooms with a single lesson pattern was problematised by the preliminary results of the Learner’s Perspective Study (Clarke & Mesiti, 2003) and this is discussed in greater detail in the next chapter. Nonetheless, the TIMSS video study data offers the opportunity to estimate the prevalence of a particular activity type identified as significant from LPS data. Similarly, activities identified in the TIMSS project as prevalent within a particular country can be evaluated from within the LPS data in relation to their capacity to stimulate specific responses in students, particularly learning outcomes. The complementarity of these two projects is acknowledged and valued by both research groups. Complementarity of accounts is an essential methodological and theoretical stance, adopted by the Learner’s Perspective Study, for the explication of mathematics teaching and learning in classroom settings, the advancement of theories relating to such settings, and the informing of practice in mathematics classrooms. THE CONSTRUCTION OF MEANING IN MATHEMATICS CLASSROOMS

Research reports and the sharing of practices among international counterparts always stimulate reflection and thoughts for improving learning and teaching in one's own country. In a country such as China, there are reports of the introduction of pedagogical ideas originating in other cultures and associated with a greater emphasis on the process of mathematics learning and an interactive atmosphere in the classroom (e.g. Mok & Morris, 2001; Zhang & Dai, 2004; and Mok, 2005). In many countries all over the world attempts to improve the teaching and learning of mathematics have been centered on the identification of standards for curriculum content and student performance. Success will not occur merely by setting standards and holding teachers accountable for their achievement. Equally, the descriptive categorization of teacher practice (as undertaken in both the TIMSS video study and the Learner’s Perspective Study) is an important preliminary to an understanding of how classrooms function, but it is not an end in itself. Lacking evidence that might associate observed teacher practice with consequent student behaviors and learning, we are ill-equipped to advocate any particular set of practices as optimal. It is essential that research address the processes leading to learning in classroom settings. Without an understanding of these processes, attempts to improve teaching practices and learning outcomes in mathematics classrooms have little chance of success. The need to improve the quality of process as an essential precursor to the improvement of product is well understood in most other professional and industrial fields. The same principle needs to guide practices in education. The improvement of mathematics teaching must be founded

15

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK

upon an understanding of both teaching and learning and the relationship of both activities to student achievement. Much of our theorizing on learning in classroom settings has centered on the negotiation of meaning (Clarke, 1996; Cobb & Bauersfeld, 1995). These meanings are not restricted to content-specific meanings but include the negotiation of the social meanings by which the practices of the classroom are constituted and enacted (Yackel & Cobb, 1993). Further, it has been argued that every aspect of classroom experience is constructed by the participants, including both the situation or classroom context as well as any contexts invoked by the tasks employed in instruction (Clarke & Helme, 1994, 1998). In the Learner’s Perspective Study, the identification of such constructed or construed meanings was a major priority. It is axiomatic that such meanings will be culturally-derived in their construction and in their interpretation by the different researchers carrying out the analyses. The participation of an international research team in the collaborative analysis of a combined data pool provides an essential informing tension between the insights and assumptions offered by insider versus outsider knowledge. The first TIMSS Video Study (Stigler & Hiebert, 1997, 1999) identified teaching as a cultural activity and sought to describe the characteristics of that activity in classrooms in Japan, Germany and the USA. One of the virtues of such international comparative research lies in its capacity to call into question practices that are so culturally ingrained as to be virtually invisible. The difficulty of such studies is the questionable legitimacy of any comparisons across cultures. If teaching is conceived in culturally specific terms, then a research strategy is required which documents local practice in a form that permits legitimate comparison. A standard approach is to ask teachers, via a questionnaire, to describe their instructional practices. This approach is fraught with difficulties even within a national sample, where terms such as "problem solving" are used in very different senses (Barnes, Clarke & Stephens, 2000; Clarke & Stephens, 1996). This lack of a shared language of teaching practice is compounded in a cross-cultural study based on questionnaire data. The responses are nearly impossible to interpret and legitimate comparisons cannot be made. Videotaping offers a form of cross-cultural documentation that, however selective, is at least partially true to the original classroom and amenable to analysis within a single coherent framework and within multiple complementary frameworks. The use of videotape in international comparative classroom research is a comparatively recent innovation, and open to the same criticisms applicable to any study in which researchers embedded in one culture interpret the practices occurring in another. Yet a common language of analysis must be found if any form of comparison is to be made. If the interpretation and classification of classroom events in each country is undertaken with the collaboration of "local" researchers then the resultant characterization is likely to be both true to the original culture and in a form that permits legitimate comparative analysis. Equally, the perspective on local practice offered by outsiders brings its own benefits and insights. 16

LEARNER’S PERSPECTIVE STUDY

Classrooms have been the subject of research for some time. Fine-grained analyses of classroom videotape data are increasingly present in the literature, but these have typically not been cross-national studies, in which the sheer scale of the samples precluded such detailed analysis. In this growing tradition of microanalysis of classroom practices, Cobb and Bauersfeld (1995), for example, have described the "culture of the classroom" in the course of an investigation of "the emergence of mathematical meaning" in one second-grade classroom. In contrast, the first TIMSS Video Study utilized a random sub-sample of the full TIMSS student achievement sample (Lokan, Ford and Greenwood, 1996). The final video sample included 231 classrooms: 100 in Germany, 50 in Japan, and 81 in the United States (Stigler & Hiebert, 1997, p.4). Data generation and analysis procedures were developed and refined by a collaborative team of researchers from each of the three countries in the study. "Only one camera was used in each classroom, and it focused on what an ideal student would be focusing on - usually the teacher" (Stigler & Hiebert, 1997, p. 5). The videotape record was supplemented by a teacher questionnaire describing such things as the goal of the lesson, its place within the current sequence of lessons, and how typical the lesson was. In the Learner’s Perspective Study, video data was much more comprehensive, permitting more fine-grained analyses, and was augmented by post-lesson interviews with students in which the students controlled the video replay and identified classroom events of personal significance, discussing these events in detail (see appendix). Entry points for analysis In our research, members of the research team interpret the data, using particular forms of analysis, consistent with the researcher’s interests and theoretical positioning. These interpretations are our “readings” of the accounts offered by the video cameras and the transcribed reconstructive accounts obtained from classroom participants. These accounts are the evidence on which our various readings are founded. Any number of interpretations, guided by any number of interests, can be built on the same foundation of evidence; but an interpretation ignoring that evidence can never be a defensible one (Vendler, 1997, p. 24). The authors of the chapters of this book have constructed different interpretations based on their analyses of various subsets of a common body of data. Helen Vendler, quoted above, is introducing her interpretations of Shakespeare’s sonnets. In her introduction, Vendler is critical of the type of literary critic who “leaves it up to the reader to construct the poem” and states her goal explicitly: “I have hoped to help the reader actively to that construction by laying out evidence that no interpretation can ignore” (Vendler, 1997, p. 24). Extrapolating this to the context of educational research, if our data generation is to anticipate multiple analyses, then there is a heightened obligation to provide a detailed account of the data

17

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK

generation, and equally to provide a detailed description and justification of the method of analysis. This transparency of data generation and analysis is all the more imperative because the two processes were not necessarily conceived together and cannot be used for mutual justification in the conventional manner. Differences between the documented practices recorded on videotape and the participants’ discrepant accounts of those practices emerged through the juxtaposition of video and interview data. Williams and Clarke (2002) have explored some of the issues related to the interpretation of video and interview data, particularly in situations where the two data sources suggest discrepant interpretations. Williams and Clarke (2002) used data from the Learner’s Perspective Study to develop independent accounts of one student’s classroom practice and associated learning during a single lesson using video and interview data separately. The resultant accounts were inconsistent in several places, but some convergence of interpretation was possible through their juxtaposition. On the other hand, some differences were irreconcilable on the basis of one lesson’s data, but could be resolved with recourse to the lessons preceding and following the incidents in question. At the heart of the synthesis of data-level accounts are the questions, “Whose perspective is being documented?” and “Whose practice do we seek to understand?” One of the propositions that occupies the first half of this book concerns the viability of Lesson Events as an entry point for data analysis. Chapters 2 through 7, outline the reasoning underlying this approach and then report the results of its application. Adopting Lesson Events as the entry point for analysis leaves open the possibility of scaling the analysis up (to the level of topic) or down (to the level of utterance or negotiative event (Clarke, 2001)), depending on the research focus. The remaining chapters 8, 9, 10, and 11, report comparative analyses of data from the classrooms of two or more countries, with a variety of foci. In chapter 8, Liljestrand and Runesson address the possibility of tension between mathematical and everyday purposes. Häggstrom, in chapter 9, compares the introduction of linear equations in classrooms in Uppsala, Shanghai and Hong Kong. Khuzwayo compares teacher beliefs in South Africa, Australia and the USA, with respect to perceptions of constraints on teaching practice, and Clarke, in chapter 11, uses the combined international data set from six countries to interrogate assumptions about classroom practice. The essential characteristic of this study of mathematics classrooms is the commitment to an integrative approach. Application of the data generation and analytical procedures developed in an Australian study (Clarke, 2001) to the study of classrooms in very different cultures anticipated a much more global examination of practice than that possible with data grounded in a single culture. The commitment to examining the interdependence of teaching and learning as related activities within an integrated body of classroom practice accepted an obligation to document (and analyse) relationships between participants’ practices as well as the occurrence of the individual practices themselves. The importance attached to the meanings that participants attributed to their actions and the actions of others and to the mathematical and social meanings that are the major products 18

LEARNER’S PERSPECTIVE STUDY

of the classroom required a methodology able to access participants’ accounts of those meanings and to integrate these within a coherent picture of the classroom. It has been an exciting exploration that is still underway. We are very pleased to be able to share some of the findings with you. REFERENCES Alton-Lee, A., Nuthall, G., & Patrick, J. (1993). Reframing classroom research: A lesson from the private world of children. Harvard Educational Review, 63(1), 50-84. Anderson, L. W., Ryan, D., & Shapiro, B. (1989). The IEA classroom environment study. NY: Pergamon. Anderson-Levitt, K. M. (2002). Teaching culture as national and transnational: A response to teachers’ work. Educational Researcher 31(3), 19-21. Atkin, J. M., & Black, P. (1996). Changing the subject: Innovations in science, mathematics, and technology education. London and Paris: Routledge and the OECD (cited in Atkin & Black, 1997). Atkin, J. M., & Black, P. (1997). Policy perils of international comparisons. Phi Delta Kappa 79(1), 22-28. Bellack, A. A., Kliebard, H. M., Hyman, R. T., & Smith, F. L. (1966). The language of the classroom. New York: Teachers College Press. Barnes, M., Clarke, D. J., & Stephens, W. M. (2000). Assessment as the engine of systemic reform. Journal of Curriculum Studies, 32(5), 623-650. Brousseau, G. (1986). Fondements et methodes de la didactique des mathematiques. Recherches en didactique des mathematiques, 7(2), 33-115. Clarke, D. J. (1996). Refraction and reflection: Modelling the classroom negotiation of meaning. RefLecT, 2(1), 46–51. Clarke, D. J. (1998). Studying the classroom negotiation of meaning: Complementary accounts methodology. In A. Teppo (Ed.), Qualitative research methods in mathematics education. Journal for Research in Mathematics Education, Monograph No. 9 (pp. 98-111). Reston, VA: NCTM. Clarke, D. J. (Ed.). (2001). Perspectives on practice and meaning in mathematics and science classrooms. Dordrecht, Netherlands: Kluwer Academic Press. Clarke, D. J. (2003). International comparative studies in mathematics education. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second International Handbook of Mathematics Education (pp. 145-186). Dordrecht: Kluwer Academic Publishers. Clarke, D. J. (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. Clarke, D. J. & Helme, S. (1994). The role of context in mathematical activity. In D. Kirschner (Ed.), Proceedings of the Sixteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 239-246). Baton Rouge, Louisiana: Louisiana State University. Clarke, D. J. & Helme, S. (1998). Context as construction. In O. Bjorkqvist (Ed.), Mathematics teaching from a constructivist point of view (pp. 129-147). Vasa, Finland: Faculty of Education, Abo Akademi University. Clarke, D. J., Keitel, C., & Shimizu, Y. (2006). Mathematics classrooms in twelve countries: The insider’s perspective. Rotterdam: Sense Publications. Clarke, D. J., & Mesiti, C. (2003). Addressing the challenge of legitimate international comparisons: Lesson structure in Australia and the USA. In L. Bragg, C. Campbell, G. Herbert, & J. Mousley (Eds.), Mathematics Education Research: Innovation, Networking, Opportunity, Proceedings of the 26th Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 230-237). Geelong, Australia: Deakin University.

19

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK Clarke, D. J., & Stephens, M. (1996). The ripple effect: The instructional impact of the systemic introduction of performance assessment in mathematics. In M. Birenbaum, & F. J. R. C. Dochy (Eds.), Alternatives in assessment of achievements, learning processes and prior knowledge (pp. 63-92). Boston, MA: Kluwer. Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13-20. Cobb, P., & Bauersfeld, H. (Eds.). (1995). The emergence of mathematical meaning: Interaction in classroom cultures. Hillsdale, NJ: Lawrence Erlbaum. Eisner, E. (1991). The enlightened eye. Qualitative inquiry and the enhancement of educational practice. New York: McMillan. Emanuelsson, J. & Clarke, D. (coordinators) (2004). Research Forum 04: Contrasting comparative research on teaching and learning in mathematics. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 197-226). Bergen: Bergen University College. Emanuelsson, J., & Sahlström, F. (in press). The price of participation – how interaction constrains and affords classroom learning of mathematics. Scandinavian Journal for Education. Erickson, F. (1986). Qualitative methods in research on teaching. In M. C. Wittrock (Ed.), The rd handbook of research on teaching (3 edition) (pp. 119-161). New York: Macmillan. Erickson, F. (2006). Definition and analysis of data from videotape: Some research procedures and their rationales. In J. L. Green, G. Camilli, & P. B. Ellmore (Eds.), Handbook of Complementary Methods in Education Research (pp. 177-192). Mahwah, NJ: Lawrence Erlbaum. Feuer, M. J., Towne, L., & Shavelson, R. J. (2002). Scientific culture and educational research. Educational Researcher, 31(8), 4-14. Fernandez, C., & Yoshida, M. (2004). Lesson study: A Japanese approach to improving mathematics teaching and learning. Mahway, NJ: Lawrence Erlbaum Associates. Hacking, I. (1999). The social construction of what? Cambridge, MA: Harvard University Press. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K., Hollingsworth, H., Jacobs, J., Chui, A., Wearne, D., Smith, M., Kersting, N., Manaster, A., Tseng, E., Etterbeck, W., Manaster, C., Gonzales, P., & Stigler, J. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study. Washington, DC: U.S. Department of Education, National Center for Education Statistics. Kawanaka, T., & Stigler, J. W. (1999). Teachers’ use of questions in eighth-grade mathematics classrooms in Germany, Japan and the United States. Mathematical Thinking and Learning, 1(4), 255-278. Keitel, C., & Kilpatrick, J. (1999). The rationality and irrationality of international comparative studies. In G. Kaiser, E. Luna, & I. Huntley (Eds.), International comparisons in mathematics education (pp. 241-256). London: Falmer Press. Lerman, S. (Ed.). (1994). Cultural perspectives on the mathematics classroom. Dordrecht: Kluwer. LeTendre, G., Baker, D., Akiba, M., Goesling, B., & Wiseman, A. (2001). Teachers’ work: Institutional isomorphism and cultural variation in the US, Germany, and Japan. Educational Researcher, 30(6), 3-15. Lokan, J., Ford, P., & Greenwood, L. (1996). Maths & science on the line: Australian junior secondary students’ performance in the Third International Mathematics and Science Study. Melbourne: ACER. Mehan, H. (1979). Learning lessons: Social organization in the classroom. Cambridge, MA: Harvard University Press. Mok, I. A. C. (2005, July). How Chinese learn mathematics – Lessons from Shanghai. Invited plenary. Mathematics Education Conference, Hong Kong Institute of Education, 6-8 July. Proceedings, pp. 25-34. Mok, I. A. C., & Morris, P. (2001). The metamorphosis of the ‘Virtuoso’: Pedagogic patterns in Hong Kong primary mathematics classrooms. Teaching and Teacher Education: An International Journal of Research and Studies, 17(4), 455-468.

20

LEARNER’S PERSPECTIVE STUDY Mullis, I., Martin, M., Gonzalez, E., Gregory, K., Garden, R., O’Connor, K., Chrostowski, S., & Smith, T. (2000). TIMSS 1999 International mathematics report. Findings from IEA’s repeat of the Third International Mathematics and Science Study at the eighth Grade. Boston, Boston College: International Study Center. Sahlström, F. & Lindblad, S. (1998). Subtexts in the science classroom - an exploration of the social construction of science lessons and school careers, Learning and Instruction, 8(3), 195-214. Schmidt, W. H., McKnight, C. C., Valverde, G. A, Houang, R. T., & Wiley, D. E. (1997). Many Visions, Many Aims Volume 1: A Cross-National Investigation of Curricular Intentions in School Mathematics. Dordrecht: Kluwer. Shavelson, R. J., & Towne, L. (Eds.). (2002). Scientific research in education. Washington, DC: National Research Council, National Academy Press. Shepard, L. A. (1991). Psychometrician’s beliefs about learning. Educational Researcher, 20(6), 216. Sinclair, J., & Coulthard, M. (1975). Towards an analysis of discourse. London. Oxford University press. Stigler, J., & Hiebert, J. (1997). Understanding and improving classroom mathematics instruction: An overview of the TIMSS video study. Phi Delta Kappan, 79(1), 14-21. Stigler, J., & Hiebert, J. (1999). The teaching gap. New York: Free Press. The Design-Based Research Collective. (2003). Design-based research: An emerging paradigm. Educational Researcher, 32(1), 5-8. Thorsten, M. (2000). Once upon a TIMSS: American and Japanese narrations of the Third International Mathematics and Science Study. Education and Society, 18(3), 45-76. Vendler, H. (1997). The art of Shakespeare’s sonnets. Cambridge, Mass: Harvard University Press. Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge: Cambridge University Press. Williams, G., & Clarke, D. J. (2002). The contribution of student voice in classroom research: A case study. In C. Malcolm & C. Lubisi (Eds.), Proceedings of the tenth annual meeting of the Southern African Association for Research in Mathematics, Science and Technology Education (Part III, pp. 398-404). Durban: University of Natal. Yackel, E., & Cobb, P. (1993). Sociomathematical norms, argumentation and autonomy in mathematics. Paper presented at the 1993 Annual Meeting of the American Educational Research Association, Atlanta, Georgia, USA. Zhang, D., & Dai, Z. (2004, July). “Two basics”: Mathematics teaching approach and open-ended problem-solving in China. Regular lecture in the 10th International Congress on Mathematics Education. July 4-11, Copenhagen, Denmark.

David Clarke International Centre for Classroom Research Faculty of Education University of Melbourne Australia Jonas Emanuelsson Department of Education Göteborgs Universitet Sweden Eva Jablonka Fachbereich Erziehungswissenschaft und Psychologie Freie Universität Berlin

21

DAVID CLARKE, JONAS EMANUELSSON, EVA JABLONKA AND IDA AH CHEE MOK

Germany Ida Ah Chee Mok Faculty of Education University of Hong Kong Hong Kong SAR, China

22

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

CHAPTER TWO Addressing the Challenge of Legitimate International Comparisons: Lesson Structure in the USA, Germany and Japan

INTRODUCTION

One of the major challenges confronting the international mathematics education community is how best to learn from each other’s classroom practices. Central to this issue is the choice of the instructional unit that will serve as the basis for any cross-cultural analysis of classroom practice. In most, probably all, countries, students interact with mathematics content via the instructional unit of the lesson. The lesson, therefore, seems a sensible place to start in the search for a viable unit of international comparative analysis of classroom practice. In this chapter, analyses of lesson structure from each of the USA, Germany and Japan are reported. These reports are based on analyses of sequences of ten lessons, documented using three video cameras, and interpreted through the reconstructive accounts of classroom participants obtained in post-lesson video-stimulated interviews. The methodological approach of conducting case studies of the classroom practices over sequences of at least ten lessons in the classes of several competent eighth grade teachers in each of the participating countries offers an informative complement to the survey-style approach of the two video studies carried out by the Third International Mathematics and Science Study (TIMSS) (Hiebert et al., 2003; Stigler & Hiebert, 1999). Perhaps it is inevitable that a research design predicated on a nationally representative sampling of individual lessons, as in TIMSS, should report a statistically-based characterization of the representative lesson. A more fine-grained study of sequences of ten lessons, informed by the reconstructive accounts of the participants, has the potential to address questions such as: – What are the recurrent pedagogical elements that might typify a teacher’s classroom practice and is there evidence of a recurrent lesson structure or sequence of such elements within the practices of a single teacher or group of teachers? – What degree of variation in lesson structure is evident in the practices of the competent teachers studied in the USA, Germany and Japan?

D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 23–45. © 2006 Sense Publishers. All rights reserved.

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

The analyses reported in this chapter reveal significant structural variation in the different lessons in any one teacher’s lesson sequence. This degree of structural variation suggests that a single lesson pattern is unlikely to be an accurate or a useful representation of either an individual teacher’s lessons or of any nationallyrepresentative sample of lessons. However, the recurrence of particular lesson components in the practices of teachers participating in the same or similar school systems suggests that some form of typification may be possible, given the correct unit of comparison. The chapter concludes with the suggestion that the comparison of lesson components (‘lesson events’) is more likely to be helpful than a lesson pattern or script as a guide to the similarities and differences between the practices of different mathematics teachers and their classrooms. Meaningful International Comparisons What are the contending bases for international comparative research and how is lesson structure situated within the logic of this research? One of the most widely reported results from studies of international assessment of student achievement such as the Third International Mathematics and Science Study (TIMSS) (Beaton & Robitaille, 1999) has been the high national mean scores for students from ‘Asian’ countries. This appears to have triggered the following (naïve) line of reasoning: If Asian countries are consistently successful on international measures of mathematics performance, then less-successful non-Asian countries would do well to adapt for their use the instructional practices of Asian classrooms. Such a line of reasoning is grounded in four key assumptions: i that the term ‘Asian’ identifies a coherent cultural conglomerate with respect to educational practice; ii that the performances valued in international tests constitute an adequate model of mathematics, appropriate to the needs of the less-successful country; iii that differences in mathematical performance are attributable to differences in instructional practice, such as lesson structure (and not to other differences in culture, societal affluence or aspiration, or curriculum); and iv that the distinctive instructional practices of more-successful countries (e.g., norms of lesson structure), should these exist, can be meaningfully adapted for use by less-successful countries. Each of these key assumptions can be problematised on a variety of grounds (e.g., Clarke, 2003; Westbury, 1992). Such cross-cultural comparisons can also be undertaken in a more introspective manner by individual countries. Wang (2001), in discussing technical concerns with TIMSS, cites Hu (2000, p. 8) as saying, “This study does not break down Americans by race, if they did, Asian Americans would likely score as high as Asians in their home countries, and Whites would rank near top of the European nations.” There are several ways to interpret this observation. It is worth comparing this quote with an analogous statement from Berliner (2001). Which America are we talking about? . . . Average scores mislead completely in a country as heterogeneous as ours . . . The TIMSS-R tells us just what is

24

LESSON STRUCTURE IN THE USA, GERMANY AND JAPAN

happening. In science, for the items common to both the TIMSS and the TIMSS-R, the scores of white students in the United States were exceeded by only three other nations. But black American school children were beaten by every single nation, and Hispanic kids were beaten by all but two nations. A similar pattern was true of mathematics scores . . . The true message of the TIMSS-R and other international assessments is that the United States will not improve in international standings until our terrible inequalities are fixed (David Berliner, Washington Post, Sunday, January 28, 2001). Hu’s statement explicitly partitions the American population by race and makes comparisons with the performance of corresponding groups internationally. Berliner also partitions the population of US school students along racial lines and locates each sub-population on an international league table of student achievement. Similar partitioning along socio-economic or gender lines is also possible. In his 2005 address to the annual conference of the American Educational Research Association, Berliner pointed to the connection in the United States between race, socio-economic status and educational access and participation. The essential point that has been made by Clarke (2003) and others is that nationally aggregated data can conceal important differences in educational outcomes, reducing the explanatory potential of international studies and, possibly, producing misleading or erroneous recommendations for the future deployment of (limited) educational resources. From several perspectives the comparison of national means of student achievement is problematic. Comparisons between sectors of the community within a given country may be more fruitful, even more so within a given state or school system. Such comparisons may at least highlight community groups who are less equal in the benefits they accrue from a school system intended to benefit all students equally. Educational policy can then be framed to address any inequalities. But, what are the implications from the perspective of cultural traditions? Wang and Lin (2005) reviewed the research literature with respect to the mathematical performance of Chinese, Chinese-American and other US student groups. Their review problematised “ambiguous cross-national categorizations of East Asian students from Japan, China, Korea, and other East Asian regions and countries” (Wang & Lin, 2005, p. 4). The extent to which such culturally-inclusive categorizations can imply possibly misleading similarities can be seen in the accounts of classroom practice provided by Clarke, Keitel, Shimizu and their colleagues in the Learner’s Perspective Study (LPS) (Clarke, Keitel & Shimizu, 2006). Elsewhere, Wang and Lin made the point “Although Chinese students showed superiority to U.S. students in symbolic and abstract thinking, Chinese students show no advantage in graphing, understanding tables, or open-process problem solving” (Wang & Lin, 2005, p. 5). This latter statement emphasizes the dangers of over-aggregation for the purpose of cross-cultural comparison, but in this case in relation to the specific mathematical content. Wang and Lin (2005) note that while there does appear to be a “widening gap between Chinese and U.S. students” (p. 5), “the performance gap between Chinese 25

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

Americans and Caucasian Americans also increases as both groups move through U.S. schools” (p. 5). Most importantly, Wang and Lin conclude “whether Chinese students actually outperform Chinese American students is still unresolved” (Wang & Lin, 2005, p. 5). All of which suggests that the cultural affiliation of the learner (whatever their geographical location) is possibly as important as the cultural alignment of the school or school system and certainly should not be simplistically identified with nationality. The previous remarks are not intended to challenge the premise that school systems enact cultural values. However, they do challenge the simplistic identification of culture with nationality. Once the identification (confusion) of nation with culture has been problematised, then the utility of international comparative research can be considered with greater cultural sensitivity. Studying Lesson Structure The analysis of video data collected in the video component of TIMSS, as reported by Stigler and Hiebert (1999), centred on the proposition that the teaching practice of a nation (at least in the case of mathematics) could be explained to a significant extent by the teacher’s adherence to a culturally-based “teacher script.” Central to the identification of these cultural scripts for teaching were the Lesson Patterns reported by Stigler and Hiebert (1999) for Germany, Japan and the USA. The contention of Stigler and Hiebert was that at the level of the lesson, teaching in each of the three countries could be described by a “simple, common pattern” (Stigler & Hiebert, 1999, p. 82). By contrast, the Learner’s Perspective Study analysed sequences of ten lessons, documented using three video cameras, and supplemented by the reconstructive accounts of classroom participants obtained in post-lesson video-stimulated interviews. A fine-grained study of sequences of ten lessons, informed by the reconstructive accounts of the participants, has the potential to identify any recurrent pedagogical elements in a teacher’s classroom practice and any evidence of regularity in the sequencing of those elements. Such regularities and recurrent elements have the potential to serve as the basis for comparative analysis. Lesson structure can be interpreted in three senses: i. At the level of the whole lesson – regularity in the presence and sequence of instructional units of which lessons are composed; ii. At the level of the topic – regularity in the occurrence of lesson elements at points in the instructional sequence associated with a curriculum topic, typically lasting several lessons; iii. At the level of the constituent lesson events – regularity in the form and function of types of lesson events from which lessons are constituted. In terms of international comparison, it is useful to consider which of these three forms of lesson structure are likely to prove useful as units of comparative analysis. In this regard, it is important to recognize that the most appropriate unit for national typification may not prove useful for international comparison. In terms of national typification, we need to address the question: Is a nation’s or a culture’s 26

LESSON STRUCTURE IN THE USA, GERMANY AND JAPAN

classroom practice most usefully characterized at the level of the whole lesson, in the manner in which a topic is constructed, delivered and experienced, or in the form and function of the specific activities from which lessons are composed? The same three alternatives are available for the purposes of international comparison, but the optimal unit of international comparison need not be the same as the optimal unit for national typification. We can conceive of the possibility of an idiosyncratic practice that might typify the classrooms of a nation, but be so unusual as to not constitute a legitimate basis for international comparison. There are two quite distinct methodological alternatives: Alternative 1. If two groups of objects are to be compared, one approach is to consider these two questions: i. Difference – What is the characteristic about which the comparison is to be made? ii. Similarity – How might each group of objects be separately typified with respect to that characteristic? The international comparison of national norms of student achievement could be described as conforming to this approach, mediated by the test instrument employed. As Keitel and Kilpatrick (1999) have pointed out, such a test is the implicit embodiment of an idealised international curriculum taken as common across cultures and school systems. The order in which the two previous questions are posed is a major methodological signature. Alternative 2. If two groups of objects are to be compared, consider these two questions: i. Similarity – Which characteristics appear to typify this collection of objects? ii. Difference – What comparisons can be made between these two groups of objects using the identified characteristics? Posing the questions as in Alternative 2 reduces the danger of constraining the data to a predetermined structure, but may lead to the typification of the two groups by different emergent characteristics, restricting the common bases on which comparison of the two groups might be made. It should be noted also that Alternative 2 assumes a domain within which comparison is sought, such as classroom practice or curricular policy. In terms of lesson structure, it might be that for one nation or culture there is no nationally characteristic structure to the lesson as a whole, but that particular types of idiosyncratic lesson events offer the most appropriate typification. For another nation or culture, there could be a high degree of regularity to the composition of lessons, or in the sequencing of particular types of instructional activity in the delivery of a topic. Such differences in the form of typification provide a basis for international comparison that reflects something more essential to each than the identification (imposition) of the same structural level as the basis for the comparison. The choice of Alternative 1 makes the basis for comparison a matter of prescription based on either theory or on the prevailing educational priorities of the country conducting the study. Choice of Alternative 2 makes the identification of possible bases for comparison an empirical result of the research. 27

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

Incommensurability of the emergent typifications becomes relevant if the comparison is intended to be evaluative. However, the identification of idiosyncratic practices, absent entirely in some classrooms, offers the teachers of those classrooms entirely new pedagogical tools, potentially valuable, since they derive from the practices of competent teachers elsewhere. Lesson Patterns In the writings of Stigler, Hiebert and their co-workers, we find an interesting shift from discussion (and advocacy) of “lesson scripts” (Stigler & Hiebert, 1998) to “lesson patterns” (Stigler & Hiebert, 1999) and via “hypothesised country models” to “lesson signatures” (Hiebert et al., 2003) as the means by which the classroom practices of countries might be usefully compared. This trend signifies an increasing recognition that meaningful comparison of teaching practice across an international sample requires a multi-dimensional framework and a greater sensitivity to variation than is possible within the confines of a ‘lesson script.’ Givvin et al. (2005) used the TIMSS-R video data to revisit the question: Are there national patterns of teaching? Their approach to the question was handicapped by four key simplifications: i. Lesson Location – The significance of the location of the lesson within the instructional (topic) sequence as a source of variability was never addressed, either in their analysis nor in the subsequent discussion; ii. Lesson as Unit of Analysis – The possibility that such teaching patterns might be manifest at a level of an instructional unit other than the lesson was not addressed; iii. Category Independence – The three dimensions (Purpose, Classroom Interaction and Content Activity)i are not independent. We would further suggest that it is in the analysis of their interaction and consideration of the participants’ intentions and interpretations that we are most likely to gain insight into the origins of each teacher’s lesson structure and the underlying pedagogical principles. iv. Over-inclusive Codes – The three dimensions (Purpose, Classroom Interaction and Content Activity) on which the comparative analysis is undertaken were defined in extremely simplistic terms. This had the effect of maximizing the possibility of cross-classroom application of the coding scheme and minimizing variability through lack of sensitivity of the coding scheme to possible variation between classrooms. Given the breadth of the categories applied, the degree of variability evident in each country’s lesson structure should be seen as quite striking. Our main point is that the inconclusiveness of the findings of Givvin et al. (2005) should not discourage those seeking national teaching patterns. It may reflect no more than the mistake of employing the lesson as the unit of comparison. In relation to lesson structure and instructional practice as national characteristics, Anderson-Levitt (2002, p. 20) juxtaposed the statement by LeTendre et al. that “Japanese, German and U.S. teachers all appear to be working 28

LESSON STRUCTURE IN THE USA, GERMANY AND JAPAN

from a very similar ‘cultural script’” (2001, p. 9) with the conclusions of Stigler and Hiebert (199A9) that US and Japanese teachers use different cultural scripts for running lessons. As noted in chapter one, this apparent conflict can be resolved by noting that both US and Japanese teachers draw on the same small repertoire of “whole-class, lecture-recitation and seatwork lessons conducted by one teacher with a group of children isolated in a classroom” (Anderson-Levitt, 2002, p.21), but they utilise their options within this repertoire differently. This interweaving of differences and similarities is an essential characteristic of international comparative research. Policy is best informed by research that examines the nature of the interconnection of the various components of classroom practice rather than simply the frequency of their occurrence (Clarke, 2003). The identification of national patterns of teaching was not one of the goals of the Learner’s Perspective Study. The research design focused on the classrooms of a small number of competent teachers in each of the participating countries. Because we selected competent teachers according to local criteria, there is a sense in which the practices of these teachers offer a representation of the pedagogical values in operation in that school system. But each teacher’s enactment of any such values would be likely to vary according to the student group, the topic to be taught and the individual teacher’s instructional inclinations. In such a study, it was the similarities and differences in the practices enacted in well-taught classrooms around the world that was of interest, rather than any national characterization of teachers or their classrooms. Nonetheless, given the visibility of the conjectured lesson patterns in various publications relating to international comparative research (see Clarke, 2003, for a more complete discussion), we felt justified in examining the postulated lesson patterns empirically. There is no logical inconsistency here: Any identified national lesson pattern might be reasonably expected to manifest itself in the set of lessons collected in the LPS project from each of the three countries. The LPS identification of sequences of ten lessons by competent teachers adds interest to this analysis. What use do competent teachers make of the postulated national pattern? And, what consistency of lesson structure is evidenced in a ten lesson sequence by a competent teacher? Our analysis also serves to interrogate the consequences of the four key simplifications listed above. These are simplifications that any study committed to national typification and international comparison will find difficult to avoid. We would argue that it is the combination of such survey-style studies with the indepth classroom analyses of the LPS project that holds greatest promise both to characterize teaching practice internationally and to inform that practice. THE LEARNER’S PERSPECTIVE STUDY

As noted earlier, the LPS analysed sequences of ten lessons, documented using three video cameras, and supplemented by the reconstructive accounts of classroom participants obtained in post-lesson video-stimulated interviews. Test, questionnaire, and student written material were also collected. This methodological approach is dealt with in detail in Clarke (1998, 2001, 2003) and 29

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

Clarke, Keitel and Shimizu (2006) (see appendix). It offers an informative complement to the survey-style approach of the TIMSS video study. The research design for the LPS was developed specifically for the purpose of producing a sufficiently complex data set to support the investigation of such interconnections within mathematics classrooms in a wide variety of cultural contexts. In particular, the research design was constructed to complement the approach taken in the TIMSS Video Study by documenting sequences of lessons, rather than single lessons; by recording ‘private’ (interpersonal) conversations as well as public utterances; and by utilizing the retrospective video-stimulated accounts of classroom participants to determine not just the overt actions documented by the three video cameras, but also the antecedent conditions, motivations and intentions that prompted the observed actions, and the consequent interpretations, meanings and learning outcomes arising from those actions. In each of the participating countries, three 8th grade classrooms in government schools in major urban settings were chosen according to the common criteria of teacher competence (as locally defined by the community), demographic diversity and the avoidance of atypicality in the student group. Of the eighty-three lessons from nine classrooms across three countries analysed in this chapter, it is useful to note the mathematical content addressed in each classroom, as shown in Table 1. Table 1. Mathematical Content of Lessons Analysed Classroom

Number of lessons

Mathematical Content

US 1

10

1. Perimeter, Area and Volume 2. Equations, Inequalities and Formulas 3. Rational Number Concepts

US 2

5 double lessons

1. Functions, Relations and Patterns 2. Equations, Inequalities and Formulas 3. Problem Solving Strategies

US 3

10

Equations, Inequalities and Formulas

Germany 1

10

Germany 2

10

Germany 3

8 lessons including 2 doubles

Integral Rational Terms and their Reformulation and Simplification 1. Common and Decimal Fractions 2. Equations of Fractional Terms Functions

Japan 1

10

1. Proportionality: Slope, Trigonometry and Interpolation 2. Functions, Relations and Patterns 3. Special Terms Used in Mathematics

Japan 2

10

Geometric Congruence and Similarity

Japan 3

10

System of Linear Equations

30

LESSON STRUCTURE IN THE USA, GERMANY AND JAPAN

We are not claiming that the 30 lessons recorded in Japan (for example) are in any way a nationally representative sample of Japanese eighth grade mathematics classroom practice. The question we addressed in our analysis concerned the extent to which the reported nationally characteristic lesson patterns for eighth grade mathematics teaching in Japan (Stigler & Hiebert, 1999) were evident in the practices of any of the three Japanese teachers’ we had studied and what might be learned from the correspondence or inconsistency in the occurrence of these patterns. Similarly, we looked for evidence of the US and German lesson patterns in the practices of the three teachers studied in each of those countries. It is essential to emphasise that in this chapter we have analysed sequences of ten or more mathematics lessons taught by nine teachers designated as competent in three different countries. We cannot characterize the teaching of a country or a culture on the basis of such a selective sample and this was never our intention. Nor do we claim to compare teaching in one country with teaching in another. The research design was developed to support analyses intended to compare and contrast teachers and their classrooms, not cultures. Of course, the choice of school systems (Germany, Japan and the USA) was not accidental. It was intended to complement any general claims of national typicality by situating identified prevalent practice in relation to the antecedent conditions and consequent outcomes that might transform description into explanation. Also, since it was the specific intention of the TIMSS Video Study (Stigler & Hiebert, 1999) to characterize national practice in mathematics classrooms in the USA, Germany and Japan, it is reasonable to expect any lesson patterns reported as nationally typical to be evident to some extent in the respective LPS data bases of lessons from the USA, Germany and Japan. The Analytical Approach The purpose of our initial analysis can be stated simply: To determine whether the sequenced activity categories reported by Stigler and Hiebert (1999) could be identified in an analysis of the corresponding LPS data in the American, German and Japanese classrooms. Consider first the reported characterisation of mathematics teaching in the United States of America. Based on the analysis of 81 single lessons, Stigler and Hiebert (1999) reported that US lessons could be generally characterized by the recurrence of four distinct classroom activities and that these activities, when placed in a particular sequence, formed the basis of a national lesson pattern. The lesson pattern for the United States was reported as: Reviewing previous material; Demonstrating how to solve problems for the day; Practicing; and Correcting seatwork and assigning homework (Stigler & Hiebert, 1999, p. 80)

31

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

Central to any reading of Stigler and Hiebert’s lesson patterns is an understanding of the distinction between their activity titles (such as “Demonstrating how to solve problems for the day”) and the brief descriptions provided of the most common or typical enactment of that activity. After homework is checked, the teacher introduces new material, or reviews previous material, by presenting a few sample problems and demonstrating how to solve them. Often the teacher engages the students in a step-by-step demonstration by asking short-answer questions along the way. (Stigler & Hiebert, 1999, p. 80) These descriptions of typicality do not constitute definitions of the relevant activity, although they did provide useful examples of the type of actions from which the activity category might be constituted. Without strict definitions that distinguished the finer characteristics of one activity from another, we interpreted the four activity categories as liberally as possible. This proved quite challenging. For example, the description of the activity “Demonstrating how to solve problems for the day” (above) includes the phrase “reviews previous material” which is itself the title of the first of the original four activities identified as classifying the structure of US lessons, yet it also appears subsumed within another activity. A minute by minute analysis was conducted of the video record of all US lessons in the LPS data set in which it was determined which of the four activities best described the classroom behaviour for each minute of every lesson. The analysis was carried out by two researchers working independently and the results were compared and discussed and a consensus coding constructed. In order to make the structure of individual lessons more readily comparable, a system was devised whereby each activity was allocated a particular colour/shade (see Figure 1) and these colours/shadings used to distinguish between lesson components and to make any structural regularities more readily apparent (see, for example, Figure 2).

Figure 1. Allocation of colour for coding purposes to each of the classroom activities found in Stigler & Hiebert’s (1999) US lesson pattern

The results that follow report the application of this same process of schematic representation to each of the LPS data sets from the USA, Germany and Japan. We begin first with the USA. 32

LESSON STRUCTURE IN THE USA, GERMANY AND JAPAN

RESULTS

Identifying the US Lesson Components within the LPS US Data Figures 2, 3 and 4 represent the coding of the videotape footage from the LPS data of US Schools 1, 2 and 3 with the classroom activities as described in the US lesson pattern reported by Stigler and Hiebert (1999) and as set out in Figure 1.

Figure 2. US lesson pattern codes as applied to LPS US School 1

The lesson pattern reported in the TIMSS Classroom Videotape Study, namely that a lesson begins with a) reviewing previous material, followed by b) demonstrating how to solve problems for the day, then c) practicing and finally ending with d) correcting seatwork and assigning homework, did not appear as the complete lesson structure in any lesson in US School 1, although it appeared as the first half of Lesson 6. In addition, not all activities were present in every lesson: Lessons 1 to 5 appear radically different in structure from Lessons 7 to 10, while the structure of Lesson 6 appears to be cyclic in nature. This progression in lesson structure across the lesson sequence (that is, across the teaching of one topic) suggests that for this teacher the deployment of the constituent classroom activities was a matter for purposeful choice according to the location of the lesson in the sequence. Most lessons began with a warm-up activity or by checking homework and this rarely appeared to happen at any other time in the lesson. The progression in lesson structure over the course of the lesson sequence is most evident in the time devoted to teacher demonstration in the early lessons, almost completely replaced by the correction of seatwork in the later lessons. The teacher administered an ungraded, ‘conceptual’ test in Lesson 7 and the three lessons following this test were spent on explaining and correcting the tasks from the test in order to expand on and develop the students’ understanding of the related concepts. It is important to note that, although the reported characteristics of the constituent activities were fairly vague, it was possible to use these categories for a comprehensive coding of all ten lessons in US School 1. In this sense, the broad categories reported by Stigler and Hiebert (1999) were evident in the LPS US data set. There was no evidence, however, of the reported sequence as a recurrent or regular pattern in the structuring of any of the documented lessons in this teacher’s classroom other than the first half of Lesson 6. 33

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

If any structural pattern could be said to be evident in this teacher’s lessons, it was the progressive shift in dominant activity from demonstration through participation to correction and explanation. This progression was enacted, however, not over the course of a single lesson, but instead over the entire lesson sequence. Prior to this analysis, Shimizu had already observed, “Japanese teachers usually plan a lesson as part of a unit, a sequence of several lessons. This means that each lesson in a unit has a different purpose for attaining the goals of the unit” (Shimizu, 1999, p. 194). The data from US Teacher 1 suggested that this American teacher structured each lesson according to its location within the topic or unit.

Figure 3. US lesson pattern codes as applied to LPS US School 2

The structure of the lessons in School 2 (see Figure 3) appeared to be closest in structure to the pattern reported by Stigler and Hiebert (1999). All School 2 lessons were double lessons taking up two timetable periods. In fact, all the lessons in School 2 began with the reported sequence of classroom activities, while the structure of Lesson 3 is completely described by the Stigler and Hiebert (1999) 34

LESSON STRUCTURE IN THE USA, GERMANY AND JAPAN

pattern. There is no doubt that the pedagogical demands of a double-period lesson influenced the teacher’s structuring of each lesson as a whole. In addition to being the closest in structure to the reported pattern, each classroom activity type appeared in every School 2 lesson. Further, the time allocated to each activity was more evenly distributed than in US School 1.

Figure 4. US lesson pattern codes as applied to LPS US School 3

The US lesson pattern, as described by Stigler and Hiebert (1999), did not appear as the complete lesson structure for any of the lessons from School 3. The earlier portions of Lessons 1 and 7 do resemble the TIMSS lesson pattern, although the lessons themselves in their entirety do not. As for LPS US School 1, not all four activity types appear in every lesson. Lessons 1, 5 and 7 had one similar structural feature: A significant period of each of these three lessons involved the repetitive alternation of student seatwork and the correction of student seatwork. By contrast, several lessons showed little pedagogical sub-structure, with only two or three classroom activities employed (particularly review and practice) and each extending for significant proportions of the lesson’s duration. Almost all the School 3 lessons began with students correcting their homework from a transparency, which the teacher placed on the overhead projector. Any student concerns about the homework were then addressed at this time. Little time was devoted to teacher demonstration, while large portions of the lesson were spent in student practice. The contrast with US Teacher 1 is striking. A few summative remarks can be made about the coding of the US data as this is displayed in Figures 2, 3 and 4. Firstly, it was possible to interpret the activity categories of the Stigler and Hiebert report sufficiently broadly to accommodate most of the documented activities in all three US classrooms. Secondly, the US lesson pattern reported by Stigler and Hiebert (1999) adequately described only one of the 25 lessons coded (US2-L3) in its entirety, although the lesson pattern appeared within the overall structure of several lessons. Thirdly, the lessons taught by any one teacher showed evidence of purposeful variation across the topic sequence for that classroom. Fourthly, the differences in lesson structure and topic 35

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

structure between teachers suggested that each teacher combined and sequenced the various activities in ways that were not only a reflection of the mathematical topic being taught, and of the location of the lesson in the topic sequence, but also of the pedagogical style of the individual teacher. Identifying the German Lesson Components within the LPS German Data The same procedure was followed in the case of the German LPS data set. Stigler and Hiebert (1999) reported the lesson pattern shown in Figure 5 and these classroom activity types were used to code the German videotape data. The results for each of the three German schools are shown in Figures 6, 7 and 8.

Figure 5. Stigler & Hiebert’s (1999) German lesson pattern

Figure 6. German lesson pattern codes as applied to LPS German. School 1

Figure 7. German lesson pattern codes as applied to LPS German School 2

36

LESSON STRUCTURE IN THE USA, GERMANY AND JAPAN

Figure 8. German lesson pattern codes as applied to LPS German School 3

Figures 6, 7 and 8 show that there was a much greater variety of lesson structure in the German lessons from the LPS study than the characterisation of a typical German lesson given by Stigler and Hiebert would suggest. As was the case for the US lessons, a generous interpretation of the reported classroom activities allowed most classroom activities to be coded consistently with the four categories shown in Figure 5. However, lessons in all three German schools included classroom activities that fell outside the predicted categories. These uncoded activities are shown in white in each of the three preceding figures. Examples of such uncoded activities included: Organisation of school activities, classroom management and disciplinary control, arranging seating or setting up equipment, getting materials or tools in preparation for the next activity, and conducting a test. The same four summative points made in relation to the US lessons apply with equal legitimacy to the German lessons: Much of the lessons could be accommodated by the activity codes; however there was no evidence of the German lesson pattern reported by Stigler and Hiebert in any of the twenty-eight lessons analysed, and there was significant variation in lesson structure both within any one teacher’s lesson sequence and between teachers. Identifying the Japanese Lesson Components within the LPS Japanese Data The same procedure was followed in the case of the Japanese LPS data set. Stigler and Hiebert (1999) reported the lesson pattern shown in Figure 9 and these classroom activity types were used to code the Japanese videotape data. The results for each of the three Japanese schools are shown below (Figures 10, 11 and 12).

37

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

Figure 9. Stigler & Hiebert’s (1999) Japanese lesson pattern

Figure 10. Japanese lesson pattern codes as applied to LPS Japanese School 1

Figure 11. Japanese lesson pattern codes as applied to LPS Japanese School 2

38

LESSON STRUCTURE IN THE USA, GERMANY AND JAPAN

Figure 12. Japanese lesson pattern codes as applied to LPS Japanese School 3

Many of the Japanese lessons can be characterized as ‘structured problem solving’ in the sense that the teacher typically intended to have the students work on a problem and discuss solution methods, and then to highlight and summarize the main points in each lesson. But Figures 10, 11 and 12 reveal a rich substructure to each lesson that could not be represented adequately by any single lesson pattern, such as that shown in Figure 9. As can been seen from Figures 10, 11 and 12, the lesson pattern varied within the instructional sequence for teaching a topic, depending on the teacher’s intentions. The coding of the classroom activities for the videotaped sequences of ten consecutive mathematics lessons in Japanese classroom in the Learners’ Perspective Study (LPS) data suggests that there are significant variations from the lesson pattern reported by Stigler and Hiebert (1999) and that a more complex categorization of the classroom activities or lesson components of Japanese mathematics lessons is needed than the simple five-component lesson pattern described by Stigler and Hiebert’s (1999) analysis of single lessons. It is interesting to note that Stigler and Hiebert referred specifically to one Japanese lesson in their data, which did not seem to fit their characterization of ‘structured problem solving.’ This lesson shows that ‘structured problem solving’ does not capture the full range of Japanese instruction. Indeed, it seems that the teaching method in this lesson is more like the methods typically used in Germany than the method typically used in Japan. If nothing else, the lesson reminds us that not all teachers within the same country use the same methods (Stigler & Hiebert, 1999, p. 51). The analysis reported above suggests that even the same teacher teaches mathematics in different ways at different stages in the instructional sequence. This finding has an obvious explanation if we consider that Japanese teachers usually plan a sequence of several lessons as part of the teaching unit (Shimizu, 1999). In other words, each lesson in a unit has a different purpose in relation to attaining the goals of the entire unit. The lesson at the introductory phase of the entire unit, for example, may look like ‘structured problem-solving’, whereas the lesson at the

39

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

final phase of the unit may have an emphasis on practicing what the students have learned. As was the case with the US and German classrooms, the TIMSS Japanese lesson pattern was not evident in the LPS lessons analysed, except in the first half of Japanese School 1, Lesson 6. The variation between teachers in the constitution of the lessons was significant. One observation, highlighted by the schematic use of colour/shading in Figures 10, 11 and 12, is the preference of each Japanese teacher for a particular activity type. Teacher 1 made much greater use of ‘Discussing solution methods’ than either of the other two teachers. Teacher 2 devoted more time to ‘Highlighting and summarizing the major points’ than either of the other two teachers, and Teacher 3 made far more consistent and extensive use of ‘Students working individually or in groups.’ LESSON EVENTS AS THE UNIT OF COMPARATIVE ANALYSIS

In coding the lessons displayed in Figures 2, 3, 4, 6, 7, 8, 10, 11 and 12, we applied the activity categories reported by Stigler and Hiebert (1999). We attempted to do this using the broadest possible interpretation of each activity category in order to maximize the capacity of the resultant coding scheme to accommodate the classroom actions documented in the eighty-three lessons analysed. Our relative success overall in this coding exercise highlighted those classroom activities that could not be successfully coded, and these exceptions have been commented upon. As a result of our analysis, we became convinced of several important points. While we are in sympathy with the goal of finding pattern and structure in teachers’ classroom practice, the activity categories employed by Stigler and Hiebert (1999) were too inclusive to usefully represent the richness of the activities we found in the nine classrooms. Shimizu conducted a reclassification of the activities in the Japanese classrooms and found a minimum of thirteen activity codes necessary to accommodate the classroom activities recorded in the three Japanese classrooms. Significant and frequent omissions in the case of the Japanese classrooms included Assigning Homework and Checking Homework. Table 2

The Thirteen Categories for Coding Japanese Lessons

Reviewing the Previous Lesson (RP)* Checking Homework (CH) Presenting the Topic (PT) Formulating the Problem for the Day (FP) Presenting the Problems for the Day (PP)* Working on Sub-problem (WS) Students Working Individually or in Groups (WP)* Presentation by Students (PS) Discussing Solution Methods (DS)* Practicing (P) Highlighting and Summarizing the Main Point (HS)* Assigning Homework (AH) Announcement of the Next Topic (AN) * Activity codes from the Japanese lesson pattern reported by Stigler & Hiebert (1999).

40

LESSON STRUCTURE IN THE USA, GERMANY AND JAPAN

Stigler and Hiebert reported that “no homework is typical” (Stigler & Hiebert, 1999, p.30) for Japanese lessons, and contrasted this with Germany and the USA. All the three classrooms in the LPS Japanese data set, however, included both ‘assigning homework’ and ‘checking homework’. In such cases, homework played the role of ‘connector’ between two lessons (Sekiguchi, 2003). Our analysis of the German and US lessons also strongly supported the need for an expanded categorisation scheme, if the richness and distinctive character of each classroom was to be documented adequately. The increased detail of the codes employed in the later TIMSS-R Video Study (Hiebert et al., 2003) provides further endorsement of this point. The teachers whose classrooms we had documented showed little evidence of a consistent lesson pattern, but instead appeared to vary the structure of their lessons purposefully across a topic sequence. The challenge for us became to decide how best to represent the teacher’s purposeful structuring of their lessons in way that offered greatest insight into the intentions and motivations of the teachers (and their students) and which captured whatever was most distinctive about the practice of each classroom. The question of whether any emergent structures might provide possible contenders as national characteristics was not a question we could address, given the focused nature of our study, but it was here that we hoped the complementarity of the LPS and TIMSS-R projects might prove mutually informing. The report of the TIMSS-R Video Study (Hiebert et al., 2003) reinterpreted the idea of nationally representative lesson patterns (by implication, if not explicit statement) and reported ‘lesson signatures’ (Hiebert et al., 2003, pp. 123-151). If there are features that characterize teaching in a particular country, there should be enough similarities across lessons within the country to reveal a particular pattern to the lessons in each country. If this were the case, then overlaying the features of all of the lessons within a country should reveal a pattern or, as labeled here, a ‘lesson signature’ (Hiebert et al., 2003, p. 123). Hiebert and his colleagues constructed an imaginative way of displaying this overlaying of their coding of each country’s lessons, representing national practice through an array of parallel bands similar to seismological charts, each band indicating the percentage of lessons that exhibited that feature at that time in the lesson. These displays could support such statements as “For US eighth grade mathematics lessons, while the activity coded as Reviewing was documented at all points in the lesson, it was most frequently recorded during the first 20% of the lesson, with steadily decreasing frequency of occurrence throughout the remainder of the lesson.” An alternative reading of these lesson signature displays would be to identify a particular period in the lesson (e.g., when 40% of the lesson time had passed) and to compare the relative frequency of occurrence of the various activity types at that stage of the lesson. The TIMSS-R lesson signature approach has the advantage of supporting statements of likelihood regarding the occurrence of particular activity types at various points in a lesson for each of the countries studied. The questions which the 41

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

LPS sought to address related more to the conditions under which teachers might decide to employ a particular activity and the consequences of that decision for both the teacher’s instruction and the students’ learning. The analysis of the LPS lessons reported earlier in this chapter suggests that a primary consideration must be where the lesson is situated in the topic sequence. Further, the evident differences in the manner in which teachers structured their lessons, suggested that another unit of analysis was needed: one that corresponded more closely to the decisions made by each teacher regarding the structure of any particular lesson. Our analysis of the LPS lessons focused therefore on the form and function of recognizable activity conglomerates that we came to call ‘lesson events.’ The original conception of lesson events was sustained by those features of a country’s classroom practice that most attracted the attention of researchers from another country. For example, non-American researchers were struck by consistencies in the way in which the three US teachers commenced each lesson. By contrast, nonAustralian researchers commented on the very attenuated content introductions provided in many of the Australian lessons. At the same time, the proportion of time that Australian teachers devoted to moving around the classroom, while their students worked semi-independently on assigned mathematical tasks, also attracted the attention of non-Australian researchers. The evident significance that the Japanese teachers attached to a summative discussion towards the end of each lesson was of interest. The manner in which each teacher posed the first substantive mathematical task, intended to introduce the lesson’s content, appeared to vary significantly from teacher to teacher and lesson to lesson. Those moments when a student was called to work on the board at the front of the classroom seemed to serve a distinctive and different purpose from classroom to classroom, and yet such moments were familiar across all participating LPS research groups. Each of these lesson events was recognizable and even familiar to all members of the research group. As a result, we adopted the lesson event as a unit of comparative analysis. A Lesson Event, as we conceived it, was characterized by a combination of form and function, both of which were subject to local variation, but with an underlying familiarity and frequency of use that suggested both cross-cultural relevance and utility. Each individual Lesson Event had a fundamentally emergent character, suggested by the classroom data as having a form (visual features and social participants) sufficiently common to be identifiable within the classroom data from each of the countries studied. In each classroom, both within a culture and between cultures, there were idiosyncratic features that distinguished each teacher’s enactment of each Lesson Event, particularly with regard to the function of the particular event (intention, action, inferred meaning and outcome). At the same time, common features could be identified in the enactment of Lesson Events across the entire international data set and across the data set specific to a country. The teacher and student post-lesson interviews offered insight into both the teacher’s intentions in the enactment of a particular Lesson Event and the significance and the meaning that the students associated with that event type.

42

LESSON STRUCTURE IN THE USA, GERMANY AND JAPAN

Each Lesson Event required separate and distinct identification and definition from within the international data set. Each of the next five chapters addresses one of these Lesson Events. Additional Lesson Events are the focus of current analysis and it is anticipated that future books in this series will report on these. CONCLUSIONS

One of the most significant challenges confronting international comparative research on classroom practice is the identification of a suitable basis for comparative analysis. Based on the analyses reported earlier in this chapter, it seems likely that, if the goal is to characterise typical national practice at the level of the lesson, in attempting to accommodate the variation evident in a national sample of lessons, the resultant lesson structure and its constituent codes must be so inclusive as to sacrifice the details that might otherwise have facilitated meaningful comparison and informed practice. An inevitable consequence of any nationally representative sample of individual lessons is to average over the distinctive lesson elements, whose location in the lesson are a direct and informative reflection of the lesson’s location in the topic sequence. The ‘typical lesson’ is consequently no more informative than the ‘typical teacher’ or ‘typical student.’ However, statements of national typicality can be a useful reference point when they are expressed as statements of relative frequency or likelihood of occurrence. An example of this can be found in the next chapter on ‘Beginning the Lesson.’ Particular consistencies of practice were identified among the US lessons analysed, specifically relating to how the teachers began the lesson. The detailed LPS data set supports a fine-grained analysis of both the form and the function of these lesson beginnings as they were enacted in the three US classrooms. It cannot, of course, make any statements regarding the national frequency of occurrence of the review function at the start of mathematics lessons in the US. For this, we can turn to the TIMSS-R data (Hiebert et al., 2003). If, however, a particular activity were identified by the TIMSS-R Video Study as occurring extremely frequently in the classrooms of a particular country, the TIMSS-R data is unlikely to be able to address such questions as: In response to what antecedent conditions (both immediate and of a more extended nature) was that activity purposefully employed? With what instructional purpose? With what consequent learning outcomes? For the answers to these questions, we require the detail provided by the LPS documentation of lesson sequences, supplemented by the post-lesson interviews with teachers and students. Another distinguishing characteristic of the LPS is the decision to focus on competent teachers. Elsewhere, Shimizu has contrasted the focus on the typical lesson versus the focus on the well-taught lesson. Japanese mathematics teachers and educators would believe that studying one excellent lesson intensively is likely to be more beneficial than studying many ‘average’ lessons. They are interested not in how eighth grade mathematics is taught in Japan but how an excellent teacher, in any one of

43

DAVID CLARKE, CARMEL MESITI, EVA JABLONKA AND YOSHINORI SHIMIZU

three countries, teaches mathematics in her classroom (Shimizu, 1999, p. 191). The essential point is that the variability in lesson structure documented in this chapter reflects the purposeful decision-making of competent teachers, who structure their lessons in recognition of the needs of their students, their priorities and strengths as teachers, and the situation and consequent purpose of the lesson in the instructional sequence (cf. Givvin et al., 2005, p. 341). If we are to inform the practices of teachers internationally, our analyses should be focused on that level of activity that is in closest correspondence to the level at which teachers are obliged to make structural instructional decisions. Rather than offering teachers a model lesson, to be universally applied, as has been the case in the past (for example, in the recommendations of the process-product generation of researchers, see Bourke (1985) or Good and Grouws (1979)), our goal is to expand a teacher’s repertoire of instructional strategies by reporting the variety of forms and functions in which particular Lesson Events are carried out in the classrooms of competent teachers around the world. NOTES i. Purpose is defined to mean Review, Introducing New Content, Practicing/Applying Content; Classroom Interaction means Public or Private Interactions; Content Activity is defined to include Independent or Concurrent Problems (Givvin et al., 2005).

REFERENCES Anderson, L. W., Ryan, D., & Shapiro, B. (1989). The IEA classroom environment study. New York: Pergamon. Anderson-Levitt, K. M. (2002). Teaching culture as national and transnational: A response to teachers’ work. Educational Researcher 31(3), 19-21. Beaton, A. E., & Robitaille, D. F. (1999). An overview of the Third International Mathematics and Science Study. In G. Kaiser, E. Luna & I. Huntley (Eds.), International comparisons in mathematics education (pp. 19-29). London: Falmer Press. Bourke, S. (1985). The teaching and learning of mathematics. ACER Research Monograph No. 25. Hawthorn: Australian Council for Educational Research. Clarke, D. J. (1998). Studying the classroom negotiation of meaning: Complementary accounts methodology. In A. Teppo (Ed.), Qualitative research methods in mathematics education. Journal for Research in Mathematics Education, Monograph No. 9 (pp. 98-111). Reston, VA: NCTM. Clarke, D. J. (Ed.). (2001). Perspectives on practice and meaning in mathematics and science classrooms. Dordrecht: Kluwer. Clarke, D. J. (2003). International comparative studies in mathematics education. In A.J. Bishop, M.A. Clements, C. Keitel, J. Kilpatrick & F.K.S. Leung (Eds.) Second international handbook of mathematics education (pp. 145-186). Dordrecht: Kluwer. Clarke, D. J., Keitel, C., & Shimizu, Y. (2006). Mathematics classrooms in twelve countries: The insider’s perspective. Rotterdam: Sense Publishers. Givvin, K. B., Hiebert, J., Jacobs, J. K., Hollingsworth, H., & Gallimore, R. (2005). Are there national patterns of teaching? Evidence from the TIMSS 1999 video study. Comparative Education Review, 49(3), 311-343.

44

LESSON STRUCTURE IN THE USA, GERMANY AND JAPAN Good, T. L., & Grouws, D. A. (1979). The Missouri mathematics effectiveness project: An experimental study in fourth grade classrooms. Journal of Educational Psychology, 71 (3), 355 362. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K., Hollingsworth, H., Jacobs, J., Chui, A., Wearne, D., Smith, M., Kersting, N., Manaster, A., Tseng, E., Etterbeck, W., Manaster, C., Gonzales, P., & Stigler, J. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study. Washington, DC: U.S. Department of Education, National Center for Education Statistics. Hu, A. (2000, December). TIMSS: Arthur Hu’s index. Retrieved December 30, 2000, from http://www.leconsulting.com/arthurhu/index/timss.htm. Cited in Wang (2001). LeTendre, G., Baker, D., Akiba, M., Goesling, B., & Wiseman, A. (2001). Teachers’ work: Institutional isomorphism and cultural variation in the U.S., Germany, and Japan. Educational Researcher, 30(6), 3-15. Sekiguchi, Y. (2003). Analysis of connections between lessons in mathematics classrooms in Japan: A case study of two classes [in Japanese]. In Y. Shimizu (Ed.), Cross-cultural study on the teaching and learning process in mathematics classrooms. Tokyo: Tokyo Gakugei University. Shimizu, Y. (1999). Studying sample lessons rather than one excellent lesson: A Japanese perspective on the TIMSS videotape classroom study. Zentralblatt für Didactik der Mathematik, 6, 191-195. Stigler, J. & Hiebert, J. (1999). The teaching gap. New York: Simon & Schuster. Wang, J. (2001). TIMSS primary and middle school data: Some technical concerns. Educational Researcher, 30(6), 17-21. Wang, J. & Lin, E. (2005). Comparative studies on U.S. and Chinese mathematics learning and the implications for standards-based mathematics teaching reform. Educational Researcher, 34(5), 3-13. Westbury, I. (1992). Comparing American and Japanese achievement: Is the United States really a low achiever? Educational Resarcher, June-July, pp. 18-24.

David Clarke International Centre for Classroom Research Faculty of Education University of Melbourne Australia Carmel Mesiti International Centre for Classroom Research Faculty of Education University of Melbourne Australia Eva Jablonka Fachbereich Erziehungswissenschaft und Psychologie Freie Universitaet Berlin Germany Yoshinori Shimizu Graduate School of Comprehensive Human Sciences University of Tsukuba Japan

45

CARMEL MESITI AND DAVID CLARKE

CHAPTER THREE Beginning the Lesson: The First Ten Minutes

INTRODUCTION

Beginnings are important. Beginnings may anticipate what is to come or connect with what has happened previously. The beginning of a lesson may serve to establish the authority of the teacher or devolve that authority to the students. The tasks may revisit skills already developed or invite students to explore new mathematical territory. The beginning of the lesson provides an opportunity to arouse the students’ interest and facilitate their engagement, to situate and introduce the lesson’s content, and to establish the subsequent work pattern for the lesson. Whether these opportunities are exploited and in what form will vary from classroom to classroom. In this chapter, we examine the classroom practice of ‘Beginning the Lesson’ over sequences of ten lessons. We define this particular lesson event as having commenced the moment the teacher undertakes the first communicative act for the whole class and as encompassing the period from this moment followed by the next ten minutes. The choice of ten minutes was not an arbitrary one. The first ten minutes constituted between twenty and twenty-five percent of most of the lessons analysed. In most cases, it was a sufficient period to include at least one transition between activity types. The patterns of practice identified by our analyses did not all occupy exactly ten minutes; rather, they represent coherent sets of actions regularly and/or effectively used specifically for the beginning of a lesson. Each identified ‘Iconic Sequence’ (see the section titled ‘Iconic Sequences’ for more detail) had its own coherent purpose, and that purpose was fundamentally introductory. Even where the activity had the character of a review, it served a purpose closely connected to its location at the beginning of the lesson. We have investigated in detail various classrooms in the USA, Australia, Japan, and Sweden in order to gain knowledge about the possible differences in function of this specific phase of the lesson. In carrying out this analysis, we devised a coding scheme that allows us to examine closely the activity characteristics, both observable and inferred, and the nature of student and teacher participation in the event. We identified verbal and non-verbal communicative acts and the differing number of activities realised within that time frame amongst the different classrooms. In some instances, the differences in function appeared not only when

D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 47–71. © 2006 Sense Publishers. All rights reserved.

CARMEL MESITI AND DAVID CLARKE

comparison was made with other classrooms but also when made within the tenlesson sequence of a single classroom. The practices of eight classrooms were analysed in detail for this chapter and specific elements were identified as recurrent features of Beginning the Lesson. An individual teacher’s practice in beginning a lesson can be characterised by the idiosyncratic ways in which that teacher selects, combines and sequences these different elements. It is not surprising that eight different teachers in different countries should begin their lessons differently. As competent, experienced professionals, several of the teachers we studied varied their lesson beginnings in ways that reflected the situation of the lesson in the topic sequence, and the focus and structure of the lesson as a whole. There were many appealing patterns that emerged from our analysis of lesson beginnings, but among the most interesting were the occasions when an element evident in one classroom could be identified in the practices of a classroom from a different country entirely. Things that might have seemed culturally-specific recurred in classrooms as culturally-distant as Melbourne (A1, A2, A3) and Sweden (SW1) or San Diego (US1, US2, US3) and Tokyo (J1). One of the immediate challenges confronting us was the question of when the lesson actually began. In several cases, the lesson commenced without any explicit verbal instructions from the teacher. This suggested to us that the teachers had established a routine of practice in which the students were complicit. The lesson was defined as commencing from the teacher’s first communicative act to the whole class, whether this was turning on a projector, writing on the board, or giving explicit verbal instructions to the class. Some lessons appeared to have a diffused starting point: Students entered over a period of time, rather than as a group, the teacher welcomed students individually, the students took their seats and prepared for work, but the commencement of whole class activity was delayed by several minutes. In such cases, for the purpose of adequately including a full ten minutes of class activity in our analysis, we considered the lesson to have begun from the teacher’s first communicative act clearly addressed to all students present. The coding of activities provided the basis from which we were able to identify patterns that appeared either characteristic of the practices of particular classrooms, or indicative of the pedagogical orientation of different teachers, or which the data suggested were of sufficient interest and effectiveness to warrant reporting. In the remainder of this chapter, we first set out the ‘Dominant Components’ for Beginning the Lesson evident in the first ten minutes of the lessons analysed, and then identify particular Iconic Sequences composed by combining some of these Dominant Components. These Iconic Sequences represent alternative ways to begin a mathematics lesson. Some will be immediately familiar, others will appear quite novel. What must be remembered is that these lesson beginnings were identified from the classroom practices of competent teachers in quite different cultural settings. Our intention is to broaden the instructional repertoire of mathematics teachers internationally, while at the same time examining the pedagogical and epistemological principles on which the various lesson beginnings are predicated. 48

BEGINNING THE LESSON: THE FIRST TEN MINUTES

Activity conglomerates such as ‘Warm-up’ have the consistency of structure and social interaction that we have elsewhere identified as ‘Patterns of Participation’ (Clarke, 2004). Rather than report the minutiae of this detailed coding, we have chosen to report the crux of our analysis of Beginning the Lesson in terms of these activity conglomerates. For example, Warm-up itself is a specific instance of a broader category that we have called ‘Review’. Review includes other patterns of participation in addition to Warm-up (such as ‘Run-through’). It is at the level of these Patterns of Participation that we feel classroom practice is most readily understood. It was the teacher’s purposeful initiation and deployment of such elements as Warm-up that most clearly revealed the structure and purpose of the various ways in which these competent and experienced teachers commenced their lessons. POSSIBILITIES AND SEQUENCES: THE DOMINANT COMPONENTS FOR BEGINNING THE LESSON

We have chosen to present these emergent patterns of participation in terms of possibilities and sequences. That is, what possible choices are open to the teacher in deciding how to begin a lesson, and how might these alternative activities be sequenced? Figure 1 sets out these Patterns of Participation and indicates the most frequent or typical relationships between them. Using Figure 1 as a navigational tool, the reader is encouraged to move freely among the self-sufficient Dominant Component descriptions in any order. The Dominant Components are seen as separate entities and just as a teacher would choose to combine them to form various sequences, given differing purposes of individual lessons, so too the reader may choose to examine the Dominant Component descriptions in any order. Following the descriptions of the Dominant Components for Beginning the Lesson, the main findings of our analysis are reported as Iconic Sequences of these components. It is in the crafting of these Iconic Sequences that we feel the expertise of the competent mathematics teachers is most visible and most readily related to the practices of other classrooms. The ‘Pre-Education’ Component Administrative activity. A good example of an Administrative Activity is rolltaking: the recording of student presence and absence. This activity occurred in every classroom we studied. Sometimes roll-taking was a dominant activity (for example, in A3 and J1). On other occasions, roll-taking was an administrative obligation that the teacher met unobtrusively while the class was engaged in some other dominant activity. Organisational activity. Organisational Activity included the distribution of equipment, such as rulers, calculators, protractors and string. It also included the distribution of worksheets, textbooks, and student workbooks.

49

CARMEL MESITI AND DAVID CLARKE

Figure 1. The dominant components for beginning the lesson

50

BEGINNING THE LESSON: THE FIRST TEN MINUTES

In Australian Classrooms 2 and 3, the distribution of equipment was frequent and time-consuming. In most other classrooms, such activities were less intrusive and less time-demanding. In US2, the responsibility for distribution of equipment was shared with the class by devolving that responsibility to the ‘Table Leaders’, who would individually collect the equipment required by their group. In US1, the job of returning corrected student tests was delegated by the teacher to a particular student, allowing the teacher to carry out other functions, such as calling the roll or writing tasks on the board. Pastoral care activity. Pastoral Care Activity included such things as a whole class discussion about students’ responsibilities, particularly as learners (A2), and the coordination of students’ participation in an extracurricular activity (SW1). The ‘Review’ Component Coding this particular Dominant Component provided some unique challenges. Certainly the very first difficulty we faced was how to establish whether a mathematical activity was indeed ‘review’ or ‘new’. Most teachers are familiar with the challenge of determining to what degree a mathematical concept or skill is familiar or unfamiliar to a cohort of students of various ability levels, and as researchers we faced similar difficulties. Indeed, whether the concept or skill has been presented previously may have little connection with the students’ current understanding and recalled experience with the suggested mathematical activity. We therefore decided to classify the activity as Review if: i) the work was related to the topic content and there was evidence that the teacher had covered the work in previous lessons; ii) in the case when the work was unrelated to the topic content, the teacher’s introduction suggested an expectation of student familiarity. Two broad categories of mathematical activity are found in the Review component: Focusing or warm-up activity. Focusing Activities were intended to be silent periods of student work and usually lasted between 5 and 10 minutes. This activity appeared more often in American lessons, but was not unique to the US data set. Three distinct types of Focusing Activities emerged: – Short answer questions unrelated to the topic content; It appeared that a teacher would assign between three and five short answer questions that were not directly related to the topic content, in order to provide students with the opportunity to practise basic skills (US1) or review previous work on other topics (US1, US2 and A2). – Short answer questions related to the topic content; Short answer questions, when related to the current topic content, were used to reinforce work from the previous lesson(s), and connect this work to the current topic under study (US1, US2, A1, A2, J1 and SW1). – Independent homework correction; The principal function of independent homework correction was to provide students with correct answers to their homework, so that the teacher could 51

CARMEL MESITI AND DAVID CLARKE

ascertain which homework problems needed further attention, and to give students an additional opportunity to solve homework questions which appeared to have been incorrectly completed (see also ‘Independent Correction’, a subcategory of the ‘Correction’ Component). The American classrooms appeared to favour some type of Focusing or Warmup Activity and each of the American teachers was very consistent in the way they chose to begin their lessons. Recap or run-through. A Recapitulation or Run-through was an educational activity designed to provide an opportunity for students to revisit previously covered skills or concepts. Unlike the Focusing or Warm-up Activity, these were not silent activities. Teachers made use of: – Routine problems (A1, A2, A3, US1, US2, US3, SW1); – Non-routine problems (J1, US3); and – Homework questions (US1, A1, A3, J1, SW1). Teachers were found to lead the consequent discussion, and sometimes recorded student responses at the: – Board (A1, A2, A3, US1, US2, US3, J1, SW1); or – Overhead projector (US3). At other times, teachers (J1, US2 and US3) coordinated the student discussion and invited selected individual students to record the responses/suggestions at the: – Board (e.g. US2); – Overhead projector (e.g. US3); and – Averkey (television monitor) (J1 only). In fact, this ‘Student At The Board’ activity appeared to be quite a popular student task, and several students were observed to call out to their teacher to express their desire to participate. When a homework problem was used in the Recap or Run-through discussion, the mathematical function of the activity was to highlight or clarify misunderstandings, as well as to document previous approaches to problems. This differs from the Dominant Component ‘Correction’ (discussed later), involving a homework problem, as in this situation the activity’s primary function is to provide a correctly worked example, in order for students to compare their solution with the one offered in the whole class discussion. Invariably, some mathematical activity has more than one function. This situation is addressed in more detail in the section titled ‘Iconic Sequences’. The ‘Instruction’ Component The Instruction Component refers to the phase, within the first ten minutes of a lesson, when an unfamiliar skill or concept was introduced to the class. Naively, one might think that the beginnings of many lessons would consist of the introduction of new, unfamiliar content. In fact, this particular Dominant Component did not commonly occur in the beginning of the lessons we analysed. Indeed, it was absent entirely from the first ten minutes of all lessons in one 52

BEGINNING THE LESSON: THE FIRST TEN MINUTES

particular classroom (US3). We found that in all the identified instances of such Instruction Components mathematical problems were posed to students. When this component did occur, the key element that distinguished one instance from another was the intended purpose of the activity. We distinguished four categories within this component: Problem posing (structured reflection, to build definitions or theory or to make connections). In this category, the problems posed were in most cases familiar to students and, at first glance, this type of activity might appear to be one that best belongs to the Review Component. The critical difference was the teacher’s intention; that is, to generalise and categorise the different mathematical situations and the possible approaches to such situations. We found that in all cases, where this activity appeared in the first ten minutes of the lesson, there was a high level of teacher orchestration of the discussion (US1, US2, A1, J1, SW1). Students provided vital verbal contributions and were encouraged to be active participants, but the discussion was highly teacher-led. Problem posing (introducing skills or concepts). This category describes mathematical activity that involved unfamiliar but routine problems. In such cases, the teacher would present a routine problem and proceed to explain or discuss possible solutions, generally involving student contributions (US1, US2, A2, A3). Problem posing (application to a simple context). In these situations, a nonroutine problem was posed involving a real-world context (US2, SW1, J1). The task may have been an extension of a previous problem (US2), had an application to the sciences (SW1), or modelled a real-world situation (J1). In one Japanese lesson (J1-L04), the teacher introduced a challenging problem, involving a familiar context, to open her lesson. The students were invited to investigate “what changes?” in the act of folding origami paper: Transcript 1: J1-L04 T T

T T T T

[Takes out origami paper] Well, this here, yes. Uhh, this paper, yes, umm I have enlarged this to make it easier for everybody to see. Today class, the thing we are going to think about is actually about this size. Yes. And, this paper, which everybody calls origami, this paper, yes, you fold and look at, is origami, right. Okay, well, this origami, today, we are, for example, going to decide where we are going to fold, like this, and we are going to fold, like this.[Folds the origami paper] Right, okay, sometimes we do this, um yes, for example, we decide where to fold like this, yes, why donʼt we fold a little more. Yes, we try folding this rectangular origami like this, yes, fold like this. Right. Yes, and today we will examine what happens when we fold the paper, like this. Yes, well, okay, when we decide where to fold, and fold, like this, uhh what changes?

53

CARMEL MESITI AND DAVID CLARKE Key to symbols used in transcripts in this chapter … A short pause of one second or less. // Marks the beginning of simultaneous/overlapping speech. Ss Indicates two or more students speaking simultaneously, saying the same words. / Indicates that one speaker cut in, interrupting another speaker before they had finished. ( ) Indecipherable speech. (text) A plausible interpretation of speech that was difficult to decipher. (text a/text b) Alternative possible readings of speech that was difficult to decipher. [text] Comments and annotations, often descriptions of non-verbal action. text Italicised text indicates emphatic speech. text Underlined text indicates emphasis added by the author. .... Indicates that a portion of the transcript has been omitted.

Problem posing (use of an elaborate context). In one Australian lesson (A2-L07), problems involving compass angle constructions were elaborately introduced with a detailed story and dramatic presentation (see Iconic Sequence Six). The ‘Student Practice’ Component Tutorial. One of the practices of the Swedish classroom involved devoting an entire lesson to student independent work: referred to as ‘A group/half lesson’ (SW1). Students were expected to work on assigned problems. Meanwhile, the classroom teacher roamed, offering assistance to individuals and small groups of students. The ‘Student Assessment’ Component One interesting consequence of an analysis involving sequences of ten lessons was the opportunity to document formal and informal assessment practices. The inclusion of formal testing in the data collection was a direct consequence of the decision to collect coherent sequences of ten or more lessons. Such sequences typically constituted the teaching of a single mathematical topic and some included the assessment of student achievement with regard to the content taught. Both the Third International Mathematics and Science Study (TIMSS) and Third International Mathematics and Science Study-Repeat (TIMSS-R) Video studies (Stigler & Hiebert, 1999; Hiebert et al., 2003) deliberately excluded such formal testing from their sampling of single lessons. The Learner’s Perspective Study (LPS) Research Team adopted the position that such assessment was an integral part of classroom practice and should not be deliberately excluded, but rather documented and analysed wherever it occurred in the lesson sequences recorded. Diagnostic purpose. In one of the US lessons (US1-L07), students were given a commercial diagnostic test (Algebridge) to complete during a full lesson. Inevitably, the first ten minutes involved some instruction regarding the administration of the test, followed by the commencement of test completion by the whole class.

54

BEGINNING THE LESSON: THE FIRST TEN MINUTES

Formal assessment purpose. Full lesson topic tests were administered in three of the LPS classrooms (US3, A3, SW1). In all cases, the beginning of the lesson involved some administration, organisation and instruction: for example, students were directed to seats, instructed to prepare for the administration of the test by rearranging desks, and given advice regarding time management. Individual assistance was offered to students in one of the Australian classrooms (discussed in detail in Chapter 4, this volume: Kikan-Shido). The ‘Correction’ Component Homework Correction has already been defined as one type of Review activity. This separate Correction Component could be distinguished from the Instruction and Review Components in two ways: – It was always preceded by another Dominant Component (for example, Correction could only occur after students had been given an opportunity to solve a problem); and – Its primary mathematical function was to provide a worked solution and/or correct answer for students to compare with their own. One teacher cycled from Review to Instruction and then Correction several times within the first ten minutes of the lesson (US2). Whole class correction. The teacher, with varying levels of student involvement, provided a worked solution intended for the entire class. In addition, it was quite common for students to be invited to share solutions with the rest of the class. In a number of classes, students were invited to the board to record their solutions (US1, US2, J1). However, in one US classroom (US3), individual students not only wrote their solutions on the board, but were also required to explain their thinking to the rest of the class. In this classroom, students effectively took on the role of the teacher. Most times, this also involved answering questions from other students about their thought processes and mathematical reasoning. Independent correction. In one of the American classrooms (US3), students would commence with homework correction from answers provided on the overhead projector. Any problems completed incorrectly were required to be undertaken again. We feel that this activity, completed by students independently, constitutes a separate category. It belongs to both the Review and Correction Components as it goes beyond the passive correction of homework, requiring students to determine whether the provided answers match their own and, if not, to generate appropriate, correct solutions. ICONIC SEQUENCES

In deciding how to begin a lesson, a teacher may choose to combine the Dominant Components (listed previously) in a number of different ways to form various sequences. From our analysis of eight classrooms, particular sequences emerged as 55

CARMEL MESITI AND DAVID CLARKE

important because of their frequent use, and because our analysis of student actions and statements recorded during class and student descriptions and explanations provided in post-lesson interviews, suggested that the particular sequence was either highly valued by students or the teacher, highly effective in promoting student learning, participation or engagement, or sufficiently prevalent to represent a type of lesson commencement that was in widespread use. In combination, the identified Iconic Sequences represent the most generalisable result of our analysis of lesson beginnings. Sequence One: Familiarity Breeds Understanding

Figure 2. The Review-Correction Sequence

In the Familiarity Breeds Understanding sequence, the first ten minutes of the lesson begin with a Focusing Activity (Warm-up) involving short answer questions which is followed by Whole Class Correction of that activity. In terms of Dominant Components REVIEW is followed by CORRECTION. This sequence was particularly prevalent in US classrooms 1 and 2. As students arrived to class, and took their seats, they prepared to work on short answer questions that were written up on the board, or on a pre-prepared transparency. In most cases, we found that the teachers took the opportunity to greet the students and direct their attention to the work. It became quite clear that a focusing activity was a common occurrence, as not only did five of the ten US1 lessons begin in this manner, but students would enquire “We’re not doing our warm-up?” (US1-L02), if the activity were omitted. One interesting aspect of this activity was the students’ immediate engagement in the learning task; minimum teacher direction was required for students to begin working. Transcript 2: US1-L01 T

Okay guys let’s go ahead and get started on today’s warm up.

We found that while the students were engaged in silent, independent work, the US teachers would use this time to: mark the roll; distribute assignments, tests, worksheets and equipment.

56

BEGINNING THE LESSON: THE FIRST TEN MINUTES

After a period of student work that ranged from a few minutes to over ten minutes, we found two common approaches to the Correction phase of this sequence. At times the teacher was found to lead the Correction phase: Transcript 3: US1-L01 T

Alright. Let’s go ahead and try these. Take a look here. Alright number one. Just a little review of some of the things we were talking about last week

At other times the Correction phase involved student sharing of solutions: Transcript 4: US2-L01 T T T T T T T

Alright, let’s see Um … Glenna and um, Letitia come up and show us number one, the two of you. You can collaborate on the answer there. Um, the second one um, how about you Sandra, and um, Ashley. The two of you decide, put your verdict up there. Mm, number three, um, Derek and um, mm … Carl. Okay, four Keegan, how about uh, with Malcolm. Wait just a minute. Uh, just a moment. Excuse me, no. Okay, last one, who’s willing to go? Who’s willing to give it a try? Alright, David, you need a collaborator? Alright, you’re on your own. Okay. Who worked on this one? Okay, [chuckles] alright, ladies, ladies. Um, I equals P, P, I over P T equals R. Um, where did this come from? Why did you make that decision?

The activities themselves were generally standard problems that reinforced i) basic mathematical skills, such as problems involving ‘order of operations’ or ii) skill problems related to the classroom topic. They appeared to be designed for completion in under ten minutes. The Familiarity Breeds Understanding sequence also appeared in classrooms in Australia and Japan: Transcript 5: A1-L09 T

T

T

Open your exercise books. First question. Circle, draw a circle. [Draws a circle and labels it Q1] First question, find the circumference of this shape. Pardon? What’s ten? [to S1] Good boy. In your exercise books find the circumference of that shape on the board. [Draws a circle and labels it Q2] Question 2 is to find the length of this curved line, please. [Draws a semicircle and labels it Q3] Question 3 is to find the perimeter of this shape. [draws quarter circle and labels it Q4] [to Iva] You’re just finding the length of that curved line. Did you do Question 1? Good girl. And Question 2? Look at Question 1. You’ve got a full circle. And Question 2, really it’s the same circle, isn’t it, but we just want the distance of that curved line.

57

CARMEL MESITI AND DAVID CLARKE

Transcript 6: J1-L10 T S T S T

Um, as we have done so much work in the last class, I’m a little bit afraid that you have already forgotten some of them. Yeah, I might have. So, let’s use this worksheet to check if/ /Wait, wait, wait, wait, wait. Okay, um, let’s start working on this worksheet today. [Passes out the worksheets]

Transcript 7: A3-L07 T T

T T T T T

Okay. Could you have a look and please … Kevin, is there any reason why your books aren’t open? … Make it quick, please. Okay. First of all, if you remember from Friday. You probably don’t. We were talking about doing some rounding off. Okay. Could you please put today’s date. I’m going to put ten questions on the board. I want you to round these numbers off for me, please. To two decimal places. If you are having problems seeing from where you are seated, then you need to move, because the board’s in a strange position here. Okay. So I would like you to round off these numbers, please, to two decimal places. [writing on board] Okay, two decimal places it is. We did on Friday. [marking roll as she speaks] Yeah. Rounding them off, so they have two decimal places. Have a look back in your notes from Friday if you can’t remember. Remember, Nat? [continues marking roll] While you’re doing that, how many people are going out on sport tomorrow? Can you just put your hands up, please?

In all three examples, once the teacher had completed assigning the Review questions, they engaged in a Monitoring/Guiding practice, elsewhere identified as Kikan-Shido (see Chapter 4, this volume). The Japanese teacher invited her students to assist in the Correction phase of this sequence by drawing their solutions on the board, while the Australian teachers led the Correction phase themselves. Sequence Two: Connected Instruction

Figure 3. The Review-Instruction Sequence

58

BEGINNING THE LESSON: THE FIRST TEN MINUTES

In the Connected Instruction sequence, the first ten minutes of the lesson begin with a Recap/Run-through involving either examples from the homework or similar problems, which is followed by an instructional sequence involving an unfamiliar problem. In terms of Dominant Components REVIEW is followed by INSTRUCTION. In the Connected Instruction sequence the teacher typically revisits previous work with a worked example. The following excerpt illustrates the structure of this sequence in an Australian classroom (A3). The Australian teacher began by reminding students how to convert a fraction to a percentage using a calculator: Transcript 8: A3-L08 T T T

T

Okay, very quickly, I just quickly want to go over a couple of things first of all. The steps, when we’re using a calculator. Shayne, watch please. Okay, your calculator steps. Okay, if I have three and twenty and I want to make it into a percentage, Jason, how am I going to do that please? . . . . Yes. Yep, good. Three over twenty times a hundred over one if you want to put the one there as a fraction. Okay, steps on the calculator then are what Shayne? . . . . Okay, so you need to watch carefully, don’t you? Nat, steps on the calculator . . . . Yep, three divided by twenty, multiplied by a hundred. And my answer will be a percentage. Do it quickly please on your calculators.

The teacher continued by communicating to her students her intention to address an unfamiliar problem, however, she needed to review certain conversion facts before introducing the unfamiliar problem: Transcript 9: A3-L08 T

T

Okay, there are some things you need to know, you need to know some conversions. Arpard, put your pen down and watch up. First of all, some things that we need to know for this task. One kilometre. Teo, tell me how many metres . . . . Good glad you knew that one. Very important. Okay, one thousand metres. Okay. A centimetre, okay, let’s do a metre. A metre is how may millimetres. Oh good.

Once the students had been alerted to the skills with which they needed to be familiar and proficient, only then do we find the teacher addressed the unfamiliar problem: Transcript 10: A3-L08 T

T

We want to make things into a percentage. We’ve got two amounts there. We’ve got six hundred metres and we’ve got two kilometres. Okay, how we go about making it into a percentage. Any suggestions? Any ideas, any suggestions? Why do you say six hundred over three thousand Cam? Don’t know. Damn good guess. I like it. Was it a guess? Good.

59

CARMEL MESITI AND DAVID CLARKE T

It’s a good guess. What did you do to this, Cam, tell me . . . . Why did you do that? You did exactly what I wanted you to do. Any suggestions why?

After some discussion involving student responses, the class had arrived at an agreed upon approach for finding solutions to similar problems: Transcript 11: A3-L08 T T

T T

Imagine an athlete has completed eight hundred metres of a twokilometre race Obviously, eight hundred metres is less than half of two kilometres, so an answer of forty thousand percent is ridiculous. Yay, I agree. The error in the above calculation was in mixing different units. Metres and kilometres. Two correct calculations are shown below, and they’ve got a correct calculation down below. It says at the very bottom paragraph, it is important when expressing one quantity as a percentage of another, to make sure that both quantities are based on the same units.

The teacher proceeded to assign similar problems for student independent work. Connected Instruction, in which focused review led into new content, was a distinctive and efficient means of beginning the lesson. Sequence Three: From the Specific to the General

Figure 4. The Review-Correction-Instruction Sequence

In the From the Specific to the General sequence, the first ten minutes of the lesson begin with a Focusing Activity (Warm-up) involving short answer questions which is followed by student sharing of solutions and answers with the entire class. A similar problem is then posed to give students an opportunity to make connections and to encourage students to arrive at an alternative solution process. In terms of Dominant Components REVIEW is followed by CORRECTION and then INSTRUCTION. This sequence was only evident in one American classroom (US2). This intriguing approach of correcting work, encouraging reflection on the worked solution, with an intention to make connections with other areas of mathematics, and increase the mathematical sophistication of the initial problem was felt to be a signature characteristic of this classroom.

60

BEGINNING THE LESSON: THE FIRST TEN MINUTES

The following excerpts illustrate the From the Specific to the General sequence in US2, Lesson 5. The three questions below were assigned as Warm-up questions: 1.

1 1 1 1 + + + = 2 4 8 16

2.

1 3 7 15 + + + = 2 4 8 16

3.

>, <, or = to zero?

a)

1

127 128

b)

1

15 16

7 8

Students completed these questions independently and mostly in silence. After five minutes the teacher began the Correction phase of this sequence, whilst encouraging students to share their solutions and explain their approach: Transcript 12: US2-L05 T J T Ss C T Ss T

Alright let’s take a look. Um. How about the first one? Let’s see, um … Jong. ( ) How many people agree with Jong? [Several students raise hands.] What did she say? //Fifteen over sixteen, fifteen-sixteenths. //Fifteen over sixteen. Alright. Um, [T writes on board].

The teacher carefully worded her remarks to encourage student reflection: Transcript 13: US2-L05 T T

C T C

You know what? Somebody in period one actually said that the answer was one. Um. That’s not true of course. You’ve told me fifteen-sixteenths, but, um . . . . I’m wondering if somebody can think of a fast way, really fast, efficient way, to do that addition. And to convince that, that person that it wasn’t one. I think that person just was lazy, maybe. Um, thought it was one. [C raises hand] But I can come up with this answer really fast. It’s not exactly fast, but to make sure all the denominators are the same and then multiply the numerators by . . . . get it.

The teacher sought a second student response: Transcript 14: US2-L05 T A A T C T

Let’s hear Abbie’s comment. I would find out the answer by, one-half plus one-fourth equals one-third, er, three-fourths. Sorry. And, one-eighth is even less then a fourth, and one-sixteenth is less then an eighth, so it can never add up to one. Okay. Okay. Ohhh. There’s always a little missing piece.

The teacher sought a further response: 61

CARMEL MESITI AND DAVID CLARKE

Transcript 15: US2-L05 S S S

Well, this problem is really easy though becauseOne half is a, um, like the next … one half of the half of that is one fourth and half of that is, is, one eighth and half of that is one sixteenth. So all I do is cut the number in half and return, because I put eight if cut it in half record it after the four, plus two, plus, so that’s just my logic

At this point the teacher felt she should share her own reflection with the class: Transcript 16: US2-L05 T T T

Okay. There is- there is something something to do with halving here, isn’t there? Now, notice something about this. How far is fifteen-sixteenths from one?

She then posed a similar problem to the one just discussed, to encourage students to find a pattern that would assist them to determine an alternative solution approach: Transcript 17: US2-L05 1 1 1 + + = 2 4 8

]

T

Let’s try this one. [writes on the board:

T T T

What’s the common denominator there? How many eighths is this [points to board]? Four-eighths. One-half is four-eighths [writes on board]. Onefourth? Okay. One-eighth [writing on board]? [Writes on board] What’s the numerator? What’s theSum of the numerators there? How far is it away from one? [Teacher points to board.] [Quietly] Ooh.

T T T T T Ss

Of interest is her decision to not articulate beyond the following: Transcript 18: US2-L05 T

Say something to you? Think about it. Okay.

After this excerpt she continued to correct the rest of the Warm-up questions. We see in this example the craft of the teacher in deciding when to elicit the students’ thoughts and methods and when to contribute her own thoughts. Both actions are undertaken with the goal of assisting the students in constructing progressively more sophisticated mathematics. As has been discussed elsewhere, a major component of a teacher’s skill lies in the balance she constructs between eliciting and initiating (Lobato, Clarke, & Ellis, 2005) or between monitoring and guiding (Clarke, 2005, and Chapter 4, this volume). From the Specific to the General illustrates this skilful practice beautifully.

62

BEGINNING THE LESSON: THE FIRST TEN MINUTES

Sequence Four: Student-Led Corrected Review In the Student-Led Corrected Review sequence, the first ten minutes of the lesson begin with a Focusing Activity (Warm-up) involving Independent Homework Correction, an activity that belongs to both the Review and Correction Components. This is followed by Whole Class Correction of particular homework questions. In terms of Dominant Components REVIEW/CORRECTION is followed by CORRECTION.

Figure 5. The Review/Correction-Correction Sequence

This sequence began nine out of ten lessons of the US3 classroom data set. While the students were involved in the Warm-up activity the teacher performed several administrative and organisational duties, such as marking the roll and recording completion of homework. In contrast to the other Warm-up activities, those involving short-answer questions, the students in this classroom arrived to class with attempted homework questions. They were then given the opportunity to ‘grade’ their work. At first glance, this appeared an opportunity for students to mark their questions correct or incorrect, however, the student discussions at their group tables offered a valuable insight into their thinking during this phase of the lesson: Transcript 19: US3-L03 J C

I got one-o-four A wrong. Hey I got one-o-four B wrong too. I don’t know why I got one B- I don’t ( ) I got B wrong on that one.

63

CARMEL MESITI AND DAVID CLARKE J Ta T J Ta J M J Ta M J M J M J

Oh, I know why I got it wrong. I made it in a negative slope. Oops. Did you get one-o-two right? Who’s still grading their homework? Yes. Thirty? Yes I did. No, wait, yes I did. Very simple answer for such a big problem. Hey, stop, it was not. Simple answer, the answer was thirty. What problem? One-o-two Hey, no, why are you showing everybody? Hey, I got it right. I’m serious. ( ). Thirty ( ). I got it right. Wow, Mark got something right? I’m surprised. That one was easy though, all you had to do was set the ratio. Oh you had- did you draw the diagram? What diagram? It said draw a diagram.

It appears that students were not only determining whether their answers were ‘correct’ but also ‘making correct’ those that were incorrectly completed. In the above example, Tanya appears to benefit from the conversation about question one-o-two. Her exclamation regarding Judy’s perceived simplicity of the problem in question reflects her own difficulty in solving the problem. This act of ‘Listening-in’ is further discussed in detail in Williams (2006). Independent Homework Correction was then followed by a phase of Whole Class Correction. Typically in this classroom, this was student-led: Transcript 20: US3-L02 T T S T R R T R T S T R T R T R

64

Who’s still grading their homework? Okay. Okay, everybody finished grading? Okay, any questions? Any questions? Oh, Yolanda. Number eighty nine. Oh, I forgot to turn this in. Shh. Rachel, were you the original author over here? Come on up. Okay, the equation is … yeah. Eight. Like you have to do the A squared plus B squared is equal to C squared. You wanna write that at the top? ( ). Who’s my audience? ( )? Yolanda, and- and Serina. Serina’s over there. Yolanda’s over there, Carter is right here. Anyway, the eventual- the equation is um, I wouldn’t- I don’t know how to say that. Like eight root two, or eight squared orEight times the square root of two, or eight root two. Plus X squared. Um, equals nine root three squared. Um, so then you find this and it’s most- like Kerri, her thing was is that she did two, umWell just show- show us how to do it right, first. Bef- so we don’t get confused. Um, you go two- like the square root of two times eight, which is like eleven point something, and then you square that. And that’s one twenty eight. Plus X squared. Equals two forty three.

BEGINNING THE LESSON: THE FIRST TEN MINUTES R C T S S

And then, you subtract one twenty eight from both sides … and have X squared equals one fifteen. And then you square root it all, you get X equals ten point seven. Sweet. Good job. Um, see the- isn’t it because eight root two is a one number, right? You’re supposed to put parentheses around the eight times the square root of two. And then you square it. That means you’re squaring the whole thing.

This teacher was most encouraging of her students and she stated during the postlesson interview that her intention was to give everyone the opportunity to publicly explain a solution, that is, effectively, to take on the role of teacher at some point in the term. Transcript 21: US3-Teacher Interview 1 I

T

T

You often have students come up to the board to share their answers. How did you- how do you decide which students to go up and sort of why do you- why do you do that? Instead of you know, maybe you doing it? I used to do it in the beginning of the year, I do it all the time, it’s always me. And then as I sensed their level of confidence in competence, then I’ll say "well who got this right that would like to come up here?" I used to give them extra credit if they did it to encourage them, you know five little points added to their homework percentage or something. And they love that, they- I think they like to play teacher or something, and they like to show the other kids that they can do it. Um, and you know, they love it when they get it. How do I choose who goes up there? I try and pick everybody um, that wants to, there’s a lot that never have and just try and see who hasn’t had a turn. There’ll be some kids that would go up there every single day if you let ‘em. . . . . but I make them explain, talk and direct their explanation to the person that was asking, not just write it and then go sit down. So I feel they do just as good a job as I do. But if it’s really, really long, hard and confusing you know, II do it too.

This sequence appears to encourage student reflection, as time is intentionally allocated so that students may ‘correct’ their homework and ‘make correct’ those that are incorrect. Students are encouraged to learn from each other; from those that are in their ‘study teams’ and from those peers who participate in the public sharing and explaining of selected examples. Student-Led Corrected Review appears to be a sophisticated and effective method of Beginning the Lesson.

65

CARMEL MESITI AND DAVID CLARKE

Sequence Five: Pastoral Care

Figure 6. The Pre-Education Sequence

The Pastoral Care sequence highlights the teacher’s role as encompassing duties beyond simply mathematics instruction. In the examples provided below, the entire first ten minutes of these lessons was devoted to activities that: i) assist students to become more responsible for their learning; and ii) involve coordinating students’ participation in an extracurricular activity. While every teacher displayed some acceptance of pastoral care responsibility, in these two specific examples this activity dominated the beginning of the lesson. In the Australian example below (A2), the teacher devoted the first ten minutes to giving students advice about their responsibility as students. The context of the discussion involved the punctual submission of a homework piece: Transcript 22: A2-L09/10 T

T

T T

T

66

[to all] Okay I have a few things I want to talk about before I get down to what’s, what we’re doing today. I know that ah, you didn’t get very far with what we did last time and I’m going to spend a fair bit of time on that, as well. So don’t panic about that, and yes I have your books to return. Alright, but, to get you guys thinking and joining the words Leah . . . . responsibility. There are some things that you can control, some things that you can’t control. And that’s what responsibility is about. You can control things that you are responsible for. You can control your own actions. And some of us are slipping in the controlling of our own actions. And I say in terms of not getting homework done, in terms of trying to find other excuses for not getting things done, or not getting our work done in class or so forth. Alright. Responsibility has got a big I in it because I have to be responsible for myself. In a big way. It’s got three big I’s in it. To remind me three times that I have to be responsible for myself. And again I’ll say it, I’m not saying it because I’m wanting to have a go at people about what they’ve done, I’m trying to give you tactics to move on from here. We took ages to get that last homework piece done. I’m about to give you the next homework piece which will be due in a week. It should not take some people three weeks before they can get there. If you do not have the book then you ensure, you are responsible to ensure that either you have the book or you see me about getting a copy. Before it is due in. If you forgot it at home, if you did it early or something like that you make sure you are responsible for getting it in. It’s the I there.

BEGINNING THE LESSON: THE FIRST TEN MINUTES T

So this next homework will be in next Monday.

This teacher interpreted his role as extending beyond the teaching of mathematics only, to include the encouraging of particular work patterns and personal attributes. At heart, is the desire for his students to do well; this involves making students responsible, encouraging them to take control of their own learning and to see the role they play in their own education. The selective and purposeful devolution of responsibility from the teacher to the students is a phenomenon documented in the practices of many of the competent teachers in this study. In Lesson 14 of the Swedish classroom, the teacher devoted the beginning of the lesson of his mathematics class to coordinating student participation in an extracurricular activity. We have included it as an example of Pastoral Care as it is clear that the teacher values his students’ participation in whole-school activities not just classroom activities. He appears to acknowledge that his role is broader than and extends beyond the challenges that belong to the mathematics classroom. Transcript 23: SW1-L14 T

er my voice isn’t what it should be today so let’s start with ( ) games and you Agneta spoke with you yeah little bit yesterday right? S yes [to T] T about it S she wrote (loads of stuff) up on the board [to T] T exactly, er, but we’re going to hand in, er, I thought we’d start with V sack race //the canoe race M //( ) Viktoria it went well last year F (hope) ( )/ T /wait the canoe race. Is there anyone who wants to row? We’ve got yeah whole load then let’s see then so there are two girls who want to row so that’s two down T shh shh last year it was Miraz and Faro wasn’t it F yes T and who else of the guys was it who wanted to do it Johan and Hannes T then it’s them it’s them it’s Johan and Hannes who are doing it (them/today) (J) yeah we won Ss [talking] T four yes it’s got to be four T good er then we have the class relay T well it’s what they usually en shh it’s what we usually end er the whole day with it’s that’s it it was there that’s where we were good so I hoping for yeah profit there this year

67

CARMEL MESITI AND DAVID CLARKE

Sequence Six: Elaborate Problem Posing

Figure 7. The Elaborate Problem Posing Sequence

The Elaborate Problem Posing sequence provides an example of a detailed story delivered dramatically by the classroom teacher. In the Australian example below (A2), the teacher devotes the first ten minutes to telling a fictional story about Captain Compass: Transcript 24: A2-L07 T T

May the twenty-fourth is Captain Compass day, and it’s a bit of a fluke that we happen to be doing things involving the compass today on Captain Compass day. You go check it up in your, it’s a little print in your calendar, Maurice, stick with me. Course, as a student I assume you’ve all heard of Captain Compass.

His story-telling involved a dramatic, costumed presentation: Transcript 25: A2-L07 T

T

T

T

68

See, that’s why we’ve got the hat. To celebrate Captain Compass day. Captain Compass was famous. He was from the Andorra Navy. That’s why we’ve got this tub here today that’s from the Andorra Navy. It’s an old relic from the twelfth century when Captain Compass was involved in the siege of Andorra. And there he was, see, Captain Compass was very smart. He was what we call a tactician. In fact, he was better than a tactician, he was a mathetactician. Which means he was a tactician that was good at maths. . . . . A tactician’s someone who’s been good at plans and things, and … you know, he stood there … Captain Compass stood there. He was on the big walls of the city of Andorra. We had walls all ‘round it. . . . . City of Andorra was surrounded, by the enemy, who were the um, probably the ( ) so that they reached all the way out to Andorra. They were surrounded up there, he was standing on the walls, they had been under siege, ( ) which meant no one on the inside of the city could actually get any food or things in, or things out ( ) and they were going nuts because they had squeaky wheels and things, and Captain Compass had to work out a way of getting everybody out.

BEGINNING THE LESSON: THE FIRST TEN MINUTES T

T

T T

And as I said he was a mathematician, he wasn’t just a, a tactician. And Captain Compass had this one wooden leg, which was very handy. [rotates on one leg] And he could draw circles. Which is probably where he got his name, Captain Compass. Or maybe that’s where the word compass comes from, the compass. And he stood there and he could draw circles. But he was cleverer than that, ‘cause he knew he needed to do more than just draw circles. ! . . . . Um, he had to come up with a whole new tactic, a whole new angle, and he came up with ways of getting the compass to draw us more angles. And that’s what we’re going to look at. So what you are going to need to do. What you need to do firstly is get out a piece of paper, and I’ll get you out the rest.

This commencement to the lesson was followed by activities involving the construction of angles with the exclusive use of compasses, pencil and paper. The use of elaborate contexts to frame lesson content has been advocated in curricular materials in Australia for many years. For example, the Mathematics Curriculum and Teaching Program (Lovitt & Clarke, 1988) used the term ‘Storyshells’ to describe such framing contexts. The justification for the instructional use of such contexts takes several forms: – Motivational – students are interested in the story/context framing the mathematics. – Cognitive – the context provides a form of scaffold to assist the student in understanding and internalising the mathematics embodied in the context. – Utilitarian – the use of ‘real-world’ contexts in particular is often justified on the grounds that a student’s subsequent use of a particular mathematical concept or skill will depend upon their capacity to recognise its relevance in a variety of contexts. One teacher’s use of such Storyshells will differ from that of another teacher according to the connectedness of the context to the mathematics that is the intended focus of the lesson. In the example provided above, the elaborate and amusing storyshell appeared to engage students’ attention, but the context itself provided little justification for the relevance and/or utility of the lesson’s particular mathematical skills and concepts. If the function of the context is only motivational, then some of the potential power of contextualised mathematics may be lost. Nonetheless, the use of such elaborated contexts represents a distinctive and potentially powerful approach to Beginning the Lesson. CONCLUSIONS

Our focus in this chapter has been the question: What possible choices are open to the teacher in deciding how to begin a lesson, and how might these options be sequenced? By examining the practices of eight competent teachers in Australia, the USA, Sweden and Japan, it has been possible to identify particular Dominant Components (see Figure 1) from which these teachers crafted the effective 69

CARMEL MESITI AND DAVID CLARKE

commencement of their lessons. However, it is in the crafting of what we have called Iconic Sequences that we feel the expertise of the competent mathematics teachers is most visible and most readily related to the practices of other classrooms. The documentation of pattern and structure in the practices of competent teachers has its own practical legitimacy and value, but the “wisdom of practice” (Shulman, 1987) embodied in such sequences also has significance for our theorising about learning and instruction. Each sequence of activities with which a lesson is commenced presumes particular attributes in the learner and in the way the learner’s interaction with the content might most usefully be orchestrated. The teacher’s selection of a particular sequence to commence a lesson is indicative of several key elements synthesised in the practices of well-taught mathematics classrooms: – The relative emphasis the teacher accords to the cognitive and affective stimulation of the students at the commencement of the lesson (e.g., Sequence Five); – The situation of the particular lesson within the topic and curricular sequence and the nature of the connections (in both directions), which the teachers feel obligated to promote (e.g., Sequence Two); – The significance accorded by the teacher to such instructional strategies as repetition (e.g., Sequence One), challenge, or the elaborate contextualising of content (e.g., Sequence Six) (at that time, for that topic, and for those students); – The relative emphasis the teacher accords to specific or general mathematical formulations, the relationship between them, and the optimal development of that relationship (e.g., Sequence Three); – The teacher’s devolution of responsibility to the students for such activities as ‘correcting their work’ (e.g., Sequence Four). The decisions regarding the use of each of these Sequences is dependent on the instructional-learning situation as the teacher perceives it at that time, for that content and that class. Some teachers were highly consistent in the way in which they began their mathematics lessons. Some teachers appeared to be more selective and vary their deployment of particular Components and/or particular Sequences from one lesson to the next. This diversity or consistency could be seen in the initial coding of Components. For example, the American teachers all made use of Warm-Up, almost to the exclusion of any other strategy for beginning a lesson. By contrast, A1 and J1 varied the structuring of the lesson’s commencement as a consequence of their perception of the needs of the class and the demands of the content. A recurrent theme of this book is that teaching competence takes many forms. This chapter sets out some of the alternative ways that good teachers commence their lessons. Our analysis has not suggested that any particular approach should be valued over any other. All of the reported Iconic Sequences catered for particular student needs and met particular instructional goals. Teachers of mathematics would do well to consider adding some of these strategies to their instructional repertoire. 70

BEGINNING THE LESSON: THE FIRST TEN MINUTES

REFERENCES Clarke, D. J. (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. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K., Hollingsworth, H., Jacobs, J., Chui, A., Wearne, D., Smith, M., Kersting, N., Manaster, A., Tseng, E., Etterbeck, W., Manaster, C., Gonzales, P., & Stigler, J. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study. Washington, DC: NCES. Lobato, J., Clarke, D. J., & Ellis, A. B. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education 36(2), 101-136. Lovitt, C., & Clarke, D. M. (1988). Mathematics curriculum and teaching program: Activity bank, Vols. 1 & 2. Canberra: Curriculum Development Centre. Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review 57(1), 1-22. Stigler, J., & Hiebert, J. (1999). The teaching gap. New York: Simon & Schuster. Williams, G. (2006). Autonomous ‘looking-in’ to support creative mathematical thinking: Capitalising on activity in Australian LPS classrooms. In D. Clarke, C. Keitel & Y. Shimizu (Eds.), Mathematics classrooms in twelve countries: The insider’s perspective (pp. 221-236). Rotterdam: Sense Publishers.

Carmel Mesiti International Centre for Classroom Research Faculty of Education University of Melbourne Australia David Clarke International Centre for Classroom Research Faculty of Education University of Melbourne Australia

71

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

CHAPTER FOUR Kikan-Shido: Between Desks Instruction

INTRODUCTION

Of all the lesson events that might be observed in mathematics classrooms around the world, one of the most immediately familiar is that moment when the teacher, having set the students independent or group work, moves around the classroom. This chapter reports a fine-grained analysis of this lesson event in a selection of well-taught mathematics classrooms located in six different cities around the world. The Lesson Event is conceived as an event type sharing certain features common across the different classrooms studied. Each individual Lesson Event had a fundamentally emergent character, suggested by the classroom data as having a form sufficiently common to be identifiable within the classroom data from each of the countries studied. In each classroom, both within a culture and between cultures, there were idiosyncratic features that distinguished each teacher’s enactment of each Lesson Event, particularly with regard to the function of the particular event. At the same time, common features could be identified in the enactment of Lesson Events across the entire international data set and across the data set specific to a country. This chapter details the differences and commonalities of ‘Kikan-Shido’ (Between Desks Instruction) in eighteen classrooms located in Berlin, Hong Kong, Melbourne, San Diego, Shanghai and Tokyo. Methods of Instruction and Patterns of Participation Greeno observed that “Methods of instruction are not only instruments for acquiring skills; they also are practices in which students learn to participate” (Greeno, 1997, p. 9). With regard to the learning of mathematics, some classroom practices will resemble those of other communities who habitually employ skills specific to mathematics (the mathematical activities of accountants or surveyors, for example) and some practices will be classroom-specific in the sense of relating to the process of learning (providing particular forms of explanation, asking particular types of questions when in doubt, seeking and offering assistance, and so on). Greeno also made reference to “patterns of participation” developed by students (Greeno, 1997, p. 9). This is a particularly apt phrase, combining the

D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 73–105. © 2006 Sense Publishers. All rights reserved.

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

fluidity of participation in a social setting with the implicit regularity of a pattern. If we are to understand what occurs in social settings, it is the patterns of participation that are likely to offer insight. As will be argued, in considering social interactions in the classroom, the teacher must be considered co-participant with the students in any practices of the classroom community. Like Wenger (1998), this analysis of patterns of participation in classroom settings stresses the multiplicity and overlapping character of communities of practice and the role of the individual in contributing to the practices of a community (the class). Clarke (2001) has discussed the acts of interpretive affiliation, whereby learners align themselves with various communities of practice and construct their participation and ultimately their practice through a customising process in which their inclinations and capabilities are expressed within the constraints and affordances of the social situation and the overlapping communities that compete for the learner’s allegiance and participation. By examining classroom practice over sequences of ten lessons, the Learner’s Perspective Study (LPS) provides data on the participation of teachers and learners in the co-construction of the possible forms of participation through which classroom practice is constituted (cf. Brousseau, 1986). But co-construction of practice and joint participation in practice do not connote commonality of purpose among the participants in that (classroom) practice. To some extent both teacher and student share a common interest in advancing the student’s learning, but they are not positioned identically within that purpose (cf. Davies & Harré, 1991), and their classroom participation will both confirm these positionings and co-construct them. In this chapter, we examine the proposition that not only can the lesson event ‘Kikan-Shido’ (Between Desks Instruction) serve as the basis for useful comparison of classroom practice across several countries, but it also provides evidence for the co-constructed nature of a particular pattern of participation. This suggests that such Lesson Events, while deriving from the teacher’s instructional intentions and reflecting structural characteristics of the mathematics lessons of that classroom, also represent the consequence of a co-constructive process by which particular patterns of participation are established in the classroom. Classroom Practice is a form of communal collaborative activity as it is constructed through the participation of both teachers and learners and only understood (and optimised) through research that accords value and voice to all participants. It is for this reason that the Learner’s Perspective Study supplements the multi-camera documentation of classroom activity with post-lesson reconstructive interviews of the participants. Teaching and Learning are not simply distinct but interdependent activities that share a common setting, rather they should be conceived as aspects of a common body of situated practice and studied as such. It is ironic that recognition of this fundamental unity is enshrined in several languages other than English and that the dichotomisation of Teaching and Learning may be, in part, an artefact of our use of English as the lingua franca of the international education community. This chapter provides evidence of the mutuality of teaching and learning and supports their interpretation as components of a single body of communally constituted practice. We are assisted in this 74

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

argument by Harré’s work on social positioning (Davies & Harré, 1991) as this gives recognition to the mutuality of social practice, where the positioning of an individual carries both rights and responsibilities and is only sustained by mutual compliance. Of course, a position can be contested, and negotiation is a constitutive element of classroom practice (see Clarke, 2001). The Data This chapter reports the results of the Learner’s Perspective Study based on analyses of sequences of ten lessons, documented using three video cameras, and supplemented by the reconstructive accounts of classroom participants obtained in post-lesson video-stimulated interviews, and by test and questionnaire data, and copies of student written material (Clarke, 1998, 2001, 2003). In each participating country, the focus of data collection was the classrooms of three teachers, identified by the local mathematics education community as competent, and situated in demographically different school communities within the one major city. This gave a data set of 30 ‘well-taught’ lessons per school system (Berlin, Hong Kong, Melbourne, San Diego, Shanghai, and Tokyo), and, for the purposes of the analyses reported here, a total of over 180 videotaped lessons, supplemented by over 50 teacher interviews, and almost 400 student interviews. The teacher and student interviews offer insight into both the teacher’s intentions in the enactment of the particular Lesson Event and the significance and the meaning that the students associated with that event. Chapter Structure In the sections that follow, Kikan-Shido is defined and then discussed from several perspectives: its form as observed on the video record of class activity; its meaning as reconstructed by teacher and students in post-lesson video-stimulated interviews; and its function (intention, action, and interpretation). Our main purpose in this chapter is to use Kikan-Shido to establish the legitimacy and utility of Lesson Events as one basis for international comparison of classroom practice. A secondary purpose is to examine the legitimacy of the characterisation of KikanShido as a whole-class pattern of participation, and to situate the actions of teacher and learners in relation to this pattern of participation. It will be argued that while engaging in Kikan-Shido, the teacher and the students participate in actions that are mutually constraining and affording, and that the resultant pattern of participation can only be understood through consideration of the actions of all participants. Comparison of the enactment of Kikan-Shido across 180 videotaped lessons in the data set provides significant insight into the pedagogical principles underlying the practices of different classrooms internationally. In making this argument, we are positing Lesson Events as a category (and Kikan-Shido as a particular instance) with the capacity to sustain useful international comparisons of classroom practice.

75

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

Japanese teachers possess an extensive vocabulary with which to describe their practice. Among the large number of terms available to them is the term ‘KikanShido,’ which means ‘between desks instruction’, in which the teacher walks around the classroom, predominantly monitoring or guiding student activity, and may or may not speak or otherwise interact with the students. Our use of ‘KikanShido’ honours the existence in one language of an established term that succinctly encapsulates an activity that could only be described in English by an extended phrase or lengthy definition. The utilisation of such terms conforms to a tradition that has seen ‘déjà vu’ and ‘Schadenfreude’ assimilated into English usage for precisely the same reasons. Whenever a particular activity (in this case, a lesson event) is succinctly and accurately designated by a local term, and no equivalent label exists in English, it is entirely appropriate for an international study such as this to acknowledge that culture’s recognition of the activity by appropriating the local term for international use. So, for the purposes of this discussion, we will use the Japanese term, ‘Kikan-Shido,’ as a signifier or cipher for a general conception of the particular activity – one that takes into account the patterns of participation of both teacher and students in the activity designated by ‘Kikan-Shido’. Kikan-Shido was clearly recognisable in a variety of mathematics classrooms internationally, both to researchers and to classroom participants (teachers and students). For all classrooms in the data set, the activity of Kikan-Shido appeared to have four mutually exclusive principal functions: (i) Monitoring Student Activity, (ii) Guiding Student Activity, (iii) Organisation of on-task activity, and, sometimes, (iv) Social Talk. Each principal function is defined in Table 1. Table 1. Definition of the Principal Functions within Kikan-Shido

Kikan-Shido Between desks instruction in which the teacher walks around the classroom, predominantly monitoring or guiding student activity and may or may not speak or otherwise interact with the students.

76

Monitoring Student Activity The process by which the teacher observes the progress of on-task activities and homework, ascertains student understanding, or selects student work, with intent to keep track of student progress, question student comprehension and record student achievement. Guiding Student Activity The process by which the teacher gives information, elicits student response in order to promote reflection, or facilitates engagement in classroom activity, with intent to actively scaffold the development of student participation and comprehension of subject matter. Organisational The process by which the teacher distributes and collects materials, or organises the physical setting in the classroom, with intent to support interactions among students and facilitate student engagement in learning activities. Social Talk The teacher engages with student(s) in conversations not related to the subject matter or current on-task activity.

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

Each principal function comprises a number of activity codes that have recurrent form across all 180 taped lessons (see Table 2).

Guiding

Monitoring

Table 2. Activity Codes Defined Selecting Work Students are chosen to share their work, methods or thinking with the whole class. This may occur immediately or later in the lesson. Monitoring Progress Teacher walks around the classroom observing student progress of on-task activity. Questioning Student An expression of inquiry that invites or calls for a reply from a student that may or may not be related to the current on-task activity. Monitoring Homework Completion While students are engaged in on-task activity, the teacher observes the completion of homework and may note student achievement or understanding of subject matter. Encouraging Student Activity pursued by the teacher intended to motivate, provide support and feedback to individuals or groups of students. Giving Instruction / Advice at Desk Teacher scaffolds the development of students’ understanding by providing information, instruction or advice, focusing on the development of a concept that addresses meaning, reasoning, relationships and connections among ideas or representations, or the demonstration of a procedure. Guiding Through Questioning A series of specific teacher questions intended to scaffold the development of student understanding of a procedure or concept during the on-task activity. Re-directing Student Activities pursued by the teacher to regulate the behaviour of student(s) who are perceived not to be paying attention to the current activity, and to support students’ on-going engagement during the lesson. Answering a Question Information given by the teacher when requested by a student.

Social

Organisational

Giving Advice at Board Instruction or advice given while an individual or group of students work at the board. The instruction or advice may be intended for those students working at the board or may be intended for the whole class. Guiding Whole Class Teacher walks around the classroom and provides information, instruction or advice intended for the whole class. Handout Materials Teacher walks around the classroom distributing materials related to on-task activity. Collect Materials Teacher walks around the classroom and collects materials from students. Arranging Room Teacher repositions furniture to enable independent, paired, group or board work. School Related Teacher engages in conversation related to school activities or curriculum. Non-School Related Teacher engages in conversations of a social nature not related to the subject matter or on-task activity.

77

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

Monitoring Student Activity is made up of four activity codes: (i) selecting work, (ii) monitoring progress, (iii) questioning student(s), and (iv) monitoring homework completion. Guiding Student Activity comprises seven activity codes: (i) encouraging student(s), (ii) giving instruction or advice at the student’s desk, (iii) guiding through questioning, (iv) re-directing student(s), (v) answering a question, (vi) giving advice at board, and (vii) guiding whole-class. Organisational consists of three activity codes: (i) handout materials, (ii) collect materials, and (iii) arranging the room. Social Talk comprises two activity codes: (i) school-related talk, and (ii) non-school-related talk. Table 2 presents the definitions for each activity code. Where it occurs in Tables 1 and 2, the term ‘scaffold’ is used to designate teacher support for student construction of knowledge. The theoretical difference between Monitoring Student Activity and Guiding Student Activity is similar to the difference between elicitation and initiation as these are theorised and discussed by Lobato, Clarke and Ellis (2005). Initiating/eliciting is not a simplistic dichotomy like “tell/not tell”—it’s not an either/or. Both categories of action are necessary and their use is interrelated . . . . Elicitation occurs when the teacher wants to learn more about students’ images, ideas, strategies, conjectures, conceptions, and ways of viewing mathematical situations. When the teacher’s communicative act functions to facilitate the expression of the student’s mathematics, then this constitutes “eliciting.” . . . . Initiating is often preceded by eliciting, so that the teacher can gather information about students’ thinking before making a judgment whether to work with and structure the students’ ideas or to introduce new information. Initiating involves the insertion of new ideas into the conversation, ideas that the teacher assumes will be interpreted in many different ways rather than passively received. Once the teacher engages in initiation, she then steps back and elicits to see what the students did with that information. Both actions have their function within the teacher’s promotion of student conceptual development (Clarke, 2005, pp. 13, 14). The distinctions between each principal function and each activity code are substantive. Each principal function and corresponding activity code are empirically grounded and the application of all principal function codes and activity codes listed in Tables 1 and 2 were subjected to inter-rater reliability checks and a level of greater than 80% was consistently maintained. Within those events classified as Kikan-Shido, both the principal codes and activity codes are coverage codes, since they are mutually exclusive and, in combination, account for all documented activities. Using StudioCode video analysis software, it was possible to code for Kikan-Shido, and its various functions, as they occurred in the video record (Figure 1). Using the coding system as shown in Figure 1, we can map the various activity codes to a timeline of a single lesson (see Figure 3). For the purpose of statistical analyses, the individual timelines from each lesson were combined to identify the frequency of each activity code.

78

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

Figure 1. Kikan-Shido code input window

While the participants in many classrooms may conspire in the enactment of Kikan-Shido, the actual functions served by Kikan-Shido help us to distinguish one classroom from another. The ways in which different teachers initiate Kikan-Shido are diverse and distinctive. This can be seen graphically in the comparison of 180 lessons across six countries in the LPS data set (see Figure 2). An essential point must be made here: we have analysed sequences of ten or more lessons taught by eighteen teachers designated as competent in six different countries. We do not presume to characterise the teaching of a country or a culture on the basis of such a selective sample. Nor do we intend to compare teaching in one country with teaching in another. Most importantly, we commenced our analysis intending to compare and contrast teachers and their classrooms, not cultures. As will be shown in the results that follow, a particular practice documented in one American classroom might also be a distinctive feature of a classroom in Japan. Where such classroom practices are found in such culturallydisparate circumstances, the particular practice assumes heightened significance. That fact that teachers situated very differently have developed similar solutions to a particular classroom challenge suggests not only the generality of the pedagogical strategy but also its cultural transferability. The occurrence of such culturallydistributed practices problematises simplistic East-West comparative cultural analyses. Figure 2 graphically illustrates both the similarities and the significant differences in the way that 18 competent, experienced teachers enacted the lesson event that we have called Kikan-Shido. For example, A-T3 and US-T3 both devoted about 45% of their class time to Kikan-Shido, but Figure 2 makes it clear that the relative weightings of monitoring versus guiding activity were completely different. If we compare G-T3 with HK-T3, we find similarity not only in the time devoted to Kikan-Shido, but even in the relative proportions of monitoring and guiding. 79

Figure 2. Comparison of Kikan-Shido across 180 lessons

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

80

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

However, at the next level of analysis, we find significant differences in the manner in which the monitoring and guiding activities were carried out. The teacher from US School 3 made extensive use of Kikan-Shido in every lesson and for extended periods of time. Generally, the teacher engaged in KikanShido during Warm Up or after setting a learning task. One-third of the teacher’s extensive Kikan-Shido activity was committed to monitoring homework completion. Interestingly, if this component (monitoring homework completion) were removed from US-T3’s Kikan-Shido record, her use of Kikan-Shido would closely resemble that of A-T1, even to the relative proportions of the activity codes. Monitoring homework completion was an administrative responsibility that clearly influenced the classroom practice of US-T3. Real understanding of the decisions and pedagogical principles underlying each teacher’s classroom practice is only evident from a fine-grained analysis of Kikan-Shido as it was enacted in each classroom. INDIVIDUAL TEACHER USE OF KIKAN-SHIDO

Differentiated Instruction for Individual Students During Kikan-Shido, Australian Teacher 1 monitored student progress with the ontask activity (11.2% of total lesson time over all sampled lessons coded as Monitoring Progress). Drawing on the insights gained from observations of students at work, the teacher appeared to adopt different strategies for individual students with the intention of facilitating student understanding. This is illustrated in the following quote. A-T1 Int A-T1

Int A-T1

Int A-T1 Int A-T1

I have a different intention for each student that I approach I think. Can you tell me a little bit more about that? For example, Earl. These boys they um … really get into it … and want to be right and want to solve it … and they will. Earl will appreciate … oh actually … oh most of them do if I come around and they want me to check their work … and so they will wait patiently until I come around- so that’s what they do. Kamahl sits there and has a lot of trouble … but doesn’t take many steps to help himself … at all unfortunately. Because he doesn’t want to appear that way. Sometimes … um … there are a couple of people I need to hint and ask them what they’re thinking … because they are not comfortable with the work. Mmh [softly] This is the sort of work that would apply to that - … about half the class would really appreciate but the other part would just wish that I just … told them how to do it and they could just repeat it. Okay mmh [softly] So it doesn't suit everyone at the moment … this sort of question. Right … who are the ones who would be enjoying it and who are the ones who wouldn’t? Sandy wouldn’t, Sandy wouldn’t enjoy this. … Mel’s a- Mel asks questions, that’s the first thing she does, she doesn’t think

81

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE … she doesn’t even take a moment to think about it. She, she’s a, she’s very verbal, she will say what she’s thinking and ask immediately so she’s not afraid to ask but I’m not going to answer her questions necessarily but she’ll say no I don’t know how to do this, she won’t take … a moment she’s very … immediate. Key to symbols used in transcripts in this chapter … A pause of one second or less ( ) Empty single parentheses represent untranscribed talk. The talk may be untranscribed because the transcriber could not hear what was said .... Omitted text // Marks the beginning of simultaneous speech. (text) A plausible interpretation of speech that was difficult to hear [text] Comments and annotations, often descriptions of non-verbal action text Italicised text indicates emphatic speech text Underlined text indicates emphasis added by the authors.

Motivational Support and Encouragement Encouragement was coded and clustered within Guiding Student Activity. It is distinguished from social categories such as Non-school Related Social Talk as it was clear, both to researchers and to classrooms participants (teachers and students), that this strategy was an instructional act related to the on-task activity and intended to motivate and to provide support and feedback to individuals or groups of students. On many occasions, Australian Teacher 1 would provide verbal encouragement to individual students (see Figure 3 for Teacher 1’s utilisation of Kikan-Shido across all ten lessons). In fact, the practices of all three teachers in Australia and those of US Teacher 3 appeared to prioritise the development of student confidence by providing motivational support and encouragement. A-T3

She needs that encouragement … she's not particularly independent and she's not well skilled and she relies heavily on a lot of other students … on this day she was by herself doing the task … and that was really pleasing … mmm.

Such explicit encouragement was much less evident in the other classrooms studied. In fact, the teachers in the Asian data set (Shanghai, Hong Kong and Tokyo), with the exception of Shanghai Teacher 2 (0.4% of total class time devoted to Encouraging student), typically did not encourage students during Kikan-Shido. On the occasions when encouragement was given, it was directed at individual students or the whole class.

82

Figure 3. Australian Teacher 1’s utilisation of Kikan-Shido across a ten-lesson sequence

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

83

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

The only instance of Encouraging the Student coded in Shanghai School 1 illustrates a unique strategy that was employed by the teacher intended to encourage, motivate and provide feedback to individual students while addressing the whole class: SH-T1

Be quick – finish the other one. Eh, [to whole class] some of you drew it very well. [Points to student 4’s work] You drew it wrongly. [To student 5] You also were wrong. [To student 6] You. You speed up [moving down the row]. You did it right [pat on the back of student 8] [taking up the paper of student 9]. Eh, he did it right. [to whole class] Student 9 also did it right.

In this example, the teacher draws the attention of the class to the student’s error. While the teacher’s intentions appear to be motivational, there is no example of this strategy (public announcement of student error) in the Australian, American, German or Japanese data. Such statements were recorded in SH1, SH2, HK1 and HK2. This suggests that encouragement and motivation in these four classrooms were predicated on a value system different from that operating in non-Chinese classrooms. The Distribution of Responsibility for Knowledge Generation Another characteristic of Kikan-Shido, as it is practiced in US School 3 and in Shanghai School 2, is the implicit devolution of the responsibility for knowledge generation from the teacher to the student, while still institutionalising the teacher’s obligation to scaffold the process of knowledge generation being enacted by the students. US-T3

And then every class they do group things, and so they are pretty used to it. And then if the whole group doesn't understand they can raise their hands and the first thing I will do is make sure that they've talked to each other. And then I say what did you try, show me what you tried first.

This echoed the statement by many students in US3. One example is cited below. S1 Int S1

And whenever we have a question we have to ask our- our table, and if no one knows then we raise our hand and ask the teacher. Oh, okay. And what does Mrs. J do when you ask her? She helps you.

‘Teacher help’ usually came in the form of instruction or advice at a student’s desk, advice at the board or answering a question. US-T3

84

Well I'll give 'em little hints or I'll tell 'em, you know look at C, or where is your part B you didn't write that out, I think if you write out and do all of part B then you'll see it. But I- they're at that point yeah, I give 'em little helps. Now if there's two and three groups and they're all having the same questions, that's when you know, you're supposed to stop, bring everybody back to the overhead and actually go through more examples or work with them through

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION the really hard part that they're not getting, so that you don't have to go say it, you know, six and eight individual times. And you just have to judge when that is, you know when they all- it's better use of your time to stop and bring them all back together and tell them all.

In Shanghai School 2, the students rarely talked directly to each other – classroom conversation was always mediated by the teacher. This high level of teacher orchestration might not appear to devolve responsibility or agency to the students; however, in one lesson, the teacher said, “Look at Shiqi’s solution! This is much better than the usual method, copy this down.” This public recognition of novel student work signifies a willingness by the teacher to assign students an active role in the generation of new mathematical knowledge. On other occasions, the same teacher would give quite explicit instruction to the class. The contemporary reform agenda in Australia and the US actively encourages teachers to elicit student mathematical understandings but is much less explicit about when the teacher should contribute their own mathematical knowledge to classroom discussion (e.g. Chazan & Ball, 1999; Wood, Nelson & Warfield, 2001). In Shanghai School 2, we see the interweaving of teacher initiation and elicitation (Lobato, Clarke & Ellis, 2005), which sustains a mutuality and complicity between teacher and students in the construction of (mathematical) knowledge. The practices of SH-T2 provided some powerful supporting evidence for the contention by Huang (2002) and Mok and Ko (2000) that the characterisation of Confucian-heritage mathematics classrooms as teacher-centred conceals important pedagogical characteristics related to the agency accorded to students; albeit an agency orchestrated and mediated by the teacher. Our analyses have demonstrated the utility of ‘the distribution of responsibility for knowledge generation’ as an explanatory framework capable of distinguishing usefully between classroom practices in both ‘Asian’ and ‘Western’ settings. Kikan-Shido as a Class-Debugging Mechanism Kikan-Shido could also serve a diagnostic role. The diagnosis could relate to particular student difficulties or to the relative effectiveness of a teacher’s explanation during a whole-class discussion. Upon identifying such difficulties or concerns during Kikan-Shido, the teacher could act in several different ways: either by giving individual students personal assistance, if the problem was not a general one, or through whole-class discussion of a misunderstanding or difficulty that appeared to be widespread. For example, German Teacher 2 provided instruction and advice intended to address and guide the whole class. One example taken from the video record is cited below: G-T2

Right, then let’s stop at this point immediately now … The … walking around has shown to me many … also small technical difficulties that you have, and now we’re going to try one by one …

85

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

This is echoed by one student: S

Int S

Then if we’re working on a problem and everybody has a different answer and we don’t know why, then we ask Mister I … then he like comes over to us and explains it … to all of together and then … and if we still don’t get it, then he usually calls all of us up to the board, or calls one of us to the board and then the whole group, I mean the whole class listens, … explains the problem, what the problem was and then he explains it to the whole class. What would you do if you couldn’t ask Mister I here? All right, then I would ask the whole class.

One of the Australian teachers (A-T1) used Kikan-Shido as an opportunity to gauge the success of her whole-class introduction of new content. In one instance, her use of Kikan-Shido revealed widespread student difficulties. A-T1

Int A-T1

Oh … this was terrible … as soon as I started going around oh I felt bad about this but it didn’t matter too much … that I hadn’t talked about- I assumed … that they knew what the base and height of a triangle … is … and how to recognise it … and … I might have gone to Kamahl first or to someone and it just sort of … was made very obvious that I hadn’t … but that that’s also another thing that I do, I do go to see them straight away so they can tell me … what they don’t understand- that that gives me a much better … understanding of whether … what I have done up the front is of is of any value at all. And then gives you a chance to (ask them). And gives me a chance yes … and then I went around and checked with … with some key students whom I know struggle and I was … feeling quite confident about it after that although … my explanation here isn’t all that crash hot I’ve … probably because I hadn’t had a chance to really think about it and I wasn’t sure how to … how to make it clear … like how you’d identify the base and the height … cause I sort of … because I hadn’t really thought about it … it was kind of a gut … what do I do how do I help them how do I help them … so it wasn’t … [sigh].

Monitoring and Guiding as Teacher Characteristics The relative frequency of Monitoring and Guiding emerged as a key characteristic of individual teachers. This characteristic and the actual form taken by each teacher’s use of Monitoring and Guiding could be used to distinguish one classroom from another. For example, the practices of Shanghai Classroom 2 and Hong Kong Classroom 3 appeared to be predicated on different pedagogical principles. Specifically, in Shanghai School 2, the teacher appeared to assume a capacity in the students to develop new mathematical knowledge. In post-lesson interviews, the teacher made comments such as: SH-T2

86

During the process, don’t teach them mechanically, don’t teach them mechanically, let them brainstorm, enhance their flexibility . . . . I was not afraid that students had all sorts of questions, I just let them appear … Sometimes if you

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION restrict them from doing this or that, their problems won’t appear, right? But the problems will appear tomorrow even if they didn’t today, right?

This teacher made reference in all three interviews to an activity that was translated as inspecting around, which was identified in our analysis as Monitoring Progress. SH-T2

I inspected around and took a passing glance. You have to discover those good points from the students. If there are any mistakes, you have to sort them out.

In several interviews, students in SH2 expressed their appreciation of teacher explanations and the correction of their errors in both whole-class and one-to-one settings. S

The teacher explained our mistakes to us. Then we became more careful when we worked on the exercises.

Consistent with the conclusion drawn by Huang (2002), the practice of Hong Kong Teacher 3 appeared predicated on different pedagogical principles from those underlying the practice of Shanghai Teacher 2. While the dominant function of Kikan-Shido in Shanghai School 2 was to Monitor Student Activity (20.5% of total class time), in Hong Kong School 3, a larger proportion of time was devoted to giving direct guidance (21.9% of total class time). The teacher would walk around the classroom in order to help students with their difficulties, and the guidance during Kikan-Shido was typically quite directive, as illustrated in this example: S HK-T3 S HK-T3 S HK-T3

[in Chinese] Come here! Come here! Hey! Hey! Come here! I don't know how to do question four! ( ) [in Chinese] A little bit different! This time … these two … Both twenty-one and twenty-four are multiples of three! [to T in Chinese] Yes! Just to simplify it? Okay. [in Chinese] It isn't to simplify it! It can't be simplified! This one no either ( ) this one is okay! This one can be simplified but this one cannot. [to T in Chinese] Then how? [in Chinese] So … this one is okay! This can be simplified! You have to divide this by seven and then multiply it by eight.

The significance of the teacher’s guidance was acknowledged by the same student in the post-lesson interview: Int S Int S

Why is it important? This part? It’s important because he came over to teach me. Mm, why is it important? I didn’t know how to do it, he came over to teach me, then I can do it.

“He came over to teach me, then I can do it” is reminiscent of student accounts of effective teaching elicited in research studies in the days before video-recording of classroom activities (e.g. Clarke, 1985). Lacking video records to assist (and to corroborate) their reconstructive accounts, descriptions of classroom practice were provided by students in response to such prompts as “Think of the best mathematics teacher you ever had. What was it she did that was so good?” 87

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

Students would routinely reply, “She explained things really well.” It is now clear, from comparison of students’ descriptions of teachers’ instructional acts with the video record of those acts, that such student accounts are constrained by the students’ lack of a technical pedagogical vocabulary and their difficulty in recognising many of the subtle and sophisticated techniques employed by their teachers to elicit the students’ mathematics and to initiate into the classroom conversation elements of the teacher’s mathematics (Clarke & Lobato, 2002; Lobato, Clarke & Ellis, 2005). The research methods employed in this study allow the student’s account to be juxtaposed with the researcher’s inferences from the video record, resulting in the constructive elaboration of both accounts. Code Switching In Hong Kong School 3, while the instructional language is English, the teacher found himself faced with students’ constant demands for Chinese explanations, which were clearly conveyed in one student’s request: 講中文啦! [“Chinese please!”] (HK3-L04). This type of teacher-student interaction was especially evident when it came to introducing a new mathematical term: HK-T3 HK-T3 S HK-T3 S HK-T3

Now, I want you to meet with the word. Now I want you to meet with the word ‘simultaneous equations’. Now, what do we mean by the word “simultaneous” -huh? Chinese meaning. 聯立方程 [Lian Li Fang Cheng – literal translation: Simultaneous Equations] 聯咩話 [Lian Mei Hua? – literal translation: Simul-what?] 立呀。聯立。聯立方程。[Li Ya. Lian Li. Lian-Li-Fang-Cheng – Literal translation: Taneous. Simultaneous. Simul-taneous Equa-tions.]

Code switching in this classroom appeared to be a normative practice that was coconstructed by both the teacher and students, and which was predominantly enacted during Kikan-Shido. The co-constructed nature of this practice was evident in the teacher’s statement quoted below, in which he clearly conveyed the challenges of choosing the instructional language to address students’ needs, while still maintaining the institutional agenda. HK-T3

88

This one, when he cannot follow he asks his neighbour. In this way you will see that when I teach I use English. But if they have questions, when they have questions, I will use Chinese. When they individually ask me questions we have to use Chinese. Because umm, the students from my observation, I mean from my observation in the past, their standard of English has not reached that level … This student, the first one, um, he always tell me that he can’t do it, he can’t understand people speaking in English. He says I have to speak in Chinese. I have told him many times that hey, here, this school is an EMI school. It’s impossible that I teach in Chinese. Therefore, you’ll see that he’s either expressionless or asking the two classmates sitting behind how to do it.

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

Selecting Work Another distinctive feature of Kikan-Shido, as enacted in Shanghai School 2, Japan School 3, and US Schools 2 and 3, is the Selecting of Student Work to be shared with the whole class, either immediately or at some later time in the lesson. The teacher in Shanghai School 2 made use of selecting work in almost every lesson, and usually more than one student was selected to present their solutions on the board. In this example, the class was attempting to solve the following systems of equations using the method of elimination by adding/subtracting.

(A)

!2x y = 5 " #3x + 4 y = 2

[Eq. 1]

8x 4 y = 20

[Eq. 3]

[Eq. 2]

(B)

!3x + 4 y = 16 " #5x 6 y = 33

[Eq. 2]

9x + 12 y = 48 10x 12 y = 66

[Eq. 3] [Eq. 4]

[Eq. 1]

19x = 144 x=6 SH-T2

SH-T2

S4 SH-T2 SH-T2

[T walks around. Picks up S1’s sheet. Problem A] Revise this bit, best if you write down equation [number] three, equation one plus equation three, oh no three plus two. Write down equation three properly here. Copy your work on the blackboard …. [to S4. Problem B] This became equation number three, eh write three down here – what does this become? Equation four. Equation three plus equation four. Um, they can be divided? Six? Is it six? Six. Yep, very good, keep working. Write it on the blackboard, you did it well today, come on and write it on the blackboard. … [to S6. Problem B] Write it on the board, that side.

In solving these two systems of equations, this teacher deliberately Selected Work from students to represent a variety of solution methods (in this example, the second problem was solved differently by S4 and S6), and these solution methods were subsequently shared among the class. It is clear that in this classroom, Selecting Work was an instructional strategy that was purposefully employed by the teacher to give voice to students' ideas as well as to share alternative solution methods with the whole class. The teacher’s deliberate use of this strategy to distribute responsibility for knowledge generation was made evident in the postlesson interview, where he emphasised the importance of allowing student misunderstandings to be made visible to the whole class. SH-T2

I think it is no big deal even if they are wrong. Just let them appear, right? Through the corrections amongst the students and the interactions between the teacher and students, you can correct them … I truly feel that, during so many years of teaching, I have many ways of solving a problem, where do they come from? From their [the students’] supplementary remarks.

89

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

The students in this classroom appeared to share the same valuing, as was explicitly acknowledged by one student in this example: S

Just then the teacher was listening to their discussions and corrected them, and then he invited a classmate to do it on the blackboard. The teacher always lets us, he listens to our opinions, and then chooses a correct one or a few and a wrong one, to make a comparison, to find out which is correct, which is wrong … to let us compare and find out where the mistake is, mainly to make it clear which one is which, which one is made incorrect more often, and so we have to correct such a habit.

The student identifies two important teacher practices: (i) The involvement of the students in identifying errors and misconceptions and (ii) the highlighting of those misconceptions that are most common. The student’s repeated reference to “our,” “us,” and “we” confirms the success of the teacher’s devolution of responsibility to the students. Orchestrating Whole-class Activity Although most of the teacher support during Guiding Student Activity was directed to individual students, teachers (particularly in A2, G1, G2, HK1, HK2 and J3) repeatedly provided information, instruction or advice intended to inform the whole class. This type of activity was coded as Guiding Whole Class within the code Guiding Student Activity. The exercise of Guiding Whole Class during Kikan-Shido suggests that the teachers attached sufficient importance to the class learning as a whole group, such that they would give guidance to the whole class, when this was judged to be appropriate, while also continuing to give assistance to individual students (Hino, 2006). Guiding Whole Class was enacted differently according to the teacher’s judgment of the situation: either upon perceiving the difficulties among students to be general, the teacher would interrupt students’ work by making clarifications to the whole class; or the teacher would provide information, instruction or advice to the whole class during Kikan-Shido as a way of orchestrating whole-class activity. On identifying the common mistakes among the students, Hong Kong Teacher 2 would give instructions to the whole class while walking around in order to remind the class of the errors they made or tended to make. In the example below, instructions to the whole class were interspersed with comments to individual students. HK-T2 HK-T2 HK-T2

90

[to VANESSA] Young lady, you've copied down the question wrongly. You are really overtaken by the twins! [to S] What's wrong? Okay. [to whole class] Hey, be careful with one thing. You've got one thing, your fatal mistake is miscopying questions. Very often you copy from your book wrongly, or you've copied the first thing correctly, but you get it wrong in the second step. Is this illusion or what? Is this a kind of 'sense dis-coordination'?

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

On many occasions, German Teacher 3 would Guide the Whole-Class Activity through a series of specific teacher questions and explicit instruction intended to scaffold the students’ understanding. For example, during an on-task activity where the students were asked to determine the surface area and volume of a rectangular prism they had made (see Figure 4), the teacher walked around the classroom Monitoring Student Progress and repeatedly Guided Student Activity by questioning and instructing the whole class.

Figure 4. Orchestrating Whole-class Activity

One example taken from the video record is quoted below: G-T3 S S G-T3 S S S G-T3 S G-T3 S G-T3 G-T3 S G-T3

If you put this together like this //then you’d need two squares too but do I have a square that big here? //Yeah then it works without it too. No. I don’t have one. And two rectangles would they fit in there? No. Yeah they would ( ). And four of those things wouldn’t fit either. When would two rectangles fit in there? If you put them in like this. When would exactly two such rectangles fit in there on each … side … when? If the rectangles … building houses again. Check this out. Is this exactly half … of this long side? Yeah. No, it’s not half, it’s more, that means I wouldn’t be able to fit two next to each other so I wouldn’t be able to stick in two rectangles next to each other. I would have to have a bigger square but we don’t have one. You should stick them together along the long edge then it works yeah. Because top surface and base are supposed to be the square. So. That’s right.

Here the Teacher alternates between questioning students and explicit instruction. The transition from questioning students to instruction appears to be predicated on the Teacher’s judgment of the level and frequency of student difficulties.

91

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

VARIATION ACROSS THE TEN-LESSON SEQUENCE

While Kikan-Shido has a recognisable structural form evident across all classrooms, the variation in the amount of time devoted to Kikan-Shido suggests that it is employed purposefully in distinctive ways in each lesson. For example, in US1 there was significant variation in the amount of time the Teacher devoted to Kikan-Shido in each lesson (see Figure 7). While Kikan-Shido represented only 8.8% of total class time across the ten-lesson sequence, the teacher made extensive use of Kikan-Shido in Lesson 7 (35.2% of total class time). Similarly, the amount of time allocated to Kikan-Shido in HK3 also varied across the ten-lesson sequence (see Figure 5). It is clear that this teacher devoted significantly more time to KikanShido during the last five lessons, especially in lessons 6 and 9. The variation in the use of Kikan-Shido across the ten lessons taught by HK-T3 (see Figure 5) confirms the point made in Chapter 2 of this book, that any attempt to characterise a teacher’s practice by a single lesson pattern ignores the teacher’s purposeful selection of structural elements according to the location of the lesson in the topic sequence. The same purposeful variation is evident in Figure 7, which documents US-T1’s selective use of Kikan-Shido across the ten-lesson sequence.

Figure 5. Variation in the utilisation of Kikan-Shido across a ten-lesson sequence

Figure 5 suggests that this variation is related to the location of the lesson in the topic sequence. The first five lessons have more direct instructional components (such as teacher demonstration) and therefore require less one-on-one student scaffolding. However, the following excerpt taken from the post-lesson teacher 92

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

interview suggests that the teacher’s utilisation of Kikan-Shido in Lesson 9 was also dependent on the degree of difficulty of the subject matter, the level of student comprehension and student willingness to learn. HK-T3

Int HK-T3

Int HK-T3 Int HK-T3

Actually, today there was one problem. I didn’t expect that there would be so many hands up. So that you see that the situation was relatively messy. And I couldn’t answer them all. Ha ha. Many students called you? Yes. It’s at the same time they suddenly … many … many of them had something to ask. It is different from the previous lesson. They didn’t have questions in the previous lesson. Maybe it was about integers and was easy. The calculation was easy. So they thought they could do it. So is today’s lesson content difficult to them? Yes, it’s more difficult. In today’s lesson so many students asked you questions. What do you think about it? Talking about asking questions they are already quite active in today’s lesson. You’ll see that in some lessons, they … comparatively the children are more talkative. They don’t ask questions, but they are talkative … Yes. They chat among themselves. But today actually they do learn. I mean, you’ll see that they are willing to learn. So they have many questions to ask. When they’re asking, perhaps … like student A asks something, student B will rebut him. He’ll rebut him even before I answer anything. It’s like this … yes, among the students. And you can see that one student asks questions and then another will stand up to see what the question is. I mean … I mean they have such a situation.

This teacher’s observation that “they didn’t have questions in the previous lesson” suggests that the variation in the amount of time devoted to Kikan-Shido was influenced by two important factors: the degree of difficulty of lesson content and the students’ willingness to learn. The practice of Kikan-Shido in this classroom thereby provides supporting evidence for our argument that all participants are able to shape the particular body of practice signified by Kikan-Shido. That is, this pattern of participation is jointly constructed. WHOLE-CLASS COMPLICITY IN PATTERNS OF PARTICIPATION

Of major interest for the purpose of this chapter is the evidence that Kikan-Shido is a co-constructed pattern of participation to which members of the classroom community subscribe. In an attempt to situate the participatory status of both the teacher and students during Kikan-Shido, each lesson was further analysed to identify whether each coded instance was initiated by the teacher or student. For this purpose, Monitoring Progress (where this was non-verbal and non-interactive) and all activity codes within Organisational were not coded for initiated interaction. Table 3 defines each interaction code.

93

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

Table 3. Student and Teacher Initiated Interaction Student Initiated Interaction The process through which the student calls for the teacher’s attention, through verbal and non-verbal acts, with the intention to confirm progress, confirm solutions to a problem or request guidance. Verbal acts by students included calling out, asking a question, or making a statement. Non-verbal student acts included hand up, approaching the teacher and showing work. Teacher Initiated Interaction The process through which the teacher instigates verbal and non-verbal communicative acts with the student(s) with the intention to Monitor or Guide Student Activity, or engage student(s) in conversations of a social nature.

Figure 6 graphically illustrates both the similarities and the significant differences in the way that both teachers and students subscribe to the pattern of participation within Kikan-Shido. For example, if we compare HK3 with A2 we find similarities in the relatively high proportions of Student-Initiated Interaction. If we compare, G3 with J3 we find similarities in the weighting of Student-Initiated Interaction and Teacher-Initiated Interaction. However, real understanding of the pattern of participation enacted within each classroom is only evident from a combined analysis of Figure 2 and Figure 6. Figure 6 suggests one particular commonality that was shared by all three Japanese classrooms: that is, the teacher initiated a large proportion of StudentTeacher Interactions. However, Figure 2 shows that the classroom practice of Japanese Teacher 3 was quite different from the other two Japanese teachers with respect to the high proportion of time devoted to Guiding Whole-class activity. This interweaving of similarity and difference was highlighted by Clarke (2003) as characteristic of international comparisons of classroom practice. The Japanese example shows that this is true of within-country comparisons as well. A phenomenon as complex as classroom practice can be characterised in many different ways. Similarity in one aspect does not mean similarity in all aspects. In fact, teachers within one culture may differ in many respects. For example, if we compare the three Hong Kong classrooms as represented in Figure 6, it is clear that HK2 evidenced more Teacher Initiated interaction, while HK1 and HK3 had a much higher frequency of Student Initiated Interactions. With respect to Guiding and Monitoring (Figure 2): HK1 had a higher proportion of Guiding Student Activity, with an emphasis on Guiding Whole Class; HK2 gave equal weight to Guiding and Monitoring Student Activity; while HK3 was similar to HK1 in the frequency of Guiding Student Activity, but very different in that the guidance took the form of Giving Instruction/Advice at Desk.

94

Figure 6. Student and Teacher Initiated Interaction during Kikan-Shido

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

95

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

Both similarity and difference can provide insights. Variation in one aspect of classroom practice suggests tolerances within the socio-cultural norms of practice affecting both teacher and student behaviour. Tolerance of such variations suggests that no highly-valued principle is being challenged by the variations. Similarity suggests that practice may be predicated on a shared pedagogical principle. Where the similarity is specific to the classrooms in one country (or one cultural grouping), then the principle may reflect a socio-cultural norm specific to that country (or cultural grouping). Any such similarities identified within the LPS data may be researchable through analysis of the TIMSS video data, if a case for national or cultural typification was to be made. Our interest, where similarities occur, is in the identification of commonalities in the practices of competent teachers. One of these characteristics is clearly the purposeful selection of instructional strategies in response to topic, lesson location, class capabilities and individual student need, within the affordances and constraints of school system and culture. Relating Teacher and Student Perspectives The post-lesson interviews from both teachers and students revealed that KikanShido has a recurrent form, recognisable to those participating in it. This is not to say that the participating teachers and students attributed corresponding meanings to the activity. The point has already been made (Clarke, 2001, p. 296, and elsewhere) that individuals can participate in a practice whilst being positioned differently within it, and whilst attributing different characteristics to the activity. That is, without being identical, the participants’ descriptions of the activity make it clear that they are talking about essentially the same form, but they may attribute quite different functions to that form. For example, in Australia school 3, the Teacher’s intention appears to be predicated on the devolution of the responsibility for knowledge generation from teacher to student(s): A-T3

Often I just enjoy sitting down with … a student and saying … “mmh well … let's think about this because I don't know the answer” … let's try and see if we can find a way [sigh] and I … will make suggestions … But, ah yeah, I think sometimes it's good that- and they can see that I don't know … as well, and I am happy to call in somebody … and … ask other students if they’ve done it and if they have … ideas.

This can be contrasted with one student’s description of Kikan-Shido that suggests this same Teacher’s activity was typically quite directive: Int Rhys Int Rhys

96

Alright- okay I- I am just interested in- when Mrs. Greeno comes … and goes … what sort of feelings you have … then? Probably makes … um sometimes it helps you to focus a bit more … Aha. most of the time probably it makes us focus a bit more … and … yeah I mean you won't know … how … how well you are doing unless she comes and … has a look and sees if you are doing alright on not … so …

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION Int Rhys

So it helps you to know what you are doing? It helps- yeah it does … cause yeah … well … I don't know if it was that lesson but I had trouble with … ah … the … w- what do you call it? … The … equation … type of thing- the way you set- like you get the two numbers and you have to like … I didn't know … which way to set it out … when she comes and tells you what to do … then you're on your way like know what to do and everything.

While each participant is talking about the same form, i.e. Kikan-Shido, it is clear that each participant attributes quite different functions to that form. Evidence that students contribute to the form taken by a pattern of participation such as KikanShido can be found in the following statement from A1-LO9: S

S S Int S

She’s a big help … it’s such a change from … last year um I had … a pretty bad teacher … and I spent most of the lesson with my hand up wanting to get help but she didn’t- and she didn’t help me … and … I failed … every maths test- I can’t remember if I passed … even one … but Mrs M … this year she explains … everything to everyone before … you do the test … or anything she explains it really really well. Can you describe what it is like being in Mrs Milano’s lesson in terms of how you sort of … feel … before you even come in to a lesson? Well after- Maths isn’t my favourite … subject … [laugh] ah … but it is good to know you have got a good teacher who can … help you … and … if you don’t know anything … she’s there.

In general, many students in this class attached a high level of significance to the co-construction of Kikan-Shido. Indeed, many students’ participation in classroom practice in general seemed to be predicated significantly on the belief that KikanShido would provide them with valued support should they need it. From the postlesson interviews, students predominantly valued individual assistance, explanation and advice and the opportunity to ask questions (see Table 4 for a breakdown of student references to Kikan-Shido). One example from A1 is cited below. S Int S I S

Oh … ah it’s really good when Mrs Milano comes around to every-one individually … it’s like so if you are not sure about anything … you just like … she’ll come around. Oh I see. Yeah. Alright … it- it’s pretty good you say, can you tell me more? Yeah, like, say if you um don’t know something when she’s talking, when she’s up the front and, yeah, of the class and she comes around to every-one to see like how you’ve been doing so you can see like, if you are doing well or not and understanding you can just ask her individually.

97

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

Table 4. Student Reference to Kikan-Shido in the post-lesson interview

Based on our analysis of the Teacher Interviews, the Lesson Event we have called Kikan-Shido was explicitly valued by all three Australian teachers, in particular Teachers 1 and 3 (A-T1 made 7 references to Kikan-Shido; A-T3 made 10). Interestingly, the number of times each teacher referred to their Kikan-Shido practice was in proportion to the time these teachers committed to Kikan-Shido. This was also the case for US Teachers 2 and 3, Hong Kong Teacher 1 and all Shanghai teachers. For example, Shanghai Teacher 1 did not commit a large proportion of the class time to Kikan-Shido (8.4%). It is not surprising that this teacher did not refer to Kikan-Shido in the post-lesson interviews. In an exception to this pattern, if we compare SH1 with US1, we find similarity in the time devoted to Kikan-Shido, but difference in each teacher’s reference to Kikan-Shido. The teacher in US School 1 referred to Kikan-Shido three times during the post-lesson interviews. However, two of these three references corresponded to the one lesson (L07), when the teacher devoted a large proportion of class time to Kikan-Shido. That is, this exception is the sort of anomaly caused when an atypical situation affects a low occurrence count. As a general pattern, teachers made reference to Kikan-Shido with a frequency largely consistent with their use of it. The teacher in Japan School 1 stressed the importance of the monitoring function of Kikan-Shido. J-T1

98

I was walking between desks and seeing how students were doing. I could see how most of the students were doing by looking from the front of the class. But I cannot see all the students, and it is hard to see how those who are most likely to be behind are doing. So I gave them the clear procedure of how to work on such problem, to make a table for a graph, and an equation from a table.

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION Int J-T1 Int J-T1

I see. Did you feel anything by then? Like, there were more students who take a bit more time to understand this than you thought? Those who wouldn’t work on the problem were most likely to be those who don’t like studying. Those students were, ah, same as usual, those I expected. And you saw what you expected, huh? Yes. So I stopped by and gave some advice to those students.

The above interview excerpt emphasises the importance attached by this teacher to monitoring students individually and at a level of detail not possible from the front of the classroom. It also establishes the teacher’s willingness to intervene or guide individual student activity during Kikan-Shido. The Use of Physical Positioning The ways in which teachers chose to position themselves physically during KikanShido was often a teaching strategy intended to influence the nature of the interactions. In Australian School 1, the teacher’s deliberate physical positioning was utilised to minimise any intimidation of the students and, implicitly, to reduce the prominence of the inevitable power difference between the teacher and the student. A-T1

I don’t want my presence to be overpowering. I don’t want them to think, “Oh she’s over me just telling me what to do.” I don’t want to come down on them, and so a lot of the time I do kneel down … and I try to get on their level.

In all three Australian schools, German Schools 1 and 3, Shanghai School 3 and Japan School 2, the positioning actions of each teacher were influenced by student behaviour and the level of student engagement in the on-task activity. Often the teacher would intentionally stand near a student, Giving Instruction or Advice at Desk or Ask a Question, in an attempt to regulate the students’ behaviour. The following example from SH3–L08 indicates that the teacher’s strategy was also evident to other participants in the class: S

Our teacher is very humorous. Um, … for example … there’s one student … there’s a student in our class, he is called Bear, he does physical exercises in class, at one time, he was moving around, but then the teacher didn’t scold him, just go over and watched him, and smiled to him, then he stood very still.

This example illustrates one student’s sensitivity to the significance of teacher positioning. Pattern of Participation During Assessment It might be thought that there is less instructional teacher activity associated with a class test and, as such, the predominant teacher activity during Kikan-Shido in such lessons is likely to involve Monitoring Student Progress or Kikan-Junshi (Between

99

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

Desks Patrolling). However, our analyses revealed that, during the only two class tests among the lessons analysed (US1 and A3), the teachers and students participated in Kikan-Shido specifically to Guide Student Activity. During the class test, both US Teacher 1 and Australian Teacher 3 devoted more time to Kikan-Shido than during any other lesson in the ten-lesson sequence. Figure 7 graphically illustrates US Teacher 1’s utilisation of Kikan-Shido across the tenlesson sequence, of which lesson 7 is the class test.

Figure 7. Variation in the utilisation of Kikan-Shido across a ten-lesson sequence

US Teacher 1 devoted 35.2% of that lesson (US1-L07) to Kikan-Shido. While the teacher did Monitor Student Activity (10.2% of total class time), more time was devoted to Guiding Student Activity (23.6% of total class time) and, in particular to: Giving Instruction / Advice at Desk (12.9% of total class time); Answering Questions (8.4% of total class time); and Guiding Whole-class Discussion (1.6% of total class time). The pattern of participation in US1-L07, as seen in Figure 8, clearly demonstrates that both students and the teacher contribute to the form of KikanShido and that this pattern of participation supports to some extent, the assertion that both the teacher and students share a common interest in advancing the students’ learning. Figure 8 illustrates the high proportion of Student-Initiated Interaction (19% of total lesson time) compared with Teacher-Initiated Interaction (5.6%) during the class test. As can be seen in Figure 8, Kikan-Shido also accounted for a further 10% of total lesson time but this did not involve either type of initiated interaction.

100

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

Figure 8. Student and Teacher Initiated Interaction During a Class Test

It was during the class test that US Teacher 1 devoted the greatest proportion of time to Kikan-Shido (35.2% of total class time) of any lessons in the US1 tenlesson sequence. Similarly in A3–L10, during another class test, the teacher also significantly increased the proportion of time devoted to Kikan-Shido (68% of total lesson time) of which the majority was teacher initiated (39.1% of total class time) but also with significant levels of student initiated interaction (21.3% of total class time) (see Figure 8). While Australian Teacher 3 did Monitor Student Activity during the test (16.1% of total class time), the teacher devoted substantial lesson time to explicitly Guiding Student Activity (48.5%). In general, this guidance took two specific forms: Giving Instruction / Advice at Desk (20.9%) and Guiding Student Activity through Questioning (15.1%). The reconstructive account from the teacher’s postlesson interview illustrates the Teacher’s use of these strategies. A-T3 Int A-T3 Int A-T3 Int A-T3

I'm trying to … encourage them to put down what they think … okay- to put down … what comes into their mind. Can you find some instances of it- and can you just go through … what you were thinking … at those times? Shayne has a short-term memory, ah, it's just, what I try to do, is try to, verbalise, go through and drop some key-, well, for him to pick up on some key words. Oh can you give … an example? Um, you know, I'll sort of often say, “Now percentage is always out of … ” Aha. “Okay so and yeah … and we've got ten percent there so … think about how you are going to set that out then … ”

101

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

The significance of the teacher’s explicit guidance was further acknowledged in her post-lesson interview: A-T3

Is it yeah because, for so many of these students that have learning … difficulties, um, and have had negative experiences all their life and if you talk to their parents they will say they've had all these negative experiences and they'd been often very negative about maths. And to me, the biggest thing I can do is take that fear away from them and to make it as approachable as possible and to emphasise to them that not everybody can do everything very well … And we've got to, um, get them thinking about 'yes it's okay to ask', but I will slowly have a bit less involvement over a period of time.

In this example, the teacher indicates her strong desire to support student learning, while acknowledging the challenges of teaching students with learning difficulties. The teacher also communicates her commitment to the development of student confidence by providing support and encouragement. It is clear that despite the constraints encountered by both students and teacher, the teacher’s activity is intended to promote student participation in and commitment to the learning process. This supports the contention of Clarke (2001), who argued that learners construct their participation and ultimately their practice through a customising process in which their inclinations and capabilities are expressed within the constraints and affordances of the social situation. The example above illustrates the way in which one teacher facilitated the incremental increase in the students’ participation. The reduction, over time, in the explicit scaffolding provided by the teacher is a fascinating example of what Lave and Wenger (1991) have termed the student’s ‘legitimate peripheral participation’ in the practices of the classroom. CONCLUDING REMARKS

This chapter embodies the aspiration to find structure in the ephemeral. To a significant extent, the realisation of this goal has been assisted by the conception of Kikan-Shido as an internationally recognisable Lesson Event: a whole-class practice, having a certain visible form, with a locally-enacted pattern of participation to which teacher and students subscribe and which both teacher and students have agency to exploit and to shape. While Kikan-Shido characterises a recurrent form evident across all the classrooms, its functions appear to be a consequence of the emerging patterns of participation in which the members of a particular classroom community engage. In order to accommodate both the fluidity of social interaction and its regularity, careful distinctions have been made to delineate the form and function that constitute each teacher’s enactment of Kikan-Shido in their classroom. Our analyses have identified the differences in teachers’ utilisation of Kikan-Shido as a signature characterising their practice. We suggest that these differences are predicated on the specific pedagogical principles that appear to underlie each teacher’s practice. Another issue central to this chapter is the distribution of the responsibility for knowledge generation. Our analyses of Kikan-Shido have

102

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

demonstrated that this perspective provides a powerful explanatory framework for identifying similarities and differences of classroom practices in both ‘Asian’ and ‘Western’ settings. In particular, analysis from this perspective problematises the simplistic characterisation of ‘Asian’ or ‘Western’ pedagogy. In this chapter, we have attempted to frame the argument that the pattern of participation designated by Kikan-Shido must be conceived as co-constructed by both teachers and students. Some examples given in this chapter have shown that Kikan-Shido can be thought of as a familiar dance done by teachers and students, where the steps are improvised according to need. The participants in the classroom, teacher and students, are complicit (co-conspirators) in this improvisation. Acceptance of this point has implications for the research designs by which we study the activities occurring in the classroom settings. But co-construction of practice and joint participation in practice do not connote commonality of purpose among the participants in that (classroom) practice. Through juxtaposing individuals’ actual participation in Kikan-Shido with the reconstructive accounts from interviews, we were able to get insight into the meanings that were attributed to that form. Our analyses of whole-class patterns of participation show that even where all participants recognised and subscribed to the same pattern of participation (Kikan-Shido), they could attribute different characteristics to the activity. If we conceive of institutionalised patterns of participation as taking on the status of bodies of practice, then their co-constructed nature has further significance. Rather than progressively increasing the competence of their participation in a culturally or socially pre-determined practice (e.g., Lave & Wenger, 1991), this conception of the origins of practice accords significant agency (however constrained by institutional or cultural norms) to the participants to shape their particular pattern of participation, and thereby to influence the nature of that practice. This approach to conceptualising classroom practice further challenges those simplistic East-West comparative cultural analyses that presume a generality of pedagogical strategy and its intact cultural transferability. The similarities and differences documented in this chapter set out the degree of variation possible in the practices of eighteen competent teachers situated in six very different school systems around the world. Among other things, our analyses suggest that the responsibility for knowledge generation can be purposefully distributed in the classrooms of competent teachers, within the institutional and cultural norms constraining that practice. The nature and practice of this distribution of responsibility is the subject of further analysis and will be reported elsewhere. Because of its prevalence, its familiarity of form, together with its diversity of function, the Lesson Event that we have designated by ‘Kikan-Shido’ provided a particularly rewarding focus for our analyses, rich with insights into the pedagogical principles on which competent teachers around the world base their practice.

103

CATHERINE O’KEEFE, LI HUA XU AND DAVID CLARKE

ACKNOWLEDGEMENTS

The authors would like to acknowledge and thank Seah Lay Hoon for her initial analyses of Kikan-Shido. REFERENCES Brousseau, G. (1986). Fondements et méthodes de la didactique des mathématiques [Foundations and methods of the didactics of mathematics]. Recherches en didactique des mathématiques 7(2), 33-115. Chazan, D. & Ball, D. (1999). Beyond being told not to tell. For the Learning of Mathematics, 19(2), 2-10. Clarke, D.J. (1985). The impact of secondary schooling and secondary mathematics on student mathematical behaviour. Educational Studies in Mathematics, 16, 231-257. Clarke, D.J. (1998). Studying the classroom negotiation of meaning: Complementary accounts methodology. In A. Teppo (Ed.), Qualitative research methods in mathematics education. Journal for Research in Mathematics Education. Monograph No. 9 (pp. 98-111). Reston, VA: NCTM. Clarke, D.J. (Ed.). (2001). Perspectives on meaning in mathematics and science classrooms. Dordrecht, Netherlands: Kluwer. Clarke, D.J. (2003). International comparative studies in mathematics education. In A.J. Bishop, M.A. Clements, C. Keitel, J. Kilpatrick & F.K.S. Leung (Eds.), Second international handbook of mathematics education (pp. 145-186). Dordrecht, Netherlands: Kluwer. Clarke, D.J. (2005). Essential complementarities: Arguing for an integrative approach to research in mathematics classrooms. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce & A. Roche (Eds.), Building connections: Research, theory and practice. Proceedings of the twenty-eighth annual conference of the Mathematics Education Research Group of Australasia (pp. 3-17). Sydney: MERGA. Clarke, D.J. & Lobato, J. (2002). To tell or not to tell: A reformulation of telling and the development of an initiating/eliciting model of teaching. In C. Malcolm & C. Lubisi (Eds.), Proceedings of the tenth annual meeting of the Southern African Association for Research in Mathematics, Science and Technology Education (pp. 15-22). Durban: University of Natal. Davies, B. & Harré, R. (1991). Positioning. Journal for the Theory of Social Behaviour 21, 1-18. Greeno, J. G. (1997). On claims that answer the wrong questions. Educational Researcher, 26(1), 5-17. Hino, K. (2006). The role of seatwork in three Japanese classrooms. In D.J. Clarke, C. Keitel & Y. Shimizu (Eds.), Mathematics classrooms in twelve countries: The insider’s perspective (pp. 59-73) Rotterdam: Sense Publishers. Huang R. (2002). Mathematics teaching in Hong Kong and Shanghai – A classroom analysis from the perspective of variation. Unpublished Ph.D. thesis, The University of Hong Kong. Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press. Lobato, J., Clarke, D.J. & Ellis, A.B. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education 36, 101-136. Mok, I.A.C. & Ko, P.Y. (2000). Beyond labels – Teacher-centred and pupil-centred activities. In B. Adamson, T. Kwan & K. K. Chan (Eds.), Changing the curriculum: The impact of reform on primary schooling in Hong Kong (pp. 175-194). Hong Kong: Hong Kong University Press. Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge: Cambridge University Press. Wood, T., Nelson, B.S. & Warfield, J. (2001). Beyond classical pedagogy: Teaching elementary school mathematics. Mahwah, NJ: Lawrence Erlbaum.

104

KIKAN-SHIDO: BETWEEN DESKS INSTRUCTION

Catherine O’Keefe, International Centre for Classroom Research Faculty of Education University of Melbourne Australia Li Hua Xu International Centre for Classroom Research Faculty of Education University of Melbourne Australia David Clarke International Centre for Classroom Research Faculty of Education University of Melbourne Australia

105

EVA JABLONKA

CHAPTER FIVE Student(s) at the Front: Forms and Functions in Six Classrooms from Germany, Hong Kong and the United States

INTRODUCTION

A comparative study of distinct aspects of classroom practice might focus on similarities within countries or regions in order to identify culture-specific components and find explanations for differences between countries in terms of cultural values, language, teaching/learning traditions and its institutionalised forms in teaching materials, teaching approaches, modes and role of assessment or examinations. On the other hand, a comparative study of distinct elements that occur in the teaching/learning practice in mathematics classrooms in different countries or regions can contribute to identifying similarities and variations in forms and functions. Similarities can be analysed with the help of theories that focus on structural features of the school setting as a consequence of similarities in physical layout, allocation of roles as teacher and students, features of social interaction in institutional settings and similarities in curriculum including examination practices. The properties of the institutional settings are the structures students and teachers conceive of as already being shaped and on which their learning and teaching intentions are predicated. Variations can be attributed to culture-specific ways in which teachers and students deal with similar problems that arise in these settings. In the study reported in this chapter, mathematics classrooms from different cultural traditions are chosen for the purpose of cross-cultural triangulation in order to distinguish aspects of classroom practice that can be attributed to local cultural traditions from those, which can be interpreted as arising from the ‘culture’ of mathematics instruction in the context of formal schooling. In all classrooms authority and power relations that are due to the allocation of roles as teacher and students play an important role. The development of shared understanding takes place in the context of an authority relationship, with the teacher representing for the students an accepted wider culture of educational ideology and mathematical knowledge. Consequently, the ways in which this relation shapes the participants’ interactions and construction of meanings are of interest. Neither students nor teachers are free in their choices between alternative

D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 107–126. © 2006 Sense Publishers. All rights reserved.

EVA JABLONKA

interpretations and actions. The interest in the activity ‘Student(s) at the Front’ is a reminder of the fact that usually most of the time it is the teacher who is found at the front of the classroom, standing, occupying the foreground. Classrooms show some commonalities in shape and physical layout; they all have a back and a front; the front is the side where the teacher’s desk and chair, the chalk/ white board and/or a screen are located. Foucault's analysis of the body in institutions shows that these arrangements are not just superficial accessories, but a thoughtful design derived from values about authority and the nature of learning in schools (Foucault, 1977). A traditional classroom including rows of chairs facing the front provides a convenient design for presenting information in a lecture format, with or without including questions to the whole class, and for tests. Communication activities that involve students being the centre of attention are often associated with reform-oriented practices, which include occasional shifts in the responsibility for knowledge production from the teacher to the students. For example, in the TIMSS-R questionnaire, students were inter alia asked to report their perception of the frequency with which some selected activities occurred in their classrooms. Among those were ‘Students use the board’ and ‘Students use the overhead projector.’ Telese (2004) found a negative relationship between students using the board/overhead and achievement in the US data from the Third International Mathematics and Science Study (TIMSS)-R. The interest in these activities was motivated by the assumption that their occurrence indicates the extent to which activities that focus on developing mathematical understanding through dialogue are incorporated into classroom practices. He used his findings to argue that spotlighting students at the board is likely to be an unproductive activity and associates this with ‘reform-oriented’ teaching. It must have been assumed that students are called upon to share ideas at the board/overhead. But it is not obvious that students at the board or the overhead always would be required to explain their thinking and discuss their ideas. For example, in the TIMSS 1999 video study, the Czech Republic showed a relatively high average percentage of lesson time devoted to student presentations, that is 18% as compared to 5% in Hong Kong and 1% in the United States. The event was defined as: “A student presents information publicly in written form, sometimes accompanied by verbal interaction between the student and the teacher or other students about the written work; other students may attend to this information or work on an assignment privately.” However, in the lessons from the Czech Republic one or two students were occasionally called upon at the beginning to publicly exhibit mastery of knowledge and skills taught previously for the purpose of grading of students. The rest of the students usually worked individually (Hiebert et al, 2003). This version of grading students was common also in German classrooms at the beginning of the 20th century. Consequently, when analysing classroom practice it is important not to lose sight of the fact that similar forms of activity can serve a variety of different functions. The focus of the analysis presented in this chapter is the ways in which authority relations are embodied in different forms of the activity ‘Student(s) at the Front’. 108

STUDENT(S) AT THE FRONT

METHOD

Students are not necessarily asked to write on the board or to use the overhead projector when called to the front. They may be asked to demonstrate something, to show a poster or to give an oral presentation. This is done at the front where the attention is hoped to converge. In the LPS, in each of the three classrooms from a participating country or region, sequences of ten lessons were documented by the use of three video cameras. The video documentation was supplemented by reconstructive accounts of classroom participants, obtained in post-lesson video-stimulated interviews, and by test and questionnaire data, as well as copies of student written material. For identification of the event ‘Student(s) at the Front’ in the 60 lessons (ten consecutive lessons in two classrooms from each country/region), the following description was used: One or more students are at the front of the classroom – this is the side of the room on which the teacher’s desk, the board, an overhead projector (OHP), a flip chart, or a screen is located. The student(s) may, for example, be writing something on the board or on the OHP, talking, or giving a demonstration using a model. A student cleaning the board does not count as an example of this activity. The event starts when the teacher announces the activity, for example by calling upon a student, and it ends when the student has taken her seat. In a first step, the video footage of the teacher camera was used to identify the events. In a second step, the transcripts of the lessons were used together with the mixed image to determine the beginning and the end of each episode. The activity occurred in all the six classrooms; a complete list of the events identified by this procedure is given in the appendix. The selection of the six classrooms is based on the principle of maximising contrast but at the same time keeping a reasonable basis for any study of variations, which presupposes a similarity in at least some aspects. The classrooms from Germany (Berlin) are labelled G1 and G3, from Hong Kong HK1 and HK3, and from the United States (San Diego, CA) US1 and US2. (The numbers indicate the order of the three classrooms videotaped one after the other in the Learner’s Perspective Study.) While the form could be differentiated by the observed and recognisable dimensions of the activity, the description (and labelling) of the functions took into account the antecedent conditions, and, as far as possible from the data obtained, the teachers’ intentions and the students’ expectations associated with it. For this purpose, the teacher and student interviews were searched for comments referring to ‘Student(s) at the Front’, be it a particular event in a lesson or the activity in general. FORMS

In the six classrooms the number of students who are at the same time at the front varies from one to eight students. When a couple of students are solving different tasks in parallel at the board, for example in order to share the solutions from some

109

EVA JABLONKA

reviewing exercises given at the start of a lesson (in the ‘warm-up’ in US1 and US2) or after a period of individual seatwork (in HK1), the students naturally cannot explain their solutions while writing. However, even if just one student is presenting a solution, this is frequently done silently in all six classrooms; it is only the teacher who comments on what the student is writing. A form observed only in one of the classrooms from the United States (US2) consists of students at the front holding up whiteboards or pinning posters from group work on the board, displaying, and sometimes also reading out what they have produced in their group. There are also very short instances in which a student is called to the front and then just points to (part of) a drawing (in US2). In US2 and G3 students also use the overhead projector. These are also the two classrooms in which the highest frequency and the greatest variety of different functions of the activity are found (see below). In accordance with the rules of behaviour and conduct in the six classrooms, there are different ways of being called to the front. The teachers in general call only on students who volunteer. In HK3 the teacher silently hands over the pen for the white board to the student whom he wants to go to the front or puts it in front of the student on the desk and claps his hands. The forms are constrained by classroom shape, physical layout and resources. For example, in one classroom (US2) the board stretches across a whole wall of the room, while in the others it is much smaller. As could have been expected, there is a variety of forms that differ in the number of students simultaneously at the front (one to eight students), in length (less than one minute to about ten minutes), in frequency, and in the extent of interference by the teacher while students are at the front. From the perspective of the students, coming up to the board may be a pleasant or a stressful experience, depending on the purpose of this activity. For example, there are two instances when students who are called without having volunteered refuse to go to the front (G3, US1). This may be linked not only to a fear of being in the centre of attention but also to a particular purpose of being called to the front. The study of ‘Student(s) at the Front’ revealed a variety of purposes, some of which were found across classrooms. Though forms and function are here reported separately, the interpretation and discussion of distinct aspects and dimensions of this activity take both into account. On the other hand, distinct forms of classroom practice can only superficially be taken as an indicator of a type of pedagogy, as shown for example by the study of Brodie, Lelliott and Davis (2002). It documents and evaluates the ways in which teachers have taken up learner-centred practices after an in-service programme in South Africa. At the beginning of the programme, the majority of teachers, who were enrolled in the programme for three years, only took up forms without substance. One element of the learner-centred practices the programme aimed to introduce was to ask learners to put up their solutions to problems on the board so that these could be publicly compared, discussed and evaluated. At the beginning of the programme, some teachers only took up the form of this activity; they immediately corrected mistakes instead of linking the students’ solutions to related difficulties and to other concepts. 110

STUDENT(S) AT THE FRONT

On the other hand, as can be seen in some instances from the study reported here, the goal of sharing and discussing work at the board is subverted by the form through which it is pursued. In addition, the students’ expectations and their shared knowledge about school mathematics and how mathematics classrooms work constrain classroom practice. If possible from the data obtained in the interviews, the students’ interpretations are incorporated into the analysis in this study. FUNCTIONS

Germany, Classroom One (G1) Publicising and explaining work. In this classroom students are called up to the board seven times in the ten lessons videotaped. The activity exclusively serves the purpose of presenting work in order to share it with the whole class. But this is not a common activity. Five of these events happen in one exceptional lesson (Lesson 7), in which two or three delegate students of a group are asked to present results from group work produced during the previous lesson, in which the students were asked to discover geometrical interpretations (‘proofs’) of binomial products. Each of three relatively large groups had to work on one of the following products: (a + b)2, (a – b)2 and (a + b)(a – b). Some looked up the corresponding drawings in the textbook. In Lesson 7 the groups prepare three drawings at the board and afterwards are called on to illuminate and explain their proofs. This is indeed a task that asks for explanation because the construction and the translation of the geometric version into the algebraic form is not obvious. However, group work is rare in this classroom according to the information given by the students in the interviews. In this respect, Lessons 6 and 7 are not typical of the practice in this classroom. The students at the board give explanations and sometimes ask the other students whether they understand. Occasionally students hand over the task of explaining their drawings to other members of the group that had been occupied with the same formula, because of an impression that their classmates are not satisfied. The teacher had not seen the students’ work before they were asked to present it. In the interview, he concedes that he was not always able to understand their argumentation immediately. This may be the reason why he does not evaluate the students’ comments and does not ask questions but only organises turn-allocation or contributes himself to the discussion. The students are involved in a lively discussion with their peers at the front who answer their questions. However, the teacher does not want to leave it without comment, even though the students eventually are convinced by the students who present the proofs; after a while the teacher switches from monitoring the discussion into a mode of guided development in order to repeat the students’ arguments. He points to parts of the drawings and poses a series of connected questions to the students that encapsulate the chain of inferences and – if answered appropriately – in sum warrant the resulting conclusion. 111

EVA JABLONKA

The teacher and all but one of the students in this classroom do not refer to the function of the activity ‘Student(s) at the Front’ in the interviews. There is no indication that the students do not like being at the front. The atmosphere in the classroom is not competitive. Germany, Classroom Three (G3) In this classroom, the ‘Student(s) at the Front’ is a constitutive element of classroom practice. Students are at the front at least once in every lesson, altogether 18 times in the course of the ten lessons. The activity serves three different functions. Publicising work. In ten cases students are called to the front to share solutions of isolated tasks that have been completed before, be it in group work, individual work (Lesson 2/3 (7)), work in pairs (Lesson 2/3 (1), (2), (3), (4), (5) and (6), Lesson 9/10) or as homework (Lesson 8 (1) and (2)). Usually the teacher is involved in the students’ presentation by a form of guided-development that has the purpose of illuminating and explaining the students’ work, that is, the steps in the calculations, to the whole class. Consequently, the students do not talk a lot when at the front. This classroom is very small in terms of number of students and shape, so that there is no clear distinction between private and public talk because all conversations are audible to all participants. This means that even when the students intend to talk only to their peers, other students or the teacher frequently get involved. On the other hand, it is impossible for the teacher to talk privately to the student at the front, as for example in the Hong Kong classrooms (see below). The weak regulation of interaction in G3 in terms of turn-allocation causes a high level of involvement, which includes arguing about coming out to the front. In one lesson (Lesson 7(2)) the student called upon refuses to do so and the teacher proposes to write on the board while the student dictates to her what to write. Solving a new task in public. In this classroom, students are also called to the front six times in order to solve a task or parts of a task within a period of whole class instruction, in which the tasks have the status of a worked example or a ‘Learning Task’ (cf. Mok and Kaur, this volume) as in HK1. This means that the students are expected to understand a new procedure to the extent that they are able to apply it to a new example and show this in public at the board. Sometimes also, the tasks from the homework happen to be new tasks to be solved in public for those who have not completed their homework. In the interviews, some students reported that solving a new task at the board is a stressful experience. The following conversation from a student interview (G3-L02/03) illustrates this point: Selin: Yes. I don’t like it if you calculate on the blackboard. Rather on paper, then I like doing it. I don’t know why but it is simply, I am prepared for that, I don’t know. Int: Mhm. And, and you, how do you feel when you go up to the blackboard?

112

STUDENT(S) AT THE FRONT Diana: Well, it is really the way Selin describes it, that it is easier on paper because on the blackboard if you make a mistake. Int: What then? Diana: I don’t know. It’s somehow embarrassing if you made a mistake when calculating or something like that. Key to symbols used in transcripts in this chapter: ... Pause of three or less seconds (roughly where it would be in the English version of the original) // Overlapping talk / Cut A hyphen at the end of a word indicates self-interruption (text) A best guess by the transcriber

Division of labour between teacher and students. There is still another purpose of calling students to the front in this classroom. In Lesson 4, the class compares the experimental volume of solids with their estimation and calculation. The students at the front fill coloured liquid into a model of a solid in order to measure the volume, while the class watches and the teacher comments. In this lesson, many students volunteer, in contrast to the situation in which students had to solve parts of an unfamiliar task at the board. This function might be similar to one found in physics lessons, where students assist the teacher in a demonstration by taking over manual tasks which are part of an experiment. Hong Kong, Classroom One (HK1) In the course of the ten lessons videotaped in HK1, students are called to the front eight times. Two distinct functions can be attributed to these events. An extra opportunity to get the teacher’s comments. In six cases when students are at the front, they are writing their versions of solutions of a couple of tasks on the board during a period of individual seatwork (Lesson 1, Lesson 3 (1) and (3), Lesson 5, Lesson 9 (1) and (2)). This could be perceived as a way of publicising the results in order to share and evaluate them. However, the students do not perceive this as an act of addressing the whole class. They do not speak while writing on the board and the majority of the other students continue with their individual seatwork without paying any attention to what their peers are producing at the front. As the tasks solved in these periods of individual seatwork are connected tasks with increasing complexity that in each lesson refer to a distinct method of solving simultaneous linear equations in two unknowns, there is no student choice of solution methods and thus not much to discuss. Only some students choose to compare their own work with the solutions written on the board; many use the book for this purpose. The teacher does not always check these solutions immediately, but looks at them later; in one lesson, for example, he continues to walk between the desks monitoring students’ individual seat work (Lesson 3 (3)). In another lesson (Lesson 1), the teacher asks some of the students

113

EVA JABLONKA

who have already taken their seats to come back to the board in order to correct their solutions. Indeed, in these events it is the perception of neither the teacher nor the students that the aim of writing their answers on the board is publication for the purpose of sharing with the whole class. It is rather sharing the solution with the teacher. The following comments from the first teacher interview afford this interpretation: T:

Int: T:

They know that I am very kind so when I go around, they would ask me if they don’t understand. Also, they are familiar that every student has a chance to come out and write out the answer. Usually they come out voluntarily or you call their names? Mm … their habits are coming out themselves. Well, because em … although some students were not taught by me previously, they have already adopted my habit in this half year.

The purpose of helping individual students with their work by calling them to the front is also reflected in the following statement of the teacher from Lesson 1 and by Peter’s comment in the student interview (Peter, HK1-L05). The student is not referring to the event in Lesson 5, but gives a general description: T:

If you get it wrong, check each step to see which step is wrong. I'll call the one who is not answering the question to do it on the blackboard. Don’t do other things. [T walking around] Int: It’s all right. Do you think the way that you learn mathematics in the last lesson is the best? Peter: It’s okay Int: Well … Why do you say so? Peter: Because Mr … Mr. M. gave us the questions and put them on the projector. Then he let us do them. I think that’s quite good. And then, he calls us to answer them. If you don’t understand, he would teach you. That’s quite nice.

Solving a new task in public. In the other two events (Lesson 3 (2), Lesson 6) when the students are called to the front, this serves a quite different function. The students’ work on the board is integrated into a period of whole class instruction, similar to the practice in G3 (see above). The students’ solutions get the status of being public. In both events the students are engaged in a task or parts of a task that they have not completed before they are called to work at the front. For example, in Lesson 3 the teacher further develops the concept of Gaussian elimination by introducing the possibility of multiplying both equations and by introducing subtraction of the equations. One student volunteers to solve a new task by using these methods immediately at the board, while the others solve the same task individually. The teacher later cleans the board without commenting on the student’s solution. In Lesson 6, equations are solved graphically. One student is asked to construct a graph at the board. The drawing becomes an object of public discussion later in the lesson.

114

STUDENT(S) AT THE FRONT

Hong Kong, Classroom Three (HK3) An extra chance for getting the teacher’s attention. In this classroom, each lesson includes extensive periods of individual seatwork for practising tasks, during which the students engage in lively conversations with their peers. These are mostly in Cantonese, although the language of instruction is English. This is an indicator of the importance of non-public talk in this classroom. According to the analysis of the Hong Kong videos from the TIMSS 1999 video study this is not typical. The work produced in these periods is not always publicised and shared with the class. Instead the teacher evaluates it while walking between the desks. In the course of the ten videotaped lessons, students are at the front five times. The function of all these events is similar to the first function in the other classroom from Hong Kong (HK1, see above). While the whole class remains engaged in individual seatwork and the teacher walks between the desks, he occasionally hands over the pen to a student who then copies her solution onto the board. The teacher continues to walk between the desks monitoring and guiding individual students’ work. The students do in general not pay attention to the work of their classmates on the board. For example, the handwriting of the student in the event in Lesson 7 is so small that the other students are unlikely to be able to read it. The teacher sometimes checks the solutions on the board and, if he detects mistakes, talks to the students who have produced it. In the interviews, the students and the teacher of this classroom do not refer to the activity ‘Student(s) at the Front’. United States, Classroom One (US1) Publicising work, explaining work. In this classroom, the students are frequently at the front. The function is similar to the first one in G3 (see above), that is, sharing work that has been completed before, be it from the warm-up or from individual seat work. A warm-up is a conventional way of organising a review. The teacher frequently comments on the students’ solutions for the purpose of illuminating and explaining their work to the whole class, sometimes in a form of guided development. In contrast to the teacher in G3, he does not do this while the students are still writing. The students do not talk much when out at the front. However, in two events from Lesson 1 ((3) and (4)) and one from Lesson 2 ((2)) the teacher asks the students to explain their solutions to the class. Another difference to the practice in G3 is that there are frequently more than one student at the front at the same time. This inhibits interference and comments by the teacher before the students have finished. Sometimes the teacher talks to individual students in this period so that the event becomes more similar to the version found in Hong Kong, especially when the tasks are from the warm-up. In the interviews, the students do not very often refer to going to the front as an important event. If students do not want to go to the board, the teacher does not try to persuade them. Most of them do not mind presenting their solutions at the board, or even like doing it.

115

EVA JABLONKA

The following comment from a student interview (Autumn, US1-L06) shows that solving a task correctly in public is taken as a criterion of mathematical proficiency. Solving the tasks from the warm-up may be associated with grading. Int: Autumn Int: Autumn Int:

What were your personal goals for that lesson? This lesson? Mm-hm. To understand it and get through it. Okay, and what does understand mean for you? When do you know you understand? Autumn When I can- I can like, go up to the board and do a problem and get it right.

United States, Classroom Two (US2) Calling students to the front is a constitutive element of the practice in this classroom, as in G3 (see above). The lessons consist of a repertoire of distinct recurring forms of interaction and talk and show a relatively uniform structure throughout the ten lessons, which actually are to be seen as five periods of a length varying from 90 through 100 minutes. The didactical purpose of calling students to the front varies according to its place in the lesson structure. Publicising work, explaining work. The students write on the board solutions to tasks they have completed in individual seatwork. These are usually the tasks from the warm-up by which the lessons start. In the interview (first teacher interview US2), the teacher explains the purpose of the warm-ups: T:

Int: T:

Uh … uh, yes well, uh … most of the warm-ups are intended to increase number sets in some way or another. And that is often our standardized testing piece. Although mine perhaps is- is not certainly naked computation. ButWhen- when you say naked computation you meanComputation without a context.

The solutions on the board seem to get public status only when the teacher starts evaluating them and the students pay attention to what she says. Students write their solutions from the warm-up quietly on the board and only talk about it when the teacher asks them to do so. This is more likely in the case where their solutions are wrong or are not clear. Usually the teacher expands on what the students have written in a teacher presentation that includes occasional questions to the class or to the presenter. In contrast to the other classrooms, the teacher continues to query the presenters after they have resumed their seats. At times the teacher summarises major points, for example, pointing out that you divide to ‘break a multiplication bond’ or that taking the square root undoes squaring. Displaying work. In many of the events the students are asked to show the results of their group work, which usually takes place in the second half of the lesson, to the class, either on white boards or on posters. In these instances the students do 116

STUDENT(S) AT THE FRONT

not explain but only read out what is written on their boards/ posters. The teacher evaluates their product by illuminating, comparing or expanding on it. Lindsey’s (Lindsey, US2-L06) comment reflects this: Int: Um … was this typical lesson for you? Lindsey: Um … usually. Cuz I think, like, we are getting to- we- we learn about something, we get into groups and we do something, display it in front of the class.

In the interviews, some students, like Shannon (below, US2-L02), give general comments on the function of going to the front. Sharing and comparing work is also one of their perceptions of the purpose of this event. Many of the students seem to enjoy presenting work. In the lessons, the class often applauds when other students have displayed their work. Int:

Shannon: Int: Shannon: Int: Int: Shannon: Int: Shannon:

Int:

So you know, when you see people, when they start going up and they- reading their answers and putting them up on the board, what- what happens for you? Like, what do you do when that’s going on? Or, what did you do today? Well … like what do I think? Yeah. I sort of wish that I was up there. Cuz I like- I like presenting stuff //in front of the class. //Oh, you do? So you’d rather go up and do- do it? Yeah. Well, if I kn- understand it. Like, um, last- well the first time that our class was being video taped/ /Yeah. And it was our group too. Um, I raised my hand like six times. But she kept calling on somebody else. And, um, cuz I knew the answers and I felt proud of that because I don’t usually know much answers in that course one. Oh!

Division of labour between teacher and students. Sometimes the students are out at the front only for a very short time in order to answer a single question related to a task discussed with the whole class. In general, the teacher tries to delegate parts of her work to the students as far as possible. Even when the students are only bringing back and picking up materials, this might be interpreted as an indicator of this intention. The events are listed in the table in the appendix, even though these are not examples of ‘Student(s) at the Front’ according to the definition. However, the third function in G3 (see above) also is an example of this principle. SUMMARY

Table 1 shows a synthesis of forms and functions found in the six classrooms. As can be seen, the two Hong Kong classrooms were more similar than any other two classrooms with respect to the repertoire of forms and functions. The function of solving (parts of) a new task in public occurred in one classroom from each of the three countries, while others were found in classrooms from Germany and the US, 117

EVA JABLONKA

including some classroom-specific variations. In US2 all functions were identified except the first, which was found exclusively in the Hong Kong classrooms Table 1: Functions and forms of ‘Student(s) at the Front’ Functions

Forms

Classrooms

An extra chance to get the teacher’s comments

Students write solutions on the board after practice HK1, HK3 while the other students continue to work on assignments privately. Especially students who have trouble are asked (HK1 only).

Solving a new task in Students are asked to solve (parts of) an unfamiliar public task in public.

G3, HK1 US2

Publicising work

One or more students write solutions on the board G1, G3 after a period of practice, after a ‘warm-up’ or from US1, US2 homework; the teacher questions students or expands on their solutions if wrong. The other students compare with their own work. Occasionally two students collaborate at the board (US2 only).

Publicising and explaining work

Students present and give a coherent account of completed work and occasionally are interrupted by the teacher. Students explain their work and answer questions posed by other students while the teacher is listening (only G1).

Division of labour between teacher and students

Students point to parts of drawings, tables or graphs G3, at the board, draw a sketch, fill in a table (US2) or US2 assist the teacher in a demonstration (G3).

Displaying work

‘Products’ from group work are displayed on posters or on small white boards; the teacher evaluates and illuminates the products.

G1 US1, US2

US2

DISCUSSION

As the categories emerge from the data, the functions differ in different dimensions. A primary distinction concerns the fact whether being at the front is to be interpreted as a public activity. In the two Hong Kong classrooms the event was mostly rather a private activity, as the board-work was not integrated into a whole class discussion and the teachers, if necessary, corrected what the students had written on the board by talking to them individually. Another dimension concerns the function in terms of the development of new knowledge. In a couple of instances students came up to the board and had to apply a new solution method to an exemplary task in order to produce a ‘worked example’ (cf. solving a new task in public). Both dimensions turned out to be important for the students in terms of the risk involved in coming up to the board. In the interviews, only a few students from the Hong Kong classrooms talked about this risk, as compared to the number of students from the German and US classrooms who said they were afraid of 118

STUDENT(S) AT THE FRONT

saying anything considered as non-appropriate in front of the class. For the Hong Kong students the risk was minimised by the general rarity of public utterances. According to the interest of the analysis, an important aspect of the activity is the way in which the responsibility for the evaluation and illumination of what is shown in front of the class is shared between students and teachers. A common feature of classroom practice in all the six classrooms is the occurrence of ‘silent presentations’. In the two Hong Kong classrooms this is due to the main function attributed to the event. As the other students continue to work individually on their assignments, it is not expected that the students at the front talk about what they have produced and it is exclusively the teacher who evaluates it (cf. an extra chance to get the teacher’s comments). If it serves the other functions identified, the activity of being at the front is a public activity and the students are required to pay attention to what their peers produce or display at the board or overhead. However, the teachers in the six classrooms show a tendency to appropriate solutions presented by the students. The extent to which this occurs is associated with the function of the activity in the six classrooms. If it is to share results from individual practice or homework, sometimes more students write solutions of different tasks in parallel in order to save time (cf. publicising work). In this form the activity obviously does not afford students’ talk. After the students have taken their seats, the teacher evaluates and illuminates their solutions. But even if only one student is at the board, the student in general does not talk. The teachers watch carefully and evaluate what they write by short comments. Only occasionally, if the solutions contain mistakes, they ask the students to explain what they did (cf. publicising, explaining work). This practice is linked to tasks, which only allow one correct solution mostly without choice of different methods. Consequently there is nothing to talk about but distinct types of error made by some students. ‘Comparing’ means to find out whether the solution one has produced is similar to the model solution, which eventually, after some corrections, is put up on the board. The teacher makes sure that there are only correct solutions. In one instance in a Hong Kong classroom (HK 1, Lesson 5), students write different solutions of the same equation on the board. This is part of a carefully planned development of the content in the lesson, in which the teacher wants to introduce examples of simultaneous equations with an infinite number of solutions. According to the teacher’s agenda, the students’ results are publicised so that he is able to refer to the variety of results in his explanation. Even though the form in which this activity of sharing work is enacted leaves no room for students’ comments, it contains an element of incorporation of their products in order to make it functionally necessary to extend or develop knowledge. If groups of students have worked on different tasks and present their results at the board, there is naturally an urge for illumination and explanation. However, in the course of the 60 lessons analysed, there is only one example of an extended period in which the students at the board try to give a coherent explanation of the results of their group work and answer questions posed by their peers and not by the teacher (cf. publicising and explaining work in G1). In US1, the results from

119

EVA JABLONKA

group work are displayed on posters or white boards which the students show to the class (cf. displaying work). Altogether, in the six classrooms in this study, students are hardly initiated into ‘talking mathematics’. The classroom practices do not afford public student argumentation for different reasons. The impact of this feature of classroom practice on students’ conceptions of mathematics in the six classrooms is reported elsewhere (Jablonka, 2005). As it has been pointed out, public utterances carry a high risk of vulnerability because the types of tasks (with the exception of the proof in G1 and the tasks in group work in US2) ask for answers without much variation between getting it right and getting it wrong. So there is nothing to discuss except the reasons for getting it wrong. Consequently, from the perspective of the students it seems to be a risk to go to the front or to make public utterances because it is easy to evaluate what they produce in terms of correctness. The interviews with the students from the six classrooms in this study show that most of the students volunteer for a turn or to go to the board only if they are completely sure about their assertions and answers. Showing work on the board might be associated with a higher risk than oral comments because in mathematics lessons the board usually is used by the teachers for setting tasks, for showing a worked example, or for documentation of the essence of the lesson. What is written on the board during a lesson is often planned by the teacher in advance, including a carefully structured layout. Consequently, what is shown on the board gains the status of indisputable truth and the students know that it is considered important information to be copied. There might be an implicit rule in operation that it is a problem to have assumptions, tentative proposals or wrong solutions put up on the board. If in addition, the classroom atmosphere is competitive, this does not afford conjecturing. The students from the two US classrooms do not know each other very well and did not talk about the group as a whole in the interviews. The atmosphere is not just a superficial feature that makes the lessons more enjoyable but has a strong impact on students’ participation. Boaler (2000) reports that many of the students interviewed in the course of a study of six English schools mentioned their relationships with other members of the class as the most important factor influencing their predilection towards mathematics. Another reason that students are not likely to feel responsible for an explanation or illumination of their own products, is the appropriation of what they produce by the teacher. As it has been pointed out, by the teachers in the six classrooms this is done by immediate evaluation, by repetition of students’ utterances, by questioning exclusively in case of mistakes and by planned incorporation into the pre-defined agenda of the lesson. In the instances categorised as division of labour between teacher and students the students can be seen as apprentices who are allowed to take over part of the responsibility for knowledge production. Knipping (2003) has studied the collective development of proofs in German and French classrooms extensively. Inter alia she points out that these processes are characterised by an asymmetric distribution of labour between students and teacher with respect to the provision of official justifications. Theories of argumentation in institutional 120

STUDENT(S) AT THE FRONT

settings might offer further clarification of the ways in which the authority relation between teacher and students constrain an investigative and communicative classroom atmosphere (e.g., Baker, 1998; Weingarten & Pansegrau, 1993). REFERENCES Baker, C. D. (1998). Ethnomethodological studies of talk in educational settings. In B. Davis & D. Corson (Eds.), Encyclopedia of language and education Vol. 3: Oral discourse and education (pp. 43-52). Dordrecht, The Netherlands: Kluwer Academic Publishers. Boaler, J. (2000). Mathematics from another world: Traditional communities and the alienation of learners. Journal of Mathematical Behaviour, 18 (4), 379-397. Brodie, K., Lelliott, T., & Davis, H. (2002). Forms and substance in learner-centred teaching: Teachers’ take-up from an in-service programme in South Africa. Teaching and Teacher Education, 18, 541-559. Foucault, M. (1977). Discipline and punish: The birth of the prison. London: Allen Lane. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., Miu-Ying Chui, A., Wearne, D., Smith, M., Kersting, N., Manaster, A., Tseng, E., Etterbeek, W., Manaster, C., Gonzales, P., & Stigler, J. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study (NCES 2003–013 Revised). Washington, DC: U.S. Department of Education, National Center for Education Statistics. Jablonka, E. (2005). Motivations and meanings of students’ actions in six classrooms from Germany, Hong Kong and the United States. Zentralblatt für Didaktik der Mathematik, 37 (5), 371-378. Knipping, C. (2003). Beweisprozesse in der Unterrichtspraxis. Vergleichende Analysen von Mathematikunterricht in Deutschland und Frankreich [Proving in classroom practice. Comparative analyses of mathematics teaching in Germany and France]. Hildesheim & Berlin: Franzbecker. Telese, J. A. (2004). Middle school mathematics classroom practices and achievement: A TIMSS-R analysis. Cambridge: Center for Teaching & Learning of Mathematics, Cambridge College. Weingarten, R., & Pansegrau, P. (1993). Argumentationsstile im Unterricht [Styles of argumentation in classrooms]. In B. Sandig & U. Püschel (Eds.), Stilistik. Band III: Argumentationsstile. HildesheimNew York: Georg Olms Verlag.

Eva Jablonka Fachbereich Erziehungswissenschaft und Psychologie Freie Universität Berlin Germany

121

EVA JABLONKA

APPENDIX Table 2: ‘Student(s) at the Front’ in G1, Lessons 1-10 Lesson TimeIn

TimeOut

Description

5

04:31

07:28

Four students are writing the results of their group work on the board. The teacher walks between the desks.

6

37:48

45:01

Three students at the board construct drawings that they have been producing in their group work. The teacher walks between the desks. One extra student assists for a short period.

7 (1)

02:20

08:10

Seven students who are delegates of three groups draw a geometrical interpretation of three binomial formulas on the board. The teacher watches them and talks to some of them.

7 (2)

08:35

13:10

Two students explain the drawings from their group to the whole class [geometrical proof of (a + b)(a + b)].

7 (3)

13:50

23:40

Two students explain the drawings to the whole class [geometrical proof of (a – b)(a – b)].

7 (4)

23:40

30:07

Three students explain the drawings to the whole class [geometrical proof of (a + b)(a – b)].

7 (5)

33:18

34:26

One student gives additional comments on the work of the last group, prompted by a question from the teacher.

Table 3: ‘Student(s) at the Front’ in G3, Lessons 1-10 Lesson TimeIn

TimeOut

Description

1 (1)

09:29

10:22

One student is asked to find and write an equation on an OHP slide for a graph of a linear function that is shown on that slide.

1 (2)

10:52

13:13

One student is asked to find and write an equation on an OHP slide for a graph of a linear function that is shown on that slide.

2/3 (1) 20:38

23:01

One student is asked to present a solution from work in pairs at the OHP.

2/3 (2) 23:23

23:55

One student is asked to present a solution from work in pairs at the OHP.

2/3 (3) 23:56

25:20

One student is asked to present a solution from work in pairs at the OHP.

2/3 (4) 28:57

30:54

One student is asked to present a solution from work in pairs at the OHP.

2/3 (5) 31:04

31:52

One student is asked to present a solution from work in pairs at the OHP.

122

STUDENT(S) AT THE FRONT

Lesson TimeIn

TimeOut

Description

2/3 (6) 31:52

34:09

One student is asked to present a solution from work in pairs at the OHP.

2/3 (7) 75:27

79:08

One student presents the solution of a task solved in individual seatwork.

4 (1)

30:04

33:05

A student fills liquid into a model of a solid in order to find out the volume.

4 (2)

34:42

35:44

A student fills liquid into a model of a solid in order to find out the volume.

5 (1)

19:21

Not identifiable

One student solves a task at the board.

6

08:20

14:33

One student solves a task at the board.

7 (1)

05:11

06:07

One student solves part of a task at the board.

7 (2)

06:41

07:27

The student called upon does not want to go to the board and instead dictates the steps to the teacher.

8 (1)

25:37

27:28

One student solves a task from the homework at the board.

8 (2)

27:34

28:41

One student solves a task from the homework at the board.

9/10

01:20

01:24

One student presents the results from work in pairs at the board.

Table 4: ‘Student(s) at the Front’ in HK1, Lessons 1-10 Lesson TimeIn

TimeOut

Description

1

21:27

30:41

The teacher asks several students to write their solutions on the board, while the rest of the class continues individual seatwork on the tasks. The students work in parallel and sit down when finished. The teacher looks at the results and asks some students to come back to the board to fill in steps they have omitted or to correct other mistakes.

3 (1)

14:12

20:05

Two students write their solutions on the board while the others continue individual seatwork. One of the students at the board is engaged in private talk with the teacher.

3 (2)

28:13

32:48

One student volunteers to solve a task immediately at the board. The others solve the task in individual seatwork. The teacher later cleans the board without commenting up on the task.

123

EVA JABLONKA

Lesson TimeIn

TimeOut

Description

3 (3)

49:50

50:14

One student writes a solution on the board while the others continue individual seatwork and the teacher walks between the desks. Later the teacher writes the answers on the board.

5

05:47

13:55

Different students write different results of the same equation on the board. The teacher refers to the variety of results in his explanations.

6

09:01

14:12

One student constructs a graph at the board while the other students do the same in individual seatwork.

9 (1)

33:40

36:04

One student writes his result on the board while the others continue individual seatwork.

9 (2)

37:10

39:20

One student writes her result on the board while the others continue individual seatwork.

Table 5: ‘Student(s) at the Front’ in HK3, Lessons 1-10 Lesson TimeIn

TimeOut

Description

7

30:45

35:25

One student volunteers to go up to the board and copies her solution from her notes. The teacher talks to individual students.

8 (1)

16:00

17:14

The teacher puts the pen for the board on a desk and claps his hands. One student goes to the board and writes the solution. The teacher talks to individual students.

8 (2)

17:14

20:25

One student points to another student sitting next to her. The latter writes her solution on the board. The teacher talks to individual students.

10 (1)

13:45

16:48

One student is solving a task on the board after the teacher tells her to do so. The teacher continues to walk between the desks.

10(2)

18:24

20:27

One student writes her answer on the board after the teacher has handed her the pen over.

Table 6: ‘Student(s) at the Front’ in US1, Lessons 1-10 Lesson TimeIn

TimeOut

Description

1 (1)

17:56

19:14

One student is called to write her solution on the board.

1 (2)

25:42

26:18

One student is called to write his solution on the board.

1 (3)

35:52

36:47

One student is asked to write his solution and to explain it.

1 (4)

41:53

44:03

One student is called and refuses. Another student volunteers and writes her solution. She is asked to explain it.

124

STUDENT(S) AT THE FRONT

Lesson TimeIn

TimeOut

Description

2 (1)

06:06

07:35

Five students solve five tasks of the warm-up in parallel at the board (tasks 13-17).

2 (2)

16:43

17:50

One student writes her solution of a task from individual work on the board. She is asked to explain it.

2 (3)

22:41

23:45

Two students write in parallel the solutions of two different tasks on the board.

3 (3)

10:10

12:28

Five students solve five tasks of the warm-up in parallel at the board (tasks 20-24). They copy from their notes.

4

05:14

07:06

One student solves one task from the warm-up exercise at the board.

6 (1)

18:10

19:06

Three students write their solutions of three different tasks from individual work on the board.

6 (2)

22:48

23:29

One student is asked to share her version of a solution at the board because her result is different.

6 (3)

27:56

28:45

Two students write their solutions of two different tasks from individual work on the board.

6 (4)

39:37

40:41

One student is asked to share her version of a solution at the board.

Table 7: ‘Student(s) at the Front’ in US2, Lessons 1-10 Lesson TimeIn

TimeOut

Description

1 (1)

05:26

07:54

The teacher calls 8 students to solve 4 different tasks from the warm-up in parallel at the board; each pair is allowed to collaborate on one task. One student volunteers to solve the fifth task on the board.

1 (2)

10:35

10:46

The teacher calls a student to show his solution (“Come show us … quick quick … ”)

1 (3)

21:05

22:00

The teacher talks to the student at the board to save time.

1 (4)

22:00

22:48

The teacher asks a student to sketch a graph on the board.

1 (5)

29:35

30:00

One student sketches two graphs on the board.

1 (6)

31:51

33:10

Two students are filling in columns of a T-chart.

2 (1)

02:46

13:47

Eight delegate students pin posters from group work on the board and read them out. The posters from the remaining groups are hold up by the teacher on the desks and read out by a student. One student stays at the front and pins the remaining posters on the board.

2 (2)

34:20

35:02

One student draws a graph on the board

3 (1)

14:03

14:55

One student is asked to explain his result of a warm-up task at the board. The teacher makes clear that his version is wrong.

125

EVA JABLONKA

Lesson TimeIn

TimeOut

Description

4 (1)

08:25

13:16

Students pin posters from group work on the board after controlled by the teacher.

4 (2)

20:25

21:25

One student pins a poster on the board.

4 (3)

30:57

32:08

One student is asked to point out distinct graphs on the posters at the board.

4 (4)

32:18

33:19

One student points at a part of one task.

5

17:44

Not identifiable

One student makes notes at a flip chart.

6

17:37

18:12

One student is asked to explain the assignment again.

7 (1)

18:50

19:27

One student writes her solution at the OHP.

7 (2)

24:45

25:40

One student draws a graph on the board.

7 (3)

29:27

30:30

One student writes the solution of an equation on the OHP. She is called again to the board to adjust a graph that is already there.

8 (1)

27:16

29:30

Students bring back the containers with materials.

8 (2)

35:30

42:26

Four students from group 1 hold up white boards from group work. The students read out what is on their boards.

8 (3)

40:15

42:26

Four students from group 2 hold up their white boards from group work.

8 (4)

42:30

47:15

Four students from group 3 hold up their white boards from group work.

8 (5)

45:10

47:15

Four students from group 4 hold up their white boards from group work.

9 (1)

07:55

09:20

Two students talk with the teacher at the teacher desk about one graph on one of the white boards from the day before.

9 (2)

11:40

13:39

One student fills in a T-chart with the numbers dictated by other students.

9 (3)

27:17

30:22

Students carry their white boards to the front.

9 (4)

31:07

37:28

Three students hold up their white boards. The teacher explains that she had made a mistake so that the task was hard to solve.

9 (5)

40:23

45:22

Seven students from two groups hold up their white boards.

9 (6)

42:10

42:48

One student points to the y-intercept on a graph.

126

YOSHINORI SHIMIZU

CHAPTER SIX How Do You Conclude Today’s Lesson? The Form and Functions of ‘Matome’ in Mathematics Lessons

INTRODUCTION

In the TIMSS 1995 Videotape Classroom Study, certain recurring features that typified many of the lessons within a country, Germany, Japan, or the United States, and distinguished the lessons among three countries were identified as “lesson patterns” (Stigler & Hiebert, 1999). The following sequence of five activities was described as the Japanese pattern: reviewing the previous lesson; presenting the problems for the day; students working individually or in groups; discussing solution methods; and highlighting and summarizing the main point. From a Japanese perspective, the final in the sequence of five activities described above, highlighting and summarizing the major points, seems to be an indispensable element in any successful lesson. How to end the lesson in accordance with its goals is the key question to those teachers who hope to conclude their lesson as a coherent and understandable learning opportunity to the students they teach. Japanese teachers traditionally share the term ‘Matome’ for describing the corresponding teacher’s activity at the end of lessons. Characterisation of the practices of a nation’s or a culture’s mathematics classrooms with a single lesson pattern was, however, problematised by the results of the Learner’s Perspective Study (Clarke, 2003; Jablonka, 2003; Mesiti, Clarke & Lobato, 2003; and Shimizu, 2003). The earlier analysis suggested that, in particular, the process of mathematics teaching and learning in Japanese classroom could not be adequately represented by a single lesson pattern by, at least, the following two reasons (Shimizu, 2003). Firstly, lesson pattern differs considerably within one teaching unit, which can be a topic or a series of topics, depending on the teacher’s intentions throughout the sequence of lessons. Secondly, elements in the pattern themselves can have different meanings and functions in the sequence of multiple lessons. Needless to say, it is an important aspect of teacher’s work not only to implement a single lesson but also to weave multiple lessons that can stretch out over several days, or even a few weeks, into a coherent body of the unit. It would not be possible for us to capture the dynamic nature of activities in the teaching and learning process if each lesson was analysed as ‘standing alone’.

D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 127–145. © 2006 Sense Publishers. All rights reserved.

YOSHINORI SHIMIZU

An alternative approach was proposed to the international comparisons of lessons by the researchers in the Learner’s Perspective Study (LPS) team. That is, a postulated ‘Lesson Event’ would be regarded to serve as the basis for comparisons of classroom practice internationally. The Lesson Event is conceived as an event type sharing certain features common across the classrooms of the different countries studied. This chapter discusses the form and functions of the particular lesson event ‘Matome’ (“summing-up” in Japanese), in eighth-grade mathematics classrooms in Australia, Germany, Hong Kong, Japan, Mainland China (Shanghai), and the USA. Firstly, the form and functions of the event were analysed within the local contexts of Japanese classrooms. Then, the Lesson Events in the classrooms in other countries were identified and compared with those in Japanese classrooms. Particular attention is given to the commonalities in its form and the variation of its functions among different classrooms in different cultures. The student post-lesson interview data from Japanese LPS schools are also analysed for exploring the significance and the meaning that the students associated with the event type. ‘MATOME’ (SUMMING UP): THE LESSON EVENT

An earlier analysis revealed that multiple lessons are interrelated and that the structure of each single lesson looks differently when we locate it in the entire teaching unit (Shimizu, 2003). It was suggested by the analysis that mathematics teaching and learning in Japan could not be adequately represented by the analysis of a set of distinct lessons. The analysis also showed that the students perceived the significant events in the lesson in a different way from the way the teacher perceived them (Shimizu, 2006). The result suggests that the units of data collection and data analysis for the study of lessons are crucial for the international comparisons. An approach to the characterisation of classroom practices has been proposed as identifying the specific Lesson Event type and the practices through which it is constituted, such that, while not necessarily a consistent element of every or even most of the lessons being analysed, the particular type of Lesson Event is frequent in occurrence, takes a consistent general form, but may be enacted with some variation at the level of actual classroom practices (Clarke, 2003). The author identified Matome, which means "sum up one’s main point in conclusion" or "pulling together", as the specific Lesson Event type for characterising classroom practices. Traditionally Japanese teachers share several pedagogical terms, Matome among others, which describe the teacher’s key roles at various phases of the lesson. Furthermore, since Japanese teachers often organise an entire lesson around just a few problems with focus on the students' various solutions to them, they think that "summing up" is indispensable to any successful lesson in which students’ solutions are shared and pulled together in light of the goals of the lesson (Shimizu, 1999). There are some findings of the international comparative studies on mathematics lessons that seem to be related to the function of this particular event. The Third 128

THE FORM AND FUNCTIONS OF ‘MATOME’ IN MATHEMATICS LESSONS

International Mathematics and Science 1999 Video Study (TIMMS), among others, identified some characteristics of Japanese lessons (Hiebert, et al., 2003). One of the outstanding differences of Japanese lessons when compared with lessons from the other six countries was the teachers’ behaviour for summarising the problem solving activities to clarify the mathematical point illustrated by the problem: About one-quarter (27 percent) of mathematics problems per lesson were summarised by the teacher to clarify the mathematical point illustrated by the problem (table 5.4), more than in any of the other countries (Hiebert, et al., 2003, p.136). These findings suggest that Matome would be a recognisable event in the classrooms in other countries and that by identifying and analysing the event we would be able to identify certain characteristics of classroom practices in different cultures. In this chapter, Matome is defined as an event in which the teacher talks to the whole class to highlight and summarise the main point of the lesson. What students engaged and discussed in the lesson is reviewed briefly in the whole-class setting and what they learned during the lesson is highlighted and summarised by the teacher. In the following sections, we look into this particular Lesson Event with a focus on its form and functions. THREE EXAMPLES FROM JAPANESE LPS CLASSROOMS

In this section, the author examines three examples of the event type from the Japanese data. The three examples were identified as the event in which the teacher talked to the whole class to highlight and summarise the main point of the lesson by referring to what the class had done. The following three examples show how each of the Japanese teachers engaged in Matome in a consistent way with various purposes. Table1. Details of the three examples examined in this chapter. Example

Classroom

Lesson Number

Start and End Times

1

J3

3

00:43:08:15 to 00:43:35:18

2

J1

3

00:24:43:06 to 00:27:17:26

3

J3

7

00:34:05:16 to 00:40:20:14

129

YOSHINORI SHIMIZU

While Example 1 was selected to describe how the teacher at the end of the lesson emphasises the importance of what they had done during the lesson, Example 2 was selected to show that Matome can take place in the middle of lesson with the teacher’s intention to summarise what the class has done in a series of lessons. Also, Example 3 illustrates an example of Matome that encompasses broader activities stretched out among several lessons. Example 1: J3-L03, (00:43:08:15 to 00:43:35:18) 00:43:08:15

00:43:22:13

00:43:35:18

T: Yes, um, today, we will end here but we did something extremely important today. Um, it will have to be next week, solving the equation from KINO's question will have to be next week. T: But if we finish up to here, I think you'll be able to solve tons of equation. Check the calculation when you need to and I'll ask you sometimes. I'll ask you to show me how much you can do but is that ok? T: I think we were able to finish just about everything, up to the important ways of thinking of equations. You should be able to solve everything. Ok? Now, I'll give you the rest of the time to jot things down.

Key to symbols used in transcripts in this chapter … A short pause of one second or less. [text] Comments and annotations, often descriptions of non-verbal action. text Italicised text indicates emphatic speech. .... Indicates that a portion of the transcript has been omitted.

In the first short excerpt of transcription from the J3-L03, in which the students were learning to solve simultaneous linear equations, the teacher summarised and highlighted what they had done in the form of general comments. The comments were made at the final minutes of the lesson. He noted that the class had done “something extremely important”(00:43:08:15), emphasising that the students “would be able to solve tons of equation”(00:43:22:13) and they “should be able to solve everything” (00:43:35:18). Also, he encouraged the students to “check the calculation when you need to.” At the end of the lesson, after some discussions on two alternative ways of check the solution to the simultaneous linear equations, the teacher strongly emphasised that what they had done was extremely important. He then asked the students to jot things down on their notebook. In this case, the teacher appeared to promote students’ reflection on what they had done and on the importance of checking the results. The teacher pointed out the part of blackboard on which an important idea was described. Example 2: J1-L03 (00:24:43:06 to 00:27:17:26) 00:24:47:00 00:25:05:12

130

T: Because they are all shown as a linear equation, they are called linear functions. Note it somewhere in your notebook. T: And please look at this type of equations, in which B is zero.

THE FORM AND FUNCTIONS OF ‘MATOME’ IN MATHEMATICS LESSONS 00:25:18:28 00:25:33:27 00:25:48:28 00:25:49:23 00:25:51:07 00:25:54:15 00:26:06:07 00:26:41:10 00:26:57:13 00:27:07:05 00:27:08:25 00:27:17:26

T: They are in the same group, but linear functions. What did we call this kind of equations in seventh grade? T: Huh? Do you all remember that? How did you call them? How did you describe the relation between x and y? S: X is directly proportional to y. T: Yes, the proportion. T: When we learned this in seventh grade, we said that they are proportional, T: but actually you already understood it as a kind of linear functions, only difference is if it has B or not. T: You can see it in the textbook. Please open it and see it yourself. It's in page fifty-seven.[Writing on the blackboard] T: Okay, look at tenth line, no I mean ninth line. Just what we talked about. It's summarised there. T: Draw an underline from ninth line. T: We can see what just we talked about in words; T: having two variables x and y, if y can be expressed with the linear equation of x, then we call it a linear function. T: And a linear function is expressed as y= x+b. Okay?

The Lesson Event in the third lesson in the school J1, on the other hand, took place in the middle of lesson. The teacher introduced the term “linear function” as a formal mathematical term, by reflecting on several examples of linear functions that appeared as a result of the previous activities in the classroom; “these are called linear function. Because they are all shown as a linear equation, they are called linear functions”(00:24:43:06 to 00:24:47:00). Then she asked the student to make a note. In this case, she tried to sum up what the students had worked on in the three consecutive lessons. She next tried to make a connection between the concept of linear function to the concept of direct proportionality, a special case of the linear function, which her students had learned in the previous year; “And please look at this type of equations, in which B is zero. They are in the same group, but linear functions. What did we call this kind of equations in seventh grade?” (00:25:05:1200:25:18:28). The event included the teacher’s explicit efforts to make connection between the current topic and the one in the previous year. After the introduction to the formal term, she asked the student to “Draw an underline from ninth line” and then wrote the point on the chalkboard using yellow chalk. Finally she repeated main point by reading the corresponding page in the textbook; “We can see what just we talked about in words; having two variables x and y, if y can be expressed with the linear equation of x, then we call it a linear function. And a linear function is expressed as y = x + b . Okay?” (00:27:07:0500:27:17:26). The Lesson Event in Example 2 took place in the middle of the third lesson. The example shows that Matome can take place not only at the end of the lesson but also in the middle of the lesson for pulling together the students’ activities in multiple lessons. The event serves in setting the stage for introducing the new 131

YOSHINORI SHIMIZU

mathematical term based on the examples of linear functions examined in the activities in the three lessons including the current one. As Example 2 illustrates, Matome includes the teacher’s effort to make connections among lessons. The next example from the school J3 shows that Matome can encompass broader activities in more extended time periods. Example 3: J3-L07 (00:34:05:16 to 00:40:20:14) This example shows that in the event the teacher and the students make connections between the current topic and previous ones in the same teaching unit. In the seventh lesson, after an extended effort for deriving the “method by addition and subtraction” (Kagen-hou) through the second lesson to the sixth lesson, the teacher introduced another method, “method of substitution” (Dainyuu-hou) by referring to the solution method proposed by a student in the second lesson. The teacher carefully repeated the method proposed by the student DOEN in the second lesson. Um, the question with the process of adding both of the expressions together, and then solving the expression with only Y as we can remove X. The way I was talking about was introduced by DOEN here (00:34:36:10). He finally referred to the summary sentences in the textbook. The answer is the same as the ones on the blackboard, and I think they are more in detail than the textbook. So, I don't need additional explanation, I think. The definition of Dainyu-hou, I mean the explanation, well, I'll read the sentence on page forty-four, the two sentences before question six. Uh, we have removed y by substituting number one into number two. This way of answering, the way of leaving only one kind of letters by substitution is called Dainyu-hou. (00:39:43:28-00:39:59:13). The Lesson Event (Example 3) in J3-L07 is an example of Matome that encompasses broader activities stretched out among several lessons. The teacher had an intention of linking the new term “Dainyu-hou” with students’ shared experiences in previous lessons. Commonalities in the Form of the Event Among the three examples described above, there are commonalities in the event type in terms of associated observable teachers’ and students’ behaviours. Teacher’s Public Talk. Matome took place in the form of teacher’s public talk to the whole class. The teacher explicitly reviewed what they have learned and what was the main point of the lesson in mathematical sense. The teacher might ask a few students to tell their classmates what he or she learned in the lesson and the points appeared from the students’ activities. In this way, Matome can take place in

132

THE FORM AND FUNCTIONS OF ‘MATOME’ IN MATHEMATICS LESSONS

the form of interactive exchanges between the teacher and the students and between the students and students. Effective Use of the Chalkboard. An important technique used by Japanese teachers during Matome was the particular use of the chalkboard, which is usually referred to as ‘Bansho’ by them. Whenever possible, teachers put everything written during the lesson on the chalkboard without erasing. By not erasing anything the students had done and placing their work on the chalkboard in an organised manner, it was much easier for them to compare the multiple solution methods proposed. Also, the chalkboard served as a written record of the entire lesson, giving both the students and the teacher a bird’s-eye view of what had happened during the lesson. All the three teachers of the Japanese LPS classrooms used the chalkboard in a similar way. The teachers capitalised on these advantages by ‘moving’ across the chalkboard. Reference to the Textbook. Another noteworthy characteristic with regard to the Lesson Event relates to both the teachers’ and students’ particular behaviours observed during Matome. At some points during the event, the teacher asked their students to take notes of the important points. In addition, all the three Japanese LPS teachers often referred to the corresponding page of the textbook to which the students responded by opening the page and taking notes or underlining the relevant texts. As illustrated by the three examples above, the Lesson Event took the following form. Typically at the final phase of the lesson, and sometimes in the middle of it, the teacher reviewed what the students engaged in and emphasised the main point of the lesson. The teacher might ask a few students to tell to the whole class what he or she learned in the lesson. She might also write on the chalkboard, by using a colored chalk occasionally, the main points or key mathematical terms and then refer to the corresponding page of the textbook. A teacher’s announcement followed, in some cases, the start of practices for applying what has just been highlighted. The Japanese lesson pattern identified by the TIMSS 1995 Video Study included "highlighting and summarizing the main point" as the final segment (Stigler & Hiebert, 1999, pp.79-80). The above example shows that, however, Matome can take place not only at the end of the lesson but also in the middle of the lesson for pulling together the students’ activities in multiple lessons. Also, when we look into sequences of consecutive lessons as embedded in the teaching unit, different functions of Matome for making connections among lessons can be identified. As Example 2 and 3 illustrates, Matome can encompass broader activities in more extended time periods. There are four key aspects to the events that emerged from the example cited above. For the Japanese teachers, the event Matome appeared to have the following principal functions: (i) highlighting and summarising the main points in the lesson, (ii) promoting students’ reflection on their experiences by reviewing what they have done, (iii) setting the stage for introducing a new mathematical concept or 133

YOSHINORI SHIMIZU

term based on the previous experiences and for applying it, and (iv) making connections between the current topic and the previous one. Students’ Perceptions of the Importance of Matome As described in the previous section, there are commonalities in teachers’ behaviours in the event type and distinct functions of it. This indicates a reflection of the teacher’s clear intention of having this particular classroom event type for promoting students’ learning. But how do students perceive the event? Do they recognise the importance of it? The post-lesson student interviews were analysed to explore the significance and the meaning that the students attached to this event type. In the post-lesson video-stimulated interviews after the tenth lesson at the school J1, one student explicitly identified the video-recorded segment in the lesson that corresponded to the event type as the point he felt to be of significance. At this point the teacher was summarising what the students had learned in this lesson and emphasised the concept of rate of change. The student, Oba, mentioned the event at the very end of lesson as follows. 01. OBA: 02. INT: 03. OBA: 04. INT: 05. OBA: 06. INT: 07. OBA:

Here, it’s number three some time ago, isn’t it? Is it here? Um, forty-three minutes and fifty-five seconds. When the teacher explained about rate of change…… Yes. It’s the explanation for rate of change some time ago. Yes. It’s here. We were told to underline here in the textbook. So, that means, as you said, rate of change is important. Yes. We went through that part here, so I was underlining and listening to the teacher carefully at the same time.

The excerpt suggests that the Lesson Event was recognised as an important moment within the entire lesson, not only by the teacher who usually initiated the event but also by the students. If the teacher keeps summarising and highlighting the main points of the lesson as a daily routine, the students may become aware of the importance of the particular Lesson Event which tends to come on the final phase of lesson in the form of teacher’s public talk together with time for notetaking. The interviewer asked Jitsu, another student in the same classroom, after the first lesson to tell what he thought that the lesson was about and what is the best thing for him to learn from it. He commented, “When I listen to teacher's talking, I always take a note and check a point.” 08. JITSU: 09. INT: 10. JITSU: 11. INT: 12. JITSU:

134

Today, I studied about a proportion. I think. OK For my opinion, a proportion is my favorite, so I could think about it easily. Ok, thanks. Question number two. YES.

THE FORM AND FUNCTIONS OF ‘MATOME’ IN MATHEMATICS LESSONS 13. INT: 14. JITSU: 15. INT:

How, do you think, you best learn something like that? What do you think, the best thing to do? At first, when I listen to teacher's talking, I always take a note and check a point. Also I always try to think something by myself at first. I don't rely on somebody. Ok, thanks.

The excerpt reveals his recognition of the importance of note-taking and checking a point of teachers’ talk at this particular event. Another student, Suzu, a student of school JP3, also responded to the question of “When you think it’s a good class?” as follows. 16. 17. 18. 19. 20. 21. 22. 23.

INT: SUZU: INT: SUZU: INT: SUZU: INT: SUZU:

24. 25. 26. 27. 28.

INT: SUZU: INT: SUZU: INT:

When you think it’s a good class? Yes. What should happen in the class? …in the class. Yeah? Uh, Do you have anything that you think is a good class? I can present my answer, and then listen to my friend’s way as well. Yeah? The teacher’s final comment, or answer, Yeah? Listen to it carefully, and to make a good note from it. It might be good

The student also clearly mentioned to the importance of listening to “The teacher’s final comment, or answer” carefully and of “making a good note from it”. The comment is consistent with the commonalities in teachers behaviours of public talk found in the three examples of Matome. These interview data suggest that, not only the teacher who usually initiated Matome, but also the students as ‘audiences’ during the period of this particular event perceived it as an event of significance to their learning. Teachers’ Perceptions of the Importance of Matome Mr. N, the teacher at school J2, mentioned that he usually tried to conclude each lesson because if he “end in the wrong spot, everything we did is often forgotten”, and “it’s a bit awkward in the beginning of the next class”. 29. MR. N: 30. INT: 31. MR. N:

32. INT: 33. MR. N:

That took up a lot of time and I was actually a bit worried about whether we’d get into the last similarity problem or not Yes And, well we couldn’t finish it but uh it was a nice, what do you call it, for the next lesson. It was a nice preview, and I introduced it hoping I’d be able to end arousing some interest in the students. I see. Of course, I can’t conclude that similarity is this or that in that little time so,

135

YOSHINORI SHIMIZU 34. INT: 35. Mr. N: 36. INT: 37. Mr. N: 38. INT: 39. Mr. N: 40. INT: 41. Mr. N: 42. INT: 43. Mr. N: 44. INT:

Yes So, I knew that we had class the next day, in the morning, so I thought they probably wouldn’t forget it all So you don’t try to conclude in just one session, in the forty-five or fifty minutes. You think you can just go on in the next lesson. Yes. But actually I do try to conclude. Yes Uh, if I end in the wrong spot, everything we did is often forgotten Yes And it’s a bit awkward in the beginning of the next class. Hmm So here, I really wanted to cut about five minutes out of the time we took for drawing figures, I see

Mr. K, the teacher at school J3 who did not referred to Matome, responded to “What has to happen for you to feel that a lesson was a ‘good’ lesson?” as follows. 45. Mr.K:

46. INT: 47. Mr.K: 48. INT: 49. Mr.K:

I think there is an important scene in the lesson but, if we do not pay close attention to the student’s activities, I think that student wouldn’t start thinking. The responses from the students are of course various, or that student think again, Yeah. If I ask a question about the situation, The students…!start responding again. Yeah. I want the lessons to teach student to figure things out and I think that to figure things out is the most important activity in a lesson.

He identified one episode in JP3-L07 as the event of significant to him. In this excerpt from post-lesson video stimulated interview, he explained why he named one student who in her writing summarised what she had learned in the lesson. 50. 51. 52. 53. 54.

Mr.K: INT: Mr.K: INT: Mr.K:

55. INT: 56. Mr.K: 57. Mr.K: 58. INT: 59. Mr.K:

136

Um… She thought that there is the way to substitute. Yes. She explained with that kind of way. Oh, I know. The reason that I picked up her was her way of writing was the best and smart. Oh, she was. So I asked her. I tried to ask them to put their thoughts together, but there were no students, who could get their ideas in shape well, in this group as yet. Oh, is that so? To my surprise, maybe they don’t get used to write about their feeling of thinking.

THE FORM AND FUNCTIONS OF ‘MATOME’ IN MATHEMATICS LESSONS

Mr. K asked the student to present her writing in order for highlighting and summarising the main points in the lesson in a smart way. This event can promote students’ reflection on their experiences by reviewing what they have done from the learner’s perspective. The excerpts from post-lesson interviews show that the teachers perceived the event Matome as an important one from both teacher’s and students’ perspectives. THE FORM AND FUNCTIONS OF MATOME IN OTHER COUNTRIES

In this section, the Lesson Events as occurred in the classrooms in other countries are compared with those found in Japanese classrooms. Matome in Shanghai Classrooms The teachers in the three Shanghai classrooms often highlighted and summarised the lesson mostly at the end of lessons. They began Matome by reviewing what the class had done in the lesson and then emphasised the main points as illustrated by the examples below. So today we have talked about using the method of substitution to solve the linear equation in two unknowns (SH1-L06, 00:41:48:00). These are the points which we should pay attention when we want solve the linear equations in two unknowns (SH1-L06, 00:43:17:11). Good. Okay, today we've talked about some concepts of system of linear equations in two unknowns. System of linear equations in two unknowns, its solutions and how to solve the system, we've talked about one of the way to solve the system, basically it is to change two unknowns into one unknown. Today, we've used method of substitution, and we will talk about the other methods later on (SH2-L03, 00:41:49:27 to 00:42:16:16). Occasionally, they had interactions with their students while summing up. They also summarised the main points using overhead projector or on the blackboard. The teacher in school SH3, in particular, quite often (in twelve lessons out of fifteen) used the slide or the blackboard while summing up. The excerpt from the transcription of SH2-L03 (00:41:49:27 to 00:44:59:00) shows that the teacher summed up the lesson by recalling the topics and by asking the students some questions by asking students to find out the answers by themselves in the textbook. For example, the teacher asked, “I want to ask, how many equations are needed as minimal, in order to solve the system?” (00:42:24:17). In response to the student who answered to the question as “two”, he then asked another question, “Two, we can find out the solutions only when there are two independent equations. Tell me, how many solutions are there for an equation?”(00:42:35:05). The teacher summed up the lesson by asking his students questions to check their understanding.

137

YOSHINORI SHIMIZU

In general, teachers’ behaviours, and the students’ behaviours in response to them, during Matome in Shanghai schools were very similar to those in Japanese classrooms. It is noteworthy that in all the examples examined, the teachers in Shanghai highlight and summarise the lesson at the end of lesson. Matome in the US Classrooms The teacher from school US2 often made summary statements, similar to those found in the Japanese examples. However, they occurred both during and near the end of activities, rather than at the end of a lesson. It was also the case that sometimes students work on a problem in small groups and share their results, and the teacher does not summarise. In the following example, US2-L05 (00:21:55:25 to 00:25:04:09), the teacher summarised and made connections via a vocabulary activity. For about 20 minutes during Lesson 05, the teacher led a type of review activity that she called the “Vocabulary for Algebraic Representations” activity. During this activity, she engaged in a lot of summing up and connection-making between the concepts addressed in Lessons 02-04. This vocabulary activity occured after an activity called the “Algebraic Meaning for Representations” learning task, on which the students had worked on during parts of Lessons 02, 03 and 04. In this activity, the students sorted 40 function cards into 10 sets so that each set would include the table, equation, graph, and verbal statement for the same function. The equations for the10 functions were as follows:

y = x2 2

y= x+2

y =x

x+y=2

2y = x

xy = 2

y=x 2

y=2

y = 2x

x=2

During this summing up event, the teacher referred to the ‘posters’ that the students created for each function. The teacher had started the vocabulary activity by writing “Vocabulary for Algebraic Representations” on the board. She also mentioned several book pages (p. 421; p. 366; pp. 229-230; pp. 385-386; pp. 672) where the students could find definitions for the following list of vocabulary: linear, non-linear, parabola, hyperbola, slope = rise/run, y-intercept, quadratic, direct variation, undefined slope, squaring function, function, vertical line test, no slope, zero slope, linear growth, linear decay. However, she quickly made it clear that she was interested in the students’ ideas about what these notions actually meant and not the textbook’s definitions: Uh, you have your own notions of what some of these terms mean, already….Whatever context, is whether or not I - I recognise them and I can

138

THE FORM AND FUNCTIONS OF ‘MATOME’ IN MATHEMATICS LESSONS

use them in some meaningful way. It's not that I can recite a definition. That's meaningless, it always has been (US2-L05, 00:14:09:09 to 00:14:30:14). The teacher then reminded students of the y = mx + b form of a linear equation, which was introduced during Lesson 3, and summed up the meaning of m and b. She went on to highlight the equations of the form y = mx or y = kx as a special class of equations that represent direct variation. She summed up that when a function represents direct variation, the y-intercept is 0, which means that the line goes through the origin. The teacher also made a new connection for students between an ‘object’ view of y = mx and a ‘process’ view of y = kx , in which k acts as a constant operator on x values to produce associated y values. Matome in the US classrooms did not appear to be a consistent structural feature of a lesson in the same way as it appears in the Japanese lessons. It did not occur in a consistent manner but it did exist. Matome in the Australian Classrooms On the whole Australian teachers did not give a specific summary at the end of (or during) each lesson, they tend to wait until they get to the end of a topic before they summarise the important issues and concepts. However they often used the beginning of the next lesson to review and reiterate the important point from the day or lesson before. The three examples discussed here relate to concepts that the teacher has been working on for several lessons. In Lesson 13/14 in the school A2, the teacher had been teaching the students simple geometry and the students had attempted many problems relating to the relationships between adjacent angles. At the end of this double lesson the teacher summarised the important things that the students should have grasped from the previous activities. He wrote a general example on the board and asked them to contribute to the summary. Some students made notes in their books. In another example from A3-L14, the teacher had been carrying out a series of lessons on decimals and percentages, she had allowed the students time for revision and was summarising important points before the next lesson when the students would sit for a test. The teacher clears the board of revision questions and writes up a couple of percentage questions she then asks specific students to assist her in solving these problems. Matome in the Hong Kong Classrooms There are only a few examples of Matome across the lesson sequences in all the three Hong Kong classrooms. In the event in HK2-L04 (00:37:21:23 to 38:57:14) the teacher summed up the lesson on simultaneous equations by emphasising the two main points of the method of elimination by addition and subtraction. He identified things that students needed to be careful about and refreshed students mind by asking them questions.

139

YOSHINORI SHIMIZU

I've taught elimination today. There are two main points for the method of elimination by addition/subtraction. You have to be aware of a few points. First, when using elimination, you have to choose either x or y (HK2-L04, 00:37:21:23 to 00:37:30:11). In another example, HK1-L04 (00:20:25:21 to 00:21:39:03), the teacher summed up the lesson by reminding students to pay attention to the change of signs in transformation of equalities in factorisation. A student has a question on the change of signs and teacher further elaborated that. Teachers’ behaviour in the video clips appeared to be consistent with those in Shanghai and in Japan. They were talking to the whole class in front of the blackboard, pointing at the blackboard occasionally. However, the teachers in Hong Kong summarised and highlighted the lesson only occasionally. Matome in the German Classrooms By analysing the data from three German LPS classrooms, Jablonka (2003) identified the activities observed in the classrooms that could not be characterised as part of one of the four components of the “typical pattern” of German lessons reported by Stigler and Hiebert (1999). Of particular interest is her analysis concerned with ‘Reviewing’ which is comprised of Checking results of students’ work, including homework, Summarizing previous work, and Review before a test. Her analysis suggests that Matome can be observed in German data, which enables us to compare the Lesson Events between two cultures. In one example (G1-L01, 00:37:23:17 to 00:39:33:06), the teacher wants to sum up how a procedure works. He does this in an interactive mode. In another school (G2-L01), the teacher lets the students work out a description of what they have done. This can also be interpreted as summing up, at least as the teacher’s intention. Another event occurs in the same lesson, when the teacher talks over something again that some students have presented as a proof at the blackboard. In these cases, teachers just made comments on what students had presented or answered. The Lesson Event in German classrooms, in which the teacher was mostly talking to the whole class in front of the blackboard, appeared to be similar to those events in other classrooms but their utterances were slightly different from those in other countries. In the Lesson Event the teachers sum something up or make some comments on students’ procedure to solve problem. But it did not seem common for them to conclude the lesson by mentioning something they did in the lesson retrospectively. DISCUSSION

Matome as an Internationally-Recognisable Event The proposed approach to the international comparisons of lessons, postulated by the LPS team, has been to use Lesson Events as the basis for comparisons of 140

THE FORM AND FUNCTIONS OF ‘MATOME’ IN MATHEMATICS LESSONS

classroom practice internationally. The Lesson Event is conceived as an event type sharing certain features common across the classrooms of the different countries studied. The analysis reported in this chapter reveals that the particular type of Lesson Event Matome is frequent in occurrence in different classrooms in different cultures, taking a similar observable form, but is enacted with variation at the level of actual classroom practices. The data analysed in this chapter revealed that there are commonalities among the Lesson Events in terms of associated observable teachers’ behaviours. Those included are teacher’s public talk in the form of ‘short lecture’ with or without interactions with the students, the effective use of the chalkboard for organising and recording what has happened in the lessons, and the particular use of the textbook as a source to which both the teacher and the students can refer. The analysis of the particular Lesson Event in eighth-grade mathematics classrooms reveals that Matome which takes a similar observable form in different classrooms can be seen as internationally-recognisable event carried out with different functional roles. There are four key aspects to the events from the teacher’s perspective that emerge from the data cited above. For the Japanese teachers, the event Matome appeared to have the following principal functions: (i) highlighting and summarising the main point in the lesson, (ii) promoting students’ reflection on what they have done, (iii) setting the context for introducing a new mathematical concept or term based on previous experiences, and (iv) making connections between the current topic and previous one. Some of these functions seemed to be shared by the teachers in different cultures. The data in LPS suggest that by attending to local interpretations of classroom practices in each country, we can cast a new light on the international comparative research. Based on their extensive observations of lessons in the elementary schools in China, Japan, and the US, for example, Stevenson and Stigler (1992) pointed out the following. We began to see how Asian teachers create coherent lessons. The lessons almost always began with a practical problem such as the example we have just given or with a word problem written on the blackboard.… Before ending the lesson, the teacher reviews what has been learned and relates it to the problem she posed at the beginning of the lesson. American teachers are much less likely than Asian teachers to begin and end lessons in this way. For example, we found that fifth-grade teachers in Beijing spent eight times as long at the end of the class period summarizing the lessons as did those in Chicago. (Stevenson & Stigler, 1992, p.179) There were both commonalities and differences among the events in the Asian classrooms in Hong Kong, Japan, and Shanghai with respect to the functional role of Matome. Matome in Hong Kong, for instance, appeared differently from those in the other two Asian cultures. Also, teachers in Shanghai and in Japan are engaged in Matome in quite similar ways but still can be different in their intentions. The result suggests that the label ‘Asian teachers’ would not be 141

YOSHINORI SHIMIZU

appropriate for describing the differences between classroom practices in different cultures. We need to have local interpretation of each classroom in each culture. The student post-lesson interview data from Japanese LPS schools shows that, not only is it the teacher who usually initiates the Lesson Event, but the students who seem to be audiences also perceive it as the event of significance to their learning. Again, having the learners’ perspectives together with teachers’ goals, intentions, and interpretations, we can approach participants’ meaning construction in the classroom. Matome as an Indispensable Element in Japanese Mathematics Lessons One of the characteristics of mathematics lessons in Japanese schools identified by several cross-cultural studies relates to the frequent exposure of students to alternative solution methods for a problem (Stevenson & Stigler, 1992; Lee, Graham & Stevenson, 1996). Japanese mathematics teachers, as was also the case in the LPS Japanese classrooms, often plan to organise an entire lesson around the multiple solutions to a single problem in a whole class instruction mode. Since the teachers place an emphasis on finding alternative ways to solve a problem, Japanese classes often consider several strategies. It would be natural for the classes to discuss the relationships among different strategies proposed from various viewpoints such as mathematical correctness, brevity, efficiency and so on. The teaching style with an emphasis on finding many ways to solve a problem naturally invites certain teacher’s behaviour for summarising. The solution methods of a problem or mathematical concepts and terms developed in the classroom are given recognition of validity and utility by the participants in the lesson, once they are summed up. In this regard, the Lesson Event, Matome, in the forms described in this chapter have much common with the function of the situation of “institutionalisation” (Brousseau, 1997). This is a situation which reveals itself by the passage of a piece of knowledge from its role as a means of resolving a situation of action, formulation or proof to a new role, that of reference for future personal or collective uses. In the classroom, for example, the solution of a problem, if it is declared typical, can become a method or a theorem. Before institutionalisation, a student can't make reference to this problem that she knows how to solve. Faced with a similar problem, she must once again produce the proof. On the other hand, after institutionalisation she can use the theorem without giving its proof again or the method without justifying it. Institutionalisation thus consists of a change of convention among the actors, a recognition (justified or not) of the validity and utility of a piece of knowledge, a modification of this knowledge – which is ‘encapsulated’ and designated – and a modification of its functioning. Thus to the institutionalisation there corresponds a certain transformation of the common repertoire accepted and used by the protagonists.

142

THE FORM AND FUNCTIONS OF ‘MATOME’ IN MATHEMATICS LESSONS

Story or Drama, as a Metaphor for Lessons There seems to be supporting conditions and shared beliefs among the Japanese teachers for having Matome often at the end of the lessons or at the end of subunits. Any lesson has parts of an opening, ‘core’, and closing. This is particularly the case for Japanese lessons which begin and end with students bowing. A lesson is regarded as a drama which has a beginning and leads to a climax by Japanese teachers. In fact, one of the characteristics of Japanese teachers’ planning of lessons is the deliberate structuring of the lesson around a climax, “Yamaba” or “Miseba” in Japanese. Most teachers think that a lesson should have a highlight or climax, just like a drama. Stigler and Perry (1988) found “reflectivity” and “coherence” in Japanese mathematics classroom. The meaning they attached to “coherence” is similar to that used in the literature on story comprehension. Stigler and Perry (1988) noted as follows. A well-formed story, which also is the most easily comprehended, consists of a protaganist, a set of goals, and a sequence of events that are causally related to each other and to the eventual realisation of the protaganist’s goals. An illformed story, by contrast, might consist of a simple list of events strung together by phrases such as “and then…,” but with no explicit reference to the relations among events….The analogy between a story and a mathematics classroom is not perfect, but it is close enough to be useful for thinking about the process by which children might construct meaning from their experience in mathematics class. A mathematics class, like a story, consists of sequences of events related to each other and, hopefully, to the goals of lesson, (p.215) The often-mentioned idea of “KI-SHO-TEN-KETSU” by Japanese teachers in Lesson Study meetings (Lewis & Tsuchida, 1998), an idea originated in the Chinese poem, further suggests that Japanese lessons have a particular structure of a flow moving toward the end (“KETSU”, summary of the whole story). The Lesson Event, Matome, appeared to promote the reflection by the teacher and the students. Stigler and Perry (1988) also found “reflectivity” in Japanese mathematics classroom. They pointed out that the Japanese teachers stress the process by which a problem is worked and exhort students to carry out procedures patiently, with care and precision. Given the fact that the schools are part of the larger society, it is worthwhile to look at how they fit into the society as a whole. The event type seems to rest on a tacit set of core beliefs about what should be valued and esteemed in the classroom. As Lewis (1995) noted, within Japanese schools, as within the larger Japanese culture, “Hansei”– self-critical reflection – is emphasised and esteemed. Of special interest is exploring a difference between cultures at this level. Sekiguchi (1998) emphasised the importance of recognising social and cultural situated-ness of mathematics education research. Research participants, settings, unit of data analysis, interpretation, educational implications are all socially and culturally constrained. The 143

YOSHINORI SHIMIZU

reliability and validity of research results are, therefore, also socially and culturally bounded. (p.394) As the citation above points out, any research activity is socially and culturally situated. This means that we can learn something about educational practices in our own country from comparisons with those in other countries only if we have relevant information and interpret the information in a sensible way. CONCLUDING REMARKS

In this chapter the form and functions of the particular Lesson Event, Matome, in the eighth-grade mathematics classrooms were examined with the data from Australia, Germany, Hong Kong, Japan, Mainland China (Shanghai), and the USA. The analysis reported in this chapter reveals that the particular type of Lesson Event, Matome, is frequent in occurrence in different classrooms of different cultures, taking a similar observable form in different classrooms but is enacted with variation at the level of actual classroom practices. Analysing the form and functions of the particular Lesson Event invites us to attend to the meaning of the event for the participants in the classroom within an educational system. An approach is needed to identify the tacit set of core beliefs about what should be valued and esteemed in the classroom. There is a Japanese proverb, “if the beginning is good, then the end is also good”, which suggests that Matome itself occurs as a part of system. Matome needs to be examined in relation to other Lesson Events. ACKNOWLEDGEMENTS

The research reported in this chapter was partially funded by 2004-2006 Grant-inAid for Scientific Research (B) of Ministry of Education, Science, Sports and Culture (Grant No. 16300249). I would like to thank Jarmila Novotná and Seah Lay-Hoon for their helpful comments on the earlier version of this chapter. REFERENCES Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer Academic Publisher. Clarke, D. J. (2003, April). The structure of mathematics lessons in Australia. In D. J. Clarke (Chair), Mathematics lessons in Germany, Japan, the USA and Australia: Structure in diversity and diversity in structure. Symposium conducted at the Annual Meeting of the American Educational Research Association, Chicago. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K.B., Hollingsworth, H., Jacobs, J., Chiu, A.M.-Y., Wearne, D., Smith, M., Kersting, N., Manaster, A., Tseng, E., Etterbeek, W., Manaster, C., Gonzales, P., & Stigler, J. (2003). Teaching mathematics in seven countries Results from the TIMSS 1999 video study. U.S. Department of Education. Washington, DC: National Center for Education Statistics. Jablonka, E. (2003, April). The structure of mathematics lessons in German classrooms: Variations on a theme. In D. J. Clarke (Chair), Mathematics lessons in Germany, Japan, the USA and Australia:

144

THE FORM AND FUNCTIONS OF ‘MATOME’ IN MATHEMATICS LESSONS Structure in diversity and diversity in structure. Symposium conducted at the Annual Meeting of the American Educational Research Association, Chicago. Lee, S. Y., Graham, T., & Stevenson, H. W. (1996). Teachers and teaching: Elementary schools in Japan and the United States. In T. P. Rohlen & G. Letendre, K. (Ed.), Teaching and learning in Japan. New York: Cambridge University Press. Lewis, C. (1995). Educating hearts and minds: Reflections on Japanese preschool and elementary education. New York: Cambridge University Press. Lewis, C., & Tsuchida, I. (1998). A lesson is like a swiftly flowing river: How research lessons improve Japanese education. American Educator, Winter, 12-17 & 50-52 Mesiti, C., Clarke, D. J., & Lobato, J. (2003, April). The structure of mathematics lessons in the United States. In D. J. Clarke (Chair), Mathematics lessons in Germany, Japan, the USA and Australia: Structure in diversity and diversity in structure. Symposium conducted at the Annual Meeting of the American Educational Research Association, Chicago. Jablonka, E. (2003, April). The structure of mathematics lessons in German classrooms: Variations on a theme. In D. J. Clarke (Chair), Mathematics lessons in Germany, Japan, the USA and Australia: Structure in diversity and diversity in structure. Symposium conducted at the Annual Meeting of the American Educational Research Association, Chicago. Sekiguchi, Y (1998). Mathematics education research as socially and culturally situated. In J. Kilpatrick & A. Sierpinska (Eds.), Mathematics education as a research domain: A search for identity, An ICMI study. Kluwer Academic Publishers. Shimizu, Y. (1999). Aspects of mathematics teacher ducation in Japan: Focusing on teachers' role. Journal of Mathematics Teacher Education. 2(1), 107-116 Shimizu, Y. (2003, April). Capturing the structure of Japanese mathematics lessons as embedded in the teaching unit. In D. J. Clarke (Chair), Mathematics lessons in Germany, Japan, the USA and Australia: Structure in diversity and diversity in structure. Symposium conducted at the Annual Meeting of the American Educational Research Association, Chicago. Shimizu, Y. (2006). Discrepancies in perceptions of mathematics lessons between the teacher and the students in a Japanese classroom. In D. Clarke, C. Keitel & Y. Shimizu (Eds.), Mathematics classrooms in twelve countries: The insider’s perspective (pp. 183-194). Rotterdam: Sense Publishers. Stevenson, H. W., & Stigler, J. W. (1992). The learning gap: Why our schools are failing and what we can learn from Japanese and Chinese education. New York: Simon and Schuster. Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York: The Free Press. Stigler, J. W., & Perry, M. (1988). Cross cultural studies of mathematics teaching and learning: Recent findings and new directions. In D. A. Grouws & T. J. Cooney (Eds.), Perspectives on research on effective mathematics teaching (pp. 194-223). Mahwah, NJ: Lawrence Erlbaum Associates & Reston, VA: National Council of Teachers of Mathematics.

Yoshinori Shimizu Graduate School of Comprehensive Human Sciences University of Tsukuba Japan

145

IDA AH CHEE MOK AND BERINDERJEET KAUR

CHAPTER SEVEN ‘Learning Task’ Lesson Events

INTRODUCTION

Every lesson has objectives which are often explained in the teacher’s goals for the lesson. In these goals, teachers normally spell out some mathematical concepts or skills which they want their students to learn in the lesson. In order to create an experience for students to learn such skills and concepts, teachers usually demonstrate and explain what they want their students to learn with examples or let their students explore some problems. We call such examples or problems, tasks or learning tasks in general. Therefore, a learning task has many possibilities. It may be as simple as writing (a + b)2 in an alternative form, a word problem or asking students to explore some patterns in a paper-folding activity. We may easily agree that a learning task is a common feature found in all mathematics lessons. The nature and content of these tasks, however, vary a great deal depending on the topic and the specific objectives of the tasks. Despite all these variations, good learning tasks are indispensable for good learning. Moreover, the effectiveness of learning is dependent not only on the task itself but also on how the task is carried out in the lesson. This chapter is not about learning tasks but about ‘learning task’ lesson events. A ‘learning task’ lesson event comprises not only the description of the task itself but also the actual event that happened during the lesson. This encompasses the interaction between the teacher and students and among the students themselves. The data discussed in this chapter consist of 18 learning tasks from Australia, Germany, Hong Kong, Japan, Shanghai, Singapore and United States. The data used include the video record and lesson transcripts. In addition, the researchers referred when necessary to the teacher questionnaires, students’ interview transcripts and lesson tables. The lesson events were chosen by the researchers in each country from their home data according to the operational definition explained below. The analysis aims to seek an understanding of the similar and different features which may be valued in the teaching of mathematics in different cultures and the possibilities of how students may be brought to an awareness of the mathematical meaning of a learning task. Events categorised with similar dimensions were further analysed in depth. The results reported in detail in this

D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 147–163. © 2006 Sense Publishers. All rights reserved.

IDA AH CHEE MOK AND BERINDERJEET KAUR

chapter belong to the three dimensions: making the mathematical procedure visible, using a realistic context and the connected nature of a sequence of tasks. WHAT IS A ‘LEARNING TASK’ LESSON EVENT?

The idea of using lesson events as a basis for legitimate comparison of mathematics classrooms was first discussed at the meeting of the American Educational Research Association in 2003. The ‘Lesson Event’ is conceived as an event type sharing certain features common across the classrooms of the different countries studied (Clarke, 2003; Mok, 2004). The first task was to establish an operational definition in terms of form and function for a lesson event so that the researchers from different countries could use this definition to select the events from their own data to compile a practicable subset of the data from the Learner’s Perspective Study (LPS) for in-depth analysis. For the ‘Learning Task’ lesson event the authors started with the Shanghai data and worked out the following definition. An Operational Definition Form. Every lesson will have an object of learning, that is, either a mathematical concept or skill which the teacher wants the students to learn in the lesson. This is often explained in the teacher’s goals for the lesson in our data (the lesson tables). Some learning tasks are usually used in the lesson to illustrate or explain the concepts or skills. Students may be engaged in the learning tasks in whole class discussion led by the teacher, individually or in groups depending on the teacher’s class arrangement. To elaborate a bit more, a learning task is differentiated from a practice item. A learning task aims to teach the students something new and the sequence of learning tasks show a coherent development of the object of learning. On the other hand, a practice item is mostly repetition of a taught skill. A common occurrence in Western classrooms, and also in Hong Kong, is for the teacher to do a worked example on the blackboard. This worked example is definitely a learning task. The students then are asked to do a set of problems that strongly resemble the worked example. According to the definition, it means that where the similarity is very high between the worked example and the problems subsequently attempted by the students, then the worked example should be seen as a learning task, but the subsequent problems are not. Function. In the case of Shanghai, we see the teacher using learning tasks, mostly in a whole class context, for different purposes such as: – For setting a background for the topic to be learned. – For demonstration or explanation, often with visual display and interactive question-and-answers between the teacher and the students. – For an in-depth investigation/discussion of a specific aspect of the object of learning. 148

‘LEARNING TASK’ LESSON EVENTS

Examples. In addition to this definition, three examples from a lesson in the Shanghai data were attached as exemplars for selection of the data. Differentiation Between a Learning Task and a Practice Item In applying this definition, there is a special effort to differentiate between a learning task and a practice item. We clarify this with an example. In learning mathematics, teachers sometimes ask students to do some practice items of a repetitive nature. For example, problems such as “Solve x + y = 4 and x – y = 5” in HK2-L01 appears very often in Hong Kong mathematics lessons in general. Similar problems are likely to be found in other parts of the world too. A common occurrence in the three Hong Kong classrooms analysed is for the teacher to do a worked example on the blackboard. The students then are asked to do a set of problems that strongly resemble the worked example. The first worked example, which is a tool to teach the students something new, will be called a learning task where as the subsequent problems attempted by the students are not. In other words, a learning task aims to teach the students something new and is different from a practice item meant for the repetition of a taught skill. THE APPROACH IN THE ANALYSIS

The 18 learning task lesson events were chosen by the researchers in each country from their home data set, based on the operational definition. This is a very small subset of the data and it has never been our aim to find some characteristics which can be claimed typical of a culture. The value of the analysis lies in the fact that it is always possible for people to disregard some features about an event despite its importance because they are too familiar with what happens. However, these features may become prominent when they are inspected in contrast with events from lessons with a very different background. The authors applied the grounded theory approach in the analysis (Strauss & Corbin, 1990). The lesson videos were watched and the lesson transcripts were read several times for open coding. The open coding stage ended when common dimensions emerged for the next level of comparison. In the first stage, different possible categories were used to classify the tasks, for example, exploratory, daily life context, algebra, graphs, equations, etc. As a result of the variation in terms of nature and content, there was some difficulty in deciding the common dimensions for the next level of analysis. Finally, three dimensions were chosen because relatively rich features could be identified from this data set and these dimensions are generally seen to be useful in learning mathematics. Learning tasks which were coded with the same dimension were further compared with each other to look for further similarity and difference for the synthesis of a coherent framework.

149

IDA AH CHEE MOK AND BERINDERJEET KAUR

THE ANALYSIS AND RESULTS

The events coded with the three dimensions – making the mathematical procedure visible, using a daily-life context, and the connected nature within and between tasks – were analysed in depth and are presented in this section. In spite of the very different cultural origin, we see some shared values in pedagogies such as using a real life context, an interactive class atmosphere and strategies to make use of students’ contributions. Making the Mathematical Procedure Visible – Four Examples in Algebra There are many algebra tasks in this set of data. The topics include algebraic expressions, equations, graphs and word problems. The work of school algebra often involves the procedural skills of symbolic computation according to the systemic rules of algebra. This is a kind of procedural knowledge which we want our students to master (Hiebert and Lefevre, 1986). In order to make learning possible, we need means to make the mathematical procedure visible. In the sample, we found examples showing different ways of making the procedural knowledge explicit in the lessons. In this section, we will describe four examples, one each from Germany, Singapore, Shanghai and Japan. They are chosen because they form a nice contrast, showing the possibility of different levels of sharing of knowledge by different teaching strategies. The German example, G1-L05. The teacher had introduced the three binomial formulas (a + b)2, (a – b)2 and (a + b)(a – b). The students work on a worksheet containing tasks such as: (2e + 3f)2 = ?. The tasks in the worksheet were of increasing complexity. The teacher did not show the students how to apply the formulas. Therefore, the students had to figure out the method for themselves. During this period, the teacher walked between the desks (See Chapter 4, this volume). Alice and Beth were the focus students. They were engaged in non-public talk when solving the task. Alice: Beth: Beth: Alice: Beth:

What have you got? Four. I’ll write that. You’ve forgotten that it’s squared. Oh yeah.

The Singapore example, SG2-L01. In this event, the teacher led the pupils into factorisation of an expression that is in the form of a difference of two squares: a2 – b2 = (a + b) (a – b). The teacher mainly demonstrated how the difference of two squares could be used to factorise expressions that take a similar form. During her demonstration she involved the whole class through chorus responses. The set of items in the task used is carefully crafted to gradually lead pupils from the simple to the complex. For the first three items of the task the teacher demonstrated factorisation on the board with inputs from the pupils, but for the last two she

150

‘LEARNING TASK’ LESSON EVENTS

asked the pupils to do them on their own and monitored student activity while walking between the desks (Kikan-Shido). T:

T: Ss: T: S: T: T: S: T: S: T:

Alright, factorisation of expression in the form of difference of two … squares. So this factorisation we are making use of … this formula … okay. [Teacher writes: a2 – b2 = (a + b) (a b)] Right, difference of two squares … you must make sure that you have this pattern, A square minus B square. And here given to you X square minus nine. Nine, is it a perfect square number? Yes. Yes. So you can make it to … X square minus what number square? (Three square). Three square. So you can apply the formula there. You will get X plus three, X minus three. Okay. [Teacher writes: (x + 3) (x – 3)] What about the next one? … Y square minus one over sixteen. How do you change it? Y over four square. Y square minus … One over four square. Ya … one over four, whole thing square … you will get Y plus one quarter, Y minus one quarter. Very simple, right. [Teacher writes: (y + ) (y - ) ]

Key to symbols used in transcripts in this chapter: … A short pause of three seconds or less. Ss Simultaneous talk by two or more students. ( ) Indecipherable words. (text) A plausible interpretation of speech that was difficult to decipher. [text] Comments and annotations, often descriptions of non-verbal actions. .... Indicates that a portion of the transcript has been omitted.

The Shanghai example, SH3-L01-3. This event is a demonstration of how to rewrite an equation in a specific format (to change the form x + 2y = 10 by writing x in terms of y and y in terms of x). The teacher asked a student Dora to suggest a way. Dora gave a wrong answer “y = 10 – 2x” orally. The teacher asked Dora to elaborate how she got the answer and she had difficulty. Another two students Denny and Eliza were invited to give their suggestions to complete the work. Analysis. The examples from Germany and Singapore show a remarkable contrast. Both tasks are very similar in nature. In both cases, the potential for learning is embedded in the design with problems with increasing complexity. When the students try to figure out the answer, they are expected to develop their understanding. Therefore, the process of thinking inside the students’ mind is crucial. However, their thinking will not be visible until they share their ideas with others, for example, the sharing between Alice and Beth in the German example. Such sharing is highly individualised and can be very divergent between different pairs of students. Comparing the events in the three places, the Singapore example represents another extreme. Whilst the German example only has the students’ own sharing of

151

IDA AH CHEE MOK AND BERINDERJEET KAUR

non-public talk, the Singapore teacher emphasises a correct demonstration of the process led by the teacher herself. There is hardly any possibility of any sidetrack and the students’ answers were mostly to fill in what the teacher asked for. The Shanghai example falls in between the German and Singapore cases. The Shanghai teacher asked a student Dora to elaborate her work for her wrong answer and invited two other students Denny and Eliza to participate. Although the teacher’s question intended to guide the students to follow a prescriptive path, Dora was given a chance to make up for her own mistake. Consequently, the task is in some sense completed jointly by Dora, Denny, Eliza and the teacher with an effort of making the process visible to the whole class. The Japanese example, J3-L03. In this event, the teacher began by asking the students to show the solution of the simultaneous equations 3x + 2y = 23 and 5x + 2y = 29. The work was supposed to be a piece of homework given on the day before. However, not all students completed their homework. Therefore, the teacher asked the students to solve the equations in their notebooks and he walked between the desks (Kikan-Shido) to monitor their progress. After a while, a student Dan volunteered to show his work on board. The teacher went on to inspect the other students’ work for a while and invited another student Ken to present his work on the board as well. After the two students had finished the teacher asked the class to compare their work. After a while, the teacher asked Dan and Ken to explain their own methods and he summarised the differences. T: T: T: T:

T:

Umm, in this classroom right now out of the many ways that I saw of checking this calculation, I asked to have the two typical ways written on the blackboard but, do you understand how the one Dan wrote and the one Ken wrote differs? First, I want you to notice their differences. Do you see the differences? Who understands? Okay, I see. Okay, well, you can discuss this with your neighbour but what and where are these two answers different, have the people who’ve written this noticed? Do you understand the differences? You may be able to explain the things you wrote but it might actually be difficult to explain what other people wrote. Please jot down some differences.

Analysis. The Japanese example is of a different kind compared with other three examples. The focus of the Japanese example is not on the process of how the students produce their work, but on looking back on what they have produced. Consequently, the invitation by the teacher to comment on the two students’ work initiates discussion, the content of which includes ideas such as the meaning of a solution for equations and the presentation of checking. This kind of teaching strategy obviously demands another level of understanding of the procedure.

152

‘LEARNING TASK’ LESSON EVENTS

Contextualisation by Making a Link to Daily Life: A3-L06 and SH3-L01-1 A realistic context refers to a context in which students can make sense of the mathematical meaning of what they are working with. It is not limited to an experience familiar in their daily life. However, daily-life contexts are sometimes used in the design of mathematical tasks with the assumption that a daily-life context will make the content look familiar to the students so that they may appreciate a stronger link to their daily life activities. Therefore, the embodiment of mathematics in a real life context sometimes is used as means for building a realistic context. In the lesson events received, there are two tasks using daily-life contexts. One is from Australia School 3 and the other is from Shanghai School 3. The Australian example, A3-L06. In this event, the teacher’s aim for this lesson was to introduce the students to the topic of ‘rounding off’. She used a humorous approach to introduce the problem, which appealed to the students and gained their interest. The problem was about a night out at a restaurant and having to split a bill of $67 dollars between three people. She wrote “$67” and “3 people” on the board and asked for suggestions of how much each person would have to pay if the bill was divided equally. Three students gave their answers, “$22.33”, “$22.33” and “$22.34”. S3, the student who was the most generous and gave $22.34 was asked to explain how he got the answer. He explained that he had used the calculator. Then the teacher put down “67 ÷ 3” on the board and asked the students to do the sum in their calculators. The teacher put the calculator-answer “22.333333” on the board and explained the idea of rounding off which was the new topic. T: T: T:

[Writing on the board] Okay. You did sixty-seven divided by three … on your calculator. Okay. Everybody get that … do that sum and get it up on your screen, please. What have you got on your screen? [writing on board] Twenty-two point three, three, three, three, three, does it?

The Shanghai example, SH3-L01-1. In this event, the teacher’s goal was to introduce the definition of linear equation in two unknowns. He showed a word problem about a child buying stamps on the screen and asked the class to solve the problem mentally. The problem is translated below: Wong Junior goes to the post office to buy several two-dollar and one-dollar stamps, at least one of each kind, costing a total of ten dollars. How many of each kind of stamps does he get? The teacher invited the students to answer the question. Under the teacher’s guidance, four students gave the answers for the cases of different number of twodollar stamps respectively. He showed the four answers in a table on the screen. Then, the teacher asked the class to solve the problem using equations. They worked out the equation 2x + y = 10. The teacher then guided the students to point out that the equation had two unknowns and introduced the definition of linear equation of two unknowns. 153

IDA AH CHEE MOK AND BERINDERJEET KAUR T:

So let’s think, I can take out a two-dollar stamp first, what about one-dollar stamps? … If a two-dollar stamp is taken out, then how many one-dollar stamps should the staff give? Dorris. Dorris: Should give out nine. T: Should give out nine? Dorris: Should give out eight. T: Eight, should give out eight, please sit down. T: We can draw a table, if a two-dollar stamp is taken out, and Dorris says eight one-dollar stamps should then be taken out. Okay, please reconsider except for this situation, are there any other possibilities? . . . . T: Let us think, if five two-dollar stamps are taken out, then how about the one-dollar stamp? Class: No. T: Does it match the requirements of the question? Class: No.

Analysis. The Australian and the Shanghai examples share many similarities. First, the teachers in both cases used a daily-life context to introduce the topic of the day. Besides giving a background for the topic of the day, the tasks also served as an example for the subsequent concept or skill to be taught in the lesson. Both teachers invited their students to suggest their own answers and both resulted in interactive classroom atmosphere (see the above class transcripts). They are similar in the fact that the students’ answers are likely to be within the teachers’ expectations. In both cases, the teacher is very clear that there is an object of learning she or he wants the students to learn. In the Australian lesson, it is the concept of rounding off, whereas in the Shanghai class it is the concept of linear equations in two unknowns. Therefore, in both cases the story context is only a means to an end. Comparing the transcripts carefully, there is an important similarity which contributes to explicate the objects of learning. In both cases, there is a critical turning point where the teacher guides the class to focus on the mathematics instead of the story. In the Australian example, this happens when the teacher asked the class to calculate “67 ÷ 3” with the calculator. In the Shanghai case, the turning point occurs at the moment when the teacher asked the class to solve the problem again with equations. In addition to these similar features, there is a major difference between the two examples. The dinner-bill problem is more authentic than the buying-stamp problem. First, sharing a bill can be a real event in the students’ real life. Consequently, the teacher can ask the students to imagine themselves actually sharing the bill. When a student gives an answer, she can emphasise the personal ownership of both the answer and the method. For example, she reminded the class the student’s generosity for paying $22.34 to make up for the extra cent and she would like to know the students’ method of getting the answers. A3-L06 T:

154

Riley has decided that he’s going to be particularly generous… and make up the one cent short. How did you work out how you were going to pay the bill? Can somebody tell me, please? Riley? What did you do?

‘LEARNING TASK’ LESSON EVENTS

In contrast, it is very unlikely for a student to buy some stamps without a clear idea of what kind of stamps she or he wants. It is more unlikely that a student may independently work out four different possible cases with the given amount of money. In this example, the teacher’s intervention in the first probing made a significant lead. He asked in the first case, “If a two-dollar stamp is taken out, then how many one-dollar stamps should the staff then give?” The first case worked as a precedent and the students worked out the other cases without difficulty. It is also important to see how the teacher ended the search for cases by reminding the class about matching the requirement in the question. The teacher’s intervention in this event is crucial to ensure that the collection of the students’ answers would be ready for the next stage of development for the topic of the day. Comparing with the dinner-bill example, the students in the buying-stamp problem obviously take a more impersonal role in solving the problem. The Connected Nature: Within and Between Tasks A mathematics object is a very complex feature. On the one hand, it can be precisely defined in a few lines. On the other hand, it is rich in characteristics and relationship demonstrating its multifaceted nature. An understanding of the concepts can only be developed via different experiences of the object. The learner’s understanding of the object is dynamic. That is, during each interaction with the mathematics, the learner may experience something new and try to make sense of it to build a coherent picture of the object. In this way, the learner’s understanding grows. The different kinds of experiences and their connected nature presented to the learners are extremely crucial in this process. The Shanghai and US data in this sample, both belong to sequences of coherent tasks given to the class by the same teacher. This gives us an opportunity to explore the connected nature of a sequence of learning tasks. The Shanghai example, SH3-L01-1 and SH3-L01-2. The two learning tasks were the results of two consecutive tasks in the same lesson. According to the teacher questionnaire, the goals of the lesson were: “To understand linear equations in two unknowns and their solutions; and the concept of solution sets. to find the parts of solutions that satisfied some certain conditions of the linear equations in two unknowns; solution set; to build the basis for understanding linear equations in two unknowns and the concerned concepts; and linear equations in multiple unknowns and the concerned concepts.” Indicated in this teacher’s description, the lesson had many objectives which were supposed to be met by many tasks. All these tasks were then supposed to be connected in certain ways in order to meet these goals. The two learning tasks discussed in this section are consecutive in the lesson and they are about the same concept, namely, linear equations in two unknowns. Therefore, they are chosen here to illustrate the connected nature between them. SH3-L01-1, a word problem about buying stamps, is described earlier in this chapter. By asking the students to solve the problem with equations, the teacher demonstrated an example of linear equations in two unknowns for which the 155

IDA AH CHEE MOK AND BERINDERJEET KAUR

teacher concluded by saying the definition of linear equations in two unknowns and showing a visual display of the definition with PowerPoint. The subsequent learning task SH3-L01-2 is translated below: Determine which of the following is a linear equation of two unknowns: 1. 2x + 3 = 0 4.

2x + 5y = z

1 2

x=

2 3

y +1

2.

x+2y 1= 0

3.

5.

x2 + 2 y = 1

6. 2xy = 5

In the analysis, we attempted to answer two questions: i. What is the mathematics shared between the two tasks? ii. How was the connection between different lesson events built? In the task SH3-L01-1, the teacher led the following discussion in the process of helping the students to write an equation. T:

Let’s think about the unknowns before the equations are set? So what do you think of the question? How to set the unknown? How many unknowns? T: Daan. Daan: Set two unknowns. T: Good. Set two unknowns. I can set two-dollar stamps as x pieces, one-dollar stamps as y pieces, so according to this question how should the equations be set? Daan: x plus y equals ten. T: x plus … y, right? x plus y equals ten, does everyone agree? Class: No. T: Why don’t you agree? T: Felix. Felix: ( ) x plus y is (equal to ten), x stamps plus y stamps equals ten, as well as, one is quantity, the other is money. T: Using ten, so what should be done? Felix Two x plus y equals ten. T: Two x plus y equals ten, in this question we can set two x plus y equals ten, right? Okay. T: Let us see. This equation and our Linear Equation in one unknown, from the number of unknowns, compare the power of the unknown x and see if there is any difference? T: Donald. Donald: Compare with the linear equation in one unknown it has one more unknown. T: One more unknown, so how many unknowns are there now? Donald: Two unknowns. T: Two unknowns, right? Okay, please sit down. T: So how about the power of the unknown? Class: One. T: One, right. So we call this kind of equations the linear equations in two unknowns. Linear equation in two unknowns; if they consist of two unknowns and the power of each unknown is one, we call this kind of equations linear equations in two unknowns.

Analysis. In the discussion, we see that the students had to inspect the stamp problem and build their own equations. When there was disagreement, they needed 156

‘LEARNING TASK’ LESSON EVENTS

to compare the different answers and looked for a rationale to support their own answers. Felix’s answer therefore showed a strong link between how he understood the meaning of the letters in the equation and what they referred to in the original word problem. After giving the equation, the teacher asked the student to compare the equation with a linear equation in one unknown, in particular compare the number of unknowns and compare the power of the unknowns. Finally, he ended the task by introducing the formal definition for linear equations for two unknowns which was the topic of the lesson which could be fairly seen as the outcome of this task. If we compare only the text description of the two tasks SH3-L01-1 and SH3L01-2, we will see a word problem about buying stamps and a very abstract problem about definitions of symbolic equations which appear to be very independent. However, in the beginning of the second lesson event, we will see that the outcome of the first lesson event is in fact the major tool for solving the second task. Below is how the teacher introduced the task: T:

In the following, use this concept to justify these equations, please pick out if any of them belongs to linear equations in two unknowns, and briefly explain the reasons why.

Moreover, this concept (namely the definition of the linear equation in two unknowns) was used by the students explicitly every time they answered each part of task two. Elsa: T: T: Class: T: Class: T: Class: T: Class: T: Class: T: T: Class: T: Class: T: Class:

T: Franc: T: T: Class:

It is not. Because it has only one unknown, it is not … it is not linear equations in two unknowns. Good, it is not. There is only one unknown right? The first is not linear equations in two unknowns. So what about the second one? It is. Yes, say together, why? It has two unknowns and the unknowns are of power one. Good. So how about the third? It is. Good, so how about the fourth? It is not. Why? It has three unknowns. Because it has three unknowns, x, y, z right? So it is not. So how about the fifth? It is not. Why? The power of the unknown is two. Which unknown has the power of two? It is x two, x square. [The sixth equation was the most complex. The teacher let the class discuss among themselves before resuming the class discussion.] Okay. Anyone thinks it is not? We let those who disagree to explain. Why is it not? Franc. It is because two x y, x y is an unknown. x y is an unknown? [Students laughing] Students laughed, is x y an unknown? x y are two unknowns.

157

IDA AH CHEE MOK AND BERINDERJEET KAUR T:

If you use this method to justify, of course your justification is wrong, right? What else? Freda. Freda: The power of each unknown should be one, while two x y is an unit, the power of it is two. T: The power of this single unit is two, so it is not. Does everyone agree? Class: Agree.

Analysis. It is necessary to point out that the six parts of the second task are not repetitive items. They guided the students to inspect the number of unknowns and the power of unknowns in different situations. Therefore, this was considered as development and application of the concept rather than a repetitive practice item. Although the text description between the two tasks did not show an obvious link, the interaction in the second lesson event shows a strong link to the outcome of the first task. The connection is implicit in the design of the lesson and such connection only becomes explicit in the interaction between the teacher and the students. We call this kind connection-building implicit to explicit. The United States examples, US2-L02 and US2-L04-1. The US teacher stated in the teacher questionnaires that her goal for the lesson in US2-L02 was for students “to see the interrelationships among equations, their graphs, tables and verbal descriptions.” and that a goal for L04 was: “I wanted students to begin to see that the ‘equation’ form of an algebraic relationship is more than a meaningless composition of numbers and letters. In fact, the equation could be ‘translated’ completely into a picture in the mind.” The sequence begins at the end of Lesson US2-L02. The class was working on the “Algebraic Meaning for Representations” problem. For the learning task, students (arranged in small groups of four) were given two worksheets which were cut out into 40 cards, representing 10 linear and non-linear functions.. Each card contained one of the following: (a) an algebraic equation; (b) a graph; (c) a verbal statement; or (d) a table. The students were asked to sort the cards according to the function that was represented. The equations for the 10 functions were as follows: y = x2 y=x+2

y2 = x x+y=2

2y = x xy = 2

y=x–2 y=2

y = 2x x=2

During this activity, the teacher let students work on the learning task in groups of four. The target group used this time to get organised and oriented to the task although it is possible that Brenda (one of the target students) solved one of the problems. The teacher stopped the students and created a sub-task, namely to find the verbal description, graph, and table for the algebraic equation x = 2. She let students work on this problem for four minutes in their small groups. Then she demonstrated the solution by calling on students to share their solutions. The discussion of the sub-task was short but the teacher brought out two ideas: i. when x = 2, y can take on any value; and ii. the graph of the equation “x equal to some number,” is a vertical line.

158

‘LEARNING TASK’ LESSON EVENTS

Rather than conducting an in-depth and open-ended discussion of this learning task, the teacher posed a new learning task in Lesson L04 (called task US2-L04-1) as a way of pulling together the ideas from the Algebraic Meaning for Representations task. US2-L04-1 began with a new warm-up problem, but during the latter part of the lesson, students returned to the Algebraic Meaning for Representations task. They worked in groups to prepare a presentation of part of the problem. During Lesson L04, the groups posted their presentation ‘posters’ and each group checked their responses to the activity. After all the ‘posters’ of the ten solutions to the Algebraic Meaning of Representations task had been displayed and students had checked their answers against the displays, the teacher discussed the findings by posing a new learning task. She posed the following five related questions and asked students to write their responses in their notebooks and on whiteboards: 1. What do these graphs have in common? [points to the graphs of y = x + 2, 2y = x, y = 2x, y = x – 2, and x + y = 2.] 2. Is the graph of an equation like x = 2 vertical or horizontal? 3. y = 2x is a model of direct variation because it a) crosses through the origin b) passes through quadrant 1. 4. Is the slope of this graph [points to the graph of y = x – 2] positive or negative? 5. This point (0, 2) [teacher points to the point (0, 2) on the graph of y = x + 2], is the x-intercept or the y-intercept? Analysis. When comparing the learning tasks from SH3 and US2, we see both similarities and differences. In terms of learning objectives, both teachers see their tasks oriented towards one object of learning: the meaning of linear equations in two unknown for Shanghai School 3 and the algebraic meaning for representations for US School 2. The classes in both places worked on the concept of equations but they experienced very different strategies. In the case of the Shanghai examples, the concept is represented by a formal definition and explicated in examples and counter-examples of equations. The two are linked repeatedly in the second task. It seems that every time the students made a comment on the equation with a reference to the definition provided an opportunity for ‘internalisation’ of the definition. Internalisation here refers to a process which permits one to be conscious of an action, to reflect on it and to combine it with other actions (Ed Dubinsky, 1991). The first task provides a path from example to formal definition whereas the second task provides a chance to reverse the process. In the US examples, the application of alternative representations is obviously a tool for learning. It is explicit in the students’ hands-on activity in the first task by which they were asked to sort out alternative representations of the same equation. Similar to the Shanghai case, it provides the opportunity for internalisation of the concepts. However, the event was concluded in a very different way. In Shanghai, the first event lasted for about 5 minutes and was ended by the teacher giving a definition of the concept. In the US, the whole task was completed on another day,

159

IDA AH CHEE MOK AND BERINDERJEET KAUR

after the students had completed their homework and posted up their posters. Apart from the difference in terms of time and pace, the Shanghai example ended in the teacher’s endorsed formal definition, whereas the US example ended in a display of students’ work and students’ own description in a less formal language such as the following example. US2-L04-1 T: T: T: Ss: T: Ss: T: Ss: T: Ss:

Let’s put it that way. Ideas about Algebraic Representations. Okay? And we’re going to look at those four areas again what are they? One of them is? Graph. Graph, another is? T-chart. T-chart, another is? Equations. And the fourth is? Verbal expressions.

In the earlier analysis, we explained that the Shanghai events built connections between the two tasks from implicit to explicit. In the US case, the second task is explicitly connected to the first task because it makes a direct use of the product of the first task. The five new questions refer to a subset of the equations in the first task and are posed after the students’ posters are posted up in the lesson. However, in the class interaction, each question may be seen as an independent question. The connection between different kinds of categorisation (linear/non-linear, vertical /horizontal, variation) becomes implicit and less visible in the public sharing. US2-L02 T: T: T:

Okay. I’m going to point to several graphs. I’m going to ask you what they have in common. In just a moment. This one. This one. This one. This one. This one. This one. I need you to respond by telling me whether those graphs were linear or non-linear. Write it down, write down your choice. Linear or non-linear.

T: . . . . T: It turns out that a graph like X equals two always creates a vertical line or a horizontal line? Put it down. T: It turns out that a graph like X equal two always creates a vertical line or a horizontal line. T: Alright. Third question. T: This graph is a model of direct variation because it, choice, lies in the first quadrant, lies in quadrant one, crosses quadrant one or passes through the origin. Passes through zero, zero.

CONCLUSION: TOWARDS A FRAMEWORK FOR THE ANALYSIS OF ‘LEARNING TASK’ LESSON EVENTS

In the study of learning task lesson events, we use a very simple and general definition for a learning task used in mathematics lessons. A task in general can be 160

‘LEARNING TASK’ LESSON EVENTS

seen as a well-defined activity which produces product in a lesson achieving a purpose (Doyle, 1988). The definition in our study adds a specific aspect to this broad definition which hinges on the purpose and the function of the task, whether it is to help students learn something new or to carry out a learned skill in a repetitive manner. The former is called a learning task and the later is called a practice item. The differentiation cannot be done by the text description of the task itself. The differentiation must take into account how the task is used and carried out in the lessons. Therefore, what we study is not the task itself but the lesson event which encompasses the task and the realisation of the task in the lessons. We call such a lesson event, a learning task lesson event. This operational definition, firstly based on the analysis of the Shanghai data and empirically confirmed in the other LPS data, provides a way to select the lesson events. The analysis of events coded with common dimensions provides grounds for the synthesis of new dimensions for evaluating specific lesson events. Though there may be many dimensions for evaluating ‘learning task’ lesson events, this chapter has shown that from a small sample of 18 learning tasks, three dimensions have been explored. These are: differentiation of the mathematical process, building a realistic context and building connection. Differentiation of the mathematical process. Repetition can never be avoided in the path of learning but it should not be over-emphasised. In the analysis shown in this chapter, we see a need to differentiate what the students actually repeat in the path of learning. Mathematics processes consist of many features: procedural skills, internalisation of concepts or mathematical objects, application to problem situations, and reflective evaluation of one’s heuristics and strategies. All these are explicated in the plethora of lesson events in this chapter to different depths as a result of teaching strategies and designs. They are all important but can never be accomplished in a small number of tasks. Some tasks achieve a single purpose in a simple and straightforward way whereas others achieve multiple purposes with complexity. It is necessary to differentiate these mathematical processes and ensure that they are built in the path of the students’ learning. Building a realistic context. Mathematics curriculum is often characterised with a kind of ‘technique-oriented’ feature (Bishop, 1991) which demands a proficiency of working with symbols at a certain level. A student’s life is often limited and seldom subscribes to a familiar environment for practising such techniques. Therefore, we rely on the activities in the lesson to establish a realistic context. We refer such contexts to problem situations which students can imagine and situations that are ‘real’ in their mind. Such contexts may include both their experiences in daily life or the formal world of mathematics (Van den Heuvel-Panhuizen, 2003). Daily life context can be a context for making the application of mathematics skills real and useful by itself or can be means to explicate the abstractness of mathematics skills and concepts. However, masking a task with a daily life context does not make this happen automatically. In the examples discussed in this chapter, the objectives are only made possible by the skilful guidance of the teachers. The 161

IDA AH CHEE MOK AND BERINDERJEET KAUR

analysis showed some shared features between the two teachers but it also showed a reminder that we cannot be over-optimistic about using everyday contexts as a resource for understanding abstract concepts. The success depends much on how such contextualisation is made (see also Chapter 8, this volume). Building connection. Mathematics is rich in connections. Understanding of mathematics often implies making connections between mathematical concepts, facts and skills, and between experiences of all kinds. However, in addition to connections between ‘impersonal’ facts and knowledge of mathematics, meaning is also about the sharing of personal connections, ideas, imagery and metaphor, examples from different experiences and significant instances in learning (Bishop, 1991). In the events discussed in detail in this chapter, there are different examples of making impersonal and personal connections. To a certain extent, all the actions such as making comments and reflections by either the teacher or the students, and the sharing of ideas either in private talk or in public, are means to add a personal component to the learning. It is not always so easy (but not totally impossible) to confirm this in the former empirical presentation. There are some acts which will comparatively make the personal component a stronger image in the action: for example, the Australian example of sharing a bill, the Japanese example of commenting on other students’ work. Seeing a range of examples helps us to evaluate critically how we may seek a balance to avoid making the mathematics education in our own culture impersonal. The findings of this chapter, based on the analysis of a small subset of the LPS data carry neither the implication that one example is better than another, nor that some features can be claimed to be a national characteristic. However, what is evident is that by comparing ‘learning task’ lesson events the role of similar tasks in differing lesson events sheds light on the nature of mathematics teaching per se. Furthermore, these lesson events make it explicit that the success of learning does not depend on the task but on the lesson event encompassing the task. ACKNOWLEDGEMENT

The work described in this chapter was supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7192/02H) and the Centre for Research in Pedagogy and Practice, National Institute of Education, Nanyang Technological University, Singapore (CRP 3/04 BK). REFERENCES Bishop, A. J. (1991). Mathematical enculturation: A cultural perspective on mathematics education. Dordrecht: Kluwer. Clarke, D. J. (2003, April). The structure of mathematics lessons in Australia. In D. J. Clarke (Chair), Mathematics lessons in Germany, Japan, the USA and Australia: Structure in diversity and diversity in structure. Symposium conducted at the Annual Meeting of the American Educational Research Association, Chicago.

162

‘LEARNING TASK’ LESSON EVENTS Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psychologist, 23(2), 167-180. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 95-123). Dordrecht: Kluwer. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Lawrence Erlbaum. Mok, I. A. C. (2004, April). Learning tasks. In D. J. Clarke (Chair), Lesson events as the basis for international comparisons of classroom practice. Symposium conducted at the Annual Meeting of the American Educational Research Association, San Diego. Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. London: Sage. Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54, 9-35.

Ida Ah Chee Mok Faculty of Education University of Hong Kong Hong Kong SAR, China Berinderjeet Kaur National Institute of Education, Nanyang Technological University, Singapore

163

JOHAN LILJESTRAND AND ULLA RUNESSON

CHAPTER EIGHT Interaction, Organisation, Tasks and Possibilities for Learning about Mathematical Relationships: A Swedish Classroom Compared with a US Classroom

INTRODUCTION

In order to understand students’ possibilities for learning school mathematics it is important to investigate their interaction in their everyday school context. All social interaction is dependent on contextual resources of different kinds. In this chapter we will focus on how students and teachers make use of different kinds of content and ways of organising teaching, trying to understand school mathematics. The question is thus how the mathematical task and the classroom organisation contribute to shape the interaction, and ultimately what is possible to learn in the mathematics lesson. In the study presented here we approach this question by contrasting LPS data from Sweden and the United States. We want to point out explicitly that the data sets used in this study are considered as examples of different ways of organising the classroom work. So, when we refer to ‘the Swedish classroom’ we do not mean Swedish classrooms in general, but the specific classroom studied. THEORETICAL DEPARTURES

Two theoretical frameworks have been used, emerging from an attempt to combine two Swedish research traditions in the CULT project.i One of the theoretical frameworks adopted – variation theory – is about how we learn to perceive and experience the world around us (Marton & Booth, 1997; Marton, Runesson & Tsui, 2004; Pang & Marton, 2005). The way we experience something, or how we learn to see an object in a particular way, is a function of those aspects we notice or discern at the same time. If different individuals experience ‘the same thing’ differently, they discern different aspects of the object in question. In order to understand or see a phenomenon or a situation in a particular way, one must discern all the critical aspects of the object in question simultaneously. Since an aspect only is noticeable if it varies, the experience of variation is a necessary condition for learning something in a specific way. The teacher can help learners to understand, for instance the meaning of a concept, by

D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 165–183. © 2006 Sense Publishers. All rights reserved.

JOHAN LILJESTRAND AND ULLA RUNESSON

designing the learning situation in a way that promotes the discernment of all the necessary aspects at the same time. How critical aspects of the object of learning are brought out in a learning situation is significant for what is possible to learn. An aspect must be experienced as a dimension of variation. So opening up variation of those aspects we want the learner to notice is, we believe, necessary to make learning possible. A similar rationale was proposed by Dienes (1960) in his theory of mathematics learning. A concept must be taught by varying essential features of the concept, he argued. However, Marton et al. (2004) take this idea of variation and invariance further and describe how different patterns of variation opened up in a learning situation are critical instances for learning. One way to offer a variation is by contrasting. To understand what something is one must experience something different but related to compare with it. For instance to understand what ‘three’ is, one must experience something that is not three: ‘two’ or ‘four’ (Marton et al., 2004). To understand what a rhombus is one must experience those features that are characteristic for that particular shape. By, for instance, contrasting a rhombus with other parallelograms, the specific features of a rhombus become visible and possible to experience. Another way to open up variation is by exposing the learner to different examples. If the aim is to help the learners to understand a general idea or principle, it is appropriate to give different examples of this principle. Thus, the principle is invariant whereas the examples vary. This way of constituting a pattern of variation and invariance, by the teacher and the learners or by the learners themselves, is actually what happens in a learning situation, (cf. Runesson, 1999; Marton & Morris, 2002; Marton et al., 2004; Runesson & Mok, 2005). Consequently a learning situation can be studied with the focus on which aspects the learner is exposed to and in what manner these are elicited. By analysing which aspects of the object of learning the learners have opportunities to experience, it is possible to describe learning opportunities in terms of what is possible to learn. Such descriptions, however, are not descriptions of what is actually learned. The learner can, to varying extents, open up variation by herself and what is learned is a lived object of learning. In a series of studies (see e.g. Marton & Morris, 2002; Marton et al., 2004) this theoretical framework has been used as an analytical tool for examining classroom learning from the point of view of the object of learning. In the classroom the object of learning is not something given from the beginning, but is co-constituted in the social interaction. What the learners experience in the classroom is a socially and jointly-constituted object of learning, thus it is an enacted object of learning. To understand how the object of learning is enacted, we have tried in this study to combine the framework of variation with another perspective that focuses on the process of interaction. This other perspective, social constructionism (Burr, 1995) takes its departure from the way in which the social world is constructed by and through social interaction. From this perspective, social interaction is not only concerned with relationships between communicating individuals, but also considered as constructing and constituting the very social world as such. Actually, without this social construction nothing in the social world, including the educational world, 166

INTERACTION, ORGANISATION AND TASKS

could exist as a social reality for the actors. To analyse educational phenomena as social constructions implies analysing how the content takes shape in a lesson. What is possible to learn about a mathematical concept in a classroom is to a great extent contingent on the social interaction process. It seems reasonable to assume that, if this interaction had been different, the outcomes would also have differed. For both of these theoretical perspectives, learning is enabled and constrained in and by the social interaction. Learning in interaction can therefore be analysed in terms of the different possibilities for the participants to understand mathematics, depending on which objects are constituted and socially enacted in the classroom. In this chapter we will focus on different possibilities for learning in events that are socially achieved or not achieved in the classroom and how these events are shaped by interaction. From such a perspective, learning is always the learning of something, and this special kind of content, implies different consequences for the possibilities or constraints that are offered to learners in the classroom interaction. Hence our analytical question is directed on how social interaction constitutes what is possible for the students to learn about relationships. DESIGNING OUR STUDY

The aim of the study was to investigate how classroom organisation and mathematical tasks shape both the interaction and what it is possible to learn. All the lessons in the LPS study were video recorded with three cameras. One camera focused on the teacher, one on a group of selected focus students and the third on the whole class. The mobile teacher-camera, together with a cordless microphone attached to the teacher, made it possible to catch the interaction between the teacher and the students at their desks. The video recordings were transcribed verbatim and the Swedish transcripts translated into English. The transcripts in the chapter are done within a slightly simplified version of the notation developed by Gail Jefferson and her colleagues (cf. Atkinson & Heritage, 1984)ii. A lesson summary table was also made for each lesson. This gave us a good overview of what happened during the lesson and also made it possible to identify special instances that were of interest for our purpose. In the Swedish transcripts and lesson tables we were able to identify specific tasks from the textbook that recurred in different teacher-student interactions. One task focused on the meaning of a well-known concept, mathematical relationships, and occurred over three different lessons with four students in different contexts. We therefore selected four sequences with discussion of this task. Our analysis of these sequences is done on the basis of our knowledge about the lessons in general, together with an analysis of the tasks in the textbook. Only two of these four sequences are presented in this chapter. For comparative purposes in this study, we decided to match the Swedish and US data by comparing the same topic taught, mathematical relationships. Our interest was in finding themes that could be compared; the comparison is thus on a general level although there are also important differences between the two sets of data. In terms of content, the closest similarity was found in two lessons which 167

JOHAN LILJESTRAND AND ULLA RUNESSON

dealt with functions (US2-L06-07). To further match the Swedish selection of data, the US data is also taken from the teacher camera. ORGANISATION OF CLASSROOM WORK

To understand how the meaning and understanding of school mathematics is constructed in the classroom we begin with a description of the general patterns of participation and interaction (see also Emanuelsson & Sahlström, in press). One easily observable difference is the overall organisation of classroom work. We begin with the Swedish lessons. From an analysis of the twelve recorded Swedish lessons in the LPS data we identified how the lessons were structured and organised. The lessons often started with a plenary session followed by individual seatwork where the pupils worked by themselves with individual assignments from the textbook. This way of organising classroom work is common in Sweden. In the Swedish National Assessment of Mathematics Education 2003 it is reported that individual seatwork has become the most common way to organise classroom work; the amount of time spent on collective instruction conducted by the teacher has decreased during the last ten years. Group work is also rare in Swedish mathematics classrooms (Skolverket, 2003). How the lesson structure is managed and how the individual participants constitute their schoolwork through those lesson frames contributes to individualisation in the Swedish lessons. The teacher in Swedish lessons first makes general presentations but the ordinary task for all the students is to work on part-tasks from the textbook, common for everyone. The students are then supposed to work at their own pace on the tasks. Different students work on different tasks during the same lesson depending on how far they have advanced in the book. An illustration of this phenomenon is given by an overview of our data from different occasions, spread over three lessons, in which a student is working on Task 16. Table 1: Task 16 occurring in four teacher-student exchanges Student Victoria Amanda Filip Rodi

Lesson SW1-L07 SW1-L07 SW1-L10 SW1-L13

Time during the lesson 14:29 33:47 58:30 09:55

As can be seen from this table, both Victoria and Amanda work on Task 16 during the same lesson, while the other two students work on this task in two later lessons. In Sweden the tasks ‘belong to’ the individual students and the interaction with the teacher is dyadic, unless students work in pairs or are able to overhear the interaction of the teacher with another student seated nearby. The teacher’s actions are public insofar as everyone is able to see him and has a chance to obtain assistance in another dyadic (or sometimes triadic) teaching event, but it is 168

INTERACTION, ORGANISATION AND TASKS

generally impossible for most students to listen to explanations of the task by the teacher. It was found that very few of the tasks in the textbook were discussed in public in a whole class setting. Task 16 (See Table 1) was not presented in the public discourse but became relevant for teacher-student interaction only by a non-public initiative from a student. In the first lesson, after Victoria asked for help, the teacher assisted Amanda who was approaching the task for the first time with no former assistance from the teacher. As can be seen from this table then, the occasions when the task was explained are spread out as independent situations from the perspective of the individual student (Liljestrand, 2004). The typical patterns of participation in the Swedish class can then be characterised with the following features: – The class as a collective does not participate in ‘dialogue’ with the teacher. Rather, the collective consists of individual students who get individual assistance from the teacher when they ask for it. – The teacher assists different students with the same task on different occasions. – The whole collection of problems that the students encounter within one task does not become public but remains a set of individualised problems. This pattern of participation also has consequences for the individual student’s opportunities to become familiar with and reflect on the task from different points of view (Piaget, 1962). The task is thus not interpreted from different perspectives. The attempt to individualise the students’ work and to enable them to find the correct answer at their own pace results in a privatisation of the teacher discourse. This way of organising classroom work differs from that in the US lessons. The tasks are, to a much greater extent, collectively shared – first in a whole class presentation of the set of tasks, then in small group work with the same set of tasks, and finally with individual students presenting their solutions in front of the whole class. In contrast to Sweden we find the following pattern in the US data: – The task is not only written down in an exercise book by individual students, but also on the whiteboard, in front of and accessible to all students in the class – The class works in groups with the same set of tasks and finally individual students present solutions at the end of the lesson. This presentation is made in public, so is interactionally accessible for all members of the class. – The whole set of problems is made public during different phases of the lesson; the introductory teacher presentation; work in groups; and the final student presentation. In contrast to the Swedish lessons, the US students have the opportunity to reflect on the task from different points of view, which opens up for them a potential to understand it in more depth. However, for various reasons, this does not guarantee that individual students actually make use of this possibility. One reason is that the interaction with the teacher becomes more complex in the US lessons because the teacher is interacting with more than one student, as we attempt to show below. The production of meaning in the US classroom is not only richer but also more complex when many students are tackling the same tasks at the same time. Generally, this picture of deskwork points to the importance of analysing the teacher-student interaction not only as group interaction, as most researchers have 169

JOHAN LILJESTRAND AND ULLA RUNESSON

done (cf. Belack et al., 1966; Mehan, 1979; Nuthall, 1998), but as dyadic activity within a large collective, which is the common pattern in the Swedish lessons. The data in the Swedish and US lessons can therefore not be analysed by focussing on the whole-class pattern as is mostly done in the teacher-student literature (Liljestrand, 2004). To understand these contexts we must begin with a description of the different kinds of content that students are handling in the different classroom cultures. We look first at the US classroom, followed by a closer analysis of two Swedish students, Victoria and Filip, who approach the task in different kinds of contexts. RELATIONSHIPS: A MATHEMATICAL THEME AND ITS CONTEXTUAL MEANINGS

We assume that the way the mathematical content is structured in a set of tasks can tell us about what is possible to learn about a mathematical concept. We will use one task (number 16) from the Swedish data and a corresponding task from the US data to illustrate how relationships were introduced. Although the content in the two classrooms is not identical, we take it as two examples of how equations and their graphical representations could be taught in two different classrooms. The textbook task in Sweden with the four students was: “Write an equation that describes the relationship between the cost and the quantity of juice”. It is the last task in a unit of four comprising a step-by-step progression of abstraction and representation about a certain relationship – direct variation (see Table 2). Table 2: Patterns of examples: relationships, their representations and the variation in the complex of tasks in terms of direct variation. Task number The relationship

Representation

Variation brought in

Instruction

Between the cost and kilograms of apples

A table of values, a graph and an equation

Representations

13

Between the cost and kilograms of bananas

A table of values and Representations a graph

14

Comparing the relationships: cost and quantity of pears with cost and quantity of apples (price and curves respectively)

Two different graphs Price of different commodities, slope of the graphs

15

Comparing the price per litre of juice/milk/lemonade

A graph

16

Between cost and litre of juice An equation

Price of different commodities, slope of the graphs

Thus Task 16 is a part of a larger context, and the students sequentially meet with different representations of a function (graphs, tables of values and equations). 170

INTERACTION, ORGANISATION AND TASKS

The overall unit of four tasks begins first with a question about costs for kilograms of apples. This example, along with instructional text, demonstrates and explains how direct variation can be represented in different ways. This introductory illustration is presented within a frame, indicating that it should function as an exemplar for the following tasks. Thus the introduction is supposed to serve as an indication of how to cope with the subsequent tasks, as a mode of representing equations and graphs. Just one particular kind of relationship is presented (direct variation), and it is contextualised within the domain of everyday mathematics (Dowling, 1998); every task involves the calculation of costs for a certain quantity of commodities (fruit and drinks). Mathematically the tasks comprise direct variation, graphs with positive slope and intercept equal to zero. The idea that a relationship can be represented by an equation only appears in the instruction at the top of the page and in Task 16. There is nothing mentioned about this in Tasks 13 to 15. If the students do not read the instruction carefully and use it as a resource, one could anticipate that they will have difficulties with this task. The set of tasks is composed in a way that guides the student step-by-step to the ‘climax’; writing the relationship as an equation. As we will develop further, such ‘step-by-step design’ is also well adapted to an individualised way of organising the lessons, as occurs in the Swedish lessons. We consider this way of structuring the tasks as an intention to guide the students’ work via the textbook and probably to decrease the teachers’ assistance to a minimum. In the instruction at the top of the page it is explained that there is a relationship between price and quantity and that the relation can be demonstrated by a table of values, an equation and a graph. It is pointed out that cost is proportional to the quantity. In Task 13 students are asked to make a table of values and a graph representing the relation cost/quantity for bananas (15 crowns per kilogram). Task 14 asks them a) by means of the graph, to tell the price per kilogram of pears and b) to explain why the slope of the curve representing pears is higher. In Task 15 there is a diagram with three curves representing different commodities. The questions asked are “Which is the cheapest?” and “What is the price of the juice per litre?” Finally, Task 16 – focused on in our classroom study – is to transform one of the curves (juice) to an equation (y = 11x). What aspects of direct variation are then possible for students to experience from this set of tasks? The inbuilt pattern of relationship and representation in the complex of tasks is shown in Table 2. Following one of the frameworks taken – only those aspects opened as a dimension of variation are possible to notice and this variation must stand out against a background of invariance – we analysed the set of tasks from the point of view of what was varying or was invariant within each task. For instance, in Task 13 the representation varies; two different ways of representing the same relation are presented (see table 2). In the same way the instruction entails the relationship between cost and quantity and three different ways to represent this (table of values, graph, equation), again the representation is varied. So, in these tasks it is possible to experience and to learn that a relationship (direct variation) could be represented in different ways. In Task 14 the prices of two commodities are compared. The relations between the price and the quantities 171

JOHAN LILJESTRAND AND ULLA RUNESSON

of pears and apples are represented by two graphs having different slopes. Thus, the variation opened (different prices entail different slopes of the curves) makes it is possible to experience that when the relation cost/quantity is different the slope of the graph representing this relationship is different too. Furthermore, different aspects (relations between two variables [13], comparison of two variables [14, 15], finding x-values [15]) are separated and presented one at a time. The task is contextualised within an everyday context: buying goods of different quantities and comparing the prices. Whereas the Swedish exercise was a set of sub-tasks contextualised within an everyday context, in contrast, the task in the US classroom was contextualised within mathematics, with no reference at all to units or to everyday situations (including such units). This task was about finding similarities and differences between a set of pairs of equations: y = 3x + 2 0x + 3y = 6 y = x2 y = 1 – 2x 2y = x

and and and and and

y = –3x – 2 2x + 0y = 6 y = 1/x y = 1 – x2 y = 2x

Figure 1. The task worked with in US: Comparing pairs of equations

The students were expected to choose appropriate x-values, make a table of values and draw a corresponding graph to find similarities and differences between the two equations. With respect to different representations, the Swedish and US tasks are similar. Just as in the Swedish tasks, the US students have opportunities to learn about the relation between the visual slope of the graph and the slope coefficient (in letters and numbers). But taking a closer look at how the set of tasks was composed; we found an interesting pattern of contrast in how the tasks were designed. Table 3. Patterns of variation in the US tasks Comparing equations

Variation brought in

y = 3x + 2 and y = –3x – 2

Positive and negative slope Simultaneous change in both coefficients and graphs

0x + 3y = 6 and 2x + 0y = 6

One (different) unknown/variable Horizontal/vertical line Slope equals zero/infinity

y =x2 and y = 1/x

Positive/negative exponent

2y = x and y = 2x

‘Reversed’ equations

The set of tasks is composed using a particular pattern of variation; keeping something invariant while varying something else, for instance positive and 172

INTERACTION, ORGANISATION AND TASKS

negative slope, slope equal to zero versus infinity, and so on. In addition to that, the opportunity to understand the concept of linear function is brought out through a contrast with non-linear functions and the case of zero slope and no slope. This comparison shows that, from the point of view of opportunities to learn about the concept, the tasks were quite different. Our interpretation is that the presence of different kinds of ‘relationships’ in the US task implies that the concept is presented in a more comprehensive and elaborated way compared to the Swedish task. Following the framework of variation theory, to understand what something is, it is necessary to understand what it is not, that is, to differentiate. This principle was used in this case when negative slope was contrasted against positive slope, intercept = 0 against intercept 0, function against non-function and direct vs. indirect function. This pattern of variation/invariance brought out several specific dimensions of the concept. Furthermore, the mathematical relationship was brought out more distinctly and on a more general level with no reference to an outside world in the US tasks compared to the Swedish ones. It is an empirical question whether or not the students were able to see the mathematical relationship when other dimensions, such as cost, price and units were involved in the tasks. But the way in which relationship is contextualised in the classrooms implies that the tasks and problems for learning are different in the Swedish material than in the US. Thus when using these tasks as resources for understanding equations, the problems and the mathematical content for the Swedish students are probably different from those in the US classrooms. However, in order to understand the meaning of the tasks we must understand the meaning of the classroom interaction. This brings us to the next part of this chapter. THE ENACTMENT OF THE DESK TASKS ON A MICRO LEVEL

In both classrooms, working with written texts at students’ desks plays a significant role. The interaction between the teacher and the students is however different in the Swedish and the US classrooms. The interaction in Sweden to a great extent was directed towards how to solve the tasks, and the teacher more or less told the students how to work it out. In the US classroom, the focus of deskwork was on the mathematical relations and attempts to discuss these relations. This was organised through small groups regularly visited by the teacher. A typical pattern of interaction from the US classroom is shown below: the content is purely mathematical; mathematical language is regularly used; the teacher takes the initiative to visit the students and the students act within groups. When the teacher approaches Leticia (S1 below) she is suggesting that one of the values should be in the negative column according to the task presented in the beginning of the lesson (see former part in this text). In order to show the interactional pattern more clearly, for instance how the teacher does not get immediate responses to her questions, we use a simplified Conversation Analysis transcript.

173

JOHAN LILJESTRAND AND ULLA RUNESSON

Excerpt 1. US, the group of Leticia 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

T T S1 T S2 T S1 T T S1 T T S1 T S3 T

28

((moves to desk)) okay (.) lets see what’s going on here ((moves further to desk)) let me see Letitia ((points to working sheet)) x is two should be there (in the negative area) (3.0) ((T looks at his sheet)) okay okay I’m- I’m not sure this is right in every deedrespect but (4.0) but isn’t it negative two (1.7) ((to S2)) the y-intercept is negative two. yea where x is zero y is (1.0) negative two (.) mmh ((nods)) (2.0) and where x is zero here y is positive two (1.0) yea so that y intercept is different (0.7) .hh so the y-intercept is different (.) one slope is (.) negative one is ((unfinished intonation)) positive okay (.) what’s the similarity they both what ((to another student)) (1.7) linear linear ((gesticulating)) you got it okay ((proceeds to next desk)) good job

Key to symbols used in transcripts in this chapter: [ ] Left and right brackets mark the beginning and end of temporal overlap among utterances. (.) A pause of less than 0.2 seconds. (1.0) A pause of 1.0 seconds. °text° Speech produced more softly than surrounding talk. ( ) Empty parentheses represent untranscribed talk. The talk may be untranscribed because the transcriber cannot hear what was said. (text) Plausible speech but difficult to hear. (( )) Comments and annotations, often non-verbal action. x Underscoring indicates stress on a word, syllable, or sound. » (At the end of a line) The utterance is continued on the next line. » (At the beginning of a line) Continuation from a previous line. A hyphen at the end of a word indicates a self-interruption or re-start .hh Indrawn breath

The teacher actively seeks out Leticia and her group (lines 1–3). The suggestion from Leticia contains no references to a world outside of mathematics (like fruit and prices). The work of identifying similarities and contrasts between mathematical units (see our presentation above) is addressed from the beginning to the end of the teacher’s visit. At lines 9 and 23 other students join the interaction between the teacher and Leticia. The first insertion, from S2 at line 9 is taken up by the teacher to correct Leticia’s first suggestion. This insertion is achieved by using the long hesitation pause in the teacher’s turn on line 8. The hesitation from the 174

INTERACTION, ORGANISATION AND TASKS

teacher is also shown for the other students when she’s saying that she is not sure (lines 7-8). The teacher and S2 are thus cooperating to correct Leticia and introduce another line of reasoning despite an initiative from the teacher. The second proposal is from student S3, who replies to the teacher’s question on lines 23–24 by suggesting that the relation is linear (line 26). This is an example of teacher-initiated group work that was first introduced by the teacher to the whole class. On this micro-level group work can be seen as complex. The different voices are both collaborating and, on line 9, negotiating a solution to the task. S2 is self selecting on the issue introduced between Leticia and the teacher, and the teacher then shifts her attention. This group/teacher interaction offers opportunities however for different individuals to contribute to a small web of suggestions that can be heard by the other students in the group. Even if collective work could facilitate understandings of the same tasks from different points of view, collective work often becomes more complicated in terms of active participation from the individual students. From the perspective of social constructionism the construction of the enacted object of learning is complex, mediated through the different opportunities for participation offered by the group. This complexity also has the potential to develop a variation of meanings and thus enable students to understand the task more in depth. But to make use of this possibility is not only a rational intentional process, but presupposes a coordinated understanding. The opportunities for individual students to discern different variations of meaning are thus dependent on the social mechanisms of the group. On the other hand in the typical Swedish interaction at students’ desks the teacher meets with only one student at a time. The teacher is assisting the student to find the correct solution for the task. The concept of relationships is contextualised within the task connected to its specific values in the co-ordinate system, without making the algebraic premises for argumentation explicit (cf. Säljö & Bergqvist 1997). Another example illustrating this feature follows in the following excerpt 2 with Victoria, who explicitly expressed confusion about the term relationships, after the teacher tried to assist her by suggesting the correct values. The sequence begins just after Victoria has asked the teacher for help. Excerpt 2. Sweden, Victoria 1 T ((T reads)) the relation between the cost and the number of 2 litres of juice (0.7) as a formula 3 V mm 4 T .hh mm (.) then we’re almost in to (.) a these here previous 5 (V) what 6 T ((T sits down)) those we did last time (.) what’s the cost 7 (2.7) ((writes in the book)) 8 T [we can say is C (2.0) 9 V [yea 10 T how much did [a litre of juice cost 11 V [twelve 12 twelve. ((T writes, probably 12)) 13 T an’ the cost’ll be you know dependent on how many 14 (0.7) 15 T litres of juice we buy 16 (.)

175

JOHAN LILJESTRAND AND ULLA RUNESSON 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

V mm ((T writes)) T [twelve times X [((T points with his pencil)) V mm T twelve X V mm T obvious (.) the cost is twelve times how many litres [you buy V [but what d’you mean relation ((irritated voice)) T a ((points out)) here you have the relation (.) the cost is twelve times >how many litres you buy< V ((sigh)) yes yes. (5.0) T was that strange V yes ((light voice)) T what was strange about that. V I don’t get the relation ((mouth almost shut)) T yes in other words a relation a mathematical relation. that if you bu- if you buy two litre then the cost will be twenty four V mm T an’ if you buy three litres then the cost will be thirty six V mm mm T an’ then it’ll be- the this here is a relation (.) that the cost is (2.0) an’ after- when you’ve written one like that you usually say- and write there what C stands for ((writes in the same beat as he talks)) cost (1.0) X (1.0) quantity (1.0) litres V mm T ’cause then you understand the formula better. V yep T did you use your own initiative or? ((proceeding to Anna H K and Marie))

This sequence illustrates a typical pattern in that the teacher is assisting the student to solve the task correctly. As a consequence, the meaning of ‘relationship’ is brought out by examples of specific relations, like the cost of different quantities of different goods like bananas and juice, through multiplication of the two variables. Different quantities (x-values) and their corresponding y-values are focused on and compared by the teacher. The product kx is described in terms of calculation. After confirming that one litre of juice costs 12 kroner (lines 10–12) the teacher writes down the equation K = 12x (probably in lines 13-17) and summing up his point by commenting “Here you have the relation” (pointing to the equation, lines 26–27). The relation is described as a specific and correct equation. However, in this sequence, different levels of interpretation are also negotiated, by another initiative of Victoria on line 25. The teacher, who is initially defining the problem and then immediately goes on to present a strategy for a solution, gets a negative evaluation from Victoria (line 25), with an irritated tone of voice. Victoria doesn’t accept the teacher’s explanation “Here you have the relation” as a sufficient explanation. This is shown by the long pause on line 29, which the teacher interprets as a display of non-understanding, and is confirmed by Victoria’s affirmative response to his question (line 31) whether his line of reasoning was strange (line 30–31). At line 33 she insists once again that she does not get the meaning of ‘relationships’. 176

INTERACTION, ORGANISATION AND TASKS

Although Victoria expressed confusion about the concept relationship (line 25, 33) this was not explained by the teacher in a more general or elaborated sense, as was expected in the US lessons shown above. Instead the task was described in terms of a product of different calculations. Victoria is focused on the general concept ‘relationships’ which is also emphasised in the wording of the written task. Her orientation to the book, however, implicitly puts the general principle of relation in the forefront, while the teacher interprets the question (quoted by himself initially; lines 1–2) in terms of his former plenary introductions, where the graphical calculation of costs, by the identification of co-ordinates, is pointed out. This is also in line with his first plenary introduction on relationships, in which he points out the practical, everyday purpose for understanding such problems. The negotiation sequence above also illustrates how relationships as a mathematical concept becomes difficult to understand for the student when contextualised differently; as a tool for calculating different costs and as a concept for talking about mathematical principles. The case of Victoria shows a general feature in the Swedish lessons. To solve the task and to search for a correct answer, without referring to general principles for this solution, becomes the main orientation in the deskwork, while conceptual understanding as such becomes the background. So, during the desk interaction supporting learning implies helping the students to get through with the tasks in the textbook. There are several instances in the data indicating that in the interaction at the desks during the lesson, relationship was not talked about in general terms, for example in terms of how an increase in x-value corresponds to increase in the y-variable. The case above illustrates thus, how aspects which are critical for solving the task, rather than aspects which are critical for understanding i.e. direct variation, were the focus of the interaction. In this way task-solving work frames the interaction and the constitution of the object of learning. One of the four cases is however different. This occurs in a situation where all the students have left the room except Filip. Now Filip gets extended access to the teacher. Some lessons after the episode with Victoria (from Lesson 7 to Lesson 10), when working with the textbook was not the main focus of attention for the students, relationship was explained in quite a different way. The camera was still running although the lesson had come to an end. The teacher is asking Filip about his work with the book, and gradually begins to explain Task 16 in terms of different slopes and their relationships, using a lot of iconic (imitating) gestures to illustrate his explanation without referring to the everyday world at all. Excerpt 3. Sweden, Filip 1 2 3 4 5 6 7 8 9

T F T F T

is everything okay Filip ) I don’t get it ( what did you say ( ) I don´t get this write the relationship between the costs and (.) the number of litres of juice (4.0) yes if it costs eleven kronor one litre F yes it does T yes F but I still don’t get it

177

JOHAN LILJESTRAND AND ULLA RUNESSON 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

F T F T

F T

((Another student interrupts the conversation, with a non-mathematical subject. Few lines omitted)) how can it be like that well (3.0) cause you’ll get to- you’ll get to this a bit later ((points)) but (.) this one is next to it- which is x (.) or L [or numbers of litres or whatever it may be it tells you how» [mm »much it slopes (.) if it had been 15 [then it would have been» [((steep-gestures)) »then the graph would have gone up quicker right [.hh so this» [((hands back)) »number there (.) it says how [quick this will increase (.) for» [((hand raised rather fast)) »[each step of x it will increase by [eleven so (1.7) so if had» [((small horizontal x-steps)) [((steep vertical y-steps)) »said two [L (1.0)» [((raise his hands and illustrate)) »then it would’ve you- then kno- then you’d know that ah the incl- then that graph there will not slope as much (.) as [eleven .hh or fifty [((gesticulating a much more steep line)) mm hm so that’s why that’s what you try that that’s what we will try to learn eventually

The conversation this time is not about how to solve the task by calculating the correct numbers. Like Victoria in excerpt 2 Filip is confused about what the term ‘relationships’ means. However, in this situation the teacher’s response is not to support the student’s task work, as was the case in the interaction in excerpt 2. Instead of pointing out different x-values and finding their corresponding y-values by a calculation, other features of relationship are brought up. The equation y = kx is not presented as a calculation, but instead as a relationship between two variables. The relation between the x- and y-variables is described in terms of how an increase in the x-variable is related to another increase in the y-variable (“For each each step of x it will increase by eleven”, line 6). The focus is thus on the visual line or slope of the graph, and the slope coefficient (k). First, the teacher explains that the slope coefficient, independently of its value, represents the slope of the graph. (There was no reference to the graph in excerpts 1 and 2.) He continues by saying that the slope of the graph would have been different if the slope coefficient had been different, e.g. 11, 15 or 50. In the encounter with Filip, just immediately after the lesson, slope becomes the focus of attention. In the episode during Excerpt 2 the x- and y-values were the dimensions that were varied. Following the theoretical framework taken in this chapter, these occasions offered different learning opportunities for the students to learn about direct variation. Later on, the teacher commented “That’s what we will try to learn”; thus an introduced problem on what should be learned successively is pointed out, not on what to do to solve the problem correctly. An explanation of this deviant case may be that Filip did not have to share the teacher’s time with the other students, and the instruction occurred after the official lesson was completed. We ask ourselves: is this competition for getting the teacher’s assistance a consequence of the individualisation in the Swedish lessons? 178

INTERACTION, ORGANISATION AND TASKS

DISCUSSION

The two lesson cases presented in this chapter are examples of two mathematics classroom cultures – not necessarily representative for each country, but probably possible to locate within one country. However, the Swedish case does not seem to be atypical for the way Swedish mathematics classrooms are organised, according to the national assessment referred to earlier. The differences can briefly be characterised as, referential versus non-referential mathematical context on the one hand and privatised versus collective interaction and on the other. In both classrooms mathematical tasks played a significant role in the learning process of the mathematical concept relationship, but they were different in character. In the Swedish lessons, the concept was contextualised within the domain of everyday mathematics. The tasks included the calculation and/or comparison of costs for a certain quantity of commodities (fruits and drinks), whereas the US task had no reference to a world outside mathematics. What this implies for students’ possibilities to learn, whether contextualising a mathematics problem within a familiar context facilitates learning or not, is an empirical question not answered here, but the two lessons offer different possibilities for learning about relationships which we will touch upon below. Previously we have described the dyadic character of the interaction in the Swedish classroom, how the teacher, in the same lesson assisted individual students with different tasks, and that the teacher-student interaction was mainly a one-to-one interaction, with less opportunity for other students to participate. This form of organisation implied that the classroom was individualised in a way that privatised student learning. From the point of view of students’ learning, one must ask what consequences this may have. Seeing learning as social and assuming the importance of experienced variation for learning, we have questioned the learner’s possibility to experience the problem from different point of views, even if group work is more complex. In the US classroom, we saw the teacher as much more assisting on a collective level. It was possible for many students in the classroom to take part in talk with her, even when she was addressing a particular group of students. And since the whole class was working with the same task the students ‘shared’ the same problem. In the data we identified situations when the students enjoyed the dyadic interaction between the teacher and another student, which demonstrates this. Hence, the learning had a more collective feature. What is Possible to Learn about Relationships? One of the aims of the study was to obtain a better understanding of content learning; in particular to find out what was possible to learn about mathematical relationships in two different classroom cultures, without evaluating them that is, without saying that one is better or more productive than the other. One of the assumptions made is that an object of learning is constituted in the social interaction in the classroom. This is an enacted object of learning and described by

179

JOHAN LILJESTRAND AND ULLA RUNESSON

the researcher as a space of variation and invariance. As such it is a space of learning, constraining and enabling what is possible to learn. Our interpretation is that different learning opportunities were presented in the two classrooms. The analysis indicates that different kinds of content and different ways of organising teaching contributed to different objects of learning. Furthermore, the enacted object of learning was socially constructed under different conditions for organising the lesson. The way the set of tasks was composed in the US classroom and the pattern of variation/invariance opened up made it possible to experience different relationships between two variables (e.g. linear/non-linear, different slopes) and different representations (equations/graphs) at the same time. The task constituted a pattern of simultaneous contrasts by which several aspects/dimensions of the concept were brought out and could be discerned. In the Swedish classroom, on the other hand, the set of tasks was designed in such a way that it was possible to experience different linear relationships as quantitative properties, and different representations (table of values, graph, equation) (cf. Emanuelsson, & Sahlström, in press). Hence, the space of dimensions of variation was different in the US classroom than in the Swedish one. Following the variation theory framework, a good learning situation affords patterns of variation where critical aspects that is, aspects necessary for understanding the object of learning in a certain way, can be discerned. One can assume that in the US classroom several critical features of the equations were elicited at the same time when a pair of equations (e.g. linear/nonlinear, horizontal/vertical line) were systematically contrasted. Therefore, it is suggested that the US students had the possibility of understanding the mathematical concept in a more comprehensive way than their Swedish counterparts. However this possibility may not be realised from the point of view of individual students but could be seen as a potential for learning. It must be noted that we have not studied what the students actually learned – only what was possible to learn. The excerpt involving Victoria may point to a difficulty when two different contexts – the concepts of relationships belonging to a pure mathematical framework and costs for buying juice – are mixed and become a discursive hybrid (Fairclough, 1992). The hybridisation consists in that the concept relationships is purely mathematical and in this case not a necessary tool for calculating the price of a certain quantity of juice. The misunderstanding in the desk situation can be understood as a problem of premises in communication; to find a common frame for understanding the task. This does not mean that students are incapable of solving such tasks, but a risk of misunderstanding can be seen to be present; to understand relationships belongs to a non-referential practice, while calculating different costs for juice is more easily achieved without such an understanding. Interaction, Organisation and Tasks Our purpose was to examine how the mathematical tasks and the organisation of the classroom shaped the interaction and ultimately determined what it was 180

INTERACTION, ORGANISATION AND TASKS

possible for students to learn about the content. The different mathematical tasks seemed to demand different ways of organising the teaching, which also facilitated different patterns of interaction and possibilities for learning. The Swedish textbook was structured in a way that made it possible for the students to work at their own pace. From our analysis of a set of tasks, we have described how the structure and sequence of the textbook tasks led the students step-by-step to a mathematical climax; in this case an equation modelling a particular relationship. The character of the instructions and the sequence of subtasks seem to reflect an intention to keep the teacher’s assistance (and maybe the students’ common work) to a minimum. The structure of the tasks enabled the individualised organisation, just as this organisation presupposed a carefully guided instruction that would facilitate students’ progress through the textbook tasks. In the highly individualised Swedish classroom the students worked on different tasks in the same lesson and the students took the initiative in seeking the teacher’s attention. In this organisation, we identified a particular pattern of interaction; a dyadic student/teacher interaction, mostly initiated by the students. Furthermore, since the students received individual assistance from the teacher, the class did not function as a collective in that the problems were not made public. In the US classroom on the other hand, with a different structure of tasks and organisation, a different pattern of interaction was found. As the whole class was working on the same task, the teacher conducted the classroom work on a collective level. The teacher visited the students mostly on her own initiative and interacted with more than one student on the same task. Our interpretation is that the structure of tasks and the classroom organisation were contextual resources that shaped the teacher-student interaction. The structure of the tasks and the organisation of classroom work seem to function as interrelated resources for classroom interaction. In sum: mathematical tasks were complex but few and collectively shared in the US classroom, while the many and less complex everyday-like tasks in the Swedish classroom were intended to be solved individually and step by step. According to the theoretical framework of variation, a space of learning (Marton et al., 2004) is opened in the joint interaction when different features of the object of learning are focused on and opened up as dimensions of variation. Those features to which the learners’ attention is drawn determine what is possible to learn. From our analysis of the data we found that the task and the classroom organisation frame the space of learning, and ultimately what is possible to learn. From our comparison of two classrooms which were organised differently, we ask ourselves: could the more complex task been possible in the Swedish classroom with a highly individualised organisation? In order to work on such tasks, this organisation seems to presuppose that the degree of difficulty and complexity of the task is not too high. Hence, it affects the character of the learning task. The US task requires a deeper collective investigation and analysis and hence, a different organisation of the participation.

181

JOHAN LILJESTRAND AND ULLA RUNESSON

NOTES i

Swedish School Culture in Comparative Illumination, a project funded by the Bank of Sweden Tercentenary Foundation ii Details of the transcription system and examples of its application can be found on the internet at a number of sites. It is discussed, for example, in the transcription module available at http://www.sscnet.ucla.edu/soc/faculty/schegloff/.

REFERENCES Atkinson, P., & Heritage, J. (Eds.). (1984). Structures of social action: studies in conversation analysis. Cambridge: Cambridge University Press. Burr, V. (1995). An introduction to social constructionism. London: Routledge. Dienes, Z. P. (1960). Building up mathematics. London: Hutchinson. Dowling, P. (1998). The sociology of mathematics education: Mathematical myths/pedagogic texts. London: The Falmer Press. Emanuelsson, J., & Sahlström, F. (in press). The price of participation – How interaction constrains and affords classroom learning of mathematics. Scandinavian Journal of Educational Research. Fairclough, N. (1992). Discourse and social change. Cambridge: Polity Press. Liljestrand, J. (2004). Skolkarriärer som social interaktion: exempel från matematiklektioner [School careers as social interaction: Examples from mathematics lessons]. In H. Melander, H. Perez Prieto, & F. Sahsltröm (Eds.). Sociala handlingar och deras innebörder: lärande och identitet. Pedagogisk forskning i Uppsala, 150, 8-28. Uppsala Universitet. Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah NJ: Erlbaum. Marton, F., & Morris, P. (Eds.). (2002). What matters? Discovering critical differences in classroom learning. Göteborg: Acta Universitatis Gothoburgensis. Marton, F., Runesson, U., & Tsui, A. B. M. (2004). The space of learning. In F. Marton, A. B. M. Tsui (Eds.), Classroom discourse and the space of learning (pp. 3-40). Mahwah, NJ: Lawrence Erlbaum. Marton, F., & Tsui, A. B. M. (Eds.). (2004). Classroom discourse and the space of learning. Mahwah: NJ: Lawrence Erlbaum. Mehan, H. (1979). Learning lessons. Social organization in the classroom. Cambridge, MA: Cambridge University Press. Nuthall, G. (1997). Understanding student thinking and learning in classrooms. In B. J. Biddle, T. L. Good & I. F. Goodson (Eds.), International handbook of teachers and teaching (Vol. II, pp. 681-768). Boston: Kluwer. Piaget, J (2000). Commentary on Vygotsky´s criticisms of language and thought of the child and judgement and reasoning in the child. New ideas in psychology, 18, 241-259. (Original work published 1962). Pang, M. F., & Marton, F. (2005). Learning theory as teaching resource: Enhancing students' understanding of economic concepts. Instructional science, 33, 159-191. Runesson, U. (1999). Variationens pedagogik. Skilda sätt att behandla ett matematisk innehåll [The pedagogy of variation. Different ways of handling a mathematical topic]. Göteborg: Acta. Runesson, U., & Mok, I. A. C. (2005). The teaching of fractions: A comparative study of a Swedish and a Hong Kong classroom. Nordisk matematikdidaktik [Nordic studies in mathematics education], 10(2), 1-15. Skolverket (2003). [Swedish National Agency for Education] Nationell utvärdering av grundskolan 2003 [National assessment of the Swedish comprehensive school]. Säljö, R. & Bergqvist, K. (1997). Seeing the light: Discourse and practice in the optics lab. In L. Resnick, R. Säljö, C. Pontecorvo & B. Burge (Eds.). Discourse, tools and reasoning: Essays on situated cognition, pp. 385-405. New York: Springer-Verlag

182

INTERACTION, ORGANISATION AND TASKS

Johan Liljestrand Department of Education, Uppsala Universitet Sweden Ulla Runesson Department of Education Göteborgs Universitet Sweden

183

JOHAN HÄGGSTRÖM

CHAPTER NINE The Introduction of New Content: What is Possible to Learn?

COMPARATIVE STUDIES

Large international comparative studies in mathematics education, such as the Trends in International Mathematics and Science Study (TIMSS) (IEA, 2005) and the Programme for International Student Assessment (PISA) (OECD, 2005), tend to have their main focus on student achievement on a national level. In order to understand and explain the differences in achievement, data concerning background and contextual variables is also collected. This includes variables stretching from mathematics curriculum and class size to teacher education and student attitudes. This approach suggests that these studies can be placed in the tradition of ‘educational input-output research’ where the teaching process to a high degree is considered as a ‘black box’. Although differences in input-variables as well as student achievement can be found on the national level it turns out to be hard to establish distinct relations between these. Teachers’ mathematical knowledge and class size are two examples of such variables that no doubt are important but not absolutely critical regarding student learning of mathematics. Gustafsson (2003) finds after reviewing a large number of studies and meta-studies in this tradition that results tend to be inconsistent and not easily interpreted. However, the resource factor labelled ‘teacher competence’ is one that Gustafsson claims could have a less ambiguous effect on student achievement. Other studies point to the existence of large national differences in teacher competence. Ma concluded from her comparison of Chinese and US elementary teachers’ competence that there is a considerable difference in their understanding of fundamental mathematics and that “the knowledge gap between the U.S. and Chinese teachers parallels the learning gap between U.S. and Chinese students revealed by other scholars” (Ma, 1999, p. 124). But teachers’ mathematical knowledge as such cannot explain differences in student learning. I argue that only if teachers with different knowledge will teach in a significantly different way may it have an impact on student learning. There are indications that this is not the general case. No clear patterns are detected in the TIMSS 1999 data between teachers’ mathematical preparation and teaching practice or student achievement. This undercuts the myth of a simple and clear connection (Hiebert, 2005).

D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 185–199. © 2006 Sense Publishers. All rights reserved.

JOHAN HÄGGSTRÖM

Class size is another background variable that might be considered as important and many teachers would surely like to reduce the number of students in their mathematics classes. However, it is simple to argue that class size, per se, cannot determine the outcome of teaching. For instance, if a strict teacher-centred lecturing style of instruction is used, it is difficult to claim that student learning will differ if there are 100, 30 or 10 students in the ‘audience’. The potential effect of class size on achievement is indirect. A smaller number of students can make it possible to teach in a different way, which in turn might make a difference concerning student achievement. There can be no guarantees however, even though some teaching styles are certainly more difficult to carry out with larger classes than with smaller. My main point here is that, however interesting it may be to study the effect of background variables like class size from a political point of view, it is of less interest to the mathematics teacher, at least from a short term perspective. The teachers do not control the background variables. The teacher mostly has to teach the number of students that actually show up, the fixed number of lessons per week, and so on. Similar arguments can be produced for most background factors. These factors can merely provide opportunities for good teaching and learning; they do not guarantee that it will take place. One possible conclusion that can be drawn from across these meta-analyses […] is that to improve school learning, we should focus on those variables that impact directly on the learning experiences of students such as teaching and feedback (Pong and Morris, 2002, p. 11). Background variables are not in themselves critical when it comes to explaining differences in results. We need to enter the classrooms and study the teaching processes if we want to understand why some students learn better than others. INSIDE THE CLASSROOM

In the series of IEA studies of mathematics education the ‘teaching process’ was finally included through the video recording of mathematics classrooms in TIMSS 1995 and 1999. The design employed in the TIMSS 1999 Video Study (Hiebert et al., 2003) is called a ‘video-survey’. About 100 randomly selected 8th grade mathematics lessons were videotaped to generate representative samples from each country. The camera had the recording focus on the teacher, who carried a microphone. The analysis involved coding different aspects of mathematics teaching to identify the features that high-achieving countries have in common and that may explain differences in achievement. This turned out to be quite difficult, despite an extensive effort. After having coded and examined more than 60 aspects of mathematics teaching in between 50 and 140 taped lessons from seven countries the research group had to acknowledge that “we had difficulty finding lesson features that correlate with differences in achievement” (Givvin, 2004, p. 208). A closer analysis was needed of how the mathematical content was dealt with, from

186

THE INTRODUCTION OF NEW CONTENT

the student perspective, in order to find something that high-achieving countries have in common. The answer does not lie in the organisation of classrooms, the kinds of technologies used, or even the types of problems presented to students, but in the way which teachers and students work on problems as the lesson unfolds (Stigler and Hiebert, 2004, p.14). It is of course no surprise that it is hard to identify any distinct features that really matter and make a clear difference when it comes to such complex phenomena as the mathematics classroom. What I find interesting is that many of the obvious features that can be used to describe mathematics teaching, such as use of textbooks, classroom organisation, teacher-led or student-centred instruction, and so on, seem not to work when it comes to distinguishing teaching in high-achieving countries. To understand differences in student learning, these variables appear to be of much less importance than how the mathematics content is treated by the teacher and the students. Stigler and Hiebert come to the same conclusion: A focus on teaching must avoid the temptation to consider only the superficial aspects of teaching: the organisation, tools, curriculum content, and textbooks. The cultural activity of teaching – the ways in which the teacher and students interact about the subject – can be more powerful than the curriculum materials that teachers use (Stigler & Hiebert, 2004, p. 15). In order to find ways to improve teaching in the long run it appears to be fruitful to try to capture the interaction about the mathematical content. This will of course demand really close examination of the mathematics classroom with a focus on how the mathematical content is treated. FOCUS ON THE MATHEMATICAL CONTENT

From the discussion above we have concluded that features of the mathematics classroom which are fairly easy to observe (e.g. mood of instruction, the number of students in the class, how students are grouped etc.) have only an indirect impact on achievement. The impact is restricted to what degree of influence these features might have on how the teacher and students interact about the mathematical content. Thus the present study will have its focus on the interaction about the mathematical content rather than looking at other, perhaps more easily observed, features. What is then needed to be able to say something about differences in how the mathematics is dealt with? First, you need to get inside the classroom and record what is going on. Second, you need to compare how the same mathematical topic is taught in different classrooms. It will be of little use to compare teaching of different topics. Concerning this second point, it is worth noting that neither the design used in TIMSS Video Study, nor the design in the Learners’ Perspective Study (Clarke, 2000; LPS study, 2005) is particularly concerned with what mathematics is taught in the lessons recorded. This study, however, aims to explore 187

JOHAN HÄGGSTRÖM

how the same mathematical topic is dealt with in different classrooms and to discuss implications for students’ opportunities to learn. THEORETICAL FRAMEWORK

One obvious assumption for the present study is that what is possible for students to learn is strongly related to how the mathematical content is handled in the classroom. Different ways of handling the content make it possible to learn different things. The analysis is based on the framework of Variation theory (Marton and Booth, 1997; Runesson and Marton, 2002; Marton, Runesson and Tsui, 2004). Variation theory offers a framework for capturing and describing differences in the classroom interaction about a certain mathematical topic. The framework has an explicit focus on the what aspect of learning. Learning always has an object, called the object of learning. You always learn something. Further, the object of learning can be experienced in qualitatively different ways. The differences are described in terms of those aspects of the object of learning an individual can discern and keep in focal awareness at the same time. This is in some cases rather straightforward in relation to the understanding of mathematics concepts as these often have elaborated and defined properties. Learning in this theoretical framework means developing a capability to see or understand the object of learning in new ways. This means acquiring an ability to discern and be aware of aspects that were not previously discerned. A prerequisite for discerning a new aspect of the object of learning, or for seeing it in a different way, is that a variation in that respect is experienced. Features of the object of learning that are taken for granted and kept invariant are less likely to be experienced. This is central to teaching. In all teaching there is a use of variation in one way or another. Variation by creating contrasts is one example of a common teaching strategy used to direct students’ attention to important features of the teacher’s intended object of learning. Runesson (1999) showed that teaching of school mathematics can be characterised by means of what aspects of the object of learning have been kept invariant and what aspects have been varied. As a mathematics lesson unfolds, this forms a pattern of variation that describes the way in which the object of learning has been dealt with and provides insights into different ways of teaching. The pattern of variation will be different in different lessons and these differences – sometimes subtle, yet important for students’ possibilities to learn – can be captured and described (see e.g. Runesson and Mok, 2005). A teacher can deliberately plan what to vary and what to keep invariant, and how to generate a certain pattern of variation in a lesson. Variation, on the other hand, may be a result of circumstances and not intentional on the part of the teacher. Whatever the case, every mathematics lesson generates a certain pattern of variation in respect of the way in which the mathematics is handled and the students’ opportunities to experience the content.

188

THE INTRODUCTION OF NEW CONTENT

THREE CLASSROOMS

The present study is comparative. Comparison is necessary to make visible the familiar, and everything that is taken for granted. To contrast the familiar with something less familiar or, even better, with something quite different, can make it possible to discern important features that are too well known and established. In the analysis for this study my ambition is to take the learner’s perspective. I will place myself in the position of ‘a student in the classroom’ and describe and compare what is possible to learn in different mathematics lessons. The lessons have been selected on the grounds that the same mathematics is taught. Keeping the object of learning invariant will increase the possibility of discerning, from a student point of view, interesting differences in how it is dealt with. The comparison is done between three classrooms in China (Hong Kong and Shanghai) and Sweden. In all three lessons the concept of a system of linear equations in two unknowns is introduced. Because of the design of the Learners’ Perspective Study, the data collected is very suitable for in-depth analysis (e.g. Häggström, 2004) but may not be used to make generalisations on a national level. The classrooms in the study provide examples of good teaching from each country but cannot, by themselves, be representative of the different cultures of mathematics teaching or claim to show what is typical. The recorded classes in the Learner’s Perspective Study are not selected with that ambition. In order to avoid the temptation of making the possible comparison of three specific classrooms into a comparison of three cultures of mathematics instruction I will label the classrooms A, B and C. Classroom A is Lesson 6 from Hong Kong School 1. Classroom B is Lesson 5 from Shanghai School 1. Classroom C is Lesson 12 from Sweden School 2. The analysis focuses on the parts of the lessons where the concept of a system of linear equations in two unknowns is introduced to the students. It means that the way the actual introduction is done is analysed with respect to which aspects of the concept are kept invariant and which aspects are varied. Classroom A There are 38 students in this class. The teaching in general can be characterised as mainly whole-class teaching, meaning that all students basically work with the same topic and problems throughout the lesson. As a student your main activities are either paying attention and taking part in the teacher-led instruction and discussion, or working on common problems and questions together with the student sitting next to you. During this latter activity the teacher walks around and talks to students at their desks. As the lessons unfold there are constant shifts between these types of activities. Typically a problem is posed on the board, and then students are given some time to work on it, followed by a whole-class discussion of results and conclusions.

189

JOHAN HÄGGSTRÖM

The lesson studied starts with a short whole-class revision of the concept of linear equations in one unknown. A student gives the example r + 97 = 107, which is put on the board. The notions of unknown, degree one and solution are brought up. After this revision part, the teacher presents a problem. It is written on the blackboard and the students are asked to solve it. That is, to find the number of rabbits and chickens. A farmer has some rabbits and some chickens. He does not know the exact number of rabbits and chickens, but in total there are ten heads, and there are twenty-six legs. In the following there are a number of shifts between teacher-led whole-class discussion and students working individually and in pairs. The problem is handled in three ways. The first method used is ‘guess-and-check’. The teacher leads the way to the solution by posing questions: “Can all of them be chickens?”, “Can all be rabbits?”, “Can there be five each?” etc. Secondly, the problem is represented by the formulation of one equation in one unknown, 2x + 4(10 – x) = 26. This is also done with firm guidance from the teacher. When asked by the teacher, only five students admit they could have come up with this equation by themselves. The equation is posed but not solved, maybe because the answer is already known from the ‘guess-and-check’ solving. The teacher then introduces simultaneous equations as the third method for solving the problem. 22:10 Teacher: I am now going to teach you an easier method. It is also about using equations, but it is simultaneous equations in two unknowns.

After some minutes of teacher-led discussion two equations are written on the blackboard: x + y = 10 !2 x + 4 y = 26 " This example is used to define the concept. The notions of simultaneous, linear, and two unknowns are mentioned at this time. 24:46 Teacher: Well, okay, well, we call it simultaneous equations. Simultaneous means the equations will be listed out together. A moment ago … Nancy has asked me why it is called linear. This is because we can draw a straight line from this kind of equation. . . . . this is called two ‘yuan’, that means how many unknowns are there? 25:22 Student: Two. Teacher: Two. It is simultaneous, that means the equations are put together. Key to symbols used in transcripts in this chapter: 24:46 Time in minutes and seconds from the start of the recording. ( ) Indecipherable words

190

THE INTRODUCTION OF NEW CONTENT [text]

Descriptions of non-verbal actions, other comments or annotations by the author or transcriber. A short pause of three seconds or less Indicates some words or utterances omitted

… ....

The system of equations is not solved and no method for solving a system of equations is demonstrated or discussed. After this introduction a worksheet is distributed to the students. The object of learning is thereby shifted from the concept of system of equations to the meaning of a solution to such a system. The first task on the worksheet consists of a system of equations and the students are asked to find the corresponding y-values, for x = 0, 1, 2, 3, to each of the two equations separately and to list the number pairs in tables. The teacher explains and demonstrates how to find the first two y-values and the students start to work to fill in the rest of the two tables. Classroom B The number of students in this class is 50. The general picture of the teaching in this class resembles that in classroom A. The students work with the same topic and the same problems in a similar cyclic shift between teacher-led whole-class discussion and students working individually or in pairs while the teacher walks around in the classroom, answering questions and talking to students at their desks. This lesson also begins with a revision. It is quite short. The first four minutes is spent on discussing five tasks on the topic of one linear equation in two unknowns and its solutions. The teacher then announces the topic of today’s lesson and shows a slide with three questions introducing the concept of a system of linear equations in two unknowns. Q1.

What is a ‘system of equations’?

Q2.

How can you tell whether a system of equations is a system of linear equations in two unknowns?

Q3.

Identify whether the given is a system of linear equations in two unknowns. 1)

!x + y = 3 "x y =1 #

4)

! "

7)

u=v=0

x/2+ y/2 = 0 x= y

2)

!( x + y ) 2 = 1 " x y=0 #

3)

x =1 !y = 1 "

5)

xy = 2 ! x =1 "

6)

! "

8)

!x + y = 4 "x m = 1 #

x + 1/ y = 1 y=2

The students are told to read a section in the textbook and to discuss the three questions in pairs. After a couple of minutes the teacher calls for attention.

191

JOHAN HÄGGSTRÖM

The students’ answers to the first two questions are obviously the same as in the textbook. 08:10 Student: A system of equations is formed by a number of equations. . . . . 09:10 Student: There are two unknowns in the equations and the indexes of the unknowns are one. This is called system of linear equations in two unknowns. Teacher: . . . . He has just mentioned the definition of system of linear equations in two unknowns.

These important points are then repeated a number of times in the following conversation before they move on to the third question. 12:04 Teacher: Okay, these two points, oh then, let us take a look at the following questions with these two points.

All eight items in the third question are discussed, one at a time. Reasons for or against them being a system of linear equations in two unknowns are given in each case. Four of the examples meet the requirement and four do not. The lesson then continues with a focus on the meaning of a solution to a system of linear equations in two unknowns and what is required of a pair of numbers (x, y) to be a solution. As in classroom A no method for solving the system of equations is showed. Classroom C There are 24 students in this class. As a student in this classroom you spend most of the time solving problems from the textbook individually. Basically all students work with problems on the same topic, but not necessary precisely same problems at the same time. Even students sitting next to each other may work with different problems. The teacher walks around, answers questions and gives instruction at the desks. There are shorter sections of teacher-led whole-class instruction. Mostly the lessons start with a teacher-led discussion on a typical problem on the actual topic. The teacher also interrupts the work in the textbooks occasionally, usually when he finds a particular problem that many students are asking questions about. Such a problem is put on the board and discussed with the whole class. In the lesson studied, the teacher begins by returning to a problem that previously has been handled by the formulation of one equation with one unknown. Some students had tried to use two unknowns. The teacher shows how that approach is also possible and that expressing one unknown by means of the other will generate the equation in one unknown. After this the teacher and the students formulate a new problem together. One student is asked to “think of a number” (x) and the teacher “thinks of another number” (y). The student tells (whispering) the teacher which his number is and based on this the teacher writes the equation, x + y = 60, on the whiteboard. After examination, the conclusion that there is not enough information to determine the two numbers, is reached. There are many 192

THE INTRODUCTION OF NEW CONTENT

possible solutions to this equation, so the teacher adds another condition and gets the following on the board: (1) ! x + y = 60 " x = 14 y (2) # [Note: · is used for multiplication] 13:09 Teacher: Now I have two conditions and two unknowns. Now we can easily calculate the whole … So, let’s do it.

The teacher uses the method of substitution and finds the value of y. The solution is written on the whiteboard. (1) ! x + y = 60 " x = 14 y (2) # (1) and (2) gives:

14y + y = 60 15y = 60 y=4

The system of equations is not solved completely since the student’s number (x) had already been revealed to many of the students by mistake (In fact, in the end of this sequence, only the teacher’s number (y) was really ‘unknown’ to most students). The students’ attention is then directed back to the two equations when the teacher points to them. A ‘definition’ of a system of equations is made from this example. 14:50 Teacher: What is this then? [T points to the equations (1) and (2) on the whiteboard] Teacher: . . . . Well, it’s two equations. A system of equations, it is called. [T writes “system of equations” beside the equations]

After this introduction the teacher gets a similar system of equations from the textbook and writes it on the board for the students to solve. After a few minutes the teacher writes the solution on the board without any comments. The students continue to work with similar tasks in their textbook (solving systems of linear equations in two unknowns by the method of substitution), while the teacher walks between the desks and talks to individual students. PATTERNS OF VARIATION

The intention of this part of the analysis is to describe how the mathematics is handled in terms of what aspects are varied and what aspects are kept invariant (pattern of variation). The comparison of the three classrooms makes it possible to discern those aspects of the concept of system of equations in two unknowns that were taken for granted and kept invariant in one classroom but were elaborated on in another. 193

JOHAN HÄGGSTRÖM

Classroom A During the introduction the rabbit-and-chicken problem is kept invariant. The same problem is dealt with in three different ways. The method of representing or handling the problem is varied. The use of one equation is contrasted to the use of a system of equations. This contrast offers a possibility to discern some features of the new concept. There are two unknowns instead of only one. There are two simultaneous equations instead of just one. The characterisation of the equations as linear is mentioned only as a comment to a student’s question but is not further discussed. This introduction stress the system of equations as a method for solving problems as it is contrasted to the guess-and-check method and the use of one equation in one unknown. The example of a system of linear equations in two unknowns derived from the rabbit-and-chicken problem is used to ‘define’ the concept. There are no counterexamples given that could point to important features of what is not included in the concept. It is taken for granted that the system of linear equations in two unknowns is the system of equations. Classroom B Already from the revision in the beginning of the lesson it is clear that some features involved in the new concept are familiar to these students. This includes notions such as the concept of one linear equation in two unknowns, the number of solutions to a linear equation in two unknowns and how to determine whether a proposed solution is correct or not. The new concept is introduced by a description of some important characteristics – two equations, two unknowns of degree one. During the seven and a half minutes when question 3 is discussed in a basically teacher-led mood of activity, a powerful pattern of variation emerges. Eight proposed systems of linear equations in two unknowns are considered. The important points that came from the first two questions – two equations, two unknowns with indexes of one – are kept invariant while question 3 is handled. Of these three points only one is really new. Already in the revision part in the beginning of the lesson when the concept of one linear equation in two unknowns was discussed, the features – two unknowns with indexes of one – were highlighted. (Strangely it is these two previous points that the teacher emphasises, when the new concept is introduced, rather than the new feature that there have to be two equations). By the choice of items in question 3 the teacher generates a specific and quite systematic pattern of variation. The contrasts that are formed between the eight examples make it possible for the students to discern some of the features that very well might be critical when it comes to understanding the concept of a system of linear equations in two unknowns. Some of these features are:

194

THE INTRODUCTION OF NEW CONTENT

The variables must be of degree one. Even students who ‘know’ this in principle may in some cases have difficulties interpreting the mathematical symbols correctly. Some of these instances are highlighted by the examples used. – (x + y)2 is not of the first degree even though x + y is. – xy is not of the first degree even though x and y are. – y/2 is of degree one but 1/y is not. This choice of items could be an indication of the teacher’s knowledge and experience of common student errors. There must be exactly two variables. In the last example (8) three letters, x, y and m, are used. Both the teacher and the students seem to interpret these as three unknowns and the example is judged as not meeting the requirements of a system of equations in two unknowns (The letter m is often used to denote a constant, not a variable or unknown, which could have made the interpretation less straightforward). Both unknowns need not be present in both equations. In example 3 the equations x = 1 and y = 1 form a system of linear equations in two unknowns. Other letters besides x can be y can be used. The letters u and v are used in one example (7), which open up for variation in this dimension. Two equations can be merged together to what might look like one equation. It is probably not evident to all students that an algebraic expression like u = v = 0 can be interpreted as two separate equations merged together. There are other systems of equations beside the linear in two unknowns. It is not taken for granted that a system of equations is a system of linear equations. Classroom C In the first short sequence the teacher keeps a previous problem invariant while the way to represent it is varied. The earlier use of an equation in one unknown when trying to solve the problem is contrasted to the possibility of using two unknowns. The students are offered the possibility to experience a variation in the number of unknowns used. In the next sequence the two numbers ‘thought of’ are kept invariant and the number of conditions is varied. At the same time as the number of conditions is increased from one to two, the number of solutions changes from many to one. The need for two equations when finding the values of two unknowns is highlighted. During the same episode the meaning or interpretation of the letters x and y is elaborated on in a sophisticated manner. They are simultaneously kept invariant – the student and the teacher keep thinking of the same two numbers – and varied when the first condition, x + y = 60, is examined – different number pairs are

195

JOHAN HÄGGSTRÖM

suggested as solutions. In this episode the letters x and y are at the same time considered as two distinct but yet unknown numbers and two variable numbers. As in Classroom A, the system of linear equations in two unknowns is introduced only by positive examples and as a method for problem solving. There are no counterexamples that may point to some of the features that are not included in the concept. Apart from the need for two equations, no real discussion of the specification or requirement for the concept is done. In this classroom the focus is almost immediately turned to the procedure of solving. The exercises in the textbook only include positive examples of systems of equations without any counterexamples. The general pattern of variation, after the introduction, is that the method of solving is kept invariant for the rest of the lesson. The same method is used to solve different examples of systems of linear equations in two unknowns, which is then what varies. DIFFERENT OPPORTUNITIES TO LEARN

It is quite clear that the students in the three classrooms come with different previous knowledge in mathematics. This is of course an important factor for teachers to consider when they plan how to introduce a new concept. Students’ prior knowledge will also be important for their opportunities to learn. Nevertheless, I claim that the way in which the new concept is introduced, and the pattern of variation offered, have strong bearing on both what is possible and what is not possible to learn. Some features of the concept of system of linear equations in two unknowns that I have found to be elaborated on in at least one of the three lessons are listed in Table 1. Those aspects that have been elaborated on and exposed to variation in the different classrooms are marked X in the table. An empty space indicates that the aspect in question was taken for granted and kept invariant during the introduction. The table will also provide an overview of the differences in how the concept of system of linear equations in two unknowns was dealt with in the three classrooms. Table 1. Pattern of variation

Aspect of the concept A method for problem solving There are different systems of equations Two equations - can be merged into one expression Two unknowns - not three unknowns First degree unknowns - (x + y)2, xy and 1/y are not of degree one Same unknowns in both equations - need not be present in both equations Different letters may be used

196

Classroom B C X X X X X X X X X X X X X X X X A X

THE INTRODUCTION OF NEW CONTENT

What can be learned about students’ opportunities to learn in the light of the three patterns of variation? The opportunities to discern and experience important aspects of the concept taught are clearly not the same in the three classrooms studied. One difference found concerned the use of a problem as a starting point in classrooms A and C. Especially in classroom A, the same problem is kept invariant for quite some time while the method of representing the problem is varied. This makes the feature ‘a method for problem solving’ exposed and possible to experience for students. The starting point in classroom B is within a more ‘pure’ mathematics context and no possible application or use of the concept of a system of equations is discussed. This aspect is taken for granted during the introduction in classroom B. The most striking differences, however, are caused by the use of examples and non-examples in classroom B, which creates distinct contrasts and opens up dimensions of variation that are not found in the other two classrooms. The students in classrooms A and C are not offered the opportunity to experience what a system of linear equations in two unknowns is not. The conclusion is that students in classrooms A and C are not provided with the same opportunities, as students in classroom B, to experience a number of features of the concept, for example, that a system of equations can be merged into one expression and that three unknowns are not allowed (Table 1). Of course these conclusions are only valid for the sequences analysed. The aspects that I have found not to be varied during the introduction may already have been known by the students or they might be treated in lessons to come. My conclusions here are based only on the quite short introduction sequences. It points to the need to enlarge the study to cover longer sequences and to involve more than just three classrooms. It is also worth noting that, when examining the classrooms in this way, class size does not seem to impose any severe restrictions on the handling of the mathematical content. In fact it was in the classroom with the largest number of students that the most extensive pattern of variation was generated. On the other hand, teacher competence in the respect of knowledge of students’ learning and understanding of the mathematics taught, will probably be vital in order to plan and handle the content in a advantageous way. Since the classrooms in this study are just examples of mathematics education in Hong Kong, Shanghai and Sweden I am not prepared to generalise the findings and make any conclusions regarding cultural differences. However, the methodical use of variation and contrasts found in the Shanghai classroom seem to be more than an isolated example. Similar ways of handling the mathematical content have been described elsewhere (Gu, Huang & Marton, 2004; Huang, Mok and Leung, 2006; Park and Leung, 2006). The results presented here points to some implications for future research. The comparison of teaching and learning of the same topic, or mathematical content, can give insights into teaching differences. Studies of differences in the ways the same topic is handled can help teachers become aware of aspects of the mathematical content that they have previously taken for granted, and point to possible new ways to bring about these aspects in their own teaching. As argued 197

JOHAN HÄGGSTRÖM

previously, the handling of the mathematical content is central to mathematics teaching and one factor of which teachers are in charge. Studies that focus on the mathematical content can be of more relevance to the mathematics teacher than studies concerning ‘background variables’ which in most cases are out of reach for teachers. REFERENCES Clarke, D. J. (2000). The Learner’s Perspective Study. Research design. Retrieved March 10, 2006 from University of Melbourne, International Centre for Classroom Research Web site: http://extranet.edfac.unimelb.edu.au/DSME/lps/subabout.shtml#Researchd. Givvin, K. (2004). Video surveys: How the TIMSS studies drew on the marriage of two research traditions and how their findings are being used to change teaching practice. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education, Vol. 1 (pp. 206-211). Bergen: Bergen University College. Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective mathematics learning. In L. Fan, N. Y. Wong, J. Cai & S. Li (Eds.), How Chinese learn mathematics. Perspectives from insiders (pp. 309-347). Singapore: World Scientific Publishing. Gustafsson, J.-E. (2003). What do we know about effects of school resources on educational results? Swedish Economic Policy Review, 10, 77-110. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K., Hollingsworth, H., Jacobs, J., Chui, A., Wearne, D., Smith, M., Kersting, N., Manaster, A., Tseng, E., Etterbeck, W., Manaster, C., Gonzales, P., & Stigler, J. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study. Washington, DC: U.S. Department of Education, National Center for Education Statistics. Hiebert, J. (2005, May). What evidence exists about the relationship between teachers’ knowledge of mathematics and student achievement? Paper presented at the MSRI Workshop, Mathematical Knowledge for Teaching (K–8): Why, What and How? Asilomar, CA. Retrieved November 15, 2005 from http://ncm.gu.se/index.php?name=nyheter-05092-ma-kunsk Huang, R., Mok, I. & Leung, F. (2006). Repetition or variation: Practising in the mathematics classroom in China. In D. Clarke, C. Keitel & Y. Shimizu (Eds.), Mathematics classrooms in twelve countries: The insider’s perspective (pp. 263-274). Rotterdam: Sense Publications. Häggström, J. (2004). KULT-projektet. Matematikundervisning i Sverige i internationell belysning [The KULT project. Mathematics teaching in Sweden in international comparison]. In C. Bergsten & B. Grevholm (Eds.), Proceedings of MADIF4, the 4th Swedish Mathematics Education Research Seminar (pp. 133-145). Linköping: Swedish Society for Research in Mathematics Education. IEA (2005). International Association for the Evaluation of Educational Achievement. Retrieved November 15, 2005 from http://www.iea.nl LPS study (2005). The Learner’s Perspective Study. Retrieved October 14, 2005 from http://www.edfac.unimelb.edu.au/DSME/lps/ Ma, L. (1999). Knowing and teaching elementary mathematics. New Jersey: Lawrence Erlbaum. Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, NJ: Lawrence Erlbaum. Marton, F., Runesson, U., & Tsui, A. (2004). The space of learning. In F. Marton & A. Tsui (Eds.), Classroom discourse and the space of learning (pp. 3-40). Mahwah, NJ: Lawrence Erlbaum. OECD (2005). PISA. Programme for International Student Assessment. Retrieved November 15, 2005 from http://www.pisa.oecd.org. Park, K., & Leung, F. (2006). Mathematics lessons in Korea: Teaching with systematic variation. In D. Clarke, C. Keitel & Y. Shimizu (Eds.), Mathematics classrooms in twelve countries: The insider’s perspective (pp. 247-261). Rotterdam: Sense Publications. Pong, W. Y., & Morris, P. (2002). Accounting for differences in achievement. In F. Marton & P. Morris (Eds.), What matters? Discovering critical conditions of classroom learning. Göteborg studies in educational sciences (pp. 9-18). Göteborg: Acta Universitatis Gothoburgensis.

198

THE INTRODUCTION OF NEW CONTENT Runesson, U. (1999). Variationens pedagogik: skilda sätt att behandla ett matematiskt innehåll [The pedagogy of variation: Different ways to handle a mathematical content]. PhD thesis. Göteborg: Acta Universitatis Gothoburgensis. Runesson, U., & Marton, F. (2002). The object of learning and the space of variation. In F. Marton & P. Morris (Eds.), What matters? Discovering critical conditions of classroom learning. Göteborg studies in educational sciences (pp. 19-37). Göteborg: Acta Universitatis Gothoburgensis. Runesson, U., & Mok, I. A. C. (2005). The teaching of fractions: A comparative study of a Swedish and a Hong Kong classroom. Nordic Studies in Mathematics Education (NOMAD), 10(2), 1-16. Stigler, J. W., & Hiebert, J. (2004). Improving mathematics teaching. Educational Leadership, 61(5), 12-17.

Johan Häggström IPD/Mathematics Education, Göteborgs Universitet Sweden

199

HERBERT BHEKI KHUZWAYO

CHAPTER TEN A Study of Mathematics Teachers’ Constraints in Changing Practices: Some Lessons from Countries Participating in the Learner’s Perspective Study

INTRODUCTION

This chapter reports on my current involvement in studying the constraints and struggles experienced by mathematics teachers in South Africa as they attempt to take on new curricula and pedagogies. Some of the obstacles, tensions and contradictions that arise for South African teachers as they attempt to transform the apartheid curriculum to a revised post-apartheid curriculum are not unique to South Africa. This conclusion has become evident from my analyses of teacher data collected in other countries as part of the Learner’s Perspective Study (LPS). The study sought to address the following questions: – What constraints, tensions and struggles do mathematics teachers experience as they attempt to change their practices by making a shift from ‘traditional’ to ‘progressive’ forms of teaching? – What lessons can be learnt from studying the constraints, tensions and struggles in teacher practices occurring in classrooms identified by the local education community as constituting sites of competent teaching? I provide some preliminary findings of this study, followed by some lessons from the study and its implications for research and teacher education. TEACHER CHANGE AND THE STUDY OF CONSTRAINTS

Studies on teacher change in mathematics education have revealed that the most significant factors that tend to impede successful implementation are those related to teachers’ beliefs about teachers and learning (e.g., Cohen, 1990; Ernest, 1989). The underlying assumption is that teaching reforms cannot take place unless teachers’ beliefs about mathematics and its teaching and learning change. As a consequence, ‘wrong’ beliefs and ‘deficient’ knowledge can be identified as the cause of ‘wrong’ and ‘deficient’ practices (Skovsmose & Valero, 2002). We need to realise however that learning and teaching occur in a social context (National Council of Teachers of Mathematics, 2000; Romberg, 1992) and that the social context exerts a powerful influence on educational processes. This influence

D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 201–214. © 2006 Sense Publishers. All rights reserved.

HERBERT BHEKI KHUZWAYO

usually results from the expectations of others, including students, parents, teachers, policy makers. These sources lead the teacher to internalise powerful sets of constraints affecting the enactment of models of teaching (Ernest, 1989). The social context therefore constrains the teachers’ freedom of choice and (aspired) action, restricting his or her autonomy. In some cases teachers resort to traditional forms of teaching, as that offers them an easy way of dealing with these constraints. The belief that it is the teacher’s view of mathematics that is responsible for classroom practice needs to be questioned and deconstructed (Gates, 2002). Gates argues: “Beliefs do not exist in a social vacuum … Beliefs have a wider and deeper dimension, rooted in cultural norms that are themselves rooted in social structure. We are the embodiment of social structure” (Gates, 2002, p. 319). I contend that teachers continually confront constraints in their practices and are therefore compelled to act in particular ways irrespective of whatever beliefs they may have. Various role-players (e.g. teachers, parents, education policy-makers and the mathematics education community) may have different visions about what constitutes quality mathematics teaching and this may constrain teachers to reform their practices. Whilst teachers envision a quality model of teaching, there are other influences, which in many cases affect teachers’ vision of quality teaching. Boaler argues: Research that has considered professional development has focused mainly upon initial teacher education [and] professional development programs. … Yet recent evidence suggests that the most important opportunities for professional development are provided by teachers’ everyday experiences in school. … Such experiences shape teachers’ practices in profound and largely unexplored ways (Boaler, 2002a, p. 1). This study is therefore premised on the need to understand the pedagogical and contextual constraints that tend to limit teachers’ efforts to reform their practices. I argue that studies on reforming practices in mathematics education have tended to focus largely on teachers’ methods, strategies and actions, without much attention given to studying the constraints that teachers confront as they attempt to reform their practices. Understanding internal processes and the tensions that teachers experience as they change is critical to researchers as well as teachers. Educational reforms which began after South Africa’s first democratic elections in 1994 have brought with them new emphases and shifts in teachers’ methods of teaching and the curriculum in general. The new definition of mathematics, for instance, shifts away from an absolutist paradigm, and the new focus is on constructing mathematical meaning in order to understand the world, and making use of that understanding (Graven, 2002). The new emphasis on mathematical learning as relational, flexible, transferable and integrated with everyday life increases the demands on teachers for mathematical competence (Adler, Pournara & Graven, 2000). The demands of the new curriculum are putting tremendous pressure on the work of even the most committed teachers. Teachers are facing tensions, constraints and contradictions as they attempt to implement these reform

202

TEACHERS’ CONSTRAINTS IN CHANGING PRACTICES

initiatives. The present study indicates, however, that teachers in South Africa are not alone in their predicament. I now provide a brief description of this study. METHODS

For the present study, I analysed transcripts from interviews of nine teachers who participated in the LPS study. They came from three countries: South Africa, Australia and the United States of America. In all the teachers’ statements I observed the presence of constraints, tensions and struggles that confronted them as they attempted to reform their practices. That observation drew my interest to study these teachers closely and to formulate the questions I wanted to address. I sought an answer to the questions by doing a careful analysis of the statements that the teachers made during the interviews. I read all the transcripts and compared the statements made by all the teachers. The different categories of constraints emerged from the data in different rounds of analysis. As more data became available, I compared data to confirm and rework categories. I identified four categories from the teachers’ statements. Since I was interested in determining what constrained teachers in their classroom practice, I focused only on what they had experienced. Each teacher was interviewed two to three times in most cases, or a long interview was conducted on the last day of the data collection in each school. The data reflected the following themes about the teachers’ practices: Experience What have they experienced themselves? Meaning What do they mean by the key terms? (South African teachers, for instance, were using new phrases such as “Programme Organizer” that came with the introduction of Outcomes Based Education.) Understanding What do they understand? Intention What are their intentions? LIMITATIONS

There were a number of limitations that I was able to identify in this study. I briefly report on some of these: – This study sought to find constraints that mathematics teachers confront in their teaching practices based on the analysis of teacher interview transcripts. However, I was always aware that the questions asked during the interview were not designed to find out about constraints. I did my own analysis of the teachers’ statements and identified some as reporting constraints. I did, however, manage to arrange an opportunity to visit Teacher 2 in the USA who had earlier participated in the study, and conducted an in-depth interview with her in order to validate some of the information she gave in the interviews. After listening carefully to the statements made by this teacher, I realised that there was not much difference between what she told me in the interview and the information I had gathered from the interview transcripts.

203

HERBERT BHEKI KHUZWAYO

– Some of the statements teachers made could not provide all the information that I needed. For example, in a statement, “I wanted them to use calculators but I realized they did not have them” (USA, Teacher 1). It is not clear whether this was due to the fact that learners lack resources or they do have calculators and have left them at home. RESULTS

A preliminary analysis of the data suggests two categories of constraints: ‘macro’ constraints, which occur at a level outside the classroom, and about which teachers usually feel powerless to do anything. A teacher may, for example, feel powerless about changing exit examinations at the school. In contrast, ‘micro’ constraints are found inside the classroom and they are usually a hindrance to a teacher’s action. It is not always easy to distinguish between macro and micro constraints. For example, the teachers’ statements reflected that they could talk generally about behavioural problems, but these were also mentioned in so far as they affected their classroom practice. They pointed out that they spent a lot of time attending to behavioural problems, which resulted in their failure to cover all the work they had intended to cover within a given period. This example illustrates the difficulty I had in distinguishing between macro and micro constraints. The attempt to make this distinction was, however, useful in terms of obtaining some sense of the constraints that affected teachers directly inside the classroom, and those that they felt were ‘bigger’ and that they felt powerless to deal with. I need to point out also that there is no theoretical foundation for my use of this distinction. I carefully read all the teachers’ statements and was able to identify categories that emerged. Table 1 (see next page) shows an example of how I organised teachers’ statements.

204

Country South Africa

Teacher

Teacher 1

Many people have expressed the view that we need to fit in mathematics. How to integrate maths as compared to English, Zulu, or Afrikaans is a problem. Even Economics will fit in anyway but maths is a problem. Integrating across topics and across learning areas

What I find in grade 12: It’s the quality of students, they are lazy and they always want to be told what to do.

Learners’ commitment to work

I wanted them to understand the problem of AIDS but using maths. That became a problem. You may find you don’t go to the higher levels of mathematics. When you choose your Programme Organizer [that is, a theme] you may find that there is no maths you can do.

Syllabus

Choice of content

But what you find in the textbooks: They help you but they don’t give you everything. You need to be creative as an educator.

I have done quite a lot but I didn’t finish the syllabus. I didn’t, for instance, explore the patterns by looking at the rational numbers, irrational numbers etc.

Sample utterances I felt that I was prioritising maths because the questions I was asking were maths stuff except for the last two, which asked about how AIDS is transmitted and what you can do to prevent it … The problem is [that in the end] they were more knowledgeable on HIV than they could handle some of the maths questions.

Micro constraint Mathematics vs. Context (what takes priority)

Sample utterances

Textbooks

Macro constraint

Table 1: An example of macro and micro constraints

TEACHERS’ CONSTRAINTS IN CHANGING PRACTICES

205

206 X

X X

SAT2

X

X X

X

X

X X

Micro constraints

X X X

SAT1

Macro constraints

XX X

XX X

X

X

XX

X

X X XX X

X

X

AT1

X X

X X X

SAT3

X

X X

X X X

AT2

X

X

X

X

X X

AT3

X

XX

XXX

X

X

XX XXX

UST1

X X X X X Note: An X means one instance of the teacher mentioning the relevant constraints. The number of Xs indicates the number of times each constraint was mentioned by the teacher.

Maths vs. context Selection of content Incorrect assumption about learners’ conceptual knowledge Difficult concepts Conceptual knowledge gaps/Cognitive jumps Lack of imagination/Not seeing connections Assessment Sequencing Inability to reach out to all learners Covering enough content Language difficulties Groups

Syllabus Textbooks Exit exams Time Behaviour problems Lack of parental cooperation

Constraints

Table 2: Distribution of macro and micro constraints by teacher

X

X

XX XX XX

X

UST2

XX

XX X

X

USAT3

HERBERT BHEKI KHUZWAYO

TEACHERS’ CONSTRAINTS IN CHANGING PRACTICES

Table 2 (see the previous page) summarises all the constraints found, and shows that some constraints are common to teachers in all three countries while others are specific to particular countries. The teachers’ statements related mainly to the observed lessons, which differed from teacher to teacher, and also to the questions raised by those who interviewed them. While several teachers may have mentioned a particular constraint, their explanations of how it affected them differed. For example, two teachers pointed to the problem of textbooks, but they had different concerns about what was wrong with them: But what you find in [the] textbook helps, but it doesn’t give you everything. None of the textbooks so far will have work on a common “Phase Organizer” (that is, a theme). It’s so broad! (South Africa, T2). Textbooks here are products of publishing companies who whip something together to see a product. I think they don’t have a lot of practice [by the learner]. They don’t really spend enough time on that. (USA, T2) Key to symbols used in transcripts in this chapter [text ] Explanatory comments inserted by transcriber or author. .... Indicates that a portion of the transcript has been omitted.

The statement by the South African teacher indicates that his concern is that books lack important information on some aspects of the curriculum, whereas the American teacher is concerned about the lack of activities which allow the learners to practice. Sometimes overlapping of constraints could not be avoided. For example, some teachers mentioned assessment as a challenging and problematic area in the teaching of mathematics. This constraint, however, appears under the section on constraints involving selection of content. Below I discuss some of the micro constraints that I identified from teachers’ statements. I have omitted a detailed discussion of all of the macro constraints and some of the micro constraints. My choice of which to discuss in greater detail was guided by the depth of the teachers’ statements. In addition, since the macro constraints are not as easily changed, it is reasonable to concentrate the discussion on the micro constraints. MICRO CONSTRAINTS

Mathematics Versus Context Many of the reform-oriented curricula that are used in different countries are replete with contexts that are intended to bring some realism into mathematics classrooms (Boaler, 2002b). For example, when outcomes-based education (OBE) was introduced in South Africa, the proponents advocated a range of progressive educational practices in the curriculum. Mathematics teachers were, for instance, required to integrate knowledge, and this integration had to take place at several levels: integration of the various components of mathematics; between 207

HERBERT BHEKI KHUZWAYO

mathematics and everyday real world knowledge; and, where appropriate, across learning areas (Adler et al., 2000). The OBE curriculum calls for an increased emphasis on contextual problem solving and promotes “teaching for understanding” (Taylor, 2000). This requirement has placed new demands on teachers. Several problems have been highlighted by researchers (Adler et al., 2000; Sethole, 2004; Naidoo, 2005). The main concern centres on the fact that mathematics tends to ‘get lost’. One of the problems presented by real-world contexts is that they often require familiarity with the situation that is described, but such familiarity cannot be assured (Boaler, 2002b). What one finds also is that “suddenly the teacher is expected to possess a broad knowledge of matters unrelated to mathematics and possibly to be an expert in other subjects” (Adler, et al. 2000, p. 6). It would appear that from the three South African teachers’ statements that the above problem turned out to be a big concern for them. It is not surprising that all three teachers mentioned this constraint. At the heart of the problem seemed to be the issue of whether to prioritise mathematics or the context, which is evident in the following statement: I wanted them to get an understanding of the problem of AIDS, but using mathematics. That became a problem. For instance, you may find, you don’t go to the higher levels of mathematics . . . . You may find there is no maths that you can do. (South Africa, T1).

Another teacher also raised his concern: How to integrate maths as compared to English, Zulu or Afrikaans is a problem for me (South Africa, T2).

Even though the problem of integrating mathematics was a concern for one Australian teacher, hers was of a different nature: How they feel about mathematics I think sometimes they don’t think mathematics can be related to a physical reality. (Australia, T1).

This teacher would like to promote integration in her class, but learners did not see the connection. Another problem closely connected to the ‘mathematics versus context’ problem was the challenge of selecting the content to go with the context. Selection of Content Mathematics content that is taught in schools is an important critical factor impacting the future success of learners in mathematics. What content one should teach and at what level one should teach, was a concern for all the teachers. They were concerned about instructional sequencing of particular content. The problem of what appropriate content to fit in with the context appeared in statements of only the South African teachers. Why this was the case must be understood in light of curriculum reform process that was in place. This is evident in the following South African teacher’s statement: 208

TEACHERS’ CONSTRAINTS IN CHANGING PRACTICES Suppose the problem is ‘substance abuse’: What mathematics topics lend themselves to the idea of ‘substance abuse’? So you start with the problem idea and then look for the mathematics that can be built into that. So it presupposes knowing all your content quite well! How are you going to do it? (South Africa, T2).

It is clear from this statement that selecting the appropriate content was a major hurdle for this South African teacher. Assumptions about Learners’ Knowledge Some of the teachers’ statements indicated that sometimes they made incorrect assumptions about learners’ prior knowledge or about what learners were capable of knowing. At least one teacher in each of these countries mentioned this constraint. This concern can be understood in light of the fact that all the participating teachers were Grade 8 teachers. Grade 8 is the beginning of secondary school in South Africa. The teachers felt that learners were unable to build deep meaning to support some of the concepts that they had learnt in primary school. But even in Australia and the US, where students were not in general new to the school, teachers were sometimes taken by surprise when they realised that students did not know certain concepts. The students did not make connections between new concepts and what they had already learnt, which could allow them to extend their prior knowledge and transfer it to new situations (National Council of Teachers of Mathematics, 2000). One teacher in South Africa expressed his frustration with the problem by stating that: Maybe we need to get primary school teachers and ask them: Do you know you have sent us kids who don’t even know how to use a protractor? (South Africa, T3).

This problem was a concern for USA and Australian teachers as well, as can be seen in the following statements. We assumed that students will just pick it . . . . It was not so easy! To me you should kind of know your multiplication tables up through ten but they don’t! (USA, T3).

An Australian teacher shared a similar frustration: Sometimes you think it’s such an easy thing . . . . What perimeter is . . . . I did not expect this to happen . . . . There are cognitive jumps. (Australia, T1).

Mechanical Learning Teachers’ statements also indicated that sometimes learners have learnt rules without understanding the rationale behind them. Sometimes learners were struggling with ill-memorised rules. This problem was particularly evident in the interviews with all three US teachers. For example, the teacher complained that when learners come to them from primary schools: 209

HERBERT BHEKI KHUZWAYO They’ve been taught so many tricks, like cross multiplication and they don’t know how to think at all times (USA, T2).

These sentiments were echoed by another teacher who pointed out that: students pass a class on mimicry . . . . the ability to produce exactly what the teacher asks for, almost meaninglessly . . . . Certainly it happened to them in the past. (USA, T3)

The important thing about both these statements is that learners’ inability to see the connections has to do with their past and the fact that the emphasis was based largely on algorithms. Another Australian teacher who highlighted the same problem pointed out that whilst learners may know how to do cancelling in equivalent fractions: Half of them would not understand the value of cancelling equivalent fractions. (Australia, T2).

Difficult Concepts Teachers in all three countries identified some difficult concepts that the learners struggled with. Teachers in the USA were very specific about what these concepts were. Table 3 shows what teachers identified as difficult concepts faced by the learners. Table 3: Difficult Concepts Identified by the Teachers. Difficult Concepts

Fractions Definitions Area Division by Zero Perimeter Interpreting Graphs

SA1

SA2

SA3

A1

A2

A3

X

X

US1

US2

US3

X X X

X

X

X X X

As the table shows, the teachers in USA mentioned difficult concepts six times, the Australian teachers mentioned them three times and the South African teachers only once. The reason why difficult concepts were least identified by teachers in South Africa, could be that teachers, at the time of data collection, were concerned with the more macro issues of the new curriculum, such as the issue of integration. One important aspect of this study is that it begins to identify in a limited way what both teachers and learners in Grade 8 classes perceive to be difficult concepts. From school to school, however, both teachers and students will perceive these difficult concepts differently, so they can never be uniform. However, there exists some level of agreement on what teachers perceive to be the difficult concepts, particularly division by zero. 210

TEACHERS’ CONSTRAINTS IN CHANGING PRACTICES

DISCUSSION AND CONCLUSION

There were also some surprises for me with respect to the above category of constraints, namely the following: Difficult concepts: None of the teachers mentioned themselves as having difficulty with some concepts. The teachers always shifted this problem to the learners. It was also very rare to find instances where teachers questioned any of their teaching methodologies or strategies. However, the following statement made by one teacher indicates some serious thinking about her teaching: [The learners] feel that it’s the technique that’s important so they just think . . . . it’s whatever comes to their minds and it’s whatever rule. They are not confident in understanding whether their solution is correct or not. I don’t know if I’m having difficulty explaining this. (Australia, T2)

This teacher is aware of the fact that she could be the one having difficulty in explaining the concept. Similarity of constraints: It is striking that many of the needs of mathematics education in schools in a less affluent country such as South Africa are similar to more affluent communities like the USA and Australia. For example, all schools need appropriate instructional materials to address curriculum goals, and stable conditions that are conducive to teaching and learning (e.g. stable class sizes). Most teachers also make incorrect assumptions about what learners know or about what learners could master within a given time. Level of commitment: the level of commitment shown by these teachers to their learners was also outstanding. This becomes clear in such teachers’ utterances as: Well, I have to make sure that everyone is involved . . . . You’ve got the responsibility of others as well (Australia, T1) You know, you want to get to everyone. (USA, T1)

It would be interesting to know whether this strong commitment to ‘taking every learner along’ with the teacher best characterises classrooms of ‘best practice’. I believe there were some important lessons to be learnt in this study. Firstly, constraints (at both levels) occurring in any mathematics classrooms today have a bearing on teachers’ practices. Any talk of reforming practice in mathematics classrooms needs to recognise the reality of the existence of these constraints. ‘Best practice’ depends on the classroom contexts as well as on the extent to which teachers are aware of, and able to deal with, constraints in their practices. There are many instances in this study which indicate that teachers are sometimes compelled, as a means of dealing with these constraints, to resort to making decisions contrary to what they would have wanted to do at a given time. A teacher might, for instance, want to spend a little more time on dealing with a 211

HERBERT BHEKI KHUZWAYO

particular topic, but because of the threat of looming exit examinations the teacher must move on the next topic. One teacher pointed out, the dilemma she faced: And then you want to get through algebra and . . . . we have high school exit exams and that’s basically all the way through geometry (USA, T1).

Unfortunately teachers cannot look elsewhere to find solutions to deal with these constraints (some have been highlighted in this study). Secondly, new ideas about practice can work only if they are accompanied by some understanding of how to deal with constraints. The reform initiatives aimed at revitalising teacher education in South Africa, for instance, must come to grips with constraints teachers confront as they try to reform their practices – constraints such as the issue of integration (within mathematics, across learning areas and between mathematics and everyday real world mathematics). Table 1 indicates that, among the three countries, this problem maybe the greatest for the teachers in South Africa. There is therefore a need for some clarity on “what integration in the curriculum means, in principle, and what does it mean for the learning and teaching of mathematics in schools” (Adler et al., 2000, p. 2). Thirdly, it must always be remembered that because reform-oriented pedagogies require that teachers reconceive their roles in mathematical activity and student learning, the implementation of an innovative curriculum can pose significant challenges to even the most committed teachers (Lloyd, 1999, p. 246). I mentioned above that I was surprised by the high level of commitment shown by the teachers in this study. This commitment, however, is under constant threat because of the fact that these teachers have not been adequately prepared to deal with constraints in their practices. This calls into question the manner in which teachers’ struggles have always been seen: Teachers’ struggles have been attributed only to their misunderstanding or lack of appreciation for the philosophies underlying the proposed curriculum activities (Lloyd, 1999, p. 246). There is a need to go beyond this thinking and begin to challenge ways in which teachers have been prepared to deal effectively with constraints that stand in their way. It is also necessary to be constantly aware and prepared to deal effectively with discrepancies between reform recommendations and their implementation. Finally, an important lesson from this study is that constraints cannot be looked at in isolation from the contextual conditions in which they occur. As this study has shown, constraints can appear to be the same but can contain different meanings for different teachers. No general interpretation and explanation of the constraints can be found for different countries. However, they do exist and ways of dealing with them should be a concern for the mathematics education community internationally. I believe that classroom research needs continually to afford a voice to teachers in order to understand what constrains them in their efforts to implement reform in their classrooms effectively. There is a continual need to take teachers’ voices into account (as well as learners’ voices) to avoid the risk of over-simplification of the 212

TEACHERS’ CONSTRAINTS IN CHANGING PRACTICES

classroom situation. Instead, we need to acknowledge the multiple potential meanings of voice through accounts both from participants and from a variety of ‘readers’ of those situations (Clarke, 2002). Listening to teachers and knowing what they face in their classrooms can provide a modest basis for understanding how far pedagogy can be altered, given the existing organisational structure. ACKNOWLEDGEMENTS

My thanks go to Professor Jeremy Kilpatrick who was my host from January to July, 2005 at the University of Georgia, USA, for his assistance and for providing the supportive environment for the writing of this chapter. I also wish to thank the Council for International Exchange of Scholars (US State Department) for granting me a Fulbright Research Scholar Award. REFERENCES Adler, J., Pournara, C., & Graven, M. (2000). Integration within and across mathematics. Pythagoras, 52, 2-13. Boaler, J. (2002a). Advancing teacher development and mathematics learning through the integration of knowledge and practice. Retrieved May 10, 2005 from http://www.stanford.edu/~joboaler/NSFprop.doc Boaler, J. (2002b). Learning from teaching: Exploring the relationship between reform curriculum and equity. Journal for Research in Mathematics Education, 33, 239-258. Clarke, D. J. (2002). International perspectives on mathematics classrooms. In C. Malcolm & C. Lubisi (Eds.), Proceedings of the 10th Annual Conference of the Southern African Association for Research in Mathematics, Science and Technology Education (pp. 7-9). Durban: University of KwaZulu Natal. Cohen, D. K. (1990). A revolution in one classroom: The case of Mrs. Oublier. Educational Evaluation and Policy Analysis, 12, 327-345. Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. In P. Ernest (Ed.), Mathematics teaching: The state of the art (pp. 249-254). London: Falmer Press. Gates, P. (2002). Excavating and mapping the social landscape of beliefs. In P. Valero & O. Skovsmose (Eds.), Proceedings of the Third International Mathematics and Society Conference (pp. 317-325). Copenhagen: Centre for Research in Learning Mathematics. Graven, M. (2002). The effect of the South African curriculum change process on mathematics teacher roles. In P. Valero & O. Skovsmose (Eds.), Proceedings of the Third International Mathematics and Society Conference (pp. 336-344). Copenhagen: Centre for Research in Learning Mathematics. Lloyd, G. M. (1999). Two teachers’ conceptions of a reform-oriented curriculum: Implications for mathematics teacher development. Journal of Mathematics Teacher Education, 2, 227-259. Naidoo, A. (2005). Pre-service mathematics teacher education: Building a future on the legacy of apartheid’s colleges of education. In R. Vithal, J. Adler & C. Keitel (Eds.), Researching mathematics education in South Africa: Perspectives, practice and possibilities (pp. 183-205). Cape Town: HSRC Press. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Romberg, T. A. (1992). Mathematics: Problem features of the school mathematics curriculum. In P. W. Jackson (Ed.), Handbook of research on curriculum: A project for the American Educational Research Association (pp. 749-788). New York: Macmillan. Sethole, G. (2004). Meaningful contexts or dead mock reality: Which will the everyday take? Pythagoras, 59, 18-25.

213

HERBERT BHEKI KHUZWAYO Skovsmose, O. & Valero, P. (2002). Mathematics education in a world apart: Where we are all together. In P. Valero & O. Skovsmose (Eds.), Proceedings of the Third International Mathematics and Society Conference (pp. 8-16). Copenhagen: Centre for Research in Learning Mathematics. Taylor, N. (2000). Mathematics in Curriculum 2005: Submissions to the Ministerial Review Committee. Pythagoras, 51, 8-10.

Herbert Bheki Khuzwayo Faculty of Education University of Zululand South Africa

214

DAVID CLARKE

CHAPTER ELEVEN Deconstructing Dichotomies: Arguing for a More Inclusive Approach

INTRODUCTION

Education and mathematics education in particular, in attempting to make sense of the world of the classroom, has pursued the established Western tradition of dichotomising all aspects of our experience. This tendency was most explicit in the division of spirit and matter by Greek philosophers of the fifth century BC and became entrenched in the Cartesian division of body and mind. Much more recently, authors such as Fritjof Capra (1976) have drawn Western attention to Eastern philosophies in which the basic assumption is unity and interdependence, and yin and yang are seen as complementary aspects of an essential unity, rather than as in opposition. My contention in this chapter is that it is in the examination of classrooms across a variety of cultural settings and school systems that we find our educational assumptions most visible and open to challenge. With the growing internationalisation of education, and as the education community gives higher priority to international research, it is timely to examine the insights that accrue from comparative analyses of classrooms that are situated in very different cultures. The contrasts and unexpected similarities offered by research in such culturally-diverse settings reveal and challenge existing assumptions and theories and make essential a reconstruction of some of our most basic dichotomies as complementary elements in more inclusive theories. This questioning of the permanency of pervasive binary opposites is central to the ‘deconstructive’ stance adopted in this chapter. The inclination to integrate rather than segregate is also at the heart of the Learner’s Perspective Study (LPS), since it was intended from the project’s inception that any documented differences in classroom practice be interpreted as local solutions to classroom situations and, as such, be viewed as complementary rather than necessarily oppositional alternatives, within a broadly international pedagogy, from which teachers in different countries might choose to draw in light of local contingencies. I am not challenging the need for categorisation, but there appears to be an inclination within the education community to dichotomise and an associated tendency to (i) ignore the connectedness of the dichotomous categories and (ii) on occasion, to privilege one category while denigrating the other. This

D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 215–236. © 2006 Sense Publishers. All rights reserved.

DAVID CLARKE

places the emphasis on the separate and oppositional application of the dichotomies, rather than on consideration of their complementarity and interrelatedness. International research in mathematics education has provided us with a wealth of detail about student achievement levels, curricular content, prevalent problem types, teacher beliefs, class size, lesson duration, homework, textbooks, teacher question types, utilisation of real-world contexts, and, more recently, fine-grained analyses of classroom practices and interactions. The descriptive documentation of similarity and difference (Clarke, 2003a, 2003b) can only take us so far. The diversity that we find in international studies of mathematics classrooms provides us with a base from which to interrogate our own practices and the assumptions on which those practices are predicated. Among the most central of these assumptions are various dichotomous categories that act to constrain our theorising about educational settings and the processes of interest there. This chapter addresses five of these dichotomies: Teaching and Learning; Abstract and Contextualised mathematical activity; Teacher-Centred and Student-Centred classrooms; Listening and Speaking as alternative student actions; and the teacher’s contemporary dilemma – To Tell or Not to Tell. Very simply, these are false dichotomies. It is my contention that unless we can integrate each pair of categories as interconnected elements of a more inclusive theoretical framework, we will remain unable to account for the diversity we find in international studies of classroom practice. It is precisely the growing body of data from such international studies that provides us with the diversity that we need to interrogate and refine our current theoretical position with regard to classroom practice. In the discussion that follows, I attempt to demonstrate the consequences of such an inclusive approach. There are other dichotomies that I will not address here. It has already been argued persuasively by Cobb, Svard and others, that if we are to move forward, we must conceive of socio-cultural and constructivist theories of learning not as competing but as complementary. That they can be constructed so as to be in competition is evident. Each theoretical frame provides coherent accounts and explanations for particular forms of learning in particular settings. Any conception of either theory that precludes the other is arguably inadequate. The identification (construction) of a theory of learning compatible with a given situation may take the social or the individual as its starting point but ultimately will be obliged to make appeal to the other if a coherent account is to be constructed. As Jere Confrey has succinctly put it: “The self is both autonomous and communal” (Confrey, 1995, p. 36). In this chapter, I have chosen to focus on five specific dichotomies that I consider central to our theorising and to our advocacy in relation to mathematics classrooms.

216

DECONSTRUCTING DICHOTOMIES

THE REIFICATION OF FUNDAMENTAL DICHOTOMIES: TEACHING AND LEARNING

Learning and teaching represent the most fundamental and pervasive dichotomy around which our understandings of classroom practice have been constructed. Stepping outside the constraints of culture and language, we find that this central distinction is conceived very differently by different communities. In fact, the distinction between teaching and learning is very much an artefact of language. Previous research and much of our theorizing, has tended to dichotomise teaching and learning as discrete activities sharing a common context. I have argued elsewhere (Clarke, 2001) that this dichotomization is a particularly insidious consequence of the constraints that language (and the English language, in particular) imposes on our theorizing and that such dichotomisation misrepresents both teaching and learning and the classroom settings in which these most frequently occur. It is not my intention to challenge the separate integrity of ‘teacher’ and ‘learner’ as labels for individuals engaged in particular practices or discourse modes. It is just that classrooms are understood more effectively as sites for bodies of mutually-sustaining practice that in combination characterise a process we might call (in English) ‘teaching/learning’. The consequences of choosing not to dichotomise teaching and learning are farreaching. Perhaps the most compelling illustration of the dangers of dichotomisation can be seen in the comparison of two translations of the same paragraph by Vygotsky. From this point of view, instruction cannot be identified as development, but properly organised instruction will result in the child's intellectual development, will bring into being an entire series of such developmental processes, which were not at all possible without instruction. (Vygotsky, 1982, p. 121, as quoted in Hedegaard, 1990, p. 350) Compare this with the following translation. From this point of view, learning is not development; however, properly organized learning results in mental development and sets in motion a variety of developmental processes that would be impossible apart from learning. (Vygotsky, 1978, p. 90) The pivotal assertion that must be understood is whether Vygotsky was asserting the impossibility of certain forms of intellectual development ‘without instruction’ (which presumes an actively interactive more competent other) or ‘apart from learning’ (which on one level seems a tautology, but which could also be interpreted as equivalent to the assertion that properly organised interaction with the environment is essential for certain forms of development to occur). This distinction is non-trivial, since it calls into question the significance of the mediation of another more able individual (the teacher/instructor). Given what we know of the significance Vygotsky attached to the role of the teacher, it would appear that the most appropriate reading of the major premise is “a variety of

217

DAVID CLARKE

developmental processes would be impossible without instruction”. This accords with the significance attached, in the passage quoted below, to the child’s interaction with ‘people in his environment’ rather than just with all aspects of that environment, with or without the mediation of others. The conflicting translations arise because of a duality of meaning in the original term employed by Vygotsky. This duality has been noted previously, but its significance seems to have been given scant consideration in the interpretation and application of Vygotsky’s work. As we have seen in the two translations above, the same term (obuchenie) is translated both as ‘instruction’ and as ‘learning’ and clearly shares with corresponding terms in other languages the capacity to invoke both teaching and learning, as these are named in English. Once this duality of meaning is recognised, our reading of Vygotsky and our theorising about the teaching/learning process are greatly enriched. For example, in one of the most famous passages from the translated Vygotsky, the word ‘learning’ can be replaced by the word ‘teaching’ and the resultant text is still meaningful – but, perhaps, with a different meaning. We propose that an essential feature of learning [teaching] is that it creates the zone of proximal development; that is, learning [teaching] awakens a variety of developmental processes that are able to interact only when the child is interacting with people in his environment and in collaboration with his peers. (Vygotsky, 1978, p. 90) If our framing of ‘instruction’ in language presumes a complicit ‘learner,’ whose ‘learning’ is inextricably entwined with an ‘instructive’ setting, then our interpretations of the activities of the classroom are more likely to identify communal practices and incremental participation in a common discourse as essential features than to fragment the classroom into teaching and learning activities undertaken separately by different individuals. Speakers of Russian are not alone in their use of a term that combines both teaching and learning. In Japanese, tagushushido combines teaching and learning in the same way. In Dutch, there is one term that means both learning and teaching: leren. To distinguish between the practices of teaching and learning, the Dutch say leren van to signify learning and leren aan to signify teaching. In French, the term didactique and particularly Brousseau’s use of that term (Brousseau, 1986), invokes a mutuality of responsibility and participation not always found in American or Australian accounts of classroom practice. In the middle of the last century, the biologist von Uexhull put forward the proposition that a spider’s web is the spider’s model of the fly. This whimsical imagery conceals a powerful reasoning technique. From the structure of a spider’s web: the spacing and strength of the strands, the location and size of the web, and from other characteristics of the spider’s web, we can deduce much about the fly. Classrooms are a little like the spider’s web. From the way in which a teacher structures the classroom, and the practices for which it is the setting, we can infer much about that teacher’s (and that society’s) model of the student. The types of resources provided, the type and duration of the various activities, the forms of 218

DECONSTRUCTING DICHOTOMIES

interaction that are encouraged and discouraged, all offer insight into the teacher’s conception of the epistemic student, the student as constructor of knowledge. Within the confines of accepted practice and available resources, teachers attempt to construct classrooms to afford and constrain particular activities. What Brousseau (1986) has brought to our attention is the reciprocality of the construction of classroom practice. Learners (that is, students) engage in practices that afford and constrain teacher actions and the actions of their classmates. Social interaction by an individual within the classroom presumes that the individual has a model of the other classroom participants and can, to some extent, anticipate their capabilities, their needs, their expectations and their responses. What is clear is the extent to which classroom practice is a jointly constituted body of negotiative social interactions that is best investigated and understood in terms of the mutuality and reciprocality of its constituent activities and of its co-construction as Teaching/Learning. Empirically, the integration of Teaching and Learning has been addressed in analyses of patterns of participation in mathematics classrooms in a variety of countries as part of the Learner’s Perspective Study (LPS). In particular, the classroom practice referred to as ‘Kikan-Shido’ (or ‘Between-Desks-Instruction’) has provided a powerful example of a whole class pattern of participation (Clarke, 2004; Chapter 4, this volume). In making the claim that Kikan-Shido could be so described, it was necessary to demonstrate that it had a recurrent form, recognisable to those participating in it. This is not to say that the meanings attributed to the activity by those participating in it were correspondent. Individuals can participate in a practice whilst being positioned differently within it, and whilst attributing different characteristics to the activity. That is, without being identical, the participants’ descriptions of the activity make it clear that they are talking about essentially the same form, but they may attribute quite different functions to that form. The other essential element in this argument is the need to demonstrate that all participants can shape the particular body of practice signified by Kikan-Shido; that is, that the pattern of participation is co-constructed. Without reproducing the argument in full here, any theory of classroom practice must conceive of the activities in the classroom as co-constructed. Kikan-shido as it has been reported (Clarke, 2004; Chapter 4, this volume) is clearly a dance done by teachers and students, where the steps are improvised according to need. The participants in the classroom, teacher and students, are complicit (co-conspirators) in this improvisation. Acceptance of this point has implications for the research designs by which we study the activities occurring in classroom settings. DICHOTOMIES OF TASK: CONTEXT AND THE ‘RELEVANCE PARADOX’

Suppose that one society seeks to develop understanding and proficiency in mathematical proof, attaching significance to the development of those forms of reasoning and argumentation idiosyncratic to mathematics, while another attaches greater priority to equipping its people with an understanding of mathematical procedures and proficiency in utilising these in everyday practical situations, while 219

DAVID CLARKE

a third society emphasises (and rewards) concept development, mathematical creativity and collaborative problem solving. There is no reason why these goals are incompatible or mutually exclusive, but they do reflect a valuing of different aspects of mathematical activity, and a curriculum that prioritised one such goal would not necessarily resemble a curriculum that prioritised another. The evaluative comparison of the consequences of such differently targeted curricula (as in international studies of student achievement) is a problematic exercise, whereas the comparative study of the methods and success of each society in addressing its local curricular goals has the potential to be mutually enriching as one community learns from the practices of the other and adopts and adapts some of its goals and methods for local use. Many countries, especially Korea and the Netherlands, emphasised solving problems . . . Japan, Sweden, and the United States emphasized ‘recalling’ mathematical information, and Hong Kong and Israel emphasized ‘justification and proof’. (Schmidt, McKnight, Valverde, Houang & Wiley, 1997, p. 136) In her analysis of LPS data from Sweden and China, Svan examined the ‘Relevance Paradox’ postulated by Niss (1994), in which the objective relevance of mathematics in society was contrasted with its subjective irrelevance as perceived by many students. Svan was not comparing ‘mathematics teaching’ in Sweden and China, but rather looking at the beliefs and values communicated and held in two very different classrooms: one in Shanghai and one in Uppsala. Both classrooms were addressing the same mathematics topic (coordinate systems and graphing linear functions). Svan’s analysis contrasted the Chinese and Swedish mathematics classrooms from the perspective of the emphasis given by the teacher and the students to the real-world relevance of the mathematics being learned. In the Swedish classroom, the students demanded that the teacher justify the relevance of what was being taught, and the teacher provided lengthy justifications on several occasions. It was clear that the Swedish teacher felt that the demonstration of relevance was a reasonable expectation and accepted responsibility for providing this. Despite the teacher’s efforts, students were outspoken in their lack of belief in the relevance of the mathematics they were studying. Both the Swedish classroom data and postlesson interview data seemed to provide a powerful illustration of Niss’s relevance paradox. By contrast, in the classroom in Shanghai, mathematics tasks tended to be very abstract in character and the teacher made no effort to demonstrate or argue for the real world applicability of the mathematics being studied. The Chinese students did not appear, either during the lesson or in interview, to require this sort of justification of the content being studied. However, in the post-lesson interviews, the Chinese students consistently expressed strong beliefs in the utility of mathematics in general and in relation to the specific mathematics they were studying. One Chinese student said:

220

DECONSTRUCTING DICHOTOMIES

I think basically, I should grasp the fundamental points that are necessary for students and also I have to use these points in my everyday life. (Shanghai School 1, Lesson 4, post-lesson student interview) Svan concluded that analysis of the interviews with 15 of the Chinese students showed that there was a shared belief that mathematics was useful not only in future work and study, but also in their current everyday lives. It is not yet clear how the students developed those beliefs as they were not introduced to anything but abstract mathematics during the lessons. Svan has christened this the ‘Expanded Relevance Paradox’ (Svan & Clarke, to appear in a future volume in this series) and means, by this term, to refer to the paradoxical character of application-oriented mathematics teaching associated with subjective irrelevance and pure mathematics-oriented mathematics teaching associated with subjective relevance. To recapitulate: The majority of the tasks in the Swedish classroom were ‘word problems’ and involved contexts from everyday life, more or less relevant to the students. Despite the teacher’s very public commitment to demonstrating the relevance of the content, the students strongly questioned its utility. The students in the Shanghai classroom experienced teaching and tasks that focused on abstract mathematics, yet the students appeared quite certain of the immediate and future relevance of the content. Clarke and Helme (1998) identified the importance of recognising context as a social construction, and distinguished the ‘Figurative Context’ invoked by the task from the ‘Social Context’ in which the task was undertaken. As reported by Clarke and Helme, students appear to attend to the figurative context to different degrees. Context in our view is neither a neutral background for the negotiation of mathematical meanings, nor merely a catalyst mediating between task content and the individual’s mathematical tool kit. Rather we should speak of the personal task context as an outcome of the realization of the figurative context within the broader social context. (Clarke & Helme, 1998, p. 130) There is a recent commitment in South Africa to contextualising the curriculum around themes of societal significance, such as substance abuse or HIV-AIDS. Analysis of student-student interactions in the South African classrooms studied in the LPS project, led Sethole, Adler and Vithal (2002) to conclude: The context AIDS, is not understood as a ‘veneer’ to mask the mathematical intentions of the lesson but a genuine context to be engaged. To this end, and drawing from Skovsmose’s notion on critical mathematics, the new practice may be seen as an inescapable consequence of blurring the boundary between the mathematics and the everyday. (Sethole, Adler & Vithal, 2002, p. 11) The Relevance Paradox proposed by Niss (1994) is based on a dichotomisation of the function of mathematics in society and in the classroom, and postulates a dislocation between these two contexts that is experienced by students as a lack of connection (subjective irrelevance). LPS data problematises this schism in two

221

DAVID CLARKE

startling ways: firstly, Chinese students appear to have constructed the missing connection independent of explicit classroom modelling or advocacy by the teacher; and, secondly, the South African initiative removes the need for connection by dissolving the distinction between the classroom and the everyday. In the terms employed by Clarke and Helme, the distinction between the figurative and the social, always tentative, has been effectively dissolved in China, through a perspective in which the significance of classroom activity derives from its situation in a broader cultural context that does not require re-fabrication at the local level of the classroom, and also in South Africa, where the minutiae of mathematical content are subordinated to a macro-social agenda that reconstructs the nature and purpose of classroom activity in socio-cultural rather than solely mathematical terms. DECONSTRUCTING THE TEACHER-CENTRED/STUDENT-CENTRED DICHOTOMY

Popular in recent educational literature as descriptors of classroom practice are the terms ‘teacher-centred’ and ‘student-centred.’ These terms vary in definition and in use, but they represent a key dichotomy driving much of contemporary Western educational (particularly pedagogical) reform. From one perspective, they appear to offer mutually exclusive alternatives with regard to the location of agency in the classroom. Western educational reform advocates student-centred classrooms, and research in Western settings confirms the value of practices associated with these classrooms (Chazan & Ball, 1999; Clarke, 2001). For example, Clarke (2001) provided examples of student-student interactions that demonstrated the potentially significant role that students might play in the collaborative generation of knowledge in the mathematics classroom. A feature of Karen’s role in the Lauren/Karen dyad was to pose questions of Lauren and of herself. Some evidence can be found to suggest that Lauren was the more mathematically capable student. Nonetheless, the successful culmination of the dyad’s problem solving efforts must be attributed, in part, to Karen’s persistent framing of task-related questions. The effectiveness of such self-scaffolding as a component of dyadic problem solving will derive significantly from the appropriateness of such questions and the extent to which one learner attends to the questions (and other contributions) of the other. (Clarke, 2001, p. 310) In a parallel analysis of student cognitive engagement, Helme and Clarke presented evidence for the significance of student-student interactions in promoting highlevel cognitive engagement and consequent learning. We would argue that student-student interactions appeared to offer more scope for high-level cognitive engagement [by students] than teacher-student interactions, both in whole-class instruction and in interactions with small groups. (Helme and Clarke, 2001, p. 191)

222

DECONSTRUCTING DICHOTOMIES

On the basis of this evidence, student agency for knowledge generation was accorded a high level of significance in the Australian classrooms analysed in this study (Clarke, 2001) and the results of this study could be interpreted as providing further support for the advocacy of the ‘student-centred’ classroom, a key element in the recent reform agenda of most Western educational systems. By contrast, Asian classrooms have been typified as teacher-centred by both Western and Asian researchers, yet the students in these classrooms are highly successful in international studies of student achievement (‘The Asian Learner Paradox’) (Leung, 2001). Recent research in Chinese classrooms suggests that classroom practice is misrepresented by such a dichotomy (Huang, 2002) and that a theoretical framework is needed by which the ‘teacher-centred’ and ‘studentcentred’ characteristics of classrooms can be more usefully characterised and investigated, without the assumption of an absolute dichotomy. How can teacher dominance and student-centeredness coexist and work well in Chinese mathematics classrooms? (Huang, 2002, p. 226) There is a general assumption in most of the educational literature that classroom discourse encompasses any form of interaction that takes place in a classroom. Nevertheless, research involving classroom interactions has tended to focus on either the teacher’s talk (Wilson, 1999; Young and Nguyen, 2002) or teacherstudents’ interactions in either whole class (e.g., Klaassen and Lijnse, 1996; Seah, 2004) or group discussion (e.g., Knuth and Peressini, 2001). There have been very few studies, if any, that took into account the role of student-student private interactions in generating knowledge in the classroom. Clarke and Seah (2005) adopted a more integrated and comprehensive approach, by analysing, within a subset of the LPS data, both public interactions in the form of whole class discussion and interpersonal interactions that took place between teacher and student and between student and student during Kikan-Shido (between-desksinstruction). Interpersonal student-student interactions available for analysis in any one lesson were restricted to a focus group of up to four students. While this approach did not allow all interactions that took place in the classroom to be studied, it provided an avenue to track the generation of knowledge that could occur in both the public and interpersonal domains. Analysis was carried out on a selection of video and post-lesson interview data related to mathematics lessons in Hong Kong, Melbourne, Shanghai and San Diego. All teacher classroom utterances and all statements by focus students, together with post-lesson interviews with teacher and students were transcribed and translated into English. The classroom transcript of each lesson was scanned for terms or phrases that expressed, represented, illustrated or explained mathematical concepts or understandings. These terms or phrases were referred to as ‘mathematics-related terms’. These might take the form of conventional mathematical terms such as ‘gradient’ or everyday expressions such as ‘slope’ or ‘steepness’.

223

Transient Terms

y-axis

Ordinate

x-axis

Abscissa

Area

Coordinate

Current Activity

Mathematical Idea/Term

Primary Terms

224

Secondary Terms

1 - 2 mins 2 - 3 mins

T (09:15)

Simon (48:17) T (50:01)

T (32:05)

Anthea (30:14) T (32:05)

T (06:26) Eve (07:15) T (09:15) T (50:01)

T (50:01): … Eve (51:04): Coordinate axis. rectangular coordinate plane.

Sam (43:17) T (52:09) Eve (52:26)

Sam (43:17) Eve (52:26) T (56:03)

T (17:15)

Anthea (18:22)P: rectangular plane

T (07:13) T (03:19)

(0:00 to 2:57) T reviewed the things learnt in the previous lesson with the class; drawing x- and y-axes (coordinate axes), locating the coordinates of a point and features of 2 pts having the same abscissa.

0 - 1 mins 4 - 5 mins 5 - 6 mins

T (24:13)

T (34:11)

T (03:19) T (34:11)

T (49:29)

T (08:15)

T (08:15)

Eve (30:12)P location

Eve (30:12)P

T (27:19)

Anthea (29:15)P

T (27:19)

T (27:19)

(2:57 to 8:19) T discussed the method of finding the coordinates of a point and marked 5 points on the blackboard: (1) find the quadrant where the point lies; (2) draw a perpendicular to the x-axis and a perpendicular to the y-axis; (3) locate the coordinates of the point.

3 - 4 mins

Table 1. The distribution of responsibility for knowledge generation

DAVID CLARKE

DECONSTRUCTING DICHOTOMIES

These mathematics-related terms were classified into three categories: Those ‘primary terms’ that corresponded to the teacher’s stated instructional goals (in lesson plan or interview), Those ‘secondary terms’ that were subordinate to or supportive of the teacher’s main instructional goals (usually previously-introduced or familiar terms which served to explicate the meaning of the terms central to the lesson’s intended focus), Those terms that appeared infrequently and fleetingly in the course of classroom discussion (in either public or interpersonal statements). These were referred to as ‘transient terms.’ The occurrence of each term was then displayed in a tabular form analogous to the resource utilisation planning charts of engineers (Table 1) (see also Barnes’ ‘flow of ideas’ (Barnes, 2004), which derives from the same source). If these mathematics-related terms are thought of as resources drawn upon during the collaborative process of classroom knowledge construction, then the analogy is not inappropriate. Table 1 has been significantly abridged for reasons of space: Only the first 6 minutes of the lesson are displayed and only a subset of the lesson’s mathematicsrelated terms are included. The terms are separated within the table by bold lines into the three categories and a brief description is provided of the classroom activity coincident with the occurrence of the various terms. Each vertical column corresponds to one minute and the occurrence of each term is designated by speaker (T = teacher; Andrea, etc = student), by time-code (eg 06:13, seconds and frames, within the designated minute) and by ‘P’ if the utterance was an ‘interpersonal’ rather than a ‘public’ utterance. The capacity of this analytical approach to distinguish between classrooms is most evident in a comparison of eighth-grade mathematics classrooms in Shanghai and Hong Kong, since both sets of classrooms could be described as being embedded in a Confucian-heritage culture. The style of teaching in the Shanghai schools analysed was such that the teachers generally provided the scaffold needed for students to reach the solution to the mathematical problems without ‘telling’ them everything. Hence, one could find quite a few mathematics-related terms, which the teacher had not taught, that were introduced by the students during public discussion. A particularly powerful example of this devolution of responsibility occurred when the teacher in SH2-L04 (Shanghai School 2, Lesson 4) drew the class’s attention to an alternative method of solving simultaneous equations being used by a student which the teacher described as more ‘elegant’ than the standard (textbook) method. Students in the Hong Kong classes studied were generally not given the same opportunities to contribute during lessons, in comparison with classes in the other three cities studied in this analysis (Shanghai, Melbourne and San Diego). The teachers generally stated very explicitly every step for solving the mathematical problems discussed. In other words, students were guided through the steps for each problem type with very little opportunity for original thought or input into class discussion. Where a new mathematics-related term was introduced into whole 225

DAVID CLARKE

class public discussion, this was either done by the teacher or by a student in response to very explicit prompting from the teacher. There were, however, mathematics-related terms that occurred for the first time in interpersonal conversation between students, but were not subsequently voiced in the public arena. As examples of ‘Asian’ classroom practice, in several respects the Hong Kong and Shanghai lessons analysed displayed more extreme differences in practice than those evident from comparison of ‘Asian’ and ‘Western’ lessons. Within the sets of lessons analysed for each city, significant variation was evident from the perspective of the distribution of responsibility for knowledge generation. The practices of the classroom in Shanghai School 2 provided some powerful supporting evidence for the contention by Huang (2002) and Mok and Ko (2000) that the characterisation of Confucian-heritage mathematics classrooms as teachercentred conceals important pedagogical characteristics related to the agency accorded to students; albeit an agency orchestrated and mediated by the teacher. A unique teaching strategy consisting of both teacher’s control and students’ engagement in the learning process emerges in Chinese classrooms. (Huang, 2002, p. 227) Once the distribution of responsibility for knowledge generation is adopted as the analytical framework, the oppositional dichotomisation of teacher-centred and student-centred classrooms can be reconceived. The deconstruction of the teachercentred/student-centred dichotomy has specific consequences for teacher practice. In particular, one of the most contentious entailments of this dichotomy can be revisited; the legitimacy of teacher ‘telling’. TO TELL OR NOT TO TELL: DICHOTOMIES OF TEACHER PRACTICE

One common interpretation of the constructivist manifesto (i.e., that “knowledge is the result of a learner’s activity rather than of the passive reception of information or instruction,” von Glasersfeld, 1991, p. xiv) has been that it became no longer legitimate for teachers to ‘tell’ students anything. This position is not a logical consequence of adherence to constructivist learning theory, which suggests that students inevitably construct their own mathematics, whatever the classroom situation (Cobb, 1995). However, Telling or Not-Telling have been constructed oppositionally with such success that publications on contemporary pedagogy (such as Wood, Nelson & Warfield, 2001), while usefully discussing many pedagogical strategies, see no need to address any strategies that might be construed as analogous to ‘telling’ and even articles that purport to address the issue (such as Chazan and Ball, 1999) offer teachers little insight into how (and, as importantly, when) their mathematical knowledge might be articulated explicitly to the benefit of their students. Definitions of ‘telling’ have been based on the form (i.e., whether or not the teacher is making a declarative statement or other type of assertion) rather than on the function of the teacher’s action. 226

Figure 1. Kikan-Shido: Monitoring and Guiding – fifteen classrooms across five countries

DECONSTRUCTING DICHOTOMIES

227

DAVID CLARKE

A teacher’s communicative act must be addressed from the related perspectives of the teacher’s intention, the nature of the act, and the interpretations of the act by the recipients or audience. By focusing on function (intention, action and interpretation) rather than form, we overcome some of the difficulties experienced in analysing the efficacy of teacher practices from a constructivist perspective. Constructivist learning theory has been extrapolated to the domain of teaching practice, and ‘constructivist teaching’ has been set up in opposition to ‘transmissive teaching’ (Richardson, 2001, for example). Criticism of transmissive teaching has an extensive history and has sometimes led to simplistic exhortations to avoid ‘telling’ without serious discussion of those teaching actions that involve introducing new ideas directly. Clarke and Lobato (2002) (and subsequently Lobato, Clarke & Ellis, 2005) have proposed a theoretical reformulation of teachers’ communicative acts in terms of function rather than form. This reformulation is founded on the distinction between ‘eliciting’ and ‘initiating’. Such a framework offers a more incisive tool for the analysis of the teacher’s contribution to classroom discourse. It is entirely analogous to the empirically-grounded distinction between ‘Monitoring’ and ‘Guiding’ reported by O’Keefe, Xu and Clarke (Chapter 4, this volume). Figure 1 displays the relative proportions of monitoring and guiding activities employed by 15 teachers from five countries during the lesson event, Kikan-Shido (BetweenDesks-Instruction). It is clear from Figure 1 that some teachers have constructed a much more interventionist instructional practice than others. Superficial similarities in teacher orchestration of classroom practice among the LPS teachers in Hong Kong and Shanghai conceal profound differences in pedagogy associated with more or less interventional instructional approaches. The distinction between eliciting and initiating teacher actions offers a language in which to frame the devolution of the responsibility for knowledge generation from the teacher to the student, or, alternatively, the concentration of that responsibility in the teacher. For example, teacher acts that take the form of a question but have the function of telling can be identified and the responsibility for the initiation of a new mathematical idea can be correctly located with the teacher rather than the responding student. Equally, as has been argued above, the capacity of the student to contribute to the generation of knowledge can be recognised, and classrooms can be compared according to the extent to which the student is accorded the opportunity to make this contribution. The fundamental consideration is the distribution of responsibility for knowledge generation. Clarke and Lobato (2002) asserted the importance of interweaving the two functions initiating and eliciting. Since it is the development of the students’ mathematics that we aspire to promote, it is the students’ mathematics that takes priority. However, the teacher’s mathematics can also find legitimate voice in the classroom in the interest of stimulating the development of the student’s mathematics. Initiating/eliciting is not a simplistic dichotomy like ‘tell/not tell’ – it is not an either/or. Both categories of action are necessary and their use is interrelated. Eliciting has typically been defined in terms of the form of the communicative act (e.g., asking questions such as “Could you explain your 228

DECONSTRUCTING DICHOTOMIES

reasoning?”) or in terms of the degree of student involvement (e.g., the use of open-ended mathematical activities). Elicitation occurs when the teacher wants to learn more about students’ images, ideas, strategies, conjectures, conceptions and ways of viewing mathematical situations. When the teacher’s communicative act functions to facilitate the expression of the student’s mathematics, then this constitutes ‘eliciting’. In order to provide experiences that might challenge students to reorganize their thinking, teachers need to develop models of their students’ mathematical realities (Simon, 1995; Steffe & Thompson, 2000). The adequacy of these models will depend on the teacher’s ability to elicit the students’ mathematics. Initiating is most profitably used in conjunction with eliciting. Initiating is often preceded by eliciting, so that the teacher can gather information about students’ thinking before making a judgment whether to work with and structure the students’ ideas or to introduce new information. Initiating involves the insertion of new ideas into the conversation, ideas that the teacher assumes will be interpreted in many different ways rather than passively received. Once the teacher engages in initiation, she then steps back and elicits to see what the students did with that information. Both actions have their function within the teacher’s promotion of student conceptual development. The mutuality and complicit nature of these interactions bring us back to the spider’s web, the epistemic student, and the coconstructed nature of teaching/learning. The agenda that frames such classroom activity is initially the teacher’s agenda, but this agenda is iteratively modified in response to the progress of the ensuing classroom discussion in order to accommodate the students’ prior and emerging understandings (see Lobato, Clarke & Ellis, 2005, for specific examples). The complicit character of the teacher’s and students’ actions can be seen in the discussion of Kikan-Shido in Chapter 4 of this volume. Where do we see the purposeful alternation of elicitation and initiation most clearly? One example can be found in the classroom in Shanghai, already referred to above. Unlike an Australian classroom, the students in this classroom rarely ever talked directly to each other – classroom conversation was always mediated by the teacher – yet the students were clearly learning most effectively. Part of the explanation came in the interview after the lesson. The teacher said. “Don’t teach them mechanically, don’t teach them mechanically, let them brainstorm, enhance their flexibility.” He added: I was not afraid that students had all sorts of questions. I just let them appear. . . . Sometimes if you restrict them from doing this or that, their problems won’t appear, right? But the problems will appear tomorrow, even if they didn’t today, right? This is an articulate summary of the heart of the contemporary reform agenda in Western education and demonstrates a commitment to the purposeful elicitation of the students’ mathematics. But, for cultural reasons, the opportunities for student discussion of the content were provided in a teacher-led whole class approach. With regard to the value attached to the students’ mathematics, once elicited, in the 229

DAVID CLARKE

lesson referred to earlier, this same teacher said to the class, “Look at Shiqi’s solution! This is much better than the usual method. Everyone copy this down.” As was evident in the analysis of the distribution of responsibility for knowledge generation in this classroom, the responsibility was shared between teacher and students and, in so far as the teacher’s intentions could be put into effect, the classroom discourse was a purposeful alternation of initiation and elicitation. It is in this manner that the utilisation of the distribution of responsibility for knowledge generation provides an explanatory framework that problematises teacher-centred and student-centred characterisations of the classroom and resolves the false opposition of dichotomous practices by replacing them with a conception of alternative interrelated (and fundamentally complementary) classroom practices. TO LISTEN OR TO SPEAK: DICHOTOMIES OF STUDENT PRACTICE

It is worth appending one final dichotomy that revisits the same classroom situations from the perspective of the student. This is the decision by the student to listen or to speak. It has already been noted that in several of the LPS classrooms (notably in Shanghai and Tokyo) students seldom spoke directly to each other. By contrast, students in the mathematics classroom in Melbourne, San Diego, Berlin and Uppsala frequently spoke directly to each other without teacher mediation. Students in these classrooms, and in Hong Kong, would also make self-initiated contributions to public discussion. In this diversity, we can see that the student decision to speak was variously enacted and variously constrained in the different classrooms. Student listening is more difficult to identify from the video record, although inferences of student attentiveness could be made wherever one student made explicit reference to a previous statement by the teacher or by a classmate. The value and significance accorded to the act of listening was a feature of many interviews with Chinese and Japanese students. Students in Berlin, Melbourne, San Diego and Uppsala were much less likely to stress the importance of attentive listening to the teacher (or to their classmates). The interplay of speaking and listening by both students and teachers can be examined from the dual perspectives of Revoicing (Ohtani, 2003) and Selective Attention (Mason, 2003). A more detailed analysis of the dynamic between student speaking and listening will be undertaken employing both revoicing and selective attention as analytical frames and reported in a subsequent LPS publication. For the purposes of this chapter, research into one particular educational setting provides sufficient illustration of the cultural groundedness and implications of this particular dichotomy. Recent educational innovations, such as Problem-Based Learning (PBL), use small group collaborative learning and the discussion of ‘authentic’ problems to promote deep learning approaches (Lloyd-Jones, Margetson & Bligh, 1998). PBL has been described as a ‘student-centred’ approach, with a major emphasis on student development of self-directed learning skills (Whitehill, Stokes & MacKinnon, 1997). Rather than carrying the responsibility for disseminating 230

DECONSTRUCTING DICHOTOMIES

content knowledge to students, PBL teachers or tutors have a role in facilitating student engagement with the PBL process. The dialogic nature of small group collaborative learning is well recognised, and collaborative learning models such as problem-based learning (PBL) require verbal contributions from students to progress individual and group learning. Remedios and her colleagues have argued that in such settings speaking is often privileged over listening as a collaborative act. An imbalance in these values can become embedded in the classroom culture. For example, listening, as a core collaborative skill has not been foregrounded in the PBL literature. In the ‘student-centred’ classrooms central to the advocacy of the Western reform agenda, ‘just listening’ can be trivialised and its value as a collaborative learning tool can be lost (Remedios, Clarke & Hawthorne, 2006). The dialogic character of Problem-Based Learning has been shown to pose significant challenges for students from some Asian countries (Remedios, 2005). However, these same analyses have demonstrated the possibility of overemphasising speaking at the expense of attentive listening (Remedios, Clarke & Hawthorne, 2006). It is possible that the optimisation of pedagogical innovations such as PBL may be best achieved by purposefully exploiting the attentive listening skills so evident in some classrooms, in combination with the skills of student-initiated articulation evident in other classrooms. Once again, the explicit promotion of student speaking in Western reform classrooms and the dominance of student listening in Asian classrooms gives the appearance of a dichotomisation of student practice into either speaking or listening. Consistent with the theme of this chapter, an inclusive approach is advocated that acknowledges the potential value of both student activities. Acceptance of the potential value of such a synthesis has the effect of shifting the debate from the separate optimisation of either speaking or listening, to the recognition of the essential interconnectedness of speaking and listening, and the more challenging goal of identifying the criteria for the optimisation of the negotiation of meaning in classroom settings (Clarke, 2001), in which the role of listening is seen as integral to the dialogic process of negotiation. Whether such synthesis can be achieved remains to be seen, but the sites for such experimentation will be the ever-increasing number of multi-cultural classrooms in schools and universities around the world. CONCLUDING REMARKS: ALTERNATIVES TO DICHOTOMISATION

In an international comparative study, any evaluative aspect is reflective of the cultural authorship of the study. If the authors make judgements of merit, whether they are about student achievement or classroom practice, they do so from the position of the authoring culture. The design of international comparative studies must implement collaborative processes through which multiple educational, philosophical and cultural positions are given voice in the interpretation of data and the reporting of the research. The OECD study of innovative programs in mathematics, science and technology education went some way towards addressing 231

DAVID CLARKE

this issue: “A nine-member writing team prepared the final cross-national report … Almost all the countries published their own case studies in the home language for internal distribution” (Atkin & Black, 1997, p. 23). International collaborative studies can implement protocols requiring that the interpretations of data to be included in published reports must be validated by the member researchers from the country providing the data. The other aspect of cultural authorship relevant here is the issue of representation and voice. In commenting on the proliferation of OECD-initiated international comparative research projects, Cohen characterised the OECD as “a club of 29 of the world’s richest countries” (Cohen, 1998, p. 4). Even when less affluent countries participate in international studies, it is frequently as the objects of investigation rather than as partners in the research. Research is frequently conducted from a ‘Western’ perspective and evaluates the practices it studies by ‘Western’ criteria. A notable and most welcome exception is the recent “insider’s perspective” on Chinese mathematics teaching and learning (Fan, Wong, Cai & Li, 2004) and the LPS project has attempted to give voice to this insider’s perspective on an international scale (Clarke, Keitel & Shimizu, 2006). Once we have achieved more equitable representation of all interested nations in international research programs, we need to ensure that the perspectives of all participating cultures inform the design and analytical frameworks employed, and that the voices of all participating cultures are evident in the reports that arise from such research. Bourdieu (1990) has argued that it is a mistake to see individuals as somehow located in a social structure that is external to them. Rather they are part of that structure, and the structure is part of them. In that sense, learning is not just socially-mediated it is fundamentally social in character and the patterns of social participation that have provided the focus of this chapter both facilitate the participants’ learning and embody it. The detailed collaborative study of international policy and practice in mathematics education, and of the products of that policy and practice, should be undertaken in anticipation of insights into the novel, interesting and adaptable practices employed in other school systems and of insights into the strange, invisible, and unquestioned routines and rituals of our own school system and our own mathematics classrooms. One important consideration in relation to cultural authorship is the situatedness of our advocacy of any particular classroom practice. Hatano and Inagaki (1998) remind us that the adaptation of pedagogical practice requires consideration of both the practicality of technical implementation and the extent to which the beliefs underlying the adapted practice are in harmony with local cultural values. Fuller and Clarke (1994) made a related point: The next generation of [research] questions pertains to how these tools are culturally situated and understood in the eyes of teachers and pupils, including how these tools help to structure the classroom’s social rules. (Fuller & Clarke, 1994, p. 144)

232

DECONSTRUCTING DICHOTOMIES

The cultural positioning of pedagogical practice is an essential precursor to its adaptation and application in other settings. Oppositional dichotomies such as teacher-centred versus student-centred classrooms, real-world versus abstract tasks, telling versus not-telling, and listening versus speaking offer mathematics educators falsely exclusive choices, sanctifying one alternative while demonising the other. International research offers insight into possible explanatory frameworks within which such choices are no longer oppositional, but rather can be seen as reflecting strategic and interrelated pedagogical decisions, dependent on purpose and context, that must be understood in cultural terms before they can be related to any setting outside their classroom of origin. The perils of oppositional dichotomies extend to research methodology, where the qualitative/quantitative divide continues to be described in terms of oppositional disconnectedness similar to the examples that have provided the focus of this chapter. Happily, the utilisation of mixed methods designs (Johnson & Onwuegbuzie, 2004) is the subject of increasing advocacy, and complementarity is replacing incommensurability (Clarke, 1998, 2001, 2006). This chapter has attempted to demonstrate the capacity of international classroom research to problematise and deconstruct some of our most fundamental dichotomies and their frequent construction as oppositional. In each case, the alternative that is being offered to the prevalent segregated practice is an integrative perspective in which such alternatives are seen as complementary and interrelated aspects of a broader conception. Further, research, in applying such inclusive frameworks, must employ similarly inclusive methodologies. Theories that embody such an interrelated complementarity of constructs will be more challenging to research and more challenging to realise in classroom practice. International comparative research challenges existing oppositional dichotomies by demonstrating the viability of less favoured approaches in the classrooms of competent teachers in different cultural settings. As importantly, cross-cultural research provides a diversity of educational settings sufficient to contest the assumed disjunction of teaching and learning, and to demonstrate individual teachers’ effective alternation of telling and not-telling. Only by problematising our existing constructs: teaching, learning, telling, real-world, abstract, teacher-centred, student-centred, speaking and listening, will we be led to develop more inclusive (non-oppositional) constructs such as eliciting, initiating, and the distribution of responsibility for knowledge generation. Educational theories that employ such constructs will be more likely to support the development of hybrid pedagogies, adapted to the demands of each classroom setting, but less constrained by culture or convention. Any such hybridisation of pedagogies will look to international research for its structure and its empirical grounding.

233

DAVID CLARKE

ACKNOWLEDGEMENTS

I would like to pay tribute to my friends, colleagues, and collaborators: Sue Helme, Joanne Lobato, Carmel Mesiti, Catherine O’Keefe, Louisa Remedios, Seah Lay Hoon, Katja Svan and Li Hua Xu, whose research has made this chapter possible. REFERENCES Atkin, J. M. & Black, P. (1997). Policy perils of international comparisons. Phi Delta Kappan 79(1), 22-28. Barnes, M. (2004). Collaborative learning in senior mathematics classrooms: Issues of gender and power in student-student interaction. Unpublished PhD thesis, The University of Melbourne. Bourdieu, P. (1990). The logic of practice. Oxford: Blackwell. Brousseau, G. (1986). Fondements et methodes de la didactique des mathematiques [Fundamentals and methods of the didactics of mathematics]. Recherches en didactique des mathematiques 7(2), 33-115. Capra, F. (1976). The tao of physics. Bungay, Suffolk: The Chaucer Press. Chazan, D. & Ball, D. (1999). Beyond being told not to tell. For the Learning of Mathematics, 19(2), 2-10. Clarke, D. J. (1998). Studying the classroom negotiation of meaning: Complementary accounts methodology. In A. Teppo (Ed.) Qualitative research methods in mathematics education. Journal for Research in Mathematics Education, Monograph No. 9 (pp. 98-111). Reston, VA: NCTM. Clarke, D. J. (Ed.) (2001). Perspectives on practice and meaning in mathematics and science classrooms. Dordrecht: Kluwer Academic Press. Clarke, D. J. (2003a). International comparative studies in mathematics education. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick & F. K. S. Leung (Eds.), Second International Handbook of Mathematics Education (pp. 145-186). Dordrecht: Kluwer Academic Publishers. Clarke, D. J. (2003b). Similarity and difference in international comparative research in mathematics education. In L. Bragg, C. Campbell, G. Herbert & J. Mousley (Eds.), Mathematics Education Research: Innovation, Networking, Opportunity. Proceedings of the 26th Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 222-229). Geelong: MERGA. Clarke, D. J. (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,. Clarke, D. J. (2006). The LPS research design. In D. J. Clarke, C. Keitel & Y. Shimizu (Eds.), Mathematics classrooms in twelve countries: The insider’s perspective (pp. 15-37). Rotterdam: Sense Publishers. Clarke, D. J., & Helme, S. (1998). Context as construction. In O. Bjorkqvist (Ed.), Mathematics Teaching from a Constructivist Point of View (pp. 129-147). Vasa, Finland: Faculty of Education, Abo Akademi University. (ISBN 952-12-0151-7). Clarke, D. J., Keitel, C., & Shimizu, Y. (Eds.). (2006). Mathematics classrooms in twelve countries: The insider’s perspective. Rotterdam: Sense Publications. Clarke, D. J., & Lobato, J. (2002). To tell or not to tell: A reformulation of telling and the development of an initiating/eliciting model of teaching. In C. Malcolm & C. Lubisi (Eds.), Proceedings of the tenth annual meeting of the Southern African Association for Research in Mathematics, Science and Technology Education (pp. 15-22). Durban: University of Natal. Clarke, D. J., & Seah, L. H. (2005). Studying the distribution of responsibility for the generation of knowledge in mathematics classrooms in Hong Kong, Melbourne, San Diego and Shanghai. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 257-264). Melbourne: University of Melbourne.

234

DECONSTRUCTING DICHOTOMIES Cobb, P. (1995). Continuing the conversation: A response to Smith. Educational Researcher, 24(7), 25-27. Cohen, D. (1998). World league tables: What’s the score? Principal Matters, 10(1), pp. 3-7. Confrey, J. (1995). A theory of intellectual development: Part III. For the Learning of Mathematics, 15(2), 36-45. Fan, L., Wong, N.-Y., Cai, J., & Li S. (Eds.). (2004). How Chinese learn mathematics: Perspectives from insiders. Singapore: World Scientific Publishing. Fuller, B., & Clarke, P. (1994). Raising school effects while ignoring culture? Local conditions and the influence of classroom tools, rules, and pedagogy. Review of Educational Research, 64(1), 119-157. Hatano, G., & Inagaki, K. (1998). Cultural contexts of schooling revisited: A review of the learning gap from a cultural psychology perspective. In S. G. Paris & H. M. Wellman (Eds.), Global prospects for education: Development, culture and schooling (pp. 79-104). Washington, D.C.: American Psychological Association. Hedegaard, M. (1990). The zone of proximal development as basis for instruction. In L. C. Moll (Ed.), Vygotsky and Education (pp. 349-371). Cambridge: Cambridge University Press. Helme, S. & Clarke, D. J. (2001). We really put our minds to it: Cognitive engagement in mathematics classrooms. Mathematics Education Research Journal, 13(2), 133-153. Huang R. (2002). Mathematics teaching in Hong Kong and Shanghai – A classroom analysis from the perspective of variation. Unpublished Ph.D. thesis. The University of Hong Kong. Johnson, R. B. & Onwuegbuzie, A. J. (2004). Mixed methods research: A research paradigm whose time has come. Educational Researcher, 33(7), 14-26. Klaassen, C. W. J. M. & Lijnse, P. L. (1996). Interpreting students' and teachers' discourse in science classes: An underestimated problem? Journal of Research in Science Teaching, 33(2), 115-134. Knuth, E. & Peressini, D. (2001). Unpacking the nature of discourse in mathematics. Mathematics Teaching in the Middle School, 6(5), 320-325. Leung, F. K. S. (2001). In search of an East Asian identity in mathematics education. Educational Studies in Mathematics, 47, 35-51. Lloyd-Jones, G., Margetson, D., & Bligh, J. G. (1998). Problem-based learning: A coat of many colours. Medical Education, 32, 492-494. Lobato, J., Clarke, D. J., & Ellis, A. B. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education 36(2), 101-136. Mason, J. (2003). Structure of attention in the learning of mathematics. In J. Novotná (Ed.), Proceedings: International Symposium on Elementary Mathematics Teaching (pp. 9-16). Prague: Charles University. Mok, I. A. C., & Ko, P. Y. (2000). Beyond labels – Teacher-centred and pupil-centred activities. In B. Adamson, T. Kwan & K. K. Chan (Eds.), Changing the curriculum: The impact of reform on primary schooling in Hong Kong (pp. 175-194). Hong Kong: Hong Kong University Press. Niss, M. (1994). Mathematics in society. In R. Biehler, R. Scholz, R. Straesser & B. Winkelmann (Eds.), The didactics of mathematics as a scientific discipline (pp. 367-378). Dordrecht: Kluwer. Ohtani, M. (2003, August). Social formation of mathematical activity in a Japanese mathematics classroom: “Revoicing” as a unit of analysis. In D.J. Clarke (Chair), Social interaction and learning in mathematics classrooms in Australia, Germany, Hong Kong, Japan, Sweden and the United States. Symposium conducted at the annual conference of the European Association for Research in Learning and Instruction (EARLI), Padova, Italy. Remedios, L. J. (2005). The experiences and responses of overseas-educated students to problem-based learning and its classroom culture in an Australian physiotherapy context. Unpublished PhD thesis, The University of Melbourne. Remedios, L. J., Clarke, D. J. & Hawthorne, L-A. (2006, April). Not learning to listen and learning not to listen: A problem in problem-based learning. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco.

235

DAVID CLARKE Richardson, V. (2001). Constructivist mathematics instruction and current trends in research on teaching. In T. Wood, B. S. Nelson & J. Warfield (Eds.), Beyond classical pedagogy: Teaching elementary school mathematics (pp. 275-293). Mahway, NJ: Lawrence Erlbaum. Schmidt, W. H., McKnight, C. C., Valverde, G. A., Houang, R. T., & Wiley, D. E. (1997). Many visions, many aims: Vol. 1. A cross-national investigation of curricular intentions in school mathematics. Dordrecht: Kluwer. Seah, L. H. (2004). A cross-disciplinary approach to analysing a secondary school science lesson in Singapore. Unpublished MEd Thesis, The University of Melbourne. Sethole, G., Adler, J., & Vithal, R. (2002, September). When AIDS goes to a mathematics class: Is it a hindrance or a source for revising the mathematics classroom culture? Paper presented at the ICMI Study, Mathematics education in different cultural traditions: A comparative study of East Asia and the West, Hong Kong. Subsequently included in Clarke, D. J., Shimizu, Y., Ulep, S., Gallos, F., Sethole, G., Adler, J., & Vithal, R. (2006). Cultural diversity and the learner’s perspective: Attending to voice and context. In F. K. S. Leung, K. D. Graf & F. J. Lopez-Real (Eds.), Mathematics education in different cultural traditions: A comparative study of East Asia and the West (pp. 353-380). Dordrecht: Springer. Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267-306). Mahwah, NJ: Lawrence Erlbaum. Svan, K. & Clarke, D. J. (in preparation). Expanding the relevance paradox. von Glasersfeld, E. (Ed.). (1991). Radical constructivism in mathematics education. Dordrecht, Netherlands: Kluwer. Vygotsky, L. S. (1978). Mind in Society. (M. Cole, V. John-Steiner, S. Scribner & E. Souberman, Eds.). Cambridge, Mass: Harvard University Press. Whitehill, T. L., Stokes, S. F., & MacKinnon, M. M. (1997, July). Problem based learning and the Chinese learner. Paper presented at the Learning and Teaching in Higher Education: Advancing International Perspectives conference, The Flinders University of South Australia, July 8-11. Wilson, J. M. (1999). Using words about thinking: Content analyses of chemistry teachers' classroom talk. International Journal of Science Education, 21(10), 1067-1084. Wood, T., Nelson, B. S., & Warfield, J. (Eds.). (2001). Beyond classical pedagogy: Teaching elementary school mathematics. Mahwah, NJ: Lawrence Erlbaum. Young, R. F., & Nguyen, H. T. (2002). Modes of meaning in high school science. Applied Linguistics, 23(3), 348-372.

David Clarke International Centre for Classroom Research Faculty of Education University of Melbourne Australia

236

DAVID CLARKE

APPENDIX A The LPS Research Design

INTRODUCTION

The originators of the LPS project, Clarke, Keitel and Shimizu, felt that the methodology developed by Clarke and known as complementary accounts (Clarke, 1998), which had already demonstrated its efficacy in a large-scale classroom study (subsequently reported in Clarke, 2001) could be adapted to meet the needs of the Learner’s Perspective Study. These needs centered on the recognition that only by seeing classroom situations from the perspectives of all participants can we come to an understanding of the motivations and meanings that underlie their participation. In terms of techniques of data generation, this translated into three key requirements: (i) the recording of interpersonal conversations between focus students during the lesson; (ii) the documentation of sequences of lessons, ideally of an entire mathematics topic; and, (iii) the identification of the intentions and interpretations underlying the participants’ statements and actions during the lesson. Miles and Huberman’s text on qualitative data analysis (Miles & Huberman, 2004) focused attention on ‘data reduction.’ Even before data are collected . . . anticipatory data reduction is occurring as the researcher decides (often without full awareness) which conceptual framework, which cases, which research questions, and which data approaches to use. As data collection proceeds, further episodes of data reduction occur (p. 10). This process of data reduction pervades any classroom video study. The choice of classroom, the number of cameras used, who is kept in view continuously and who appears only given particular circumstances, all contribute to a process that might better be called ‘data construction’ or ‘data generation’ than ‘data reduction.’ Every decision to zoom in for a closer shot or to pull back for a wide angle view represents a purposeful act by the researcher to selectively construct a data set optimally amenable to the type of analysis anticipated and maximally aligned with the particular research questions of interest to the researcher. The process of data construction does not stop with the video record, since which statements (or whose voices) are transcribed, and which actions, objects or statements are coded, all

D. J. Clarke, J. Emanuelsson, E. Jablonka & I. A. C. Mok (Eds.), Making connections: Comparing mathematics classrooms around the world, 237–251. © 2006 Sense Publishers. All rights reserved.

DAVID CLARKE

constitute further decisions made by the researcher, more or less explicitly justified in terms of the project’s conceptual framework or the focus of the researcher’s interest. The researcher is the principle agent in this process of data construction. As such, the researcher must accept responsibility for decisions made and data constructed, and place on public record a transparent account of the decisions made in the process of data generation and analysis. In the case of the Learner’s Perspective Study: Research guided by a theory of learning that accords significance to both individual subjectivities and to the constraints of setting and community practice must frame its conclusions (and collect its data) accordingly. Such a theory must accommodate complementarity rather than require convergence and accord both subjectivity and agency to individuals not just to participate in social practice but to shape that practice. The assumption that each social situation is constituted through (and in) the multiple lived realities of the participants in that situation aligns the Learner’s Perspective Study with the broad field of interpretivist research. DATA GENERATION IN THE LEARNER’S PERSPECTIVE STUDY

Data generation in the Learner’s Perspective Study (LPS) used a three-camera approach (Teacher camera, Student camera, Whole Class camera) that included the onsite mixing of the Teacher and Student camera images into a picture-in-picture video record (see Figure 1, teacher in top right-hand corner) that was then used in post-lesson interviews to stimulate participant reconstructive accounts of classroom events. These data were generated for sequences of at least ten consecutive lessons occurring in the “well-taught” eighth grade mathematics classrooms of teachers in Australia, the Czech Republic, Germany, Hong Kong and mainland China, Israel, Japan, Korea, The Philippines, Singapore, South Africa, Sweden and the USA. This combination of countries gives good representation to European and Asian educational traditions, affluent and less affluent school systems, and mono-cultural and multi-cultural societies. Each participating country used the same research design to generate videotaped classroom data for at least ten consecutive mathematics lessons and post-lesson video-stimulated interviews with at least twenty students in each of three participating 8th grade classrooms. The three mathematics teachers in each country were identified for their locally-defined ‘teaching competence’ and for their situation in demographically diverse government schools in major urban settings. Rather than attempt to apply the same definition of teaching competence across a dozen countries, which would have required teachers in Uppsala and Shanghai, for instance, to meet the same eligibility criteria, teacher selection was made by each local research group according to local criteria. These local criteria included such things as status within the profession, respect of peers or the school community, or visibility in presenting at teacher conferences or contributing to teacher professional development programs. As a result, the diverse enactment of teaching competence is one of the most interesting aspects of the project.

238

THE LPS RESEARCH DESIGN

In most countries, the three lesson sequences were spread across the academic year in order to gain maximum diversity within local curricular content. In Sweden, China and Korea, it was decided to focus specifically on algebra, reflecting the anticipated analytical emphases of those three research groups. Algebra forms a significant part of the 8th grade mathematics curriculum in most participating LPS countries, with some variation regarding the sophistication of the content dealt with at 8th grade. As a result, the data set from most of the LPS countries included at least one algebra lesson sequence. In the key element of the post-lesson student interviews, in which a picture-inpicture video record was used as stimulus for student reconstructions of classroom events (see Figure 1), students were given control of the video replay and asked to identify and comment upon classroom events of personal importance. The postlesson student interviews were conducted as individual interviews in all countries except Germany, Israel and South Africa, where student preference for group interviews was sufficiently strong to make that approach essential. Each teacher was interviewed at least three times using a similar protocol.

Figure. 1 Picture-in-picture video display

With regard to both classroom videotaping and the post-lesson interviews, the principles governing data generation were the minimization of atypical classroom activity (caused by the data generation activity) and the maximization of respondent control in the interview context. To achieve this, each videotaped lesson sequence was preceded by a one-week familiarization period in which all aspects of data generation were conducted until the teacher indicated that the class was functioning as normally as might reasonably be expected. In interviews, the location of control of the video player with the student ensured that the reconstructive accounts focused primarily on the student’s parsing of the lesson. Only after the student’s selection of significant events had been exhausted did the interviewer ask for reconstructive accounts of other events of interest to the research team. Documentation of the participant’s perspective (learner or teacher) remained the priority. 239

DAVID CLARKE

In every facet of this data generation, technical quality was a priority. The technical capacity to visually juxtapose the teacher’s actions with the physical and oral responses of the children was matched by the capacity to replay both the public statements by teacher or student and the private conversations of students as they struggled to construct meaning. Students could be confronted, immediately after the lesson, with a video record of their actions and the actions of their classmates. In the picture-in-picture video record generated on-site in the classroom (Figure 1), students could see both their actions and the actions of those students around them, and, in the inset (top right-hand corner), the actions of the teacher at that time. This combined video record captured the classroom world of the student. The video record captured through the whole-class camera allowed the actions of the focus students to be seen in relation to the actions of the rest of the class. CLASSROOM DATA GENERATION

Camera Configuration Data generation employed three cameras in the classroom – a “Teacher Camera”, a “Student Camera” and a “Whole Class Camera”. The protocol below was written primarily for a single research assistant/videographer, but brief notes were provided suggesting variations possible if a second videographer was available. In order to ensure consistency of data generation across all schools in several countries, the protocol was written as a low inference protocol, requiring as few decisions by the videographer as possible. One or two possible anomalous cases were specifically discussed – such as when a student presents to the entire class. However, the general principles were constant for each camera: The Teacher Camera maintained a continuous record of the teacher’s statements and actions. The Student Camera maintained a continuous record of the statements and actions of a group of four students. The Whole Class Camera was set up in the front of the classroom to capture, as far as was possible, the actions of every student – that is, of the “Whole Class.” The Whole Class Camera can also be thought of as the “Teacher View Camera.” While no teacher can see exactly what every individual student is doing, the teacher will have a sense of the general level of activity and types of behaviors of the whole class at any time – this is what was intended to be captured on the Whole Class Camera. Camera One: The Teacher Camera The “Teacher Camera” maintained the teacher in centre screen as large as possible provided that all gestures and all tools or equipment used could be seen – if overhead transparencies or boardwork or other visual aids were used then these had to be captured fully at the point at which they were generated or employed in the first instance or subsequently amended – but did not need to be kept in view at the expense of keeping the teacher in frame (provided at least one full image was 240

THE LPS RESEARCH DESIGN

recorded, this could be retrieved for later analysis – the priority was to keep the teacher in view). The sole exception to this protocol occurred when a student worked at the board or presented to the whole class. In this case, the Teacher Camera focused on the “student as teacher.” The actions of the Teacher during such occasions should have been recorded by the Whole Class Camera. If the teacher was positioned out of view of the Whole Class Camera (eg front of classroom, at the side), then the Teacher Camera might “zoom out” to keep both the student and teacher on view, but documentation of the gestures, statements, and any written or drawn work by the student at the board should be kept clearly visible. Note: Although the teacher was radio-miked, in the simulated situations we trialled it was not necessary for the teacher to hand the lapel microphone to the student. The student’s public statements to the class could be adequately captured on the student microphone connected to the Student Camera. The first few lessons in a particular classroom (during the familiarization period) provided an opportunity to learn to “read” the teacher’s teaching style, level of mobility, types of whole class discussion employed, and so on. A variety of practical decisions about the optimal camera locations could be made during the familiarization period and as events dictated during videotaping. Camera Two: The Student Camera Where only a single videographer was used, the “Student Camera” was set up prior to the commencement of the lesson to include at least two adjacent students and was re-focussed in the first two minutes of the lesson during the teacher’s introductory comments – during this time the Teacher Camera could be set up to record a sufficiently wide image to include most likely positions of the teacher during these opening minutes. Once the Student Camera was adequately focused on the focus students for that lesson, it remained fixed unless student movement necessitated its realignment. After aligning the Student Camera, the videographer returned to the Teacher Camera and maintained focus on the teacher, subject to the above guidelines. If two research assistants (“videographers”) were available (and this was frequently the case), then it became possible for the Student Camera to “zoom in” on each student’s written work every five minutes or so, to maintain an on-going record of the student’s progress on any written tasks. This “zooming in” was done sufficiently briefly to provide visual cues as to the progress of the student’s written work, but any such zooming in had to be done without losing the continuity of the video record of all focus students, since that would be needed for the subsequent interviews. Since it was Learner Practices that were the priority in this study, the continuous documentation of the actions of the focus students and their interactions (including non-verbal interactions) was most important. A copy of the students’ written work was obtained at the end of the lesson. The video record generated by this camera served to display each student’s activities in relation to the teacher’s actions, the tasks assigned, and the activities of their nearby classmates.

241

DAVID CLARKE

Camera Three: The Whole Class Camera The “Whole Class (or Teacher-View) Camera” was set up to one side of whichever part of the room the teacher spoke from (typically, to one side at the “front” of the classroom). All students should be within the field of view of this camera (it is necessary to use a wide-angle lens). Apart from capturing the “corporate” behavior of the class, this camera provided an approximation to a “teacher’s-eye view” of the class. It was also this camera that documented teacher actions during any periods when a student was working at the board or making a presentation to the entire class. Microphone Position The teacher was radio-miked to the Teacher Camera. The focus student group was recorded with a microphone placed as centrally as possible in relation to the focus students and recorded through the Student Camera (use of a radio microphone minimized intrusive cables). The Whole Class Camera audio was recorded through that camera’s internal microphone. Fieldnotes Depending on the available research personnel, fieldnotes were maintained to record the time and type of all changes in instructional activity. Such field notes could be very simple, for example: 00:00 Teacher Introduction 09:50 Students do Chalkboard Problem 17:45 Whole Class Discussion 24:30 Individual Textbook Work 41:45 Teacher Summation Specific events of interest to the researcher could be included as annotations to such field notes. Where a third researcher was available, in addition to the operators of the Teacher and Student cameras, this person was able to take more detailed field notes, including detail of possible moments of significance for the progress of the lesson (eg public or private negotiations of meaning). In such cases, the field notes became a useful aid in the post-lesson interview, and the interviewee could be asked to comment on particular events, if these had not been already identified by the interviewee earlier in the interview. Student Written Work All written work produced by the focus students “in camera” during any lesson was photocopied together with any text materials or handouts used during the lesson. Students brought with them to the interview their textbook and all written material

242

THE LPS RESEARCH DESIGN

produced in class. This material (textbook pages, worksheets, and student written work) was photocopied immediately after the interview and returned to the student. INTERVIEWS

In this study, students were interviewed after each lesson using the video record as stimulus for their reconstructions of classroom events. It is a feature of this study that students were given control of the video replay and asked to identify and comment upon classroom events of personal importance. Because of the significance of interviews within the study, the validity of students' and teachers' verbal reconstructions of their motivations, feelings and thoughts was given significant thought. The circumstances under which such verbal accounts may provide legitimate data have been detailed in two seminal papers (Ericsson & Simon, 1980; Nisbett & Wilson, 1977). It is our contention that videotapes of classroom interactions constitute salient stimuli for interviewing purposes, and that individuals' verbal reports of their thoughts and feelings during classroom interactions, when prompted by videos of the particular associated events, can provide useful insights into those individuals' learning behaviour. Videotapes provide a specific and immediate stimulus that optimizes the conditions for effective recall of associated feelings and thoughts. Nonetheless, an individual's video-stimulated account will be prone to the same potential for unintentional misrepresentation and deliberate distortion that apply in any social situation in which individuals are obliged to explain their actions. A significant part of the power of video-stimulated recall resides in the juxtaposition of the interviewee's account and the video record to which it is related. Any apparent discrepancies revealed by such a comparison warrant particular scrutiny and careful interpretation by the researcher. Having relinquished the positivist commitment to identifying 'what really happened,’ both correspondence and contradiction can be exploited. The interview protocols for student and teacher interviews were prescribed in the LPS Research Design and are reproduced below. Individual student interviews Prompt One: Please tell me what you think that lesson was about (lesson content/lesson purpose). Prompt Two: How, do you think, you best learn something like that? Prompt Three: What were your personal goals for that lesson? What did you hope to achieve? Do you have similar goals for every lesson? Prompt Four: Here is the remote control for the videoplayer. Do you understand how it works? (Allow time for a short familiarisation with the control). I would like you to comment on the videotape for me. You do not need to comment on all of the lesson. Fast forward the videotape until you find sections of the lesson that you think were important. Play these sections at normal speed and describe for me what you were doing, thinking and feeling during each of these videotape sequences. You can comment while the videotape is 243

DAVID CLARKE

playing, but pause the tape if there is something that you want to talk about in detail. Prompt Five: After watching the videotape, is there anything you would like to add to your description of what the lesson was about? Prompt Six: What did you learn during that lesson? [Whenever a claim is made to new mathematical knowledge, this should be probed. Suitable probing cues would be a request for examples of tasks or methods of solution that are now understood or the posing by the interviewer of succinct probing questions related to common misconceptions in the content domain.] Prompt Seven: Would you describe that lesson as a good* one for you? What has to happen for you to feel that a lesson was a “good” lesson? Did you achieve your goals? What are the important things you should learn in a mathematics lesson? [*“Good” may be not be a sufficiently neutral prompt in some countries – the specific term used should be chosen to be as neutral as possible in order to obtain data on those outcomes of the lesson which the student values. It is possible that these valued outcomes may have little connection to “knowing”, “learning” or “understanding”, and that students may have very localized or personal ways to describe lesson outcomes. These personalized and possibly culturally-specific conceptions of lesson outcomes constitute important data.] Prompt Eight: Was this lesson a typical [geometry, algebra, etc] lesson? What was not typical about it? Prompt Nine: How would you generally assess your own achievement in mathematics? Prompt Ten: Do you enjoy mathematics and mathematics classes? Prompt Eleven: Why do you think you are good [or not so good] at mathematics? Prompt Twelve: Do you do very much mathematical work at home? Have you ever had private tutoring in mathematics or attended additional mathematics classes outside normal school hours? Prompts 9 through 12 could be covered in a student questionnaire – the choice of method may be made locally, provided the data is collected. Student Group Interviews Prompt One: Please tell me what you think that lesson was about (lesson content/lesson purpose) (Discuss with the group – identify points of agreement and disagreement – there is NO need to achieve consensus). Prompt Two: Here is the remote control for the videoplayer. I would like you to comment on the videotape for me. You do not need to comment on all of the lesson. I will fast forward the videotape until anyone tells me to stop. I want you to find sections of the lesson that you think were important. We will play these sections at normal speed and I 244

THE LPS RESEARCH DESIGN

Prompt Prompt

Prompt

Prompt

would like each of you to describe for me what you were doing, thinking and feeling during each of these videotape sequences. You can comment while the videotape is playing, but tell me to pause the tape if there is something that you want to talk about in detail. Three: After watching the videotape, is there anything anyone would like to add to the description of what the lesson was about? Four: What did you learn during that lesson? (Discuss) [As for the individual interview protocol, all claims to new mathematical knowledge should be probed. BUT, before probing an individual’s responses directly, the interviewer should ask other members of the group to comment.] Five: Would you describe that lesson as a good* one for you? (Discuss) What has to happen for you to feel that a lesson was a “good” lesson? (Discuss) What are the important things you should learn in a mathematics lesson? [*As for the student individual interviews, ”good” may be not be a sufficiently neutral prompt in some countries – the specific term used should be chosen to be as neutral as possible in order to obtain data on those outcomes of the lesson which the student values ]. Six: Was this lesson a typical [geometry, algebra, etc] lesson? What was not typical about it?

The Teacher Interview The goal was to complete one interview per week, according to teacher availability. The Whole Class Camera image was used as the stimulus. In selecting the lesson about which to seek teacher comment, choose either (1) the lesson with the greatest diversity of classroom activities, or (2) the lesson with the most evident student interactions. Should the teacher express a strong preference to discuss a particular lesson, then this lesson should take priority. Tapes of the other lessons should be available in the interview, in case the teacher should indicate an interest in any aspect of a particular lesson. Prompt One: Please tell me what were your goals in that lesson (lesson content/lesson purpose). Prompt Two: In relation to your content goal(s), why do you think this content is important for students to learn? What do you think your students might have answered to this question? Prompt Three: Here is the remote control for the videoplayer. Do you understand how it works? (Allow time for a short familiarization with the control). I would like you to comment on the videotape for me. You do not need to comment on all of the lesson. Fast forward the videotape until you find sections of the lesson that you think were important. Play these sections at normal speed and describe for me what you were doing, thinking and feeling during each of these videotape sequences. You can comment while the videotape is 245

DAVID CLARKE

playing, but pause the tape if there is something that you want to talk about in detail. In particular, I would like you to comment on: (a) Why you said or did a particular thing (for example, conducting a particular activity, using a particular example, asking a question, or making a statement). (b) What you were thinking at key points during each video excerpt (for example, I was confused, I was wondering what to do next, I was trying to think of a good example). (c) How you were feeling? (for example, I was worried that we would not cover all the content) (d) Students’ actions or statements that you consider to be significant and explain why you feel the action or statement was significant. (e) How typical that lesson was of the sort of lesson you would normally teach? What do you see as the features of that lesson that are most typical of the way you teach? Were there any aspects of your behavior or the students’ behavior that were unusual? Prompt Four: Would you describe that lesson as a good lesson for you? What has to happen for you to feel that a lesson is a “good” lesson? Prompt Five: Do your students work a lot at home? Do they have private tutors? OTHER SOURCES OF DATA

Student tests were used to situate each student group and each student in relation to student performance on eighth-grade mathematics tasks. Student mathematics achievement was assessed in three ways: Student written work in class. Analyses of student written work were undertaken both during and after the period of videotaping. For this purpose, the written work of all “focus students” in each lesson was photocopied, clearly labelled with the student’s name, the class, and the date, and filed. Additional data on student achievement was also collected, where this was available. In particular, student scores were obtained on any topic tests administered by the teacher, in relation to mathematical content dealt with in the videotaped lesson sequence. Student performance to place the class in relation to the national 8th grade population. In Australia, Japan, Korea, China and the USA, this was done by using the International Benchmark Test for Mathematics (administered immediately after the completion of videotaping). The International Benchmark Test (IBT) was developed by the Australian Council for Educational Research (ACER) by combining a selection of items from the TIMSS Student Achievement test. In the case of this project, the test for Population Two was used, since this was in closest correspondence with the grade level of the students taking part in the LPS project. In administering the IBT, the local research group in each country constructed an equivalent test using the corresponding version of each of the 246

THE LPS RESEARCH DESIGN

TIMSS items, as administered in that country. In some countries, where this was not possible (Germany, for example), the typical school performance was characterized in relation to other schools by comparison of the senior secondary mathematics performance with national norms. Student performance in relation to other students in that class. Since studentstudent interactions may be influenced by perceptions of peer competence, it was advantageous to collect recent performance data on all students in the class. Two forms of student mathematics achievement at class level were accessed, where available: (a) student scores from recent mathematics tests administered by the teacher, and (b) brief annotated comments by the teacher on a list of all students in the class – commenting on the mathematics achievement and competence of each student. Teacher Goals and Perceptions Teacher questionnaires were used to establish teacher beliefs and purposes related to the lesson sequence studied. Three questionnaires were administered to each participating teacher: – A preliminary teacher questionnaire about each teacher’s goals in the teaching of mathematics (TQ1); – A post-lesson questionnaire (TQ2 – either the short TQ2S or the long TQ2 L version – if the short version was used, the researcher’s field notes provided as much as possible of the additional detail sought in the long version); – A post-videotaping questionnaire (TQ3) (also employed by some research groups as the basis of a final teacher interview). DATA CONFIGURATION AND STORAGE

Transcription and Translation A detailed Technical Guide was developed to provide guidelines for the transcription and translation of classroom and interview, video and audiotape data. It was essential that all research groups transcribe their own data. Local language variants (eg the Berliner dialect) required a “local ear” for accurate transcription. Translation into English was also the responsibility of the local research group. The Technical Guide specified both transcription conventions, such as how to represent pauses or overlapping statements, and translation conventions, such as how to represent colloquialisms. In the case of local colloquial expressions in a language other than English, the translator was presented with a major challenge. A literal English translation of the colloquialism may convey no meaning at all to a reader from another country, while the replacement of the colloquialism by a similar English colloquialism may capture the essence and spirit of the expression, but sacrifice the semantic connotations of the particular words used. And there is a third problem: If no precise English equivalent can be found, then the translation 247

DAVID CLARKE

inevitably misrepresents the communicative exchange. In such instances, the original language, as transcribed, was included together with its literal English translation. Any researcher experiencing difficulties of interpretation in analyzing the data could contact a member of the research group responsible for the generation of those data and request additional detail. Data Storage To carry out serious systematic empirical work in classroom research, there is a need for both close and detailed analysis of selected event sequences, and for more general descriptions of the material from within which the analysed sample has been chosen. To be able to perform this work with good-quality multiple-source video and audio data, video and audio materials have to be compressed and stored in a form accessible by desktop computers. Software tools such as Final Cut Pro are essential for the efficient and economical storage of the very large video data files. Compression decisions are dictated by current storage and back-up alternatives and change as these change. For example, when the Learner’s Perspective Study was established in 1999, it was anticipated that data would be exchanged between research teams by CD-ROM and compression ratios were set at 20:1 in order to get maximum data quality within a file size that would allow one video record of one lesson to be stored on a single CD. As a result, the complete US data set in 2001 took the form of a set of over fifty separate CDs. Later, it was possible to store all the data related to a single lesson (including four compressed video records) on a single DVD. The contemporary availability of pocket drives with capacities of 60 gigabytes and higher, has made data sharing both more efficient and cheaper. It is possible to store all the data from a single school in compressed form on such a pocket drive, making secure data transfer between international research groups much more cost-effective.

Figure 2. Structure of the LPS database at the ICCR circa 2004

248

THE LPS RESEARCH DESIGN

The materials on the database have to be represented in a searchable fashion. In Figure 2, the configuration of the LPS database is displayed as a stratified hierarchy of: Country (column 1), school (column 2), lesson (column 3), data source (column 4), specific file (column 5). Any particular file, such as the teacher camera view of lesson 4 at school 2 in Japan, can then be uniquely located. Setting up data in this way enables researchers to move between different layers of data, without losing sight of the way they are related to each other. Further, data can be made accessible to other researchers. This is a sharp contrast to more traditional ways of storing video data on tapes, with little or no searchable record available, and with data access limited to very small numbers of people. At the International Centre for Classroom Research (ICCR) at the University of Melbourne, for example, several researchers can simultaneously access the full range of classroom data. This capacity for the simultaneous analysis of a common body of classroom data is the technical realisation of the methodological and theoretical commitment to complementary analyses proposed by Clarke (2001, 1998) as essential to any research attempting to characterise social phenomena as complex as those found in classrooms. ANALYTICAL TOOLS CAPABLE OF SUPPORTING SOPHISTICATED ANALYSES OF SUCH COMPLEX DATABASES.

Research along the lines argued for above requires the development of software tools for analysing video efficiently. The reasons for this are, in short, that video editing software (such as Final Cut Pro) is not analytically resourceful enough, whereas qualitative analysis software (such as Nudist or nVivo) is not well enough adapted to video and audio work. Early examples of video analysis software (such as vPrism) have been hampered by problems arising from their project-specific origins, leading to a lack of flexibility in customizing the analysis to the demands of each particular project or research focus. Collaboration with the Australian software company, Sportstec, was carried out to adapt the video analysis software Studiocode for use with classroom video data. These adaptations were driven by specific methodological, theoretical and practical needs. For example, the commitment to the capturing and juxtaposition of multiple perspectives on classroom events was partially addressed with the onsite capture of the picture-in-picture display shown in Figure 1, but the need to ‘calibrate’ the actions of the focus students against the actions of the rest of the class required multiple viewing windows. Figure 3 displays the key analytical elements provided within Studiocode: video window, time-line, transcript window, and coding scheme. The researcher has the option of analyzing and coding the events shown in the video window, or the utterances shown in the transcript window, or both. The resultant codes can be displayed in timelines (as shown in Figure 3) or in frequency tables. Once coded, single lessons, events within single lessons, or combinations of lessons can be merged into a single analysis.

249

DAVID CLARKE

Figure 3. Sample analytical display (Studiocode) – video window (top left), time-line (top right), transcript window (bottom-left) and coding facility (bottom-right)

The continual addition of new countries to the Learner’s Perspective Study community required that video data already coded should not need to be recoded when additional data (eg from a different country) were incrementally added to the database. Only the new data should require coding and the newly-coded data should be accessible for analysis as part of the growing pool of classroom data. This flexibility is ideally suited to a project such as the Learner’s Perspective Study, with many collaborating researchers adopting a wide range of different analytical approaches to a commonly held body of classroom data. The Studiocode software described above is only one of the many analytical tools available to the classroom researcher. Increasingly sophisticated public access software tools are being developed continually. Most of the chapters in this book and in the companion volume (Clarke, Keitel, & Shimizu, 2006) report specific analyses of different subsets of the large body of LPS classroom data. Each analysis is distinctive and interrogates and interprets the data consistent with the purpose of the authoring researcher(s). Analytical tools such as nVivo and Studiocode can support the researcher’s analysis but ideally should not constrain the consequent interpretation of the data. In reality, all such tools, including statistical procedures, constrain the researcher’s possible interpretations by limiting the type of data compatible with the analytical tool being used, by restricting the variety of codes, categories or values that can be managed, and by constraining the range of possible results able to be generated by the particular analytical tool.

250

THE LPS RESEARCH DESIGN

REFERENCES Clarke, D. J. (1998). Studying the classroom negotiation of meaning: Complementary accounts methodology. Chapter 7 in A. Teppo (Ed.) Qualitative research methods in mathematics education, monograph number 9 of the Journal for Research in Mathematics Education, Reston, VA: NCTM, 98-111. Clarke, D. J. (Ed.) (2001). Perspectives on practice and meaning in mathematics and science classrooms. Dordrecht, Netherlands: Kluwer Academic Press. Clarke, D. J., Keitel, C., & Shimizu, Y. (Eds.) (2006). Mathematics classrooms in twelve countries: The insider’s perspective. Rotterdam: Sense Publishers. Ericsson, K. A., & Simon, H. A. (1980). Verbal reports as data. Psychological Review, 87(3), 215-251. Miles, M. B., & Huberman, A. M. (2004). Qualitative data analysis (2nd edition). Thousand Oaks, CA: Sage Publications. Nisbett, R. E., & Wilson, T. D. (1977). Telling more than we can know: Verbal reports on mental processes. Psychological Review, 84(3), 231-259.

David Clarke International Centre for Classroom Research Faculty of Education University of Melbourne Australia

251

AUTHOR INDEX

A Adamson, B. Adler, J.

104, 235 202, 208, 212, 213, 221, 236 Akiba, M. 2, 7, 20, 45 Alton-Lee, A. 10, 19 Anderson, L. W. 2, 19, 28, 44 Anderson-Levitt, K. M. 2, 19, 28, 44 Atkin, J. M. 7, 19, 232, 234 Atkinson, P. 167, 182

B Baker, C. D. Baker, D. Ball, D. L. Barnes, M. Bauersfeld, H. Beaton, A. E. Bellack, A. A. Bergqvist, K. Bergsten, C. Biddle, B. J. Biehler, R. Birenbaum, M. Bishop, A. J. Bjorkqvist, O. Black, P. Bligh, J. H. Boaler, J. Booth, S. Bourdieu, P. Bourke, S. Bragg, L. Brodie, K. Brousseau, G.

121, 121 2, 7, 20, 45 85, 104, 222, 226, 234 16, 19, 225, 234 16, 17, 20 24, 44 10, 19 175, 182 198 182 235 20 19, 44, 104, 161, 162, 162, 234 19, 234 7, 19, 232, 234 230, 235 120, 121, 202, 207, 213 165, 182, 188, 198 232, 234 44, 44 19, 234 110, 121 13, 19, 74, 104, 142, 144, 218,

Burge, B. Burr, V.

219, 234 182 166, 182

C Cai, J. Camilli, G. Campbell, C. Capra, F. Chan, K. K. Chazan, D.

198, 232, 235 20 19, 234 215, 234 104, 235 85, 104, 222, 226, 234 Chick, H. L. 234 Chrostowski, S. J. 21 Chui, A. 20, 45, 71, 121, 198 Clarke, D. J. 1, 2, 6, 7, 8, 10, 13, 14, 15, 16, 18, 19, 20, 21, 24, 25, 29, 30, 44, 49, 62, 71, 74, 75, 78, 85, 87, 94, 96, 102, 104, 105, 127, 128, 144, 145, 148, 162, 163, 187, 198, 213, 213, 216, 217, 219, 221, 222, 223, 228, 229, 231, 232, 233, 234, 235, 236, 237, 249, 250, 251 Clarke, D. M. 69, 71 Clarke, P. 232, 235 Clarkson, P. 104 Clements, M. A. 19, 44, 104, 234 Cobb, P. 11, 16, 17, 20, 21, 216, 226, 235 Cohen, D. 201, 213, 232, 235 Cole, M. 236 253

AUTHOR INDEX

Confrey, J. Cooney, T. J. Corbin, J. Coulthard, M.

216, 235 145 149, 163 10, 21

D Dai, Z. 15, 21 Davies, B. 74, 75, 104 Davis, H. 110, 121 Dienes, Z. P. 166, 182 Dochy, F. J. R. C. 20 Dowling, P. 171, 182 Downton, A. 104 Doyle, W. 161, 163 Dubinsky, E. 159, 163

E Eisner, E. Ellis, A. B.

12, 20 62, 71, 78, 85, 88, 104, 228, 229, 235 Ellmore, P. B. 20 Emanuelsson, J. 2, 13, 20, 168, 180, 182 Erickson, F. 9, 10, 11, 20 Ericsson, K. A. 243, 251 Ernest, P. 201, 213 Etterbeck, W. 20, 45, 71, 198

F Fairclough, N. Fan, L. Fernandez, C. Feuer, M. J. Ford, P. Foucault, M. Fuglestad, A. B. Fuller, B.

180, 182 198, 232, 235 6, 20 12, 20 17, 20 108, 121 19, 20, 71, 198, 234 232, 235

G Gallimore, R.

20, 44, 45, 71,

254

121, 144, 198 236 21 20, 45, 71, 121, 144, 198 Gates, P. 202, 213 Givvin, K. B. 20, 28, 44, 44, 45, 71, 121, 144, 186, 198 Goesling, B. 2, 7, 20, 45 Gonzales, P. 20, 45, 71, 121, 144, 198 Gonzalez, E. J. 21 Good, T. L. 44, 45, 57, 59, 65, 137, 156, 157, 182, 244 Goodson, I. F. 182 Graf, K. D. 236 Graham, T. 142, 145 Graven, M. 202, 213 Green, J. L. 20 Greeno, J. 73, 96, 104 Greenwood, L. 17, 20 Gregory, K. 21 Grevholm, B. 198 Gronn, D. 104 Grouws, D. 44, 45, 145 Gu, L. 197, 198 Gustafsson, J-E. 185, 198 Gallos, F. Garden, R. Garnier, H.

H Hacking, I. Häggström, J. Hatano, G. Hawthorne, L-A. Hedegaard, M. Helme, S. Herbert, G. Heritage, J. Hiebert, J.

4, 20 189, 198 232, 235 231, 235 217, 235 16, 19, 221, 222, 234, 235 19, 201, 214, 234 167, 182 2, 5, 10, 16, 17, 20, 21, 23, 26, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39,

AUTHOR INDEX

40, 41, 43, 44, 45, 54, 71, 108, 121, 127, 129, 133, 140, 144, 145, 150, 163, 185, 186, 187, 198, 199 Hino, K. 90, 104 Høines, M. 19, 20, 71, 198, 234 Hollingsworth, H. 20, 44, 45, 71, 121, 144, 198 Horne, M. 104 Houang, R. T. 7, 21, 220, 236 Hu, A. 24, 25, 45 Huang, R. 85, 87, 104, 197, 198, 223, 226, 235 Huberman, A. M. 237, 251 Huntley, I. 20, 44 Hyman, R. T. 19

I Inagaki, K.

232, 235

J Jablonka, E.

120, 121, 127, 140, 144, 145 Jackson, P. W. 213 Jacobs, J. K. 20, 44, 45, 71, 121, 144, 198 Johnson, R. B. 233, 235 John-Steiner, V. 236

K Kaiser, G. Kawanaka, T. Keitel, C

Kelly, A. E. Kersting, N.

20, 44 10, 20 1, 4, 19, 20, 25, 27, 30, 44, 71, 104, 145, 198, 213, 232, 234, 250, 251 236 20, 45, 71, 121,

144, 198 4, 19, 20, 27, 44, 104, 145, 213, 234 Klaassen, C. W. J. M. 223, 235 Kliebard, H. M. 19 Knipping, C. 120, 121 Knuth, E. 223, 235 Ko, P. Y. 85, 104, 226, 235 Kwan, T. 104, 235 Kilpatrick, J.

L Lave, J. Lee, S. Y. Lefevre, P. Lelliott, T. Lerman, S. Lesh, R. LeTendre, G. Letendre, K. Leung, F. K. S.

102, 103, 104 19, 142, 145 150, 163 110, 121 2, 20 236 2, 7, 20, 28, 45 145 19, 44, 104, 197, 198, 223, 234, 235, 236 Lewis, C. 143, 145 Li, S. 88, 105, 198, 232, 234, 235 Lijnse, P. L. 223, 235 Liljestrand, J. 169, 170, 182 Lin, E. 25, 45 Lindblad, S. 10, 21 Lloyd, G. M. 212, 213, 230, 235 Lloyd-Jones, G. 230, 235 Lobato, J. 62, 71, 78, 85, 88, 104, 127, 145, 228, 229, 235 Lokan, J. 17, 20 Lopez-Real, F. J. 236 Lovitt, C. 69, 71 Lubisi, C. 21, 104, 213, 234 Luna, E. 20, 44

M Ma, L. 185, 198 MacKinnon, M. M. 231, 236 255

AUTHOR INDEX

Malcolm, C.

21, 57, 104, 213, 234 Manaster, A. 20, 45, 71, 121, 144, 198 Manaster, C. 20, 45, 71, 121, 144, 198 Margetson, D. 230, 235 Martin, M. O. 21 Marton, F. 14, 165, 166, 181, 182, 188, 197, 198, 199 Mason, J. 230, 235 McDonough, A. 104 McKnight, C. C. 7, 21, 220, 236 Mehan, H. 10, 20, 170, 182 Melander, H. M. 182 Mesiti, C. 15, 19, 127, 145 Miles, M. B. 237, 251 Mok, I. A. C. 15, 20, 22, 85, 104, 112, 148, 163, 166, 182, 188, 197, 198, 199, 226, 235 Moll, L. C. 235 Morris, P. 15, 20, 166, 182, 186, 198, 199 Mousley, J. 19, 234 Mullis, I. V. S. 5, 21

P Pang, M. F. Pansegrau, P. Paris, S. Park, K. Patrick, J. Peressini, D. Perez Prieto, H. Perry, M. Piaget, J. Pierce, R. Pong, W. Y. Pontecorvo, C. Pournara, C. Püschel, U.

R Remedios, L. J. Richardson, V. Robitaille, D. F. Roche, A. Romberg, T. A. Runesson, U. Ryan, D.

S Sahlström, F. N Naidoo, A. Nelson, B. S. Nguyen, H. T. Nisbett, R. E. Niss, M. Novotná, J. Nuthall, G.

208, 213 85, 104, 226, 236 223, 236 243, 251 220, 221, 235 235 10, 19, 170, 182

O O'Connor 21 Ohtani, M. 230, 235 Onwuegbuzie, A. J. 233, 235

256

165, 182 121, 121 19, 235 197, 198 10, 19 223, 235 182 143, 145 169, 182 104 186, 198 182 202, 213 121

231, 235 228, 236 24, 44 104 201, 213 165, 166, 182, 188, 198, 199 2, 19, 29, 44

10, 13, 20, 21, 168, 180, 182 Säljö, R. 175, 182 Sandig, B. 121 Schmidt, W. H. 7, 21, 220, 236 Scholz, R. 235 Scribner, S. 236 Seah, L. H. 104, 144, 223, 234, 236 Sekiguchi, Y. 41, 45, 143, 145 Sethole, G. 208, 213, 221, 236 Shapiro, B. 2, 19, 29, 44 Shavelson, R. J. 12, 20, 21 Shepard, L. A. 9, 21 Shimizu, Y. 1, 19, 25, 30, 34, 39, 40, 43, 44, 44,

AUTHOR INDEX

45, 71, 104, 127, 128, 145, 198, 232, 234, 236, 250, 251 Shulman, L. 70, 71 Simon, H. A. 45, 71, 145, 224, 229, 236, 243, 251 Simon, M. 45, 71, 145, 224, 229, 236, 243, 251 Sinclair, J. 10, 21 Skovsmose, O. 201, 213, 214, 221 Smith, F. L. 19, 20, 21, 45, 71, 121, 144, 198, 235 Smith, M. 19, 20, 21, 45, 71, 121, 144, 198, 235 Smith, T. 19, 20, 21, 45, 71, 121, 144, 198, 235 Souberman, E. 236 Steffe, L. P. 229, 236 Stephens, M. 16, 19, 20 Stephens, S. W. 16, 19, 20 Stevenson, H. W. 141, 142, 145 Stigler, J. W. 2, 10, 16, 17, 20, 21, 23, 26, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 45, 54, 71, 121, 127, 133, 140, 141, 142, 143, 144, 145, 187, 198, 199 Stokes, S. F. 230, 236 Straesser, R. 235 Strauss, A. 149, 163 Svan, K. 220, 221, 236

T Tall, D. Taylor, N. Telese, J. A. Teppo, A.

163 208, 214 108, 121 19, 44, 104, 234, 251 Thompson, P. W. 229, 236 Thorsten, M. 2, 21

Towne, L. Tseng, E. Tsuchida, I. Tsui, A. B. M.

12, 20, 21 20, 45, 71, 121, 144, 198 143, 145 165, 182, 188, 198

U Ulep, S.

236

V Valero, P. 201, 213, 214 Valverde, G. A. 7, 21, 220, 236 Van den Heuvel-Panhuizen, M. 161, 163 Vendler, H. 17, 21 Vincent, J. L. 234 Vithal, R. 213, 221, 236 von Glasersfeld, E. 226, 236 Vygotsky, L. S. 182, 217, 218, 235, 236

W Wang, J. Warfield, J. Wearne, D. Weingarten, R. Wellman. H. M. Wenger, E. Westbury, I. Whitehill, T. L. Wiley, D. E. Williams, G. Wilson, J. M. Wilson, T. D. Winkelmann, B. Wiseman, A. Wittrock, M. C. Wong, N. Y. Wood, T.

24, 25, 45 85, 104, 226, 236 20, 45, 71, 121, 144, 198 121 235 13, 21, 74, 102, 103, 104 24, 45 230, 236 7, 21, 220, 236 18, 21, 64, 71 223, 236 243, 251 235 2, 7, 20, 45 20 153, 198, 232, 235 85, 104, 226, 236

257

AUTHOR INDEX

Y Yackel, E. Yoshida, M. Young, R. F.

16, 21 6, 20 90, 223, 236

Z Zhang, D.

15, 21

258

SUBJECT INDEX

A Accounts 12, 218, 243 Complementary 12-15, 237 Participant 19, 87, 213 Reconstructive 8, 18, 23, 25-26, 29, 75, 87, 109, 238-239 Achievement 2-6, 15-17, 24-25, 27, 54, 76, 108, 185-187, 216, 220, 223, 231, 246-247

Autonomy/Autonomous 202

B Behaviours 11. 132-138, 141 Beginning of the Lesson 43, 47-71, 108, 135-136, 139, 141, 144, 173, 194

Activities (see Classroom) Agency 3, 85, 102-103, 222-223, 226, 238 Aggregration 25 Algebra 111, 138, 149-150, 158-160, 175, 195, 212, 239 System of Linear Equations 30, 113, 130, 137, 154-159, 185199 Appropriation by the Teacher 120 Argumentation 111, 120, 175, 219 Assessment 4-5, 24, 54-55, 99, 107, 168, 179, 185, 206-207

C China, Chinese 2, 9, 15, 25-26, 84, 87-88, 128, 141, 143-144, 185, 189, 220-223, 226, 230, 232, 238-239, 246 Class size 3, 5, 185-186, 197, 211, 216 Classroom Activities 31-40, 67, 87, 245 Management 37 Practice 1-29, 31, 40, 42, 43, 47-49, 54, 73-75, 79, 81, 85, 87, 94, 96-97, 103, 107-108, 110, 112, 119-120, 128-129, 141142, 144, 202-204, 215-219, 222-223, 226, 228, 230-233 Cognition 11, 69-70, 206, 209, 222

Australia, Australian 9, 18, 47, 57, 69, 82, 85, 96, 128, 144, 147, 153, 203, 208-211, 238, 246

Communication 108, 180

Authentic, Authenticity 154, 230

Competence 2, 8-9, 30, 65, 70, 103, 185, 197, 202, 238, 247

Authority 47, 107-108, 121

Competition 178, 216 259

SUBJECT INDEX

Complementarity 12-15, 41, 216, 233, 238 Congruence 30 Conjecture 6, 29, 78, 229 Connections 3, 5, 25, 51, 53, 60, 70, 77, 131-134, 138-139, 141, 156, 158, 160-162, 185, 206, 208210, 221-222

Dichotomies 215-236 Discourse 9, 11, 169, 217-218, 223, 228, 230 Discussion 3, 28-29, 42, 51-53, 60, 63, 66, 76, 85, 90, 100, 110111, 114, 118, 130, 148, 152, 156-159, 167, 187-196, 207, 216, 223-230, 241

Constant Comparison Method (see Grounded Theory)

E Encouragement 82, 84, 102

Constraints 1, 3, 8, 13, 18, 74, 96, 102, 167, 201-214, 217, 238

Evaluation 3, 119-120, 161, 176

Content 2, 5-6, 7, 9, 15, 23, 25, 28, 30, 45, 51-52, 54, 60, 69-70, 86, 93, 119, 143, 149, 152-153, 163, 167, 170, 179-181, 185199 Context 5, 9, 11, 16-17, 53-54, 66, 69-70, 141, 148-150, 153-154, 161, 165, 170, 175-180 Conversations 30, 64, 76-78, 85, 88, 94, 112, 115, 173, 178, 229, 237, 240 Culture 1, 3, 7-18, 26-27, 31, 42, 73, 76, 79, 94, 96, 107, 127-129, 140-144, 147, 149, 170, 179, 189, 215, 217, 231-233 Czech Republic 9, 108, 238

D Definitions 32, 53, 78, 138, 157, 210, 226 Dialogue 11, 108, 169 260

Expectations 57, 109, 111, 154, 202, 219 Expression 77-78, 229, 247 Mathematical 132, 150-151, 160, 195-197

F Function 6, 14-15, 26-27, 42-44, 47, 51-52, 55, 69, 73-79, 96-98, 102-103, 107-121, 127-145, 147-148, 161, 219, 221, 226229 Functions 30, 52, 122, 127-145, 158, 168, 170, 173, 220

G Geometry 139, 212 Germany, German 2, 9, 16-17, 23, 26, 28, 30-32, 36-37, 39, 41, 84-85, 91, 99, 107-109, 111112, 117-118, 120, 127-128, 140, 144, 147, 150-151, 238239, 247

SUBJECT INDEX

Graphs, Graphing 25, 98, 114, 118, 122-126, 138, 149-150, 158160, 170-172, 177-180, 210, 220 Grounded Theory 149 Guiding 58, 62, 73-105, 115, 228

H Homework 5, 31-35, 40-41, 51-52, 55, 59, 63-67, 76-78, 81, 112, 118-119, 123, 140, 152, 160, 216 Hong Kong 5, 6, 9, 13, 14, 18, 73, 75, 82, 86-88, 90, 94, 98, 107109, 112-113, 115, 117-119, 128, 139-141, 14, 147-149, 189, 197, 220, 223, 225-226, 228, 230, 238

I Iconic Sequences (see Sequences)

132, 134, 140-142, 171, 175, 181, 188, 203, 221, 225, 230, 237 Interaction 5-6, 10, 28, 49, 70, 76, 88-89, 99, 207-208, 112, 116, 158, 160, 165-183, 187-188, 216, 218, 222-223, 241, 247 Student-initiated 94, 100 Teacher-initiated 94 International Comparisons 1, 4, 24, 75, 94, 128, 140 Israel 9, 220, 238-239

J Japan, Japanese 2, 5-6, 9, 16-17, 23, 25-26, 28-32, 34, 37-43, 47, 53, 57-58, 69, 76, 79, 84, 89, 94, 98-99, 127-129, 133, 140-144, 147, 150, 152, 162, 218, 220, 230, 238, 246, 249 Justification 17-18, 69, 120, 158, 220

Individualisation 168, 178 Input 79, 150 Input-Output 185, 225 Instruction 6-7, 10-14, 16, 27-29, 39, 42-44, 47-71, 73-105, 112, 114115, 142, 168, 170-171, 178, 181, 186-187, 189, 192, 208, 211, 217-219, 222-223, 225226, 228 Instructional Practice 4, 10, 16, 24, 28, 228 Instructional Unit 14, 23, 26, 28

K Kikan-Shido 13, 58, 73-105, 151152, 219, 228 Korea, Korean 9, 25, 220, 238-239, 246 Knowledge 9, 12-14, 16, 78, 84-86, 89, 96, 103, 107-108, 111, 118120, 142, 150, 162, 185, 195197, 201, 206-209, 219, 222233

Intention 6, 28, 30, 39, 41-42, 53, 59-60, 65, 74-75, 81, 84, 94, 96, 99, 107, 109, 117, 127, 130, 261

SUBJECT INDEX

L Language 5, 217-218, 228, 247 Difficulties 206 Instructional 88, 115 Mathematical 160, 173

Meaning Construction of 15, 107 Negotiation of 16, 231

Learning Tasks 147-163

Mistakes 87, 90, 110, 119-120, 123

Lesson Beginning (see Beginning of the Lesson) Climax (Yamaba) 143 Components 47-71 Events (see below) Pattern 23-45 Structure 23-45

Monitoring 73-105, 111, 113, 115, 228

Lesson Events Assigning Homework 31, 33, 40, 41 Beginning the Lesson (see Beginning of the Lesson) Demonstration 32-33, 45, 77, 92, 109, 113, 118, 148, 150-152 Exploratory 149 Kikan-Shido (see Kikan-Shido) Matome (see Matome) Presenting 32, 40, 110-111, 115, 117, 127, 169 Review 31-35, 40-43, 47, 49, 51-65, 110, 115, 127, 129, 133, 138-141 Seatwork (see Seatwork) Student(s) at the Front 107-126 Warm-Up 33, 49, 51-52, 56, 60-63, 70, 110, 115-116, 118, 159 Listening 64, 90, 118, 134-135, 203, 213, 216, 230-233

M Matome 127-145

262

Misconceptions 90, 244

Motivation 10-11, 30, 41, 69, 82, 84, 237, 243

N Norms 24, 27, 96, 103, 202, 247 Notebooks 152, 159

O Object of Learning 148, 154, 159, 175, 177, 179, 180-181, 188191 Objectives 147, 155, 159, 161 Organisation 49, 55, 63, 76, 78 Behavioural 9 Classroom, Classroom Work 2, 6, 165, 167-168, 180-182 School 2

P Participation Patterns of Participation 1, 3, 4, 13, 49, 73-74, 76, 93, 102-103, 168-169, 219 Student 13, 67, 76, 102 Teacher 47

SUBJECT INDEX

Patterns of Interaction 181 of Participation (see Participation) Pedagogy 9, 103, 110, 215, 226, 228

Risk 8, 118-120, 180, 212 Roles of Teacher 66-67, 107, 128, 255, 202, 212, 217 of Student 55, 65, 67, 85, 107, 202, 222

PISA 4-5, 185 Perception 5, 11, 18, 70, 108, 114, 117, 134-135, 247

S Seatwork 2, 29, 31, 33, 35, 110, 113, 115-11, 123-124, 168

Perspective, Student 96, 187 Selecting Work 77, 78, 89 Philippines 9, 238 Sequences, iconic 47-71 Proof 5, 111, 119, 122, 140, 142, 219-220

R Race 24-25 Rationale 6, 8, 157, 166, 209

Shanghai 9, 13-14, 18, 73, 75, 82, 84-87, 89, 98-99, 128, 132, 138, 140-141, 144, 147-155, 159161, 189, 197, 220-221, 223, 225-226, 228-230, 238 Sharing 7, 60, 65, 111, 144-115, 117, 119, 151, 160, 162

Reflection 15, 53, 60-62, 65, 76, 130, 133-134, 137, 141, 143, 162

Singapore 9, 147, 150-152, 162, 238

Reform 85, 108, 201-203, 207-208, 211-212, 222-223, 229, 231

Social Constructionism 166, 175 Interaction 8-9, 11, 49, 74, 102, 107, 165-167, 179, 219

Relationships Mathematical 167, 173, 179 Relevance 42, 69, 198, 219-221 Representations 77, 138, 158-160, 170, 172, 180

Socio-economic Status 25 South Africa 9, 18, 110, 201-203, 205, 207-212, 221-222, 238239 Space of Learning 180-181

Research Design 3-4, 12, 14, 23, 2931, 103, 219, 237-251

Student-centred 187, 216, 222-223, 226, 230-231, 233

Revision 10, 139, 190-191, 194

263

SUBJECT INDEX

Sweden 2, 9, 14, 47-48, 69, 165, 168-170, 173, 175, 177, 189, 197, 220, 238-239

T Teacher-centred 85, 186, 216, 222223, 226, 230, 233 Teaching Approach 107 Method 39, 211 Patterns 28 Practice 2, 7, 10, 15-16, 18, 26, 28-29, 185, 203, 228 Process 5, 185-186 Strategies 150, 161 Style 142, 186, 241 Unit 39, 127-128, 132-133 Textbook 49, 111, 131-134, 137138, 141, 167-171, 177, 181, 187, 191-193, 196, 205-207, 216, 225 TIMSS 2, 4-5, 7, 10, 15-17, 23-26, 28, 30-31, 33, 35, 40-41, 43, 54, 96, 108, 115, 127, 133, 185187, 246-247 Topic Sequence 35-36, 41-43, 48, 92 Triangulation 107 Typification 24, 26-29, 96

U Understanding Conceptual 177 Mathematical 85, 108 Shared Understanding 107 through Dialogue 108 US, USA 2, 9, 16, 18, 23 25-26, 29264

35, 37, 40-43, 47-48, 51-57, 6064, 69, 79, 81-82, 84, 85, 89, 98, 100-101, 108-110, 115-120, 124-125, 128, 138-139, 141, 144, 155, 158-160, 165, 167170, 172-174, 177, 179-181, 185, 203-204, 206-207, 209212, 238, 246, 248

V Variation Theory 165, 173, 180, 188

W West, Western 79, 85, 103, 148, 215, 222-223, 226, 229, 231-232 Word Problems 141, 147, 150, 153, 155, 157, 221

Y Yamaba (see Lesson Climax)

Further Reading: Mathematics Classrooms In Twelve Countries The Insider’s Perspective Editors:

David Clarke, University of Melbourne, Australia Christine Keitel, Freie Universitat Berlin, Germany Yoshinori Shimizu, University of Tsubuka, Japan

Paperback ISBN 90-77874-95-X Hardback ISBN 90-77874-99-2

This book reports the accounts of researchers investigating the eighth grade mathematics classrooms of teachers in Australia, China, the Czech Republic, Germany, Israel, Japan, Korea, The Philippines, Singapore, South Africa, Sweden and the USA. This combination of countries gives good representation to different European and Asian educational traditions, affluent and less affluent school systems, and mono-cultural and multi-cultural societies. Researchers within each local group focused their analyses on those aspects of practice and meaning most closely aligned with the concerns of the local school system and the theoretical orientation of the researchers. Within any particular educational system, the possibilities for experimentation and innovation are limited by more than just methodological and ethical considerations: they are limited by our capacity to conceive possible alternatives. They are also limited by our assumptions regarding acceptable practice. These assumptions are the result of a long local history of educational practice, in which every development was a response to emergent local need and reflective of changing local values. Well-entrenched practices sublimate this history of development. The Learner’s Perspective Study is guided by a belief that we need to learn from each other. The resulting chapters offer deeply situated insights into the practices of mathematics classrooms in twelve countries: an insider’s perspective.

Other Series of Interest: New Directions in Mathematics and Science Education Series Editors:

Wolff-Micheal Roth, University of Victoria, Canada Lieven Verschaffel, University of Leuven, Belgium

Editorial Board: Angie Calabrese-Barton, Teachers College, New York, USA Pauline Chinn, University of Hawaii, USA Lyn English, Queensland University of Technology, Australia Brian Greer, Portland State University, USA Terezinha Nunes, University of Oxford, UK Peter Taylor, Curtin University, Perth, Australia Dina Tirosh, Tel Aviv University, Israel Manuela Welzel, University of Education, Heidelberg, Germany Rationale: Mathematics and science education are in a state of change. Received models of teaching, curriculum, and researching in the two fields are adopting and developing new ways of thinking about how people of all ages know, learn, and develop. The recent literature in both fields includes articles focusing on issues and using theoretical frames that were unthinkable a decade ago. For example, we see an increase in the use of semiotics as a theoretical tool to understand how students learn, how textbooks are written, and how different forms of knowledge are interconnected. Science and mathematics educators also have turned to issues such as identity and emotion as salient to the way in which people of all ages display and develop knowledge ability. And they use dialectical or phenomenological approaches to answer ever arising questions about learning and development in science and mathematics. The purpose of this series is to invite and encourage the publication of books that are close to the cutting edge of both fields. The series will be a leader in contributing cutting edge work—rather than out-of-date reproductions of past states of the art—shaping (producing) both fields as much as reproducing them, thereby closing the traditional gap that exists between journal articles and books in terms of their salience about what is new. The series is intended not only to foster books concerned with knowing, learning, and teaching in schools but also with learning in the two fields across the lifespan (e.g., science in kindergarten; mathematics at work); and it is to be a vehicle for publishing books that fall between the two domains—such as when scientists learn about graphs and graphing as part of their work.

Other Titles in Mathematics Education: Theorems in School From History, Epistomology and Cognition to Classroom Practice Paolo Boero, Università di Genova, Italy (ed.) Paperback ISBN 9077874-21-6 Hardback ISBN 90-77874-22-4 Handbook of Research on the Psychology of Mathematics Education Past, Present and Future Angel Gutiérrez, Universidad de Valencia, Spain Paolo Boero, Università di Genova, Italy (eds.) Paperback ISBN 9077874-19-4 Hardback ISBN 90-77874-66-6 Traveling Through Education Uncertainty, Mathematics, Responsibility Ole Skovsmose, Aalborg University, Denmark Paperback ISBN 9077874-03-8 Hardback ISBN 90-77874-67-4