Researching Mathematics Classrooms: A Critical Examination of Methodology

RESEARCHING MATHEMATICS CLASSROOMS

Recent Titles in International Perspectives on Mathematics Education Leone Burton, Series Editor Multiple Perspectives on Mathematics Teaching and Learning

Jo Boaler, editor

RESEARCHING MATHEMATICS CLASSROOMS A Critical Examination of Methodology Edited by Simon Goodchild and Lyn English

International Perspectives on Mathematics Education Leone Burton, Series Editor

Library of Congress Cataloging-in-Publication Data Researching mathematics classrooms : a critical examination of methodology / edited by Simon Goodchild and Lyn English. p. cm.—(International perspectives on mathematics education, ISSN 1530-3993) Includes bibliographical references and index. ISBN 1–56750–666–6 (alk. paper) 1. Mathematics—Study and teaching—Research—Methodology. I. Goodchild, Simon, 1950– II. English, Lyn D. III. Series. QA11.2 .R47 2002 510.71—dc21 2002072544 British Library Cataloguing in Publication Data is available Copyright © 2002 by Simon Goodchild and Lyn English All rights reserved. No portion of this book may be reproduced, by any process or technique, without the express written consent of the publisher. Library of Congress Catalog Card Number: 2002072544 ISBN: 1–56750–666–6 ISSN: 1530–3993 First published in 2002 Praeger Publishers, 88 Post Road West, Westport CT 06881 An imprint of Greenwood Publishing Group, Inc. www.praeger.com Printed in the United States of America The paper used in this book complies with the Permanent Paper Standard issued by the National Information Standards Organization (Z39.48–1984). 10

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Contents

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Tables and Figures

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Series Foreword Leone Burton

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Introduction Simon Goodchild and Lyn English

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Methodology and Methods in Mathematics Education Research: Where Is “The Why”? Leone Burton Students-Teacher Interactions and the Development of Students’ Mathematical Thinking Maria Manuela David and Maria da Penha Lopes Reaction by Anne Watson

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Exploring Students’ Goals in Classroom Activity Simon Goodchild Reaction by Barbara Jaworski

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Special Educational Needs in Mathematics—A Problem Developed in School? Nora Linden Reaction by Cyril Julie

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A Methodology of Classroom Teaching Experiments Kay McClain Reaction by Marja Van den Heuvel-Panhuizen

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Context and Content: What Are Student Teachers Learning about Teaching Mathematics? Ruth Shane Reaction by Gilah C. Leder Longitudinal Measurement of Student Motivation and Achievement in Mathematics Shirley M. Yates Reaction by Wendy Keys

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Afterword: Applying the Theory to the “Practice” in This Book Leone Burton

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Index

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About the Contributors

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Tables and Figures

TABLES 6.1 6.2 6.3 6.4 7.1 7.2

Example of Categories of the Nature of Children’s Responses Children’s Interview, Sample Scoring Sample Questions from the Lesson-Specific Interview Protocol Sample Questions from the Summary Interview Protocol Numbers of Students by Grade Level and Gender at Time 1 and Time 3 Latent Variable Order and Acronyms for the PLSPATH Analysis

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FIGURES 2.1 2.2 2.3 2.4 5.1 5.2 5.3 5.4 5.5 5.6 5.7 7.1 7.2

Interactions upon Which the Lesson Observation Focused Teacher’s Example of a Quadrilateral Jair’s Example of a “Very Crazy Quadrilateral” Jair’s Second Example of a Quadrilateral The Design Research Cycle The Interpretive Framework Elaborated Interpretive Framework A Conjectured Learning Trajectory for Three-Digit Addition and Subtraction Symbolizing Unpacking a Roll The Inventory Form for the Candy Factory An Addition Problem Posed on the Inventory Form An Overview of the Longitudinal Plan of the Study Significant Paths in the PLSPATH Model

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Series Foreword

This is a new undertaking for books in mathematics education and, as such, will, I hope, make a major contribution to the field. Mathematics education has always been a very eclectic discipline when it comes to the research methods that it utilizes, but a focus upon these methods and the rationale behind their choice and use, the researcher’s methodology, is a new venture. Indeed, the differentiation of methodology and methods, so important to the researcher’s approach to their study, but also to a reader’s engagement with a research report and the trust that they can place in the results, has not been at the forefront of publications in mathematics education. It is therefore with delight that I welcome this volume as one in the series, International Perspectives on Mathematics Education. The editors, Simon Goodchild and Lyn English, have devised a very supportive and interactive way of compiling their book, a strategy that I know has been appreciated by their authors. To implement their approach required the collaboration of a number of colleagues across the world, which they secured, and the book is stronger for their contributions. I am sure that Simon and Lyn are extremely grateful, as am I, for their willingness to engage with what has been a novel way of compiling an edited collection. But the interaction portrayed in the pages of the book is also consistent with its focus and the dialogue that Simon and Lyn, and their contributors, are hoping to stimulate into this aspect of research and an open and challenging critique of methodology and method. It is, of itself, a model of the kind of exchange that would be so productive in our discipline. Consistent with the series, this book has an international group of authors and commentators, including those from countries less frequently encountered in the mathematics education literature. It is my hope that the book will be ap-

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preciated by those whose research, until now, has not involved them in a reconsideration of methodology, as well as by less-experienced researchers who are setting out on a new career in mathematics education research and who might find in these pages some arguments to challenge and engage them. Leone Burton Emeritus Professor of Mathematics Education University of Birmingham, United Kingdom

Introduction

This book is a product of the Classroom Research Group (CRG) of the International Group for the Psychology of Mathematics Education (PME). Each year at its annual conference, PME brings together researchers in mathematics education from across the globe. The conference program includes the presentation of research reports, discussion groups, working groups, and lectures. The Classroom Research Group was, until very recently, a feature of the annual conference and for many years had supported mathematics education researchers from new researchers to those more experienced. The CRG provided a forum for sharing ideas and approaches to researching classrooms in a critical and supportive context. Many, not least the contributors to this book, can attest to the personal benefit they have gained from the group. Researching Mathematics Classrooms: A Critical Examination of Methodology was conceived when PME held its annual conference in Stellenbosch, South Africa, in 1998. The aim of the book is to communicate ideas from the CRG to the broader mathematics education community. The book is different from others in the field because the editors and contributors have invited other researchers to review the enclosed chapters and have then included their reviews within the chapter. Some of the authors chose to respond to the critiques and their responses are included as the final component of their chapter. Consequently, each chapter comprises two, or sometimes three, parts: first, the description of a piece of research; second, a critical review; and, in some cases, this is followed by the author’s response to the review. The members of the CRG who have contributed to this book took a courageous step when they decided to present an account of their own classroom research alongside informed critique by fellow researchers. This at-

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titude reflects the spirit of open inquiry, support, and critique that is characteristic of PME and the CRG. The authors describe their research in a self-critical manner, presenting it, in the words of Oliver Cromwell, “warts and all.” The authors claim, first of all, that they are being honest with the reader; our attitude is that the “perfect research project” has not yet been conceived. Second, the authors are concerned to inform and educate, and our hope is that this very public critique of our work will help to illuminate many of the subtleties of the methodology of classroom research. Underlying our approach to this book is the assumption that research methodology is not merely a matter of choosing methods and research design, as one would select commodities from a supermarket shelf, so to speak. Methodology is about the underlying basis for the choices that are being made; it includes a consideration of the researcher’s beliefs, attitudes and values, the research questions being explored, the answers being sought, and crucially, the nature of the key informants together with their social and cultural environments. In the opening chapter Leone Burton develops this articulation of methodology and explains why its disclosure by researchers in reports of their work is so important. Leone Burton then provides a brief afterword in which she reviews the chapters from the position she has presented in the opening chapter. The chapters bring together a wide variety of methods that are applied to exploring what happens in classrooms. In some cases the research activity takes place within the classroom; in others, the research activity is remote from the regular activity of the classroom. The chapters describe the use of interviews, conversations, observation, and questionnaires as means of gathering data. Some chapters describe approaches that aim at minimal disturbance of regular classrooms, while others describe intervention approaches that focus on changing the teachers’ strategies or the design of the teaching and learning program. The chapters include examples of the use of both qualitative and quantitative methods. Maria Manuela David and Maria da Penha Lopes provide, in chapter 2, a window into their ongoing inquiry into the way in which students’ understanding of mathematics and thinking skills develop as a result of their experiences in Brazilian mathematics classrooms. The authors of this chapter offer readers an insight into how the research process changes their own understanding and knowledge and how this development is accommodated in the emergent design of the study. The chapter exposes the development of the research in response to the challenges the researchers faced as a result of their concurrent engagement in the study and related literature. The authors do not present a discrete and conclusive piece of research, and they do not claim this. The value of the chapter lies in its description of the dialectic between emerging knowledge and development of the research focus and methods adopted in fieldwork and data analysis. In chapter 3 Simon Goodchild describes his attempt to explore activity in a mathematics classroom in the United Kingdom. He does so in a way that recognizes the dialectic that exists between the individual/psychological and

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social/cultural contexts of the classroom. He adopts an ethnographic-style approach that is intended to cause as little disturbance as possible to the routine of a regular classroom. The account is worthy of attention for a number of reasons. These include the way that theory is developed to construct a framework for exploring the classroom and the careful manner in which Simon Goodchild tries to demonstrate the congruence among theory, the research focus, the characteristics of the learners, the approach adopted and, significantly, the collection of values, beliefs, and experiences that he brings to the inquiry. In chapter 4 Nora Linden describes her investigation into the early learning experiences of young Norwegian children who, in their first year of school, have been identified as in need of “special education.” She explains why, in her study, it is necessary to collect data through conversations outside the classroom. Eliciting information from children who have learning difficulties presents special challenges in her research, and hence the account is of value because it explains how these challenges were met. The chapter is also interesting because Nora Linden recognizes that different informants, namely, teachers, special needs teachers, parents, and of course the children themselves each construct their own understanding of what is happening in the classroom. An essential part of the account of the methods adopted in this study is the justification of using conversation as a means of gathering data and its subsequent analysis that can be claimed as enabling a trustworthy interpretation of the way participants construe their classroom experiences. A classroom teaching experiment in the United States is presented by Kay McClain in chapter 5. Identifying her approach as “design research,” Kay McClain describes how the research team collaborates with the teacher to transform learning within the classroom. This transformation is engineered through the development of students’ behavioral norms as well as the materials used in lessons. The research team therefore starts with an idealized view of the classroom and an appropriate route to the students’ attainment of the learning objectives. The ideal is then realized through an iterative cycle of implementation and modification in response to concurrent monitoring and evaluation. There is a well-developed genre of this type of research, and its usefulness in the development of curriculum materials is widely recognized. Kay McClain’s chapter makes an important contribution to the book because of its particular focus on the research methodology, which is rarely discussed to any depth in the research reports arising from classroom teaching experiments. In chapter 6 Ruth Shane also describes an approach where the researcher set out to modify the practice within classrooms. Her focus is on the development of students during initial teacher education courses at a teachers’ college in Israel. Ruth Shane is concerned about the impact of school teaching experiences on the development of students’ pedagogical awareness. She starts from her conjecture that the enlightened messages about reformed practices of teaching and learning mathematics may be swamped by students’ experiences in traditional classrooms. The researcher then attempts to influence the practice in classrooms

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by providing the regular teachers and the student teachers with a short inservice workshop in which materials for communicating mathematics are developed. From this point on the researcher observes and monitors the practice and students’ emerging awareness without further intervention. The chapter thus makes a special contribution to this book because of its focus on the role, values, attitudes, and beliefs of teachers in the classroom. An approach to classroom research that contrasts sharply with the other contributions is provided by Shirley M. Yates in chapter 7. Her research is in response to an appeal from an Australian primary school principal to reveal the relationship between students’ attitudes and their attainment in mathematics. This approach contrasts with others in this book, where students, classrooms, and schools respond to the researchers’ interests, motivation, and initiative. Shirley Yates’s research relies on the use of psychometric instruments, including tests and questionnaires, and powerful statistical analysis. The chapter represents an important strand of educational research. Within the context of the other chapters of this book, it provides an opportunity for readers to compare and contrast the arguments presented by authors espousing, on the one hand, naturalistic and qualitative methods and, on the other, positivist and quantitative methods. The chapters in this book provide an opportunity to consider claims for different research methods. The book is intended to be provocative rather than authoritative: we hope it will lead readers to reflect on the key issues that are at the heart of research methodology—at least the methodology of classroom research. We would, however, advise readers to begin with a careful study of Leone Burton’s opening chapter, then read each of the other chapters and form an evaluation of the methodologies described before reading either the reactors’ comments or Leone Burton’s afterword. If this book helps to illuminate the nature of the methodology of classroom research and how it might be reported, if it helps others to develop their own research activity and contribute to the knowledge of the processes of teaching and learning mathematics, if it helps those who are required to critically evaluate research, then the book will have achieved its purpose. Simon Goodchild and Lyn English

Chapter 1

Methodology and Methods in Mathematics Education Research:Where Is “The Why”? Leone Burton Discussions of method (techniques for gathering evidence) and methodology (a theory and analysis of how research should proceed) have been intertwined with each other and with epistemological issues (issues about an adequate theory of knowledge or justificatory strategy). (Harding, 1987, p. 2)

INTRODUCTION In many of the Ph.D. theses that I have read, the chapter headed Methodology has dealt, in fact, with the methods used by the researcher to undertake their research. Likewise, in the majority of articles in journals and chapters in books, a description is provided of “how” the research was done but rarely is an analysis given of “why” and, more particularly, out of all the methods that could have been used, what influenced the researcher to choose to do the research in the manner described. It comes as a refreshing change when one reads an author’s reflections on what impact such choices might have had on the research outcomes. This is despite a growing literature and recognition of the implication of values and beliefs in all research. With Kathleen Lynch, I agree that “the idea that we accept or reject theories on purely rational terms is untenable; many theories are either rejected or accepted before the evidence is presented” (1999, p. 5) and “many of the values and assumptions which we hold most dearly are those that are least likely to be subjected to critical reflection” (p. 5); nonetheless, we can “name the conditions and contexts in which we write. By identifying the parameters of our intellectual domain, we can at least identify some of the limiting conditions of our own analysis” (p. 5). As a reader of research reports, however, I rarely know anything of the researcher’s methodology, in the sense outlined in the quote that opens this chapter; it is consequently very difficult for me to untangle and evaluate, in terms of

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methodology, the author’s description of their study. For example, to quote Sandra Harding again: “Feminist researchers have argued that traditional theories have been applied in ways that make it difficult to understand women’s participation in social life, or to understand men’s activities as gendered (vs. as representing ‘the human’)” (1987, p. 3). Examples of the methodological insertion of values that distort outcomes can be found in a number of studies that used a single-sex population to arrive at unwarranted population generalizations. (See, for example, the critique by Gilligan, 1982). The result has been that those gendered research results presented as species-wide would have been more robust, indeed authentic in the sense of male-focused, if the authors reporting on them had been required to provide the kind of theoretical contextualization that informed of the researcher’s methodology. This might have impelled such researchers into awareness of the limitations imposed by their study on their claims. From a different field, Evelyn Fox Keller (1983), in her biography of Nobel Prize winner Barbara McClintock, made it clear how, by adopting a noninterventionist approach to the study of growth in corn, observations were possible which had not been available to those pursuing traditional modes of study. This approach sprang from Barbara McClintock’s “identification” with the corn she was studying; that is, she chose a methodological approach to her work that contrasted with the strategies associated with “scientific objectivity,” which had been conventional within her discipline. Even the august U.S. National Academy of Sciences has acknowledged that “human values cannot be eliminated from science and they can subtly influence scientific investigations” (National Academy of Sciences, quoted in Harding, 1993, p. 342). Furthermore, “values do contribute to the motivation and conceptual outlook of scientists”(p. 343). However, Patti Lather pointed out that: Research programs that disclose their value-base have been typically discounted . . . as overly subjective and, hence, “non-scientific.” Such views do not recognize that scientific neutrality is always problematic; they arise from an objectivism premised on the belief that scientific knowledge is free from social construction. (1991, p. 52)

This case, it seems to me, should no longer need arguing. Indeed, Rom Harré emphasized that: Even in the natural sciences the content of a theory is not expressed fully in the explicit discourse of that science. There are tacit beliefs, principles and assumptions, which are essential both to the way in which the theory is understood by its users and to the analysis of that theory by those who wish to comment upon it. (1981, p. 4)

Methodology, in the sense that I am using it, might have come only recently into mathematics education, but the arguments for such an approach have been present in many other disciplines for some time. These include the sociology, history, and philosophy of science, which have responded, especially, to the critiques of feminist scholars. Indeed, in writing about the impact of recent legislation on

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teaching and teacher education, Pat Mahony and Ian Hextall said “it would have been illuminating had the originators of the policies which form the substance of our book made their value positions explicit, described the contexts in which these were formed and the grounds on which they were founded” (2000, p. 2). In my view, this statement is equally applicable to reports of research in mathematics education. I do not believe that there is ever a case where the researcher’s beliefs, attitudes, and values have not influenced a study, nor do I believe that it is possible for a researcher ever to assume that values can be assumed as shared within a “scientific community.” I acknowledge that this, in itself, is a statement of my beliefs and values. In these circumstances, it surprises me, therefore, that the implications of such statements as I have quoted above fail to be noticeable in research publications, both in the lack of transparency and clarity with which researchers provide access to the choices that they make and the reasons for these choices and, indeed, to the language with which they report on their study. The continued prevalence of the third person passive voice in writing ensures that the “I” of the researcher is lost in a presentation that aspires to a pseudo objectivity, but newcomers to research labor under the delusion that such writing is the only acceptable form in which research can be presented. In this chapter, I will set out my own position on methodology and then outline how I see methodology, design, methods, and analysis as fitting together in mathematics education studies. As an afterword, which concludes this book, I will apply this analysis to the chapters in this volume pointing out their strengths, as I see them, as well as where their authors sometimes fail to provide the reader with enough methodological context to enable sound judgments to be made. I will conclude with an appeal to the authors of future research reports.

A POSITION ON METHODOLOGY Let us undertake a thought experiment. Imagine that I am sent a paper to referee that reports on a study done on the relationship between the color of socks worn by pupils in all the schools in Tristan da Cunha on a particular day and their attitudes to learning mathematics. The researcher explains that the conjecture that sock color is a good indicator of attitudes stems from a substantial local psychological literature linking dress choices to attitudes and behaviors. In the case of this paper, the author has tied this work to a conjecture about school mathematics learning. All pupils in schools throughout Tristan da Cunha were asked to complete a very brief questionnaire asking the color of socks they were wearing that day and asking them to indicate, on a Likert scale, their responses to a number of questions on how they felt about learning mathematics. Their mathematics teachers were also asked to provide a rating of the pupils’ expected performance. The researcher could, of course, have interviewed a number of the pupils. Observations could have been done in classrooms. Pupils could have been asked to keep a diary over a week, recording their reactions to activ-

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ities in the mathematics class. Hypothetical story lines could have been presented, orally or in writing, to pupils. Many other strategies could have been chosen to explore attitudes to learning mathematics. In order to come to some conclusions about the results, I need to know:

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why the researcher chose that focus; why the study was designed by the researcher in that way; why alternatives were rejected; what were the questions the researcher was asking; and how the researcher ensured that confidence could be felt in the data gathered and in their analysis of those data.

For example, the researcher does not explain why sock color should be significant in terms of dress choice, whether some of the schools have restrictions on the choice of sock color allowed, whether there has been any attempt to ascertain if the choice of sock that day might have been influenced by other factors, such as the alternative choice yesterday developing holes. But, most of all, I am not told how a Likert-style response on pre-established statements can be confidently taken as representing what the pupils actually feel about learning mathematics, nor indeed, what they understand by the statements they are offered and the choices they are asked to make between them. If it cannot, what has the researcher done to increase my level of confidence? So, the attempt at correlating these two factors is, for me, problematic in the extreme and, of course, a statistical correlation, even if it exists, does not ensure an interpretation of causality. Some might interpret these questions as falling within the orbit of operationalization, of which Lincoln and Guba (1985) have been critical. However, I would claim that to understand the demands of methodology as able to be met through operationalization, is not to understand that I am asking for something far more complex and demanding of the researcher. In the first instance, I am asking for researchers to be clear to themselves about the values, beliefs, and attitudes that are driving the study that they propose to do and to make that clarity visible to the reader. Such clarity is necessary to meet the requirements placed on pursuing research utilizing that methodological stance and helps to reinforce flexibility in coping with changing circumstances. Second, I am making the assumption that research in mathematics education is emancipatory in that its intentions are to empower—pupils, teachers, curriculum designers, policymakers—those who could be users or affected by use of the research. In her discussion of research design, Patti Lather (1991) said: In an attempt to reveal the implications that the quest for empowerment holds for research design, I will focus on three interwoven issues: the need for reciprocity, the stance of dialectical theory-building versus theoretical imposition, and the question of validity in praxis-oriented research. (p. 56)

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Reciprocity is to do with power in the research relationship and transparency in how that power is recognized and curbed in the research design. In a reflexive article on “informed consent,” David, Edwards, & Alldred (2001) exposed the contradictions they encountered in undertaking research on children’s understandings of the parental involvement in school. They concluded: A straightforward notion of children and young people’s right or freedom to choose to participate in social research on the basis of the provision of adequate and appropriate information in the school setting especially seems naïve. On reflection, our understandings of research with children and young people, and as conducted in the school context, were altogether too simplistic at the outset. Their responses to our topics and methodologies, however, forced reappraisal such that we are now able to see the complex relationships between, and contradictory natures of, notions of children and young people’s competence and age, and information and education. We are left with difficult questions about whether consent should, or can, be “informed” or “educated.” (p. 364)

In discussing dialectical theory-building, Patti Lather said: Building empirically grounded theory requires a reciprocal relationship between data and theory. Data must be allowed to generate propositions in a dialectical manner that permits use of a priori theoretical frameworks, but which keeps a particular framework from becoming the container into which the data must be poured. (1991, p. 62)

With respect to validity, Patti Lather warned that: If illuminating and resonant theory grounded in trustworthy data is desired, we must formulate self corrective techniques that check the credibility of data and minimize the distorting effect of personal bias upon the logic of evidence. (1991, p. 66)

Finally, there is the issue of research reporting. Even with restrictions of space often imposed by journals, it is always possible for the author to include a brief statement of their methodology in order to explain the choice of methods, the research design, and indeed to clarify their conclusions and stated implications. To return to my mythical review, as it happens, I “know” that in the 1930s a powerful psychologist spent some time in Tristan da Cunha following up some work he had done previously on the relationship between dress codes and their influence on behavioral choices and that the small but supportive body of work being quoted in the article arose from that original input and the influence it had locally. However, no work is quoted from elsewhere, nor is there any indication that the existence of this local work provoked searches, for example, in the wider English, never mind the non–English language literature. As a referee, I can examine the statistics, establish that the size of sample is, just, large enough to warrant being so called and that the correlations being claimed are justified in terms of the numerical manipulations. But what I cannot do is place confidence in the assertions being made unless the author provides me with a methodological rationale that allows me to feel that sense of confidence in the reasons for the research design and methods that the research

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has used, in the ways described, and that alternative interpretations have, at least, been considered and rejected for, apparently, sensible reasons. In the case of the imaginary study I have described, I think that such confidence would be unlikely to be generated, but I am posing this hypothetical situation as being not too far distant from the situation that lies behind many publications in mathematics education. If my imaginary colleague had explained the values, beliefs, and attitudes that provoked the study and that consequently bounded and drove the choice and rejection of the many different research methods that were available, as a reader I would be in a much stronger position to decide whether, as a result, the argument being offered convinced me, or not. However, without that context, I am not in a position to make such a judgment. I prefer to “see” the researcher in the writing that I am reading, but I recognize that that view is not shared by everyone. Nonetheless, I am unimpressed by authors who hide motivations or choices inside third person, passive voice sentences because of their beliefs that they are not “significant variables in their research; thus in testing an hypothesis, they expect other researchers handling similar data to come to the same conclusion that they find” (Bassey, 1995, p. 12). By contrast: to the interpretive researcher the purpose of research is to describe and interpret the phenomena of the world in attempts to get shared meanings with others. Interpretation is a search for deep perspectives on particular events and for theoretical insights. It may offer possibilities, but no certainties, as to the outcome of future events. (p. 14)

It can be seen from these two quotes that the methodological stance of the researcher who adopts the first, or the second, approach to their study is bound not only to be fundamentally different but to carry with it cultural baggage that will influence everything about the study from the original hypotheses, through method to data analysis and reporting. In the discussion of methodology, I have incorporated a view on epistemology that is central to the approach that I am taking. How knowledge is constructed is a function of values and, indeed, is also about the community that can define those values and establish the gatekeeping criteria for maintaining them. Inevitably, therefore, I see epistemology as interlocked with methodology.

FITTING TOGETHER METHODOLOGY, DESIGN, METHODS, AND ANALYSIS In describing her methodology in undertaking a study of sexual harassment in schools, Carrie Herbert wrote: it was necessary to construct a specific methodology in order to cope with the problems of silence, suppression, misinterpretation, guilt, embarrassment, intimacy, privacy, trust, and the language of naming concepts and behaviour. (1989, p. 40)

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These problems helped her to identify her methodological starting points. She went on to explain the four possible sources of research methodology from which I could draw in designing this project, all coming under the broad category of ethnographic research: traditional ethnography (Lacey, 1970; Ball, 1981); action research (Kemmis, 1982; Davis, 1983); democratic research (MacDonald, 1974; Walker, 1974); and research conducted by feminists (Mies, 1983; Oakley, 1981). (pp. 40–41)

She continued by outlining why she rejected traditional ethnography and how she came to construct her personal research style, reviewing and explaining the reasons for the methods that she chose and those she rejected. As Carrie Herbert demonstrated, having been guided by the methodological stance, it is impossible to separate choice of methods, design, and analysis since the choices available for these are a function of the values, beliefs, and attitudes of the researcher. She sums up, for example, her approach to “validity.” The “normal” ways which have been used to “validate” data in the past are clearly unhelpful in research of this kind. But there are other ways of “knowing” if the experiences are credible. . . . But this “normal” way of creating “truths” must not be allowed to persist and dominate research where corroborative evidence is impossible. If society does not want to know about the extent of sexual abuse, one powerful way to remain ignorant is to invalidate, and thus silence and suppress, research findings of this nature. (pp. 174–175)

William Torbert (1981) suggested that: The reason why neither current practice nor current research helps us to identify and move towards good educational practice is that both are based on a model of reality that emphasizes unilateral control for gaining information from, or having effects on, others. . . . the effort at unilateral control presumes that the initial actor (whether researcher or practitioner) knows what is significant from the outset and that this knowledge is to be put to the service of controlling the situation outside the actor, in order to implement the pre-defined design as efficiently as possible. . . . If everyone in a given situation acts in accord with this model, then no-one is open to learning new strategies or to examining their own assumptions. Moreover, to the extent that the different actors’ substantive assumptions and strategies differ at the outset, then they won’t even succeed in “teaching” one another the “facts” of the situation, since the relevant facts will differ according to the particular assumptions and strategies of particular actors. (1981, p. 142)

What this reinforces is that it is not the method, data collection, nor analysis that should dictate how a study is described or even pigeonholed. Quantitative and qualitative methods are equally available, have their different uses, and lend themselves to different analytical techniques. However, the choice of which method(s) is best, in order to gather the data necessary to the exploration of a particular question, is always a function of the theoretical stance adopted by the researcher(s) together with, of course, the research context and the related research

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questions, the informants, and so forth. This, in itself, is a product of the attitudes, beliefs, and values underlying that stance. For example, for many feminists, the values underlying their research are the need to address invisibility and distortion of female experience and the hope that the results of the research will empower those who engaged with it, as participants, to understand, and possibly change, their world. For others, such as those adopting the stance of general emancipatory research, similar values might apply, but there could be others that are invoked by their study. The research methods chosen by someone espousing such values must be consistent with “voice,” “empowerment,” and becoming a catalyst for change. Within such a value system, those who engage with the research as participants cannot be “subjects” as the choice of word is representative of attitudes that are inconsistent with the values underpinning the research. As researchers in the field of mathematics education, we are part of a social science in which we are researching aspects of human experience. Such experience is embedded in social and cultural contexts that cannot be ignored whether the researcher shares, or is alien to, those contexts. There are certain products of such a stance. The behavior that is being researched can only be understood within its environment, which needs to be explored and explained. Furthermore, acknowledging social and cultural dependency requires a recognition of diversity and the rejection of the possibility of universality. An outcome is the need for epistemological transparency by which I mean that the researcher is open and accessible about the influences on the processes through which they came to know about the focus of their study and, possibly, how their participants also came to know, where this is entwined with the object of study. Epistemology is a way of describing theories about how knowledge is constructed, achieved, and legitimized. It leads to questions about who is empowered to do this, for whom, and by what criteria. It can help to problematize the adequacy of the descriptions of educational cultures and those who inhabit them, especially through a recognition of the role of complexity and the difficulties raised by it. Often, the epistemology held about mathematics can be in conflict with the social science approach to behavior preferred by the researcher. This is a particular problem for those researching in mathematics education. We have been educated to believe in the “objectivity” of mathematics; at the same time, we might be trying to hold a belief in learning being socially negotiated. Thus we might end up with internally contradictory stances. Finally, a word on objectivity: Jayaratne and Stewart (1991, p. 98) raised three important caveats: 1. Apparently “objective” science has often been sexist (hence, not “objective”). 2. Glorification of “objectivity” has imposed a hierarchical and controlling relationship upon the researcher/researched dyad. 3. Idealization of objectivity has excluded from science significant personal subjectively-based knowledge and has left that knowledge outside of “science”. . . leaving the subjective outside of science also leaves it unexamined.

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Utilizing so-called objective methods does not make a research study objective. Nor does the inclusion of subjective information make the study subjective. “Objectivity” is gained through the internal consistency and coherence with which the story is told and the robustness with which that story resonates across the different perspectives related to it, and to which it relates. At the end of the day, the researcher/author must be able to convince the reader of their trustworthiness and of the authenticity not only of what they have done, but of the conclusion that they have reached and the resultant implications that they have drawn. In conclusion, a brief word on how this book has been put together and how that reflects the stance that I have been taking in this chapter, which is that methodology is about values, beliefs, and attitudes. Each of the authors who submitted a chapter for the book was open to their chapter being read and comments being made as a reaction. In some cases, these comments became a vehicle for a more extended communication between authors and reactors. In other cases, the author(s) did not want to respond to the comments of the reactor. But in all cases, the chapter authors were open not only to receive comments but for these comments to become public within the book. The book itself, therefore, is an example of a discussion that has begun and that readers can now take further. Why is that methodological? Because this openness and reflection is a necessary part of a methodological discourse that assumes that, through interaction, work can only be questioned, reflected upon, and improved. Those attitudes are praiseworthy and distinctive, I believe, and substantially different from the attitudes that underlie a sense of closure when a piece of work is submitted and published and the author moves on to other challenges.

CONCLUSION In this chapter, I have raised distinctions between methodology and method. The history of research in the discipline of mathematics education is that it has, until recently, followed the well-worn paths of psychology and sociology as they apply to education. Inevitably, this has led to methodology being treated as the unproblematic statement of the methods used in the research being reported. More recently, two developments have caused this to be reassessed. One of these is the growing challenge, particularly from those in the history, sociology, and philosophy of science and most particularly from feminists, to the assumption that research conducted according to conventional (quantitative) methods is objective and that it is unnecessary to confront the underlying assumptions of the researcher. The second development is the strength and range of research methods being used to gather data in the social sciences, many of which have been adopted by those in mathematics education or, indeed, invented and then used by them. The multiplicity of ways of conducting research leads, legitimately, to questions about which methods are best to do which job. Inevitably, therefore,

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issues about “the why” are being raised to challenge researchers who, formerly, presented only “the how” and “the what.”

REFERENCES Bassey, M. (1995). Creating education through research: A global perspective of educational research for the 21st century. Kirklington, UK: Kirklington Moor Press, together with the British Educational Research Association. David, M., Edwards, R., & Alldred, P. (2001). Children and school-based research: ”Informed consent” or ”educated consent”? British Educational Research Journal, 27(3), 347–365. Gilligan, C. (1982). In a different voice. Cambridge: Harvard University Press. Harding, S. (1987). Feminism and methodology. Milton Keynes, UK: Open University Press. Harré, R. (1981). The positivist-empiricist approach and its alternative. In P. Reason & J. Rowan (Eds.), Human inquiry (pp. 3–17). Chichester, UK: Wiley. Herbert, C. (1989). Talking of silence. London: Falmer Press. Jayaratne, T., & Stewart, A. (1991). Methods in the social sciences. In M. M. Fonow & J. A. Cook (Eds.), Beyond methodology: Feminist scholarship as lived research (pp. 85–106). Bloomington: Indiana University Press. Keller, E. F. (1983). A feeling for the organism: The life and work of Barbara McClintock. New York: W. H. Freeman. Lather, P. (1991). Getting smart. London: Routledge. Lincoln, Y., & Guba, E. (1985). Naturalistic inquiry. Beverly Hills, CA: Sage. Lynch, K. (1999). Equality in education. Dublin, Ireland: Gill & Macmillan. Mahony, P., & Hextall, I. (2000). Reconstructing teaching. London: Routledge Falmer. National Academy of Sciences. (1993). Methods and values in science. In S. Harding, The “racial” economy of science (pp. 341–343). Bloomington: Indiana University Press. (Reprinted from On being a scientist, 1989, Washington, DC: National Academy of Sciences Press.) Torbert, W. R. (1981). Why educational research has been so uneducational: The case for a new model of social science based on collaborative inquiry. In P. Reason & J. Rowan (Eds.), Human inquiry (pp. 141–151). Chichester, UK: Wiley.

ACKNOWLEDGEMENTS I wish to thank all of those who have been kind enough to give me comments on this chapter but also those who have been engaged with me in a methodological debate over a long period of time. These include all my Ph.D. students, past and present, but I would like to mention by name Marja van den HeuvelPanhuizen and Simon Goodchild.

Chapter 2

Students-Teacher Interactions and the Development of Students’ Mathematical Thinking Maria Manuela David and Maria da Penha Lopes

INTRODUCTION This chapter describes a phase in our ongoing research into the ways in which teachers communicate and encourage the development of students’ mathematical thinking. In the phase upon which we focus here, we describe methods that aimed at minimum intervention and employed minimum technology. Research assistants were used as classroom observers, and we took their observations and field notes as the main source of data for the interpretation of classroom interactions. The research arose out of our interest in Gray and Tall’s (1993) work on flexible thinking, which we developed within a social constructivist framework (Ernest, 1991) to explore the nature of students-teacher interactions. We adopted a phenomenological approach because this was consistent with the aim of the inquiry, which was to interpret incidents and interactions that arise in classroom situations. In this account of our research, it will be necessary first to address the notion of flexible thinking as described by Gray and Tall (1993) as this gave our inquiry its initial impetus. We will also consider the wider construct of mathematical thinking because our inquiry convinced us that the notion of flexible thinking was too narrow for our purposes; here we will draw on the review of related research produced by Schoenfeld (1992). We will then present our approach to classroom observation as a means of exposing evidence of teachers’ and students’ mathematical thinking and explore the influence that the teacher may have on developing students’ mathematical thinking. Our account also includes a description of our development of a model for characterizing the nature of mathematical thinking, which we produced to support the analysis of classroom dialogues and interventions. The analysis was intended to expose evidence of the

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ways in which teachers could influence, either positively or negatively, students’ development of mathematical thinking skills. The discussion will then explain how the observations led us to notice ways of thinking that are complementary to flexible thinking, hence the need to extend the concept of flexible thinking to the wider notion of mathematical thinking (National Research Council NRC, 1989; Schoenfeld, 1992). We now believe these other forms of thinking, like flexible thinking, are important if students are to develop a comprehensive, sound, and robust understanding of mathematical concepts.

THEORETICAL FRAMEWORK Mathematical Thinking Gray and Tall (1993) identify flexible thinking as a feature of mathematical thinking that, they argue, is necessary for long-range success in mathematics. Underpinning their description of flexible thinking is the notion of what Gray and Tall (1993) chose to call “procepts,” and to properly understand their notion of flexible thinking it is necessary to explain what they mean by procepts. They explain that procepts are mental objects that consist of a combination of a process and a concept produced by the process, together with a symbol that may be used to represent either or both process and concept. For example, the expression 32 can represent either the process of adding 3 to 2 or the concept of addition of 3 and 2. According to Gray and Tall the notion of procept applies to those concepts in arithmetic, algebra, and analysis that are initially learned through a process but not to concepts learned by definitions nor to the majority of geometrical concepts that are introduced through visual perception. Gray and Tall then approach the issue of success and failure in mathematics by exploring how students work with procepts. They claim that the students who are successful in mathematics are those who are able to think flexibly, that is, those who can master mathematical symbolism, dealing flexibly with the symbol as a process and realizing that underlying the symbol there is a mathematical concept, as in the example above, hence their term “flexible thinking.” The work of Gray and Tall helped us understand and make more explicit some issues in the teaching of mathematics about which we had been thinking for some time, in particular, the emphasis given to the teaching of rules and algorithms unconnected to their underlying concepts. We are of the opinion that excess attention given to training rules and algorithms does not stimulate students’ flexible use of concepts that, as Gray and Tall have argued, lie at the heart of a knowledge of mathematics that is versatile, extensible, and available for solving nonroutine problems. In our previous work (David & Machado, 1996; David, Machado, & Moren, 1992), the analysis of students’ errors led us to notice that the approaches adopted by teachers can contribute to error and failure in mathematics. For example, excessive training in routine procedures sometimes leads the student to produce

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false rules. The work of Gray and Tall (1993), interpreted in the context of our own research, exposed the need to examine more thoroughly a number of issues that stimulated our thinking in this area. These issues are evident in the questions we considered as we planned the next phase of our inquiry, in particular:

. . . .

If didactical procedures can contribute to failure in mathematics, what are the teaching approaches that lead to success? Do the attitudes that teachers reveal in the classroom influence students’ development of thinking skills that facilitate success in mathematics? In what ways can the teacher contribute toward students’ development of thinking skills that facilitate the acquisition of a comprehensive, sound, and robust understanding of mathematics? Do the approaches regularly used in the classroom more often inhibit or encourage the development of these thinking skills?

Starting from these broad and rather ill defined questions we needed to identify and focus upon research questions that we could pursue in the continuation of our work. As we have indicated above, Gray and Tall’s work, while stimulating us to embark on this study, proved to be insufficient because it did not describe adequately the full range of mathematical thinking we have observed in classrooms. An essential part of our research was to recognize and characterize what we believe to be important features of mathematical thinking. This necessary extension of Gray and Tall’s work finds support in the literature of mathematics education and in recent trends in teaching mathematics. New curricula and new pedagogy promote the belief that mathematics teaching should “mirror” the activity of mathematicians when they are solving problems (Lampert & Blunk, 1998; NRC, 1989; Schoenfeld, 1992). Developing mathematical thinking in the classroom, therefore, entails the socialization of ways and processes of thinking that are typical of mathematicians (Schoenfeld, 1992). There are a number of thinking skills or “modes of cognition” that we consider to be forms of mathematical thinking. These extend the notion of flexible thinking and include modeling, optimization, symbolism, inference, logical analysis, and abstraction (NRC, 1989). The development of these skills and the inclination or disposition to use them in appropriate situations contribute to the development of what we refer to as mathematical sense making, that is, the disposition for “acting like mathematicians, at the limits of their own community’s (the classroom’s) knowledge” (Schoenfeld, 1992, p. 362). Schoenfeld asserts that an emerging point of view about the conceptualization of mathematical thinking is that aspects related to the socialization of mathematical thinking, beliefs, attitudes, affect, and metacognition should be regarded as important. These are in addition to the fundamental aspect of modes of cognition associated with problem solving. The aspects of thinking related to beliefs, attitudes, and affect are not the central concern of our research, but we do regard those aspects related

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to metacognition and the socialization of mathematical thinking as very important in our analysis. Being a mental process, mathematical thinking is not directly observable, but it reveals itself through cognitive or metacognitive performance that can be observed in the relation between the student and mathematical objects, at the level of “savoir faire.” Following the exposition of the National Research Council (NRC, 1989), we considered the following cognitive skills as manifestations of mathematical thinking to which we append our own working definitions:

. .

. . .

Modeling—the student builds an idea that captures important aspects of phenomena described by words or figures. Autonomous and flexible thinking—the student presents noncanonical solutions (i.e., solutions that do not conform to standard procedures), explores different solutions for the same problem (some concept-based, others of a more procedural nature; some more economical and others more time consuming), and/or finds the best solution by asking “what if?” and exploring other possibilities. Inference—the student reasons from data, from premises, from graphs, from incomplete and inconsistent sources and/or establishes analogies between parallel cases. Proofs and demonstrations—the student searches logical explanations grounded in previously studied facts to justify a statement. Generalization and abstraction—the student draws out a general conclusion from particular occurrences or selects common aspects to different situations.

. Symbolism—the student uses the symbolic language of mathematics in an appropriate manner, knows how to translate an event or relationship experienced concretely into mathematical language or, conversely, ascribes meaning to mathematical symbolism.

These characteristic forms of mathematical thinking were inspired by the modes of cognition outlined by the National Research Council (1989) but we adapted them, as indicated by our working definitions, to fit our own classroom observations. That is, they were selected to achieve a compromise between the existing literature and what we found appropriate to analyze the evidence arising from our observations. In this manner, besides some changes in the naming or in the characterization of some skills, we have included in the above list “autonomous and flexible thinking” because it appeared to us as an important category of analysis that extends the idea of “optimization” (NRC, 1989). It should be noted that here we use the idea of flexible thinking in a more extended sense than that ascribed to it by Gray and Tall (1993). Furthermore, Gray and Tall’s meaning, which is related to the ability to cross the different levels of association between a symbol and the process or concept represented by the symbol, is covered by a combination of some of the skills in the list above, namely by modeling, autonomous and flexible thinking, and symbolism. The boundaries between these modes of thinking are ill defined. They all are important for the development of mathematical sense making, which, however, does not comprise them alone.

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Beyond these cognitive skills we also consider as manifestations of mathematical thinking other aspects that are related to metacognition, such as having knowledge and consciousness of these forms of thinking and knowing how to regulate their application (Brown, 1987). We believe these metacognitive aspects to be part of the normal activity of mathematicians and, therefore, they are considered to be a necessary condition for the development of mathematical thinking. Being a mathematician is more than knowing how to do mathematics. It is also about being conscious of the cognitive processes that permeate knowing “how to do,” knowing the appropriate moment to make use of these processes, and knowing how to speak about them. A teacher who is conscious of these processes may socialize them by talking about them with her/his students, such as by making it clear the moment when they have made a generalization, or when they have made use of a logical inference or deductive method. Accepting that the construction of knowledge is essentially a social process entails the recognition that language acts as a necessary mediator and a fundamental element of the socialization of mathematical thinking in the classroom (Luria, 1990; Vygotsky, 1979, 1996). Thus, developing mathematical thinking necessarily has to be accompanied by the concurrent development of the specialized language of mathematics. According to Vygotsky (1979), language has two functions: first, social interchange, when language is used to communicate an idea, and second, “generalizing thought,” when language is used as an instrument of thought. These two functions are important in the classroom; both contribute to give meaning to the words used. Furthermore, according to Vygotsky (1979), the relation among word and thought and meaning is so close that it is almost impossible to distinguish between a phenomenon of talk and a phenomenon of thought. In the same way, it is almost impossible to distinguish between mathematical thinking, mathematical language, and talk about mathematics and mathematical processes. It may be helpful to interpret the notion that language is an instrument of thought and essential for the development of mathematical thinking in the context of a mathematics class. Most children learn, quite early in their school experiences, to be able to recognize a triangle. However, a student may have built a narrow or partial understanding of this concept, and only if he/she makes use of generalizing thought will he/she develop a meaning for triangle as a category of figures with particular characteristics that distinguish them from other figures. In this more advanced stage of the concept, the student may start thinking about the idea of triangle in a generic/abstract way and may be able to use it in such a way as to establish relations with other mathematical ideas. The characterization of mathematical thinking outlined above indicates that the best environment to study its development is within the arena in which this development is the goal, that is, within the mathematics classroom. In the classroom it is possible to observe mathematical thinking being socialized and exposed in the students-teacher interactions through talk and the use of specialized mathematical language. In the next section we make reference to some

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classroom-based studies that have been influential to us in the design of this study particularly because of what we perceive as limitations in their scope or design.

THE DEVELOPMENT OF MATHEMATICAL THINKING IN THE CLASSROOM Schoenfeld reviews a wide and varied body of research into the development of mathematical thinking (Schoenfeld, 1992). However, we identified limitations in the research, which we wanted to address in our own work. In the discussion above we drew attention to Schoenfeld’s (1992) remark that we already know quite well what mathematical thinking means; however, we still have many questions regarding how and when children develop, or fail to develop, the cognitive skills that facilitate mathematical thinking. Schoenfeld reports a number of studies, some designed to teach heuristics or problem-solving strategies, others “designed to help students develop self-regulatory skills during mathematical problem solving” (p. 355). Other studies carried out within “classroom environments were designed to be consonant with the instructor’s epistemological sense of mathematics as an ongoing, dynamic discipline of sense making through the dialectic of conjecture and argumentation” (p. 363). In spite of the care and effort put into these studies, which developed with the direct participation of the researcher in specially designed settings, the outcomes arising from them are somewhat disappointing. In general, their results draw attention to the great difficulty in conclusively demonstrating the possibility of enabling or motivating the development of metacognition and problem-solving skills. Coles (1993) is somewhat more encouraging; he analyzes programs that were designed to teach thinking in a more general context, that is, not restricted to mathematical thinking, involving such aspects as formal and informal logic, critical thinking, fluency and flexibility of thought, reading, interpersonal relationships, and social skills. He concludes: We do have evidence, partly gained from evaluations of the programmes, that it is possible, in schools, to improve the abilities and attitudes associated with thinking. There is evidence to show that teaching thinking programmes will, in some measure, increase such things as pupil’s powers of judgement, reasoning, memory, attention and motivation. (p. 341)

Most of the research related to the development of thinking or the development of mathematical thinking, analyzed respectively by Coles (1993) and Schoenfeld (1992), reports on situations where there is direct influence and participation of the researcher; that is, either the researcher takes on the role of the teacher or it is the teacher who is doing the research. These are situations where the teacher/researcher intervenes deliberately as mediator between the student and the mathematical thinking or the more general thinking process. With this

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type of intervention by the researchers one can expect that they will promote these forms of thinking in the classroom. However, what we aim at with our observations in regular classrooms, that is nonspecifically designed environments, is to identify those students who demonstrate characteristics of mathematical thinking, and to identify teachers’ actions that might be promoting or inhibiting this type of thinking. Thus, out of the list of broad and ill-defined questions listed above, our main research question became: what characteristics of mathematical thinking do students show in the classroom in their interactions with the teacher?

RESEARCH DESIGN Beliefs and Values That Permeate the Study Although our own beliefs and values are clearly not part of the research design, they do provide a basis for the rationality of the design and the underlying choices made in its formulation. Furthermore, since we adopt qualitative methods, there is a need to set out the beliefs and values that we bring to the work. Thus we provide the following before describing the design of the study. Given our concern with mathematics classes and with the way mathematics is being taught in our country, in this research we have assumed a mathematical/ pedagogical bias, focusing our attention on the mathematical activity of teachers and students in regular classroom situations. We believe that one of the functions of school is to give students the opportunity to access that knowledge which is recognized as “scientific.” This kind of knowledge is historically and socially organized and is part of human culture. We value this kind of abstract and scientific knowledge but at the same time do not devalue everyday knowledge, which, in some cases, is essential to give meaning to scientific knowledge. In Brazil, in recent years, the teaching of mathematics has oscillated between approaches that we refer to as “traditional” and “innovative.” Traditional teaching is characterized by a formal presentation of the subject that follows very closely the logical structure that is used to present mathematics within the scientific community. Innovative teaching is based on problem solving and the development of interdisciplinary projects or ethnomathematical approaches, where the issue of the logical organization and the systematization of knowledge is not the primary consideration. The latter teaching approaches are more influential at elementary levels of teaching and often, for social reasons, do not continue to more advanced levels. Beyond the elementary levels a more traditional approach is adopted to ensure coverage of content necessary for further education and/or employment. Thus, it is reasonable to conclude that mathematical knowledge, as defined by the scientific community, is still an important reference for the teaching of mathematics in our country, and we believe that this should continue. However, as in most countries, notwithstanding some significant differences between alternative teaching approaches, the act of learning school math-

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ematics presents some difficulties for our students because, often, they do not succeed at making it meaningful. Students do not deduce any meaning from their mathematical activity, neither intrinsically within the activity itself, nor extrinsically by relating it to everyday life or practical applications. We do not believe that all students should develop the competencies of professional mathematicians, but we do believe that if we want them to learn school mathematics in a more meaningful way, they should learn how to think mathematically. This will be achieved by socializing them into the processes and ways of thinking that are characteristic of the activity of mathematicians. A mathematician engaged in mathematics is challenged by a problem for which he/she seeks a solution and engages in a number of activities to reach his/her goal. He/she analyzes the problem, represents it symbolically, makes conjectures, establishes relations, and explores other possibilities of solution. He/she tries to find logical arguments to prove his/her solutions, discusses these solutions with his/her colleagues aiming at a better understanding of the problem, and this may lead him/her to develop new mathematical concepts. When we began this research we expected to find some of these activities being carried out, at least in some schools. As we have remarked above, being a mathematician is more than knowing how to do mathematics: a mathematician’s activity includes other important dimensions related to socialization and metacognition. Thus, we also believe that these dimensions should be sought in an exploration of the teaching of school mathematics, and when we began our study we were not sure if we would be able to observe them being encouraged and developed. Although students may experience mathematics in a variety of informal yet socially significant contexts outside school, except for some rare exceptions, students only encounter formal and systematized mathematics inside a school setting. The main mediator between the students and this kind of mathematics is the teacher, whom we consider, therefore, as a key element in students’ mathematical development. Our belief about the potential of the teacher’s influence alerted us to the importance of studying, not only students’ mathematical activity in isolation, but the students’ mathematical activity as it develops through interactions between teacher and students, and between students themselves, and additionally, to examine the approaches used by the teachers when they are teaching mathematics. We wanted to identify any relationship between the teacher’s approaches and the students’ use of flexible thinking, and to do this it was necessary to observe teachers and students at work in their regular activity. We recognized the need to go to the classroom to explore how students’ flexibility of thinking was being promoted, or suppressed. We chose to observe teachers and students at work in three classrooms. Our experience in the context of the professional development of mathematics teachers leads us to believe that classroom observation can facilitate the identification of factors concerning the influence of teachers on the development of students’ cognitive processes. We also concur with Vygotsky’s assertion that these cognitive processes or modes “emerge on the intramental

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Figure 2.1 Interactions upon Which the Lesson Observation Focused

plane from recurrent interactions that take place between actors on the intermental plane” (O’Connor, 1998, p. 22). Within this perspective, it is possible to conceive how a deliberate attitude of the teacher can contribute toward developing these cognitive processes.

Classroom Research We wanted to explore students’ activity and expose evidence of mathematical thinking, and we wished to identify the approaches used by the teacher that might encourage or inhibit this form of thinking. Thus it was necessary to use classroom observation to gather evidence of the teacher’s actions and the interactions between teacher and students and between students themselves. The observation would also expose interactions between teacher and students with the mathematical knowledge upon which the lesson focused. These interactions are illustrated in figure 2.1. The interpretation of the data arising from the observations would then be based on a description of the classroom reality. We believed that answers to the questions that motivated the research would emerge from this interpretation. We were not seeking to make general statements or prove a hypothesis that had emerged from an analysis of substantive theory. The observation and interpretation of the interactions outlined above allows us to identify general questions and make tentative hypotheses that can be reformulated as the research progresses. This was the emphasis we sought in the research and, therefore, it was appropriate to establish the basis of the theoretical framework for the research design on ethnographic and phenomenological research traditions (Bogdan & Biklen, 1994; Jaworski, 1994). All the characteristics of phenomenological research identified by Bogdan and Biklen are present in the approach we took. It is qualitative research of an

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interpretative character. The subjects, situations, and events are not endowed with any preconceived or conventional meaning; on the contrary, the researcher attributes the meaning to them. The results are obtained through the interpretation of episodes/narratives from the natural classroom environment. However, to minimize the inevitable risk of subjectivity, the researcher spends a considerable amount of time in the empirical setting, gathering and reexamining a large amount of data, continuously confronting his/her own opinions and beliefs with these data. Thus, the importance of this type of study lies in its potential to elucidate, describe, or even generate theory about a phenomenon. Our central preoccupation is the immediate circumstance of the interactions among teacher, student, and mathematics. The significance assigned to these interactions, which is inevitably permeated by our personal views and life experience, serves to develop, inductively, a set of categories that allows us to characterize mathematical thinking. Furthermore, we develop the theoretical dimension by identifying the roles a teacher can play in a classroom that are intended to develop mathematical thinking (Jaworski, 1994; Strauss, 1993; Strauss & Corbin, 1990). Thus, we construct a model that can be used in the analysis of mathematics lessons. In this research we focus on descriptions of interactions among teacher, students, and mathematical knowledge as exposed by students-teacher talk because the meaning of these interactions is exposed through this talk. Furthermore, the interactions will provide evidence of the promotion (or otherwise) of students’ mathematical thinking. The written description of those verbal manifestations and of the teachers’ and students’ attitudes is the starting point for the search for meaning and explanation of the observed facts. We chose to make our observations in a school where the students are known for showing an independent way of thinking and for having an attitude of autonomy regarding the learning process. Thus, in this school, we expected to find teachers using methods of instruction that evoke flexible ways of thinking. That is to say, we thought we might find teachers who encouraged their students to think, plan their own work unaided, and discuss their work with other students and the teacher. We selected three teachers whose approach would favor the dynamics of their classes to be centered on dialogue with the students. We made observations of three classes comprising pupils varying in ages from eleven to fourteen years, over a period of approximately one month, with an average of four lessons being observed each week, in each class. We finished the observations when we believed that no new information relating to the focus of interest of our research was emerging, that is, when no new forms of mathematical thinking exposed by the students were appearing and when the approaches used by the teacher that appeared to influence those forms of thinking were being repeated. We wanted to examine the use of mathematical thinking as it was revealed in regular lessons; hence, we reduced our interference in these classes to simple observation. Graduate students who were involved with our research made the observations. This approach was an important feature of the research design that

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was chosen because we did not want to disturb the natural setting of the classroom. We believed that graduate student observers would exert less pressure and be less threatening to both teacher and students because the observed teachers would not “read into the minds” of the graduate students the same degree of expectations about “lesson quality” that we believed they might in trained teacher-researchers. In this way we believe the research design caused minimal, we hope negligible, disturbance to the activity and interactions that regularly occur within the classroom. To prepare the research assistants for this task we first reviewed and discussed with them the literature related to the study. They were then asked to make exploratory observations in classrooms as part of their training program. They were instructed to record in field notes in the most thorough way possible, the interactions among teacher, students, and mathematics, paying special attention to those interactions where any form of mathematical thinking was exposed, either by the teacher or by the students. The records obtained in this initial training phase were then discussed in the research group, comprising the research assistants and ourselves. The notes made by the research assistants during lessons concurrent with students’ and teachers’ mathematical activity provided the data that were analyzed to reveal manifestations of mathematical thinking. However, in the event, some of the notes made during this training phase were much more of the form of a commentary of students-teacher dialogues than the verbatim record of the dialogues we wanted. Additionally, the research assistants appeared to be unduly influenced by our initial characterization of mathematical thinking. Thus it was necessary to give further direction so that the field notes would include as many dialogues recorded in a trustworthy manner and transcribed as carefully as possible, together with brief comments made by the observer in which they highlighted significant moments of the dialogues. This procedure allowed us to identify cognitive skills that we had not considered previously, and this led us to extend our characterization of mathematical thinking. An audiotape record was also made of some classes and our research assistants transcribed the tapes while at the same time they identified and outlined a characterization of the forms of mathematical thinking present in these recorded episodes. Following an initial phase of the data collection that comprised a description of the activities that were being developed in the classroom and the transcription of the dialogues, we initiated a more reflexive phase. In this reflexive phase we, the two principal researchers, first selected the classes that we thought most fruitful in terms of the interactions among teacher, students, and mathematics. Then we selected for each class those dialogues in which we perceived evidence, in the activity or articulation of either the teacher or the students, of the use of any of the cognitive skills we had already identified from literature reviewed and prior observations. Having thus selected the episodes that we considered significant in the context of our research we then identified the attitudes and approaches of the teachers that might be contributing to the manifestation of

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mathematical thinking by the students. The analysis of the episodes then proceeded by setting these attitudes and approaches alongside the cognitive skills identified. We, the principal researchers, mainly performed this part of the analysis of data while the research assistants continued to be involved in a supporting role. Inevitably the observations and their interpretation are imbued with the researchers’ view of world and teaching conceptions. This is an unavoidable limitation of this kind of research because it is not possible to eliminate the influence of the researchers’ feelings and perceptions. The validity of the conclusions depends, therefore, on the consistency and strength of the theoretical framework used by the researchers (Bogdan & Biklen, 1994; Jaworski, 1994) as well as on the rigor of the analysis and reporting. Despite this limitation, the observation of interactions among teacher, students, and mathematics, as we can demonstrate from the analysis of the classroom episodes/dialogues, provides unambiguous evidence of the importance of the role of the teacher in the development of a productive dialogue. Furthermore, we can clearly distinguish those dialogues that lead a student to re-elaborate the mathematical arguments and mathematical thinking from those that do not contribute to this re-elaboration. This affirms the appropriateness of the methods used.

ANALYSIS OF DATA At the beginning of our research, we focused on flexibility of thinking as the main category of analysis; our aim was to verify if the use of flexible thinking by the teacher could contribute to the development of this kind of thinking by the students (David & Lopes, 1998a, 1998b). As the work developed we became aware of the impossibility of demonstrating the teacher’s contribution toward the students’ development of flexible thinking in the classroom. Simply, it appeared the teacher was not making use of this form of thinking. We did notice, however, some interactions among the teacher, the students, and the mathematics that attracted our attention because they revealed other forms of thinking that are important for developing understanding of mathematical concepts and procedures. At this point, the literature of mathematics education led us to reinterpret our data and construct a revised framework of analysis focusing on the wider notion of mathematical thinking in place of the more restricted notion of flexible thinking. We thus selected those classroom episodes where we could find evidence of the use, either by the teacher or by the students, of those cognitive and metacognitive skills considered to be important for the development of mathematical thinking, thus extending the notion of flexibility of thinking. In this way, we exposed the significance of the teacher in promoting a productive dialogue with the students. Consequently, a major outcome from our research is a model for characterizing the nature of mathematical thinking that can be applied to the

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analysis of classroom dialogues and interactions and provide evidence of the promotion or inhibition of students’ mathematical thinking. It follows that this account of the research methodology needs to include an explanation about how this model was developed, and it is to this end that we now turn. Jaworski (1994) and Rittenhouse (1998) each develop models for analyzing classroom episodes supported by observations in classrooms with contrasting goals. In the first case Jaworski’s objective was to encourage students to develop mathematical process skills and strategies (investigation), and in the second case Rittenhouse sought to talk with the students about mathematical processes and strategies (mathematical discourse). These two models informed us in the analysis of “our” classes because we were concerned to identify manifestations of mathematical thinking, which includes both investigation and management of mathematical discourse. Jaworski (1994) identifies three categories that, according to her, “describe the complex classroom environment” (p. 108). These categories arise from a consideration of observations in classrooms where the teachers were determined to work in an investigative way, that is, a way that “encourages critical construction in the classroom” (p. 12). The three categories are:

. Management of Learning—This category relates to the strategies used by the teacher

.

.

that create opportunities “for thinking and learning to take place” (Jaworski, 1994, p. 132). These strategies include a range of behaviors such as organizing the class in such a way as to facilitate an investigative way of working and interventions in the form of questions or suggestions of procedures that can influence the learning. Sensitivity to Students—This relates to the teacher’s attitudes evident in teacherstudent relations. These are characterized by the teacher’s knowledge of the difficulties students experience and the skills they develop. It is manifest in the teacher’s respect and care for their students, and it shows through the teacher’s valuation of the work done by students and the encouragement given to them to continue. Mathematical Challenge—This arises from the way the teacher deals with the mathematical knowledge in his/her interactions with the students, “depending on their individual needs and levels of progress” (p. 108). It relates to the way in which the teacher “elicits knowledge, without saying what knowledge he wishes to elicit” (p. 123). We could call it the management of mathematical knowledge from “the teacher’s own epistemological standpoint” (p. 108).

On the other hand, Rittenhouse (1998) analyzes the role of the teacher “in helping students learn mathematics” (p. 163), taking as a starting point the description of the classes taught by Lampert, whose aim was “that students should learn how to talk mathematics” (p. 9). This study was of interest to us because, as we have indicated above, we believe that knowing how to talk about mathematics is an important dimension of mathematical thinking. Rittenhouse identifies two roles that Lampert plays when she is “engaged in mathematical discussions with her students, . . . that of a participant in the discussion and that of

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a commentator about the discussion.” She describes these as “stepping-in” and “stepping-out” (p. 173), which are characterized in the following ways:

. .

Stepping In—when the teacher participates in the discussion with the students, listening to their ideas, asking questions and contributing to the discussion by providing insights according to her own mathematical knowledge Stepping Out—when the teacher offers commentary and formally teaches “the rules and norms of the discourse she wished her students to use as they talked about mathematics” (p. 173). For example, in some circumstances, the teacher stresses and gives value to features of the students’ articulations when they get closer to the mathematical discourse that she intends to teach.

We use the models of both Jaworski and Rittenhouse as complementary instruments in the analysis of data arising from the lessons observed in our study. The three categories from Jaworski describe particular aspects that we identified in the role of participating in the discussion (stepping in) and in the role of commentator (stepping out). Although Jaworski’s categories can help to make the characteristics of these roles clearer, they are insufficient to characterize them completely. For example, the role of commentator about the discussion emphasized in the study from Rittenhouse, in which the teacher wants to develop the dimension of “talk about mathematics,” goes beyond those categories. Using her own words, “as a commentator, Lampert was able to stop the conversational action to draw her students’ attention to what they were doing in order to help them understand how their talk was supporting their mathematics learning” (Rittenhouse, 1998, p. 180). Thus, Lampert goes beyond the level of cognition and acts at the level of metacognition. Having located our analysis in classroom dialogue, we must take into account both teacher and students’ participation in these dialogues. With regard to the actions of the teacher, he/she will assume either the role of participant in the dialogue or the role of commentator/organizer of the mathematical knowledge. In both cases, the teacher needs sensitivity to perceive the points of view expressed by the students and to present issues that will lead the students to progress in terms of their knowledge. When we analyze the actions of the teacher in his/her interactions with the students, we identify two types of intervention according to its effect on the students’ talk. We consider as a positive intervention by the teacher those situations in which he/she enables the manifestation of any of the cognitive or metacognitive skills characteristic of mathematical thinking by the student. In our observations some of those moments were provoked by a challenge made by the teacher that led students to elaborate a thought, or re-elaborate an earlier idea, or else to interpret a definition or to make a generalization. Alternatively, we consider as a negative intervention by the teacher those situations in which he/she does not contribute to the re-elaboration of the students’ mathematical thinking or when, contrary to stimulating the students’ mathematical thinking, the teacher appears to inhibit it. The most significant negative interventions by the teachers in our observations were those that took place to-

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gether with a positive manifestation of mathematical thinking by a student. These were moments when the student was showing some characteristics of mathematical thinking, which the teacher appears to ignore. This suggested to us a classification for students-teacher dialogues as productive dialogues, which encourage the development of students’ mathematical thinking, and nonproductive dialogues, which may not encourage, or possibly inhibit, manifestations of mathematical thinking. To demonstrate the application of these categories to the data collected by our research assistants we include two examples of our analysis of episodes from one of the three classes we have observed. We then outline our analysis of the episodes and manifestation of mathematical thinking. The aim is to expose the nature and effect of the interactions between teacher, students, and mathematics. In these episodes it is possible to identify both moments when there is a positive and a negative intervention by the teacher.

EXAMPLES The two examples below are extracted from the observations of a fifth grade class. In Brazil, basic education corresponds to eight years of schooling, starting at seven to eight years of age and continuing until the age of fourteen to fifteen. The fifth grade is a period of transition between the first four years of schooling, when students have one or at most two teachers who are responsible for all the subjects, and the following four years, when each subject is taught by a specialist teacher. The observed class has five periods of mathematics each week; two of these are devoted to geometry. During the period of observation the subject in algebra was equivalence, comparison, and operations with fractions. The subject in geometry was quadrilaterals, their properties, perimeters, and areas. The teacher maintains a continuous dialogue with the class by questioning and challenging students; this contributes toward a class dynamic that favors the interaction between the teacher and the class as well as between the students themselves. We have noticed moments of students’ responding to a challenge from the teacher and other moments when they were spontaneously asking or making comments about the subject being studied. Generally, the class is conducted with an agreeable and pleasant atmosphere.

Episode 1. (All students mentioned in the episodes below have been given fictitious names) Teacher: Marta, what are parallel sides? Marta: They are sides that don’t cross. I’ve learned last year. (The teacher shows the

chalk box and the students say that it is a parallelepiped whose faces are rectangles. The teacher asks for some characteristics of rectangles.)

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Editors’ note: This dialogue is translated from Portuguese. Brazilian children would be familiar with the Portuguese word “paralelepípedo” as it is used to name the roughly cuboid stones used in old cobbled roads. Joana: Two equal sides. Marcio: Four right angles. Teacher: How many rectangles to make the box? All: Six. Teacher: Tania, give me an example of a parallelepiped in your home or in the class. Tania: The cupboard. (The teacher asks them to list ten objects that are

parallelepipeds.) Marcio: Soap powder box. Berta: Toothpaste box. Emilia: Medicine box. (All want to participate and when somebody mentions the

milk box the teacher reminds the class that not all are parallelepipeds.) Berta: The TV box is a square. Teacher: A square? All: It’s a cube. Teacher: A cube is formed by how many squares? All: Six. (Someone shows a toothpaste box and the teacher opens it showing that

like this we obtain a plane figure. Making use of this box, he shows that there exist lines that do not cross and are not parallel.) Teacher: Marcio, what is a vertex? Marcio: It is a point. Teacher: Every point is a vertex? Marcio: No. Vertex is a meeting point of two sides. (The teacher shows one face of the

box and asks the name.) Henrique: Face. Teacher: What is an edge? Marcio: It’s the intersection of two faces. (The teacher shows two skew edges and asks

if they have a common point. All agree that they do not have a common point and that they are not parallel.)

In this episode, when Marta answers that parallel sides “are sides that don’t cross,” she gives an answer that reveals a particular view of “parallel sides,” probably imagining a plane figure and finite line segments. There follows a clear exposition from the teacher apparently intended to give Marta the chance to reformulate her answer and provoke her to infer a definition for parallel lines. Using a toothpaste box as an example, the teacher calls attention to the fact that in this object there are edges that do not cross and are not parallel, so that she can form a more generic representation of the idea of parallelism. Al-

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though the aim of the teacher was to lead the student to a generalization, he does not make explicit his intent, and he does not formalize the more general definition of “parallel sides.” In our classroom notes there is no evidence that Marta reformulates her initial idea. The teacher takes the toothpaste box and begins to explore its characteristics—How many faces? How many angles? Which other forms look like a parallelepiped?—moving away from the central issue, without continuing the dialogue with the student. In this way he does not encourage the student to review her contribution to the mathematics and construct a general definition of parallel lines in space. However, in Marcio’s case, the intervention of the teacher leads the student not only to make more precise his definition of “vertex” but also to extend it to the definition of “edge.” When Marcio answers that a vertex is a point, the question posed by the teacher “every point is a vertex?” directs attention to the central issue, maintaining the dialogue around it. We observe a crucial difference between the ways the teacher interacts with the two students and in the ways he uses language in each of the interactions. With Marta the dialogue is interrupted, which makes it impossible for the teacher to verify if she or her colleagues have understood his intent. In this case, although the teacher has shown sensitivity to the student, he does not take on adequately the role of participant in the discussion. In the dialogue with Marcio it becomes quite clear that he takes possession of the discourse to present an appropriate definition of a vertex and to extend this newly introduced language to define an edge. In this case the teacher takes on and adequately fills the role of participant in the discussion and his intervention causes the student to reformulate his definition of a vertex and provide a definition of an edge. In these two elements of this episode, with Marta and Marcio, the interventions of the teacher provide evidence of the use of mathematical thinking. His questioning demonstrates his concern to develop mathematical thinking skills (inference, generalization/abstraction); however, in Marcio’s case, he developed a productive dialogue, but in Marta’s case he did not succeed at this.

Episode 2. (This extract arises from the same class and teacher as above). The teacher begins the class reminding students of what they have already studied about quadrilaterals. He recalls the definition of parallelogram and says that the most important quadrilaterals in the fifth grade are squares and rectangles. The teacher asks Jair, ”What is a rectangle?” and he does not receive any answer. Marcio says that it is a quadrilateral with four right angles. Then the teacher writes the definition of trapezoid on the blackboard and says that they will not be very important in the fifth grade. Trapezoids: are all quadrilaterals which have no parallel sides.

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Figure 2.2 Teacher’s Example of a Quadrilateral

Figure 2.3 Jair’s Example of a “Very Crazy Quadrilateral”

Then begins the following dialogue: Teacher: Marcos, what is perimeter? Marcos: It is the sum of the angles. Teacher: No! It is the sum of the sides. How much does it measure the sum of the

angles in this triangle? (He draws a triangle on the blackboard.) Marcio: One hundred and eighty degrees. Teacher: How much does it measure, the sum of the angles in a quadrilateral?

Some students answer 360 degrees, and Marcio asks if it applies to every quadrilateral. The teacher says yes and asks someone to prove this statement. Miguel intervenes saying that it will be enough to divide the quadrilateral in two triangles with 180 degrees each. (The teacher corroborates and draws a figure on the blackboard [figure 2.2].) (Then follow two exercises to apply the calculation for the perimeter and the sum of the interior angles of two quadrilaterals.) Then, with the purpose to persuade the students that the sum of the interior angles of any quadrilateral is 360 degrees, the teacher asks Jair to draw a very crazy quadrilateral.

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Figure 2.4 Jair’s Second Example of a Quadrilateral

Jair draws a figure (figure 2.3) with four sides non-convex. The teacher praises the pupil’s creativity and says it is a quadrilateral but not a convex one and that therefore it is not going to be studied at the moment. He asks the student to draw another one. Jair makes another drawing (figure 2.4). The teacher marks the angles, asking the value of x: In the beginning of this episode the teacher explores the wrong answer from Marcos to introduce the subject: sum of the interior angles of a quadrilateral. For this purpose the teacher asks the students to recall that the sum of the interior angles of a triangle is 180 degrees. Stimulated by the teacher, the students use this information to find a logical explanation to justify the value for the sum of the interior angles of a quadrilateral, and Miguel presents a proof. As with the first episode above, the teacher’s intervention appears to be directed toward provoking the students to generalize the value of the sum of the interior angles for any quadrilateral. This provides evidence that the teacher, himself, is making use of mathematical thinking; otherwise he would not have been able to maintain the dialogue at the level of generalization. Marcio’s question, “Does it apply to every quadrilateral?” shows that he has made an inference interpreting the assertion and he had an insight about the generalization. This would have been the kind of question a mathematician would have made in similar circumstances. The teacher succeeds in maintaining a dialogue at an advanced level of mathematical knowledge for these fifth grade students. It is possible that he would have attained an even greater degree of generalization if he had considered Jair’s drawing of a “very crazy quadrilateral.” Challenged by the teacher, who asked him to draw a “very crazy quadrilateral,” Jair draws a noncanonical figure, correctly interpreting the definition and modeling the idea of a quadrilateral. Jair provides an intelligent response to the teacher’s challenge to draw a very crazy quadrilateral. However, at this point the teacher loses the opportunity to sustain a productive dialogue, and, furthermore, he inhibits Jair’s positive inference.

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In general, this teacher undertakes the role of participant in the discussions, but we have not identified any occasion in the episodes recorded when he has acted as a commentator. That is, the teacher maintained a discussion about mathematics with the students, but he did not systematize this discussion by making comments about the main ideas and mathematical processes involved in the discussion. We have noticed as a key characteristic of this teacher’s class the constant questioning addressed to each student in the classroom with the objective of making explicit their ideas about the mathematical content. He was concerned to involve everyone in the discussion, but, most of the time, interactions occurred between the teacher and some students, and there was no deliberate intention to stimulate the dialogue between students. Our analysis led us to the conclusion that in some moments, as a participant in the dialogue, the teacher shows that he is attentive to what the students are saying. However, at other moments he does not make an appropriate use of students’ interventions. In the same manner, it is possible to identify some moments when the teacher maintains a dialogue with the students that challenges them to reconsider and revise their statements and to reformulate their concepts. These interventions appear to be successful according to the mathematical knowledge revealed by the students.

FINAL CONSIDERATIONS The teachers we observed were selected because we believed the dynamics of their classes would favor “productive” interactions among teacher, students, and mathematics. In the selected episodes we have identified productive dialogues, that is, dialogues in which the students’ talk shows the use of elements of mathematical thinking. Sometimes the dialogue, although led by the teacher, leads the students to rethink and to reformulate their answers. In other dialogues, although the students show evidence of skills that are characteristic of mathematical thinking, the teacher does not appear to perceive, or does not reinforce, the skills demonstrated, and on some occasions he/she even inhibits them. Our research demonstrates that it is possible to identify in a mathematics class the manifestation of certain skills that we consider important for the development of mathematical thinking. This has been made possible through the analysis of teacher-students-mathematics interactions/dialogues, which allowed us to classify dialogues as either productive or nonproductive. Our interpretation of the observations exposes the importance of the role of the teacher in guiding productive dialogue. He/she participates in this dialogue challenging students with questions appropriate to the situation. The teacher is conscious about what he/she is aiming at; the teacher is attentive and with his/her intervention contributes to the process of elaboration and reelaboration of the students’ mathematical thinking. As a participant in the dialogue, the teacher introduces the use of proper mathematical language and

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influences the students to appropriate this special language, although the pace at which this occurs is rather slow. We consider conscious use of certain cognitive skills to be important for the development of mathematical thinking. However, we have not observed or identified moments when the teacher makes explicit and comments on the different manifestations of those skills and of the mathematical processes used; that is, they have not been observed to act at the level of metacognition. These teachers are concerned with the teacher-students-mathematics interactions in their classes at the cognitive level, but they seem to be unaware of the importance of their role as “commentator” for the development of students’ metacognitive skills. In other words, their concerns are directed more to doing mathematics than toward talking about mathematics. In this phase of our work we were interested to expose what is going on in a natural mathematics classroom setting, that is, unaffected by the direct influence of researchers. This has enabled us to characterize in a more complete way what we understand by mathematical thinking and develop a model of analysis for the students-teacher dialogues. Additionally it has enabled us to introduce a classification of interventions as productive or nonproductive according to whether or not they contribute to the development of the students’ mathematical thinking skills. The observations confirmed that some teachers are attempting to work toward the development of some cognitive skills that we consider to be characteristic of mathematical activity. Now that we are equipped with this more elaborate characterization of mathematical thinking and model of analysis, we believe that in subsequent phases of our work we can approach our more immediate concerns regarding the influence of the teacher. In the next phase the plan is to engage with the teachers in their activity. We will want to make them conscious of their role of encouraging students to talk about mathematics and about the procedures used in mathematics. Then, by observing classes for regular periods throughout a whole year and marking possible changes in the students’ responses, we will no longer be mere observers. We will become participant researchers. Our adoption of methods that avoided direct interference in regular classroom activity, while appropriate and helpful for making a portrait of the kind of teacher-students-mathematics interactions as they occur at present, created another limitation of our study. If by chance, during the progress of a lesson, a student makes explicit use of the skills we are investigating, then we have useful evidence of his/her mathematical thinking. However, if he/she simply does not reveal any evidence of mathematical thinking, it is not possible to conclude that it is not present; that is, no evidence of the object cannot be interpreted to mean the object is absent. Thus, at some later stage, it will be interesting to develop assessment instruments that enable more explicit exposure of students’ use of forms of mathematical thinking, giving us a means to verify if changes in the teacher’s actions can significantly influence the development of the students’ mathematical thinking. We are conscious of the difficulties involved in develop-

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ing such instruments, especially because they should include a very wide range of skills. However, we believe that, in spite of the inevitable limitations these instruments may entail, they should have the potential to provide us with some important and original elements for our analysis. Related to the adoption of a noninterference strategy was our decision to rely on our research assistants for the great majority of the classroom observations. Their collaboration was of fundamental importance during this phase of our work, not only for the research itself but also regarding their own development as researchers. However, greater contact with teachers, as outlined for the next phase of our research, may establish a teacher-researcher relationship based on greater confidence and open up the possibility of classroom observations to be made, on some occasions, by ourselves, rather than, exclusively, by the research assistants. In spite of the limitations it is possible to identify in this research, such as the small number of schools/teachers investigated, or the limited use of electronic recording, we want to stress the potential of observational and interpretative methodology to describe students-teacher dialogues/interactions. In some episodes these dialogues clearly expose occasions when the intervention of the teacher results in the reformulation of students’ mathematical thinking; this would not be perceived without a careful reading of those dialogues.This interpretative perspective should be maintained in later phases of our research, associated to new perspectives that aim at remedying some of the limitations identified in the present work. We do not want to conclude this chapter without declaring our warm gratitude to Anne Watson, who has collaborated with us as critical friend and reviewer of this chapter. She fulfilled this role with great competence, respect, and sensitivity to our beliefs, values, and perspectives. Her comments and observations have been crucial in shaping this chapter in its present form. Anne has been more than a critical friend and reviewer, and the collaboration has contributed toward the development of our present and future work.

REACTION BY ANNE WATSON Editors’ note: Whereas the other reactors writing in this volume were presented with finished chapters, Anne Watson was invited, with the agreement of Manuela and Penha, to act as “critical friend” in the production of this chapter. The purpose here is to demonstrate the potential for productive collaboration among researchers whose only contact is via E-mail.

Humility For several reasons it was with humility I approached the task of working with two authors I had never met and who were working in a country I had never visited. First, being an English speaker, I do not have to work in a second or third language in order to make my writing acceptable to the international community.

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One of the aspects of writing I have contemplated, while working with Manuela and Penha, is the role that organizing ideas in order to communicate plays in helping us sort out what we really think. I wonder whether that role is enhanced or inhibited by reading and writing in a language other than one’s own first choice. The second reason for humility is that offering our work to others always makes us vulnerable. My role in the trio was not only to critique their work post facto, but to work alongside them during the writing. Manuela and Penha have not had the protection of thinking “I have sent that off, and now I can forget it and do something new.” Their work goes to and fro between us; they cannot ignore my comments or use the evasive arguments we all use when faced with negative reviews. They have had to respond to my comments and to treat me as if I know what I am talking about, when perhaps I do not. Third, I have been conscious that the standards by which research writing is judged have been created by the rationalistic, relatively stable, European and North American academic cultures, and that I, as a British academic, might be expected to promulgate those standards. I cannot be said to represent those standards because I can only offer my interpretation of them. In my reading and commenting on this chapter, I have responded according to my perceptions of a range of issues, which I shall summarize here.

Generality It is generally felt that research has to be meaningful, or applicable, in situations other than the researched arena. In their work, Manuela and Penha are constrained to work in depth in a few classrooms. They may be assuming common practices across the classrooms of their experience or that what can be found out from any teacher is true for all teachers. Small scale is sometimes chosen due to lack of funding and sometimes due to other constraints, such as what can be achieved by individuals or part-time researchers. However, it can also be a deliberate choice as it gives opportunity to look in depth at classrooms, building up the personal relationships and presence that allow more to be learned. Rarely do researchers try to generalize from such situations; the data is situation-specific and cannot tell us anything about teaching in general, but it can alert us to issues, stories which might help others make sense of their work, possible relationships, sensible hypotheses, and give us enough data to develop tentative characterizations and classifications. I would like to substitute the criterion “usefulness” for generality in educational research, so that readers are encouraged to ask “does this research enable me to think differently, or more deeply, with more insight, about my work?” These authors use small-scale research in a limited range of classrooms to augment the literature on what teachers might do to encourage mathematical thinking. They say “This is what our teachers did; what we found fits, in part, with existing literature, but we found out a little more and gave some more details. Are our findings useful for others?”

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Validity All I can know from the research in this chapter is that these authors tell us that certain teachers did certain things and mathematical thinking appeared to be prompted as a result. The details of technical validity are beyond me to judge. As a mathematics educator I can ask myself: “Is this credible?” If it is credible and useful, then it has value in the mathematics education community. It is the experience of many researchers that they go through a phase of rediscovering aspects of teaching and learning that have been anecdotally known by practitioners for a long time. In fact, anecdotes and professional commonplaces are often the starting point for our hypotheses. Credibility has the important feature that we can make a judgment about it informed by our own experiences rather than having to ask very detailed questions about someone else’s events and actions. Does it matter that they say nothing about what the teacher thinks about how her actions and reactions promote or stifle mathematical thinking? It does not matter to me because I am still free to make my own interpretations of the vignettes provided in the paper.

Interpretation of Data In my reading I have to be able to see a relationship between the data and the findings; I want to believe that if I took a similar analytical route, I might find the same things, or at least I would not contradict the findings. In order to do that, I need to know how the researchers would have made decisions, and what beliefs they would have brought to bear on the work. Usefulness and credibility, as criteria for acceptance of research, depend on a common knowledge of classrooms, mathematics, learning, and research. How can I assess these from my very foreign standpoint? In my own career I learned what aspects of my position need to be stated, in order for others to access my work, by being questioned in seminars all over my own country and elsewhere in the world, and from the referees of journals. This way I have become enculturated into the practices of a Eurocentric research community with which I can expect to share values and standards of usefulness and credibility. Outside that community, in other parts of the world or in the professional world of practicing teachers, other values and standards might operate. I question whether European and U.S. research standards are an appropriate way to judge research done within other cultural systems, or within the world of professional practice, and control the dissemination of its findings.

Problems I started reading the chapter with a belief that I would share little with the experience of these writers; therefore, I would have to be given a great deal of

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detail before I could understand and find the conclusions credible. I found many aspects problematic. For example, why did they use research students to observe in classrooms when the manifestations of mathematical thinking they hoped to observe were difficult even for the researchers to identify? Would it not be the case that the students would be interpreting classroom observations rather differently than the researchers might? Should we know more about their training and their perspectives and beliefs? This matter is exacerbated by the lack of electronic evidence of classrooms, so that everything we are told comes from the students’ notebook evidence. In Europe and North America videos and audiotapes of classrooms are commonly used, sometimes excessively, and we return to them again and again, constructing multiple interpretations of incidents with different audiences, or even on our own as our theorizing proceeds. Manuela and Penha chose to use written notes and describe how note taking had to develop to include transcriptions of dialogue as well as other features of the classroom. In their case this was a conscious choice, but for some researchers it is a constraint of resourcing. The use of notebooks requires interpretation and selection of data in the classroom as a first stage of the research. We do not know enough about this process to judge how the conclusions in this chapter follow from the classroom events. Were the researchers aware of this difficulty, and if not, why not? I would argue that research training, awareness of the effects of interpretation, access to literature about qualitative methods, and access to advice are resources that may not be available to everyone in mathematics education. Unless one starts as a sociologist or anthropologist, it is all too easy to wander into qualitative, interpretative research with only a mathematical and pedagogical background and assume that observation is objective, particularly when one has taken care to describe behavior and not intentions as the objects of observation. When reading this chapter, I found myself assuming that there was a difference in material and intellectual resourcing between what was possible in Brazil and what might be possible in Europe. After learning that audio and video were available but were rejected, I rethought my assumptions about intellectual resourcing too. I realized that all mathematics education researchers who enter research through mathematics and professional qualifications are at a disadvantage when they learn that their research has to use humanities disciplines in which they may have no grounding. Another problem has been how to carry out my task. I am neither a friend, nor colleague, nor critic, nor supervisor, yet the emotions of my engagement with their work have encompassed all these roles. I asked them lots of questions, and told them what puzzled me, and suggested some reading that related to observing what teachers do to promote thinking. At times I have reviewed what I have written to the authors and felt awkward about the tone of my writing. They have been dismissive of my apologies, reporting instead how valuable they find the contact. Then I find myself asking why they should value the contact,

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hoping that it is not misplaced, hoping that they do not ascribe expertness to me, nor see me as representing a research hegemony. Despite these hesitancies we have formed a working relationship that I believe to be based on common purpose.

Common Experience The common purpose we share is to identify what teachers do and can do to prompt mathematical thinking. I was interested in this as a development of the work published in Watson and Mason (1998), in which we write about the kinds of questions and prompts that might promote mathematical thinking. Manuela and Penha go further than this and suggest that some actions and utterances can also inhibit mathematical thinking. Their characterization of productive and nonproductive dialogues, exemplified in the paper, is a useful addition to the field. It turns out that our beliefs about mathematical thinking are similar: I recognize what they say about it, and they recognize what I say. There are some who would see this as an example of the imperialism of European views of mathematics and classrooms and might suggest that Manuela and Penha could be more concerned with what is appropriate for students in a Brazilian context, or with what might be construed as mathematical thinking in a Brazilian classroom. I would argue that the ease with which we have communicated about the purpose of their study, and our shared passion for a particular view of mathematical thinking, provide evidence for the universality of human experience when thinking about what is conventionally known as mathematics.

RESPONSE TO ANNE WATSON In her reaction Anne Watson has fairly interpreted our expectation regarding the generality of our research. In the phase of our work described in this chapter, we were not endeavoring to validate our interpretation of data by the investigated teachers’ points of view about their role in teacher-students-mathematics interactions. Instead, we intended to validate them through the coherence of our interpretations, based not only in our beliefs but also in the theoretical discussion about the characterization of mathematical thinking and about its development in the classroom. Anne Watson played the role of critical reviewer making her standpoint quite clear: that of a European mathematics education researcher. We consider that she has played this role with great competence, making an analysis of our work that, in spite of the possible differences between the cultural communities we belong to, was very productive and made a lot of sense for us. She gave us the chance to see our work passed through the sieve of her standards of evaluation, thus strengthening its coherence and consistency as a scientific work. It is in accordance with this spirit that we want to address some of the issues she raises.

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We are entirely conscious of the difficulty of interpreting data that, to a great extent, was collected and transcribed by our research assistants and, perhaps, we should have given more details about this in our text. For example, we could have mentioned that in the beginning of the research the two principal researchers were together with one of the research assistants in the classroom, taking notes in at least one of the classes of each teacher investigated. The notes of both were later compiled. Furthermore, we have, in fact, worked as a team all the time, discussing, together and frequently, all the stages of the research and trying to match our perspectives and beliefs. In spite of this, we are not claiming that principal researchers and assistant researchers share exactly the same perspectives and beliefs as we cannot claim that this is true even for the two principal researchers. We have tried to diminish this limitation of our work by collecting and discussing a considerable number of observations and dialogues and trying to extract a meaning from these dialogues constrained by its coherence with the observations made. In spite of the humility Anne has expressed in the way she describes her participation in our work, she played various and important roles in it, acting at times as a friend, as a colleague, as a critic, or as a supervisor. We agree that one of the aspects that has facilitated our communication, besides the mutual respect between us, was the affinity of our beliefs and values about “the universality of human experience when thinking about what is conventionally known as mathematics.” This leaves us with the expectation that our collaboration will not come to an end with the conclusion of this chapter, but that this should be followed by other opportunities for new collaborative ventures.

REFERENCES Bogdan, R., & Biklen, S. (1994). Investigação qualitativa em educação: Uma introdução à teoria e aos métidos. (M. J. Alvarez, S. B. Santos, & T. M. Baptista, Trans.). Porto, Portugal: Porto Editora Lda. Brown, A. (1987). Metacognition, executive control, self-regulation, and other more mysterious mechanisms. In F. E. Weinert & R. H. Kluwe (Eds.), Metacognition, motivation and understanding (pp. 65–116). Hillsdale, New Jersey: Erlbaum. Coles, M. J. (1993). Teaching thinking: Principles, problems and programmes. Educational Psychology, 13, 333–344. David, M. M., & Lopes, M. P. (1998a). Professores que explicitam a utilização de formas de pensamento flexível podem estar contribuindo para o sucesso em Matemática de alguns de seus alunos. Zetetiké, 6(9), 31–57. David, M. M., & Lopes, M. P. (1998b). Teacher and students flexible thinking in mathematics: Some relations. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education, 2 (pp. 232–239). Stellenbosch, South Africa: University of Stellenbosch. David, M. M., & Machado, M. P. (1996). Como alguns procedimentos de ensino estão contribuindo para o erro e o fracasso em Matemática. Educação e Matemática, 40, 25–29.

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David, M. M., Machado, M. P., & Moren, E. B. (1992). Diagnóstico e análise de erros em matemática: Subsídios para o processo ensino-aprendizagem. Cadernos de Pesquisa, 83, 43–51. Ernest, P. (1991). The philosophy of mathematics education. New York: Falmer Press. Gray, E. M., & Tall, D. O. (1993). Success and failure in mathematics: The flexible meaning of symbols as process and concept. Mathematics Teaching, 142, 6–10. Jaworski, B. (1994). Investigating mathematics teaching: A constructivist enquiry. London: Falmer Press. Lampert, M., & Blunk, M. L. (Eds.). (1998). Talking mathematics in school. Cambridge: Cambridge University Press. Luria, A. R. (1990). Desenvolvimento cognitivo. São Paulo, Brazil: Ícone. National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: National Academy Press. O’Connor, M. C. (1998). Language socialization in the mathematics classroom: Discourse practices and mathematical thinking. In M. Lampert & M. L. Blunk (Eds.), Talking mathematics in school (pp. 17–55). Cambridge: Cambridge University Press. Rittenhouse, P. S. (1998). The teacher’s role in mathematical conversation: Stepping in and stepping out. In M. Lampert & M. L. Blunk (Eds.), Talking mathematics in school (pp. 163–189). Cambridge: Cambridge University Press. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan. Strauss, A. (1993). Qualitative analysis for social scientists. Cambridge: Cambridge University Press. Strauss, A., & Corbin, J. (1990). Basics of qualitative research. Newbury Park, CA: Sage. Vygotsky, L. S. (1979). Pensamento e linguagem. (M. Resende, Trans.). Lisboa, Portugal: Edições Antídoto. Vygotsky, L. S. (1996). A formação social da mente. (J. C. Neto, L. S. Barreto, & S. C. Afeche, Trans.). São Paulo, Brazil: Martins Fontes. Watson, A., & Mason, J. (1998). Questions and prompts for mathematical thinking. Derby, UK: Association of Teachers of Mathematics.

Chapter 3

Exploring Students’ Goals in Classroom Activity Simon Goodchild

INTRODUCTION This chapter describes an ethnographic-style inquiry that set out to explore students’ activity in a mathematics classroom. The underlying purpose was to expose the nature of students’ interpretation of their activity and the tasks set by their teacher. The aim was to gain fresh insights into the characteristics of regular classroom activity, rather than students’ responses within a context modified by the research process. Hence, the inquiry was designed to avoid, as far as possible, any disturbance to the natural flow of activity. The inquiry followed the spontaneous development of activity in the classroom and avoided the imposition of a systematic structure designed to facilitate the collection of evidence. It was necessary, therefore, to compensate for the lack of structure, organization, and system in the research design and methods through the development of sound substantive theory that could be used as a lens, or combination of lenses, through which activity in the classroom could be viewed. Consequently, this chapter outlines a complex theoretical framework based on the juxtaposition, rather than integration, of different theories of cognition. It is argued that this framework is necessary because the researcher is familiar with the classroom context and must be primed to observe from alternative theoretical positions in an attempt to gain fresh insight. Into this framework a theory of goals is developed from the work of academics who have made significant contributions to such theories in their respective accounts of cognition. It should be evident from the foregoing that existing theory occupies a significant position in the study, and this is presented, as far as possible, from original sources rather than relying on more recent applications or interpretations.

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It might be anticipated that a study that relies as heavily on existing welldeveloped theory as described here would only serve to confirm theory rather than provide the fresh insights sought. Indeed the study did expose evidence consistent with the theoretical framework constructed from existing theory and thus supported claims regarding the trustworthiness and generality of the research. However, the research also exposed events that were complementary to the a priori framework, and the analysis and interpretation of these events led to insights that were not anticipated from the literature used to build the framework.

Researcher’s Background In the study described here the researcher takes a central role in the creation, interpretation, and reporting of evidence. Consequently, it is necessary to make the researcher known to the reader and thus, consistent with widespread practice in reporting qualitative studies, this account is written using first person pronouns. When I embarked on the research described here I had taught mathematics for over 20 years. After teaching for 16 years in secondary schools, I moved into teacher education and spent much time observing other people teach with varying degrees of success. Thus, I was well acquainted with the activity of mathematics classrooms and, one would suppose, I had gained some expertise in the processes of teaching and learning. However, despite my familiarity with these processes, the characteristics of effective teaching and learning left me puzzled. The longer I spent in classrooms, the less I could perceive any direct relationship between teaching and learning. Existing research contributed to the enigma. Denvir and Brown (1986), for example, concluded after a careful study in which hierarchies of arithmetic were identified and a teaching program was “tailored” to the assessed needs of students, “the children did not always learn precisely what they were taught so attempts to match exactly the task to the child may not always have the expected outcome” (p. 163). I found myself in agreement with Bishop (1985), who observes, “From the point of view of most theories of learning, the mathematics classroom with its noisy atmosphere, with its multiple objectives, with its fixed-time lessons and with its atmosphere of mutual evaluation, is not a very good place in which to learn mathematics” (p. 25). Yet, my experience revealed that despite all these defects, students did make progress in their development of mathematical knowledge. I was also intrigued by Howson and Mellin-Olsen’s (1986) assertion that “The pupils’ development of a proper meta-concept. . .relieves the teacher of the necessity of doing ‘interesting’ things every lesson: it makes teaching mathematics within a school system possible italics added” (Howson & Mellin-Olsen, 1986, p. 30). I found this tantalizing because they appear to describe something fundamental to successful classroom activity, and within limits I believed my

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own practice to be successful, yet, quite frankly, I felt I did not truly understand what they meant by “a proper metaconcept.” To address these uncertainties and questions it was necessary to shift my position in the classroom again; I had already done this twice, first from student to teacher, then to teacher-educator. I now wanted to reenter the familiar arena1 of the mathematics classroom as researcher and attempt to gain a deeper understanding of what, despite my familiarity, I perceived as a complex and mysterious place. The preceding paragraphs not only provide my rationale for engaging in classroom research but also expose the underlying threats and opportunities of the inquiry. The possession of extensive experience of mathematics classrooms made it necessary to confront my own beliefs and values. Otherwise it is possible I would only observe what I expected and report what I considered significant. It is possible that I would only succeed in revealing evidence that justified or supported my own beliefs and values, and I would not begin to unravel what I perceived to be the mysteries of the classroom. Conversely, as an accustomed participant in mathematics classrooms, I was confident that I could enter into the routine life of a class and cause minimal disturbance therein. The methodology described here is an account of the approach taken to mitigate the threats and capitalize on the opportunities in an attempt to engage in systematic inquiry and trustworthy interpretation of mathematics classroom activity.

A THEORETICAL FRAMEWORK Theories of Learning Experience of teaching mathematics and browsing professional journals had furnished me with a “philosophy” of mathematics education that was sufficient for my needs. My principal informants were Skemp (1971) and the Concepts of Secondary Mathematics and Science (CSMS)2 research (Hart, 1981). These led me to adopt a “pseudo-constructivist” position that appeared to explain my experience as a learner, and which I tried to translate into my classroom practice. This position might have been adequate to inform my professional practice, although I would now question this; it certainly lacked the depth and definition to inform the research I intended. As the reader will be aware, there are many contrasting accounts of learning. If I had focused only on constructivists’ accounts of learning, then I would be in danger, as noted above, of seeing only that which I expected. Furthermore, as noted above, I was intrigued by Howson and Mellin-Olsen’s (1986) notion of “metaconcept,” which arises from the perspective of activity theory that they adopt in their discussion. To understand the notion of “metaconcept” it would be necessary to explore the tenets of this sociocultural theory that arises from the work of Vygotsky (1978) and Leont’ev (1979). Different theories focus on particular aspects of cognition, and each addresses its own set of questions. Here is not the place to provide a substantive treatment

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of any theory. The following observations are superficial but necessary to explain the theoretical framework eventually adopted. There is a danger in this approach: brevity could be interpreted as naïveté; more significantly, it could appear to diminish thoroughly worked and respected positions. I hope it is possible to avoid both charges.

Constructivism Radical constructivism takes the individual learner as the subject of inquiry; it accepts that the learner is solely responsible for constructing her/his own knowledge. Knowledge is organized into structures, or schemas, that provide a context for interpreting new experiences as they occur. In this way each person constructs her/his own model of the world and acts in the world according to that model; when the model does not fit with experience, it is adapted to regain “viability.” The principal mechanism or mental function that enables adaptation, hence learning, is the individual person’s own reflection (or “interpretation,” Glasersfeld, 1989). Constructivism does not entail a theory of teaching (Confrey, 1994; Pirie & Kieren, 1992). The role of the teacher is to provide students with experiences that will lead them to reflect on the subject matter to be learned.

Activity Theory The foregoing contrasts with activity theory, which posits learning to be a process whereby the learner internalizes socially mediated tools, in particular, language and symbol systems. Cognition is inseparable from activity, which is argued to be the “nonadditive, molar unit of life” (Leont’ev, 1979, p. 46). Learning occurs when a person is supported by a more knowledgeable other to achieve ends that were unattainable on her/his own. Thus, learning occurs within the “zone of proximal development” (Vygotsky, 1978, p. 84), which Vygotsky describes as the distance between, on one level, the current state of a subject’s development and what they can achieve without help and, on another level, what the subject can achieve with the support of a more experienced, knowledgeable other. Activity theory also recognizes the role of reflection in learning. Raeithel (1990), for example, conjectures that reflection occurs “when the flow of action is broken by events that were not anticipated” (p. 36). Thus the teacher’s role is to work with students in creating a zone of proximal development and present them with challenges that will interrupt their flow of activity, which it is hoped, will lead them to reflection. Whereas a theory of teaching has to be inferred from constructivism, such a theory is explicit in the more socially focused activity theory. The above paragraphs have not made explicit the underlying inconsistency between the individualism of constructivism and the social nature of cognition in the theory that emerges from Vygotsky’s work. Cobb (1994) makes the root

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of the inconsistency clear when he poses and subsequently addresses the question, “Where is the mind?” Constructivism places the mind entirely within the individual, whereas sociocultural theories argue that “mind” extends beyond the thinking person.

Situated Cognition A fully worked cultural theory of cognition, referred to as “situated cognition,” arises from anthropological studies (e.g., Lave, 1988; Scribner & Cole, 1981; Wenger, 1998) and post-structuralist psychology (Henriques, Hollway, Urwin, Venn, & Walkerdine, 1984; Walkerdine, 1988). From this position the cognizing subject is perceived as being inseparable from the social arena in which they are placed and the social practice in which they are engaged. From this perspective Lave (1988) explains cognition as being “distributed—stretched over, not divided among—mind, body, activity and culturally organized settings (which include other actors)” (p. 1), and she proposes that “cognition is constituted in dialectical relations among people acting, the contexts of their activity, and the activity itself” (p. 148). In developing a theory of situated cognition Lave casts doubt on the evidence of cognitive psychology that arises from studying subjects’ mental functioning in “laboratory” situations through carefully controlled experiments framed within a scientific paradigm. Lave (1988) argues that if the subject is relocated into the artificial context of the psychological laboratory their cognition, the meaning they take from the situation, will be significantly transformed.

Assembling the Theories These three theoretical perspectives open the possibility of studying classroom activity from a variety of perspectives: the individual learner, the classroom with all its constituents as a social organism, and the interaction of students in that space between the public realm of the classroom and the private realm of their own conceptions. Let me remind the reader that I come to the research as a mathematics teacher concerned to explore and understand better the mathematics classroom. I have at no stage considered it my place or responsibility to define or refine a theory of cognition. The different theories provide a set of complementary positions from which it is possible to study students’ activity. In particular, the articulation of situated cognition provides a structure that draws attention to dialectical relationships among constitutive order, arena, setting,3 and student (Lave, 1988). Activity theory provides an approach to consider the student in relationship to her/his arena and setting, and constructivism considers the student’s individual interpretation in response to the experiences that arise from the setting. Additionally, although working within a naturalis-

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tic paradigm (Lincoln & Guba, 1985) leads me to eschew notions of cause and effect that underpin much of the empirical work of cognitive psychologists, I cannot ignore their discussion of cognition completely. Any contradiction between the results of cognitive psychology and those theories upon which the theoretical framework used here is based will require careful investigation. The deliberate, purposeful, and critical occupation of different positions that some would argue are “inconsistent,” but that I prefer to describe as “complementary,” is one means through which it is possible to overcome the problem of the research being saturated by a personal belief system. The work of Lave, and others articulating a theory of situated cognition, provides a theoretical justification for adopting an ethnographic style inquiry. Little encouragement is needed to do this since, as noted above, a strength I can bring to classroom research is an ability to “fit in” with the routine of a classroom and cause minimal disturbance. The validity of this claim will be considered later. The concern here is to understand better students’ classroom activity, and therefore it is appropriate to base any inquiry to this end within a regular class. This is not to deny the contribution of researchers who have chosen to collect evidence outside the classroom (e.g., Hoyles 1982; Linden, this volume) or those who have chosen to explore students’ activity by implementing intervention programs that transform the activity within the classroom (e.g., Cobb, Yackel, & Wood, 1992; McClain, this volume). It must be recognized, however, that because of my familiarity with the classroom I cannot engage in ethnography as an “ignorant stranger,” which is Lave’s (1988) characterization of anthropologists who engage in ethnographic research.

Goals To overcome any tendency to be blinded by expectations arising from familiarity with the arena of study it is also necessary to be clear about the focus of the inquiry, that is, to have a well-defined research question. This arose in various forms from the different theoretical perspectives being adopted. In particular, Skemp (1979), writing from the perspective of constructivism, asserts “If we want to know what someone is doing, we attach at least as much importance to their goals as to their outwardly observable actions” (p. 2). It follows that if I wanted to gain a better understanding of students’ activity then it would be appropriate to focus the inquiry on the goals students hold. The issue of students’ goals is well developed in the literature of the different theoretical positions mentioned above. Adopting the perspective of activity theory, Mellin-Olsen (1981; 1987) argues a case for the significance of a student’s rationale for engaging in classroom activity. He identifies two rationales for “learning”: an S-rationale (socially significant) and an I-rationale (instrumental), which he explains in the following manner. First, a student in possession of an S-rationale for learning identifies the subject matter as being of intrinsic value, that is of use, relevance, or significance in the context of her/his own life: “The S-rationale. This rationale for school learning I have called the

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S-rationale to indicate its social importance. It is the rationale for learning evoked in the pupil by a synthesis of his self-concept, his cognition of school and schooling, and his concept of what is significant knowledge and a valuable future, as developed in his social setting” (Mellin-Olsen, 1981, p. 357). Second, an Irationale exists when the student perceives the activity has some extrinsic value: “In its purest form the I-rationale will tell the pupil that he has to learn, because it will pay out in terms of marks, exams, certificates and so forth” (Mellin-Olsen, 1987, p. 157). Personal experience leads me to conjecture a third rationale that students may hold when they engage in classroom activity. They behave in a particular way, not for any intrinsic value they perceive in the activity, nor to achieve a goal extrinsic to the activity, but because they have learned to behave in that way in the classroom. I call this a P-rationale because in possession of this, students are merely conforming to the practice of the classroom. It is possible to identify the P-rationale in the cultural psychology of Bruner (1990) and Barker (1978), who describes people as “behaving Post Office,” referring to the fact that when entering a post office a person knows exactly how to behave (e.g., queuing patiently, etc.) and does so without necessarily being aware of their conforming behavior. In the theoretical framework constructed for this inquiry, students’ rationales for their engagement are placed at the level of the shared, public arena of the classroom. As a student enters the classroom, I was interested to expose what provides the rationale for her/him to attend to, and engage with the activity prescribed by, the teacher. The next level of classroom activity defined by Lave is the student’s setting. This is an intermediate stage between the shared, public arena and the student’s own private individual activity. It is possible to identify goals at this level in the influential work of Doyle (1983, 1986), who contrasts students’ attitudes to academic activities as being about the production of artifacts, such as pages of writing, or about learning. In the theoretical framework students’ purpose is located at the level of setting; once engaged in activity, do students perceive their purpose to be to understand, make sense, create meaning, or generally, to learn? Alternatively, do they see their purpose as fulfilling a production target imposed by the teacher with some admonition to “work hard”? It will be recalled from the above that setting and arena are levels ident– ified by Lave (1988) in her model of “cognition in practice.”4 Whereas Lave does not identify a “level” below (or within) the setting, the constructivist position focuses on the private conceptions of the individual cognizing subject. Without paying attention to activity at this level, the exploration would not be complete; therefore, it is necessary to identify “goals” at this level also. As indicated above, from both constructivism and activity theory a significant mechanism of learning is reflection and thus evidence of students’ reflective activity, or conversely, evidence of the absence of this is sought. An absence of reflective activity is widely referred to in the literature of both constructivism and activity theory as “blind activity” (e.g., Carr, 1996; Christiansen & Walther, 1986). Thus at the level of students’ conception, evidence of

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reflection and blind activity was to be exposed. These are described here as different types of interpretation. In summary then, students’ goals are identified within the context of four levels of analysis (i.e., constitutive order, arena, setting, and individual). At the level of arena, goals are described as students’ rationales, with three complementary “types” of rationale defined: S-, I-, and P-. At the level of the setting, interest focuses on students’ purpose, with two types defined: learning and production. Third, at the level of students’ conception, interest focuses on the nature of their interpretation, whether they were reflecting upon their activity or their activity was “blind.” The research was then framed to address two principal questions: 1. What are the goals, rationales, purposes, and interpretations, toward which students work in their regular mathematical activity? 2. What are the features of the classroom, arena, and setting, which constitute the sociocultural context in which students engage in mathematical activity?

Before moving on to describe the methods employed to explore these questions, a number of points, significant in the research but less so in this account of its methodology, need to be addressed. First, in her account of situated cognition, Lave (1988) dismisses the notion of “goal,” drawing attention to the dialectical relationships that, she argues, exist between the components of her model of cognition. Lave asserts that goals imply influence in a single direction and thus she prefers to refer to the acting person’s “expectations” that arise from the practice in which they are engaged. I do not share Lave’s difficulty with the notion of goals because I believe they can be, and are, modified in the light of experience, thus allowing the possibility of dialectical relationship. This belief finds support in some of the research carried out with students working in a computer-programming environment. Second, the foregoing discussion of theory that underpins this research has focused on cognition and students’ goals. However, as the inquiry is placed in a mathematics classroom, it is also necessary to consider the nature of mathematics, students’ conceptions of mathematics and mathematical activity, and the mathematics content experienced by the students. Here again, it is necessary to be aware of the personal set of values and beliefs that I carried into the classroom with me and attempt as far as possible to “immunize” my observations from these. This immunization is achieved through the appropriation of well-researched and widely respected theoretical positions such as those articulated, for example, by Skemp (1971, 1976, 1982); Bell, Costello, and Küchemann (1983); and Ernest (1991, 1993a,b).

METHOD An Ethnographic Style of Inquiry To this point, the main concern has been to establish that I, as an experienced mathematics teacher steeped in my own belief and value system, should be able

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to produce a trustworthy account of activity in a mathematics classroom. My response has been the formation of a theoretical framework based on multiple perspectives of cognition and different articulations of students’ goals; these, I assert, open the possibility of leading me to a fresh interpretation. The subsequent challenge lies in eliciting “authentic data” (Cooper & McIntyre, 1996), that is, accurate representations of students’ rationales, purposes, and interpretations; this challenge is the main theme of this section. The challenge is, perhaps, best understood by considering that students, although frequently ready to seek justification from their teacher for a particular activity, are unaccustomed to considering or reflecting on their own goals. Furthermore, there may well be a discrepancy between the goals held and the goals students admit after a moment’s consideration. For example, a student may, in terms of the theory expounded above, hold a P-rationale, that is, “going through the motions” conforming to classroom practice in a rather aimless manner, albeit producing the “work” expected. However, when called upon to articulate a rationale, the student would claim that she/he is “learning” because this is, after all, the generally accepted purpose of classroom activity (Linchevski & Williams, 1999). The adoption of an ethnographic style of inquiry was predicated by my espousal of situated cognition as one of the positions from which I would explore the classroom. To conduct the inquiry remote from the classroom or in a context removed from the regular activity of the classroom would immediately call into question the authenticity of the data. The unity of subject and social context is also recognized in activity theory, and again any approach that did not simultaneously attend to the social context would be open to criticism. Indeed, the second research question, above, is concerned with the sociocultural features of the classroom, and to be addressed properly substantial time needs to be spent within the classroom. In this acceptance of a particular style of inquiry, I was following Eisenhart (1988), who provides a compelling argument for an ethnographic approach to classroom research. I also realized that the inquiry would have to be prolonged and sustained if it was to expose evidence of students’ goals; this is a requirement of naturalistic inquiry (Lincoln & Guba, 1985). However, beyond the need to ensure trustworthiness, I was aware that eliciting the evidence required would take time and patience. Boaler (1998), for example, found that it was “generally very difficult” (p. 50) to elicit students’ explanations about their activity. Alongside these positive reasons for adopting an ethnographic approach were reasons for rejecting more traditional approaches such as interviews and questionnaires, or more complex methods that relied on these instruments such as a theory of personal constructs (Kelly, 1955).

Choosing the Appropriate Research Instrument Other researchers have made effective use of students’ stories (e.g., Hoyles, 1982), questionnaires (e.g., Jaworski, 1994), or interviews (e.g., Kloosterman, 1996) as tools to elicit evidence in their research. I did not believe these were appropriate

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methods to answer the questions I had posed. First, there is the problem entailed in formulating questions so that their meaning is clear without leading subjects to a particular type of response. Making this point, Jaworski (1994) paraphrases Brousseau’s account of the “topaz” effect. “The more explicit you are about what you want, the more likely you are to get that because it’s perceived that you want it, not because it is actually the case” (Jaworski, 1994, p. 139). Jaworski only used questionnaires as a minor part of her study, which is principally concerned with the teaching of mathematics, and she reflects on the quality of data gained from questionnaires and remarks. “I regretted not having asked more direct questions of students throughout the observation, rather than trying to gather this information retrospectively” (Jaworski, 1994, p. 107). Moreover, given my own familiarity with mathematics classrooms, there is an increased danger that the questions I posed might only serve to confirm the set of beliefs I held at the outset and not lead to any deepened understanding. A further complication with questionnaires arises because students may give answers they believe are in some way “politically correct,” as observed by Forgasz, Leder, and Gardner (1996), who note the discrepancy between questionnaire responses and classroom observation. It is also argued that evidence arising from questionnaires may be strongly tainted by respondents’ subjectivities, as Marshall, Reay, and Wiliam (1999) opine,“The emotive responses of respondents, the partialities, distortions, misrecognitions and misrepresentations ... always permeate the process of filling in a questionnaire... ” ( p. 4). Eliciting students’ retrospective accounts of their activity in the classroom also has inherent problems. My concern was that students would be selective in their memories, recalling events that were significant for them rather than their regular activity in which I was interested.A possible means of overcoming this is “stimulated recall.” Here, the researcher observes classroom activity without intervention, possibly recording events on audio or videotape. As soon as possible after the lesson, the researcher confronts students with the recorded evidence and questions them about their thinking and activity (as in Clarke, 1998). Beyond the fact that this approach was not open to me, given the limited resources that I had available, it is possible that it would not elicit authentic data for the purposes of the research described here. For example, Eisenhart and Borko (1993) draw attention to the criticisms of stimulated recall made by the cognitive psychologists Ericcson and Simon (1980) based on an information processing model of cognition: Ericcson and Simon hypothesized that data retrieved retrospectively from the long term memory (LTM) are considerably more limited than data retrieved from short-term memory (STM) because: 1) some of the contents of STM are lost in the transfer process to LTM, 2) data retrieved from LTM depend on the adequacy of search of LTM, and 3) persons may fill gaps in information retrieved from LTM by reconstructing or inventing data. (Eisenhart & Borko, 1993, p. 32)

More recently Convery (1999) has discussed the validity of teachers’ stories and reveals how teachers account for events that have shaped their practice, beliefs, and values in ways that will present themselves as consistent, morally up-

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right, rational professionals who have struggled and overcome adverse circumstances. It is very likely that students would construe events and activities in the classroom that would present themselves in a particular “light” that justifies themselves in their own eyes. This possibility is also considered by Cooper and McIntyre (1996), who expose the possibility of students’ “post-hoc rationalisations.” A major concern was to collect data in the course of regular activity without disturbing the very activity that was being explored, an impossible task it might be argued. It was necessary to adopt an approach that would minimize the effect of any interference and later take into account any influence resulting from the intervention at the data analysis stage. It was necessary to “follow” students’ activity rather than recast it to my own agenda. The approach, therefore, was to engage students in “conversation” about their activity in the course of their engagement with the activity. The “conversations” would not be scripted or structured in any way, such as with a prepared schedule of questions, although I would, during the course of the conversation, question students about what they perceived to be the purpose of their activity. Before initiating a conversation students would be engaged in their activity, and it would be possible to observe their activity and listen to their conversations with other students to gain some idea about what they were doing. Thus at the outset of the conversation, it was possible to start from their position in the activity. Even so, it could be argued that my very presence in the proximity of a student would influence the nature of their activity. Although it was not possible to disguise the fact that I was an adult in the room, I was careful not to be seen as a teacher or authority figure within the class. Thus, if students requested help or claimed they were stuck, in the course of a conversation, they would be referred to their teacher for help. Furthermore, I would not intervene as “teacher” in terms of monitoring and attempting to modify or correct behavior, or motivate activity. This lack of intervention might have been problematic if it were not for the sound relationship that existed between the regular teacher and the class. The criteria I applied in choosing the class for the study helped to ensure this positive teacher-student relationship existed. A further measure to ensure the authenticity of the data arising from the conversations was the duration of my time with the class, almost a complete year, and the repeated questioning and probing of students during this time. In particular, I was looking for evidence of students’ rationales for engaging in the activity of the arena, and of students’ purposes as they engaged in the tasks set by their teacher. I was also interested in exploring and probing the nature of the students’ conceptions and interpretations of the tasks. In addition, I wanted to illuminate and describe the classroom culture.

Implementation In the school term immediately preceding the main study, I conducted a small pilot study that was intended to test the feasibility of the methods that were to

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be employed. The pilot study was carried out in a variety of schools and classes and served as a period of training in ethnographic methods. I joined the class for the main study in October, five weeks after the start of the academic year. It was a year 10 class (students 14 to 15 years old) that had been formed at the beginning of the academic year. During the first three weeks of my visits, I did not interact with any student. My role was to observe and maintain field notes. This delayed introduction to the class ensured that the teacher was able to develop her regular relationship with the students; it also meant that she would be able to identify any disturbing effect I might have on the class when I joined it. The three weeks of nonintervention enabled me to learn the class routine and the names of students, and become a familiar figure. Although recordings of the conversations formed the main core of the data, evidence was collected from other sources; this was used to corroborate the evidence arising from the conversations. The additional data also made a significant contribution to the construction of a consistent and coherent account of activity in the class. During periods when the teacher was addressing the whole class, I would observe, maintain an audiotape record, and write field notes, which included a copy of board work and other incidents that would not be recorded on the audiotape. In addition to recording my own conversations with individual students, when I sat near students who were collaborating and discussing their work, I would record their conversation. After some weeks with the class, I realized that I was missing an important feature of the students’ experience, that is, the occasions when the teacher intervened with individuals and small groups of students. To fill this gap in the data, I asked the teacher to take the tape recorder5 on several occasions and thus collected a record of these interventions. Meanwhile, during these lessons I maintained a record of her movement about the class, and thus I was able to construct a full account of the interventions after the event. I made photocopies of pages from students’ workbooks (that is, of students’ work completed at the time of conversation). This, together with pages from the textbooks and audio recordings, allowed me to describe in detail students’ activity at the time of the conversations. This detailed description was an important part of the analysis and preceded any attempt to interpret the students’ activity. The class was regularly tested as part of the school’s scheme of work for mathematics. After each test, when the papers were marked and returned, students were required to complete a personal record of achievement that included a summary of what they had achieved and any personal reflections on their test result. I was able to make copies of the completed test papers and the records of achievement. Both provided an important source of evidence that was independent of my own activity, and these were used to support the account that I constructed from my own data. The only request I made that disturbed the normal routine of the class (apart from the conversations) was to ask that the students complete a test at the beginning and end of the year. For this purpose, Chelsea Diagnostic Tests (Hart, Brown, Kerslake, Küchemann, & Ruddock, 1985) were

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used. Some students completed an algebra paper, some a paper on fractions, and some a paper on decimals and place value. Students sat the same test on both occasions. The purpose of this was to obtain independent data about the attainment of the class that could be compared with a wider (national) cohort of students and, I hoped, would provide some evidence of students’ development over the course of the year. I made notes at the regular parents’ meeting which I was able to attend. The documentary material I collected also included student texts, scheme of work, the teacher’s record book, the schools’ Pre-Inspection Context and School Indicator Report,6 and the school brochure. At the outset of the research I wanted to expose the teacher’s goals in addition to those of the students. However, in the event this did not prove to be realizable without undue disturbance because I sensed that the act of exposing the information I sought would have influenced the teacher in her routine approach to her work. The teacher was aware that I did not want to exert any such influence on her routine and thus acceded to my request not to discuss my work during the course of the year. I did, however, record in my field notes any occasional remarks regarding the class or lessons which the teacher made to me and, as a matter of courtesy, I invited her to examine my field notes whenever she wished; this invitation was not taken up. At the end of the year with the class, I held an extended interview with the teacher in her own home to explore with her some of the key issues that had arisen during the time of my inquiry. In this interview I was able to explore the teacher’s beliefs about mathematics, teaching and learning, and the difficulties and challenges she had faced during the year. The content of this interview became an important source of information in the subsequent description of the classroom arena, and it was possible to present this through the teacher’s own, edited, words. Member checking, that is, providing informants with the opportunity to respond to the researcher’s interpretation of their contributions, was not open to me within the research design I had adopted. To return to students in subsequent lessons and “pick over” the earlier conversation with them would have caused significant interference in their routine, which I wanted to avoid. However, the teacher read the completed report and remarked that she thought it was a fair representation of the class although she believed that the students understood more than I claimed. Gallagher (1995) discusses the issue of differences between researcher and informants’ interpretations that emerge through “member checking” and argues that such differences do not automatically imply the researcher’s view is wrong. In my own case I observe that my conversations with individual students generally extended over 10 minutes and occasionally beyond 30 minutes. This is perhaps not surprising, given that I wanted to expose students’ underlying sense of rationale for the activity and their purpose in doing the task. I also wished to investigate whether the students had a sense of a learning goal or whether they were merely producing output to the teacher’s specification. Students’ understanding of the mathematics of the task was also of interest. Conse-

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quently, it could not be a simple matter of merely asking the “obvious” questions, such as, “Why are you doing this?” or “What are you doing?” Of course, these “obvious questions” needed to be asked, but the student’s responses also had to be set within the context of their observed activity and their own freely expressed accounts of their activity. Thus the students might be asked to teach me how to do it, and I would prompt their explanations by asking “Why?” at appropriate moments. My goal was to try to see the student’s activity through her/his own perception. Contrast this with the teacher’s interventions with individual students that lasted on average less than two minutes.7 Also, I was able to study in depth all the written work of students that was available to the teacher, and so I stand by my interpretation despite the difference of opinion. Ethnographic approaches are often associated with a grounded theory of knowledge generation (Glaser & Strauss, 1967); this was not open to me. First, because of my own experience in the field of inquiry, I had to expose a strong theoretical basis at the outset. Second, grounded theory approaches require the researcher to conjecture and develop theory during the course of the study, and on subsequent visits to the “field,” test the emerging theories through specifically refocused enquiries. This was not viable in my own situation with only limited time alongside the regular demands of full-time employment. Furthermore, as I was responding to the regular and changing activity of the class, I was not in a position to create a context in which I could choose a particular focus for conversations to test emerging theories. However, Glaser and Strauss (1967) admit the possibility of the production of grounded theory at a secondary level, at the time of data analysis, when there is a large body of data. The research described here generated over 1,000 pages of transcript and other data; hence this secondary level of grounded theory was feasible. It was through this process that I was able to reach an understanding of the notion of metaconcept, which was indicated above as one of the concerns I had before embarking on this research.

ANALYSIS Searching the Data The approaches to gathering evidence outlined above were intended to ensure that the data could be authenticated against other sources. Authentic data on its own does not ensure trustworthy interpretation. Once collected, the analysis of data needs to be as careful and systematic as the data collection process. The core of the main body of data collected was the audio record of conversations with students. These were transcribed and then used initially to reconstruct the occasion of the conversation. Following this, they were interpreted in the context of the theoretical framework outlined above. The process of analysis, therefore, went through several stages, and the conversations were treated in different orders in each stage of the analysis. First, the transcriptions were prepared and refined by working through the audio recordings sequentially on a lesson-by-lesson basis.8

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The second stage comprised the detailed description of students’ activity at the time of conversations and the application of the theoretical framework. At this stage I considered all the conversations with a student before moving on to the next student, thus working through the data on a student-by-student basis. After each conversation was analyzed in this way, a short summary of the main points was written, and then once all the conversations with a student had been considered, a “case history” of that student was compiled. Computer software (Q.S.R., 1994) was used to support the coding process. This software enables the electronic coding of data; the data can then be sorted and recalled on the basis of the codes applied. This facilitated the next stage of the analysis in which sections of transcripts that had been similarly coded were retrieved and subjected to reanalysis alongside each other. This process enabled the sharpening and validating of coding decisions and provided the basis for the resultant discussion of students’ goals and the other features of students’ activity exposed by the research.

Validating the Evidence A number of steps were taken to ensure the trustworthiness of the interpretation. These included, first, the archiving of 10 percent of the conversations, selected at random. The archived data were analyzed after the interpretation from the main set of data had been formulated. The purpose of this was to ”test” the conjectures drawn. Second, a sample of 10 transcripts (about 7 percent) was chosen9 from the main data set for independent analysis by research assistants— two newly qualified teachers. I gave these assistants instruction in the theoretical framework; they were then provided with conversation transcripts, pages from student workbooks and relevant texts, and required to analyze these using the theoretical framework constructed for the inquiry. The assistants were unaware of my own interpretation. It was only when they had completed their work on a transcript that we put their interpretations alongside my own, and we looked for consistencies and contradictions. As agreement in excess of 80 percent of coding decisions was achieved, the analysis is believed to be “reliable” (Eighty percent is stated as the best that can be expected to be achieved across a team of researchers by Miles and Huberman, 1994). Third, as the research continued10 I presented examples of data with my interpretation in a number of local, national, and international conferences and seminars to research students, experienced mathematics teachers, and mathematics educators and researchers, thus subjecting myself to regular scrutiny of “peer review.”

PERSONAL CRITIQUE As the research progressed, I was concerned about the quality and authenticity of the data I was collecting through the conversations, the impact of my own

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presence and activity in the classroom, and fundamental ethical questions about the propriety of the research. These issues are the subject of this section. I have explained above that locating the research in the classroom, following the course of regular activity, and engaging students in unstructured conversations concurrent with their activity were features of the approach intended to ensure the collection of authentic evidence. I have also indicated that multiple “types” of corroborating evidence were being sought from different sources. Thus I was able to expose the level of consistency between students’ utterances to me, their activity including any conversation with their peers and written work, and the sociocultural context of the class. From this variety of evidence, it was possible to produce accounts of activity within the class that were consistent within each of the theoretical positions and complementary across the different perspectives. Space prevents me from providing an example of this in sufficient depth to be worthwhile; the interested reader can find a full account of the research in the published monograph (Goodchild, 2001).

Disturbance It is not easy to assess the disturbance I might have made either in the classroom or in individual students’ activity. On a number of occasions the teacher remarked, without any prompting from me, that she did not notice my presence during a lesson. I was reassured that any disturbance was unnoticeable, or at least slight. I was interested in the conceptual interpretation students made about their activity, and I believed that any intervention might make some form of reflective activity more likely. In the subsequent analysis I was very concerned to find corroborating evidence from other sources before inferring a student’s reflective activity. As a consequence it was not possible to report many incidents of this type of activity;“blind activity” was less problematic, and there was proportionately more evidence of this. Interestingly, it was the analysis of archived data that drew my attention to what might have been a significant effect of my presence in the class. One of the first conversations I had with a student was included with the archived data. As I analyzed this, it appeared, within the context of the theoretical framework I was applying, that the student had “no rationale” (this should be distinguished from no evidence of a rationale). No allowance was made for the possibility of an absence of rationale. I had assumed that students were engaging, and therefore, a priori, there would be a rationale. However, in this case the student was not engaging. Such an event did not recur in the data and it can be conjectured that the students were more attentive to the set activity when it was likely that I would engage them in conversation.

Concern about Lack of Structure The unstructured nature of the conversations was an important feature of the approach that I used to facilitate students’ authentic responses. However, this

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also became a cause of some unease as I reflected upon the body of data collected and realized that it was, as a result of this unstructured approach, unsystematic, and it was apparent that I “lost my way” in some of the conversations. At the end of the research, I felt that if it were to be repeated, I would want to attempt to bring more direction into the conversations. It is, however, possible that such an attempt might undermine the spontaneity that was so important in exposing authentic evidence of students’ goals.

Researcher Fatigue When choosing the method of data collection, it will be recalled that I did not use students’ stories or accounts made after the event because I wanted to avoid “post hoc rationalisations.” In a very real sense any report of the research I make will fall into this trap. Even in this account I want to present the methodology in a positive light, drawing attention to everything I did as being well thought through before any implementation. I want the research, and myself, to be taken seriously, so I avoid writing about the conversations where I find myself slipping from the role of researcher into the role of teacher. I do not dwell on the period during the year of classroom visits when I found myself getting bored with the enterprise and thus paying less attention to the research goals. Also, I do not want to admit to the time when my own position was compromised by the school principal, who was substituting for the regular teacher and told the class they were lucky to have “two mathematics teachers” in the room to help them. Unfortunately, these things did happen, and, of course, they affect the trustworthiness of the account. I do not want to admit to the flaws in my research, so I try to conceal them and challenge the critical reviewer to find them! Having just dug a hole and jumped in it, I will attempt to scramble out by asserting that the final account of the class was, I believe, based only on authentic evidence that was reported with sufficient detail for the interpretation to be properly tested.

The Ethical Dimension The major ethical dilemma is whether it is appropriate to enter a classroom for research purposes and intentionally withhold from the teacher information about the class that might contribute positively to the effectiveness of teaching and learning. It will have been observed that this is exactly what I did. The teacher did not invite me into her class; she acceded to my request. The students did not ask to be research subjects; they politely responded positively to my request. Is it possible to justify this approach to research that aims to expose teacher and students to scrutiny without discussing observations that could possibly influence, for the better, the quality of teaching and learning in the class? Alternatively, can I claim that my research activity considered the rights of the teacher and students? Burgess (1989) writes about the necessity for an ethical relationship between research and subjects, one that “implies a respect for the rights of the individual whose privacy

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is not invaded and who is not harmed, deceived, betrayed or exploited” (p. 60). I believe the research can be justified on the grounds that I did share the completed report with the teacher, thus giving her a unique view of her practice. I believe the students gained from the experience, as other evidence demonstrates the value of discussion in learning (e.g., Hoyles, 1985). In the conversations I held with students, I may have challenged their thinking, but I do not believe I misled them, particularly into the construction of misconceptions. In this respect my own background as a mathematics teacher was essential. My background also meant that there was a bond of collegiality between the teacher and me. When one’s practice is observed so consistently and closely as I did that of the class teacher, and then that practice is held up against theoretical models and research evidence, it is, perhaps, inevitable that not all of the work observed will be of the highest quality. The research was intended to explore students’ goals, not the quality or effectiveness of the teacher’s work. I have attempted, in any report arising from the research, first to avoid making value judgments and second to avoid exposing an anonymous teacher to criticism. Anonymity of all the subjects is crucial. This is achieved by concealing names; gender-sensitive pseudonyms for students are used; and the teacher is referred to using female pronouns although the teacher’s gender may not be inferred from this. It is merely a literary device to aid the flow of text. I do not want to suggest that the end justifies the means, although if the outcome is of dubious value, then the research would be difficult to justify. As the research findings are briefly summarized below it will be demonstrated that the study is of value, and it exposes features of classroom activity that should be recognized by every mathematics teacher and curriculum planner. During the course of the research, I was continually aware of the possible conflict between my research goals and responding in an ethical manner if a student should ask me for help. I also reasoned that if for the sake of research purity I was to ignore students’ requests for help, I would be in danger of losing their cooperation with my goals.As Cooper and McIntyre (1996) observe, the researcher should have a disposition of “unconditional positive regard” toward her/his subjects. Gallagher (1995) recounts her struggle with this problem, and she expresses her regret that she resolved, after one “slip,” to maintain the purity of her research.When, on occasions, students asked for help, I explained that I was only allowed to ask questions and then sat with the student and engaged in the type of conversation I would normally hold. These occasions clearly produced rather “tainted” evidence that had to be carefully used, if at all, in the analysis. In any event there were very few students who did ask for help, and I tried, quite successfully, not to appear idle in the vicinity of those who were likely to do this, so the problem did not arise often.

SUMMARY The combination of a novel theoretical framework that adopts complementary perspectives informed by theories of goals developed from each of the perspec-

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tives described at the outset of this chapter, together with an ethnographic style of inquiry, led to a rich and deep understanding of activity in the classroom. It might be argued that I am claiming too much to suggest that the insight was richer or deeper than other researchers have gained with other methodologies; but as my approach was novel, so it contributes to our knowledge of students’ activity in a sociocultural learning context. Perhaps the major contribution of the research is the development of a methodology that can be demonstrated to be fruitful in exploring classroom situations. Inevitably, if the methodology were to be applied in other classrooms, the account of activity would be different, After all, the approach is predicated on the individuality of students and the unique combination of historical and social events that contribute to the creation of a class, its students, the teacher, and “structuring resources” (Lave, 1988). However, it is worthwhile to provide a brief account of how the symbiotic relationship between theory and research design contributed to some of the main observations arising from the study. The time I spent with the class, the frequency of visits, and consistency of attendance not only worked to ensure I witnessed the class in its natural state, as noted above, but also facilitated a critical account of the class from each of the perspectives. The resulting accounts provided complementary explanations of the sociocultural arena, the semiprivate setting of students engaged in their work, and evidence of students’ private conceptions. Significantly, these accounts revealed an underlying coherence despite the inherent inconsistencies between the theories. I anticipated, and privately hoped for, evidence of associations between goals at the different levels; that is, a student espousing an S-rationale would be more likely to hold a learning purpose and reveal signs of reflection or metacognitive activity. No such relationships were exposed. However, engaging students repeatedly in prolonged conversations made it possible to observe that students were neither exclusive nor consistent in their apprehension of goals. That is, in one conversation it was possible to identify students espousing more than one goal at a particular level and varying goal adoption between conversations. On the other hand, the theory had led to the expectation of goals at different levels, and the conversational approach did prove to be successful in exposing these. Again the reader is referred to the published monograph (Goodchild, 2001). Mellin-Olsen’s development of S- and I- rationales (1981, 1987) led me to expect these would be influenced by the levels of success students met in tests, examinations, and other assessments. This proved not to be the case. In particular, contrary to my expectation, students who experienced a succession of poor results in assessments were no less likely to reveal an S-rationale than others were. Thus I was led to infer that these are resilient, long-term constructs, and I conjecture that they are associated with students’ belief systems. This observation was specifically enabled by the integration of theory and methods developed in this research. The notion of observing students in a class setting may suggest the creation of the “typical” student or the “typical” learning event in that situation. The

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underlying theory provided the tools to explore the classroom as a social entity simultaneously with exposing individual differences between students. The research design provided opportunities to observe the community as both collective and individuals, and the relationship between these levels. The approach to subsequent analysis ensured this quality in the data was not lost. I would challenge any assertion that the research added little if anything to what is already available in the literature. The theoretical framework was constructed from existing literature and was found to be an effective context within which to pursue the inquiry (Goodchild, 1995). The teacher set “work,” and this met the students’ expectations (Woods, 1990). The students expected the teacher to make the work easy, and the teacher complied by familiarizing novel tasks (Doyle, 1983, 1986). In so doing tasks were stripped of challenge, and there was little to provoke students’ reflection by interruptions to their activity that were not anticipated. The students’ view of mathematics appeared to be consistent with those of one of the schools, which was following a similar scheme of work, in Boaler’s study: “A large number of students also appeared to be influenced by an extremely set view of mathematics that they essentially regarded as a vast collection of exercises, rules, and equations that needed to be learned” (Boaler, 1998, p. 49). I was struck by the fragmented nature of mathematics that students actually experienced despite the apparent attempts of the scheme’s writers to present a coherent, spiral development of the subject. My analysis of students’ activity revealed only fleeting and unrepeated contact with a large number of apparently unconnected facts and operations. This may provide some explanation for Boaler’s observation. Although I had hoped to gain some understanding of the term “metaconcept,” I was not fully prepared for it when it was illumined by the research! In the conversations, and later the analysis of transcripts, I became increasingly conscious of a construct that I referred to as the students’ “awareness” (cf. Mason, 1987). This construct appeared to comprise three strands: awareness of the nature of mathematics, awareness of the nature of mathematical activity, and awareness of the nature of learning mathematics. Students’ “awareness” refers to the mindset with which they approach their studies in mathematics; it is the basis upon which they make decisions about the relative significance of objects and events in the classroom and how these should be interpreted. For example, consider a student who does not expect mathematics to be characterized by consistency, regularity, and meaning. She/he may be content to memorize unrelated routines and procedures and not reflect upon unexpected anomalies that might indicate her/his possession of a misconception or something new to learn. These attitudes toward work that emerge from students’ awareness now seem to me to be crucial in the way students respond to their regular activity. “Awareness” is not taught to students directly, but it is learned. Bateson (1973) referred to learning that occurs coincidentally, and possibly unintentionally, as meta-learning. Bateson describes meta-learning in terms of learning a variety of context markers that elicit particular responses to cues that arise

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in regular classroom discourse. This awareness had not been taught to students directly but had developed as a result of meta-learning (Bateson, 1973), that is, their learning a variety of context markers that elicit particular responses to cues that arise in regular classroom discourse. The result of students’ meta-learning is their metaconcept of mathematics. I want to conjecture, on the basis of the evidence I collected from the class, that most of the students had developed a metaconcept of classroom mathematical activity that deleteriously interfered with their learning experiences. Furthermore, I am led to assert that the teaching of mathematics in school, from the earliest stages, should pay as much attention to the development of a proper metaconcept as to the mathematical content itself. Thus I want to echo the words of Howson and Mellin-Olsen. “The pupils development of a proper metaconcept . . . makes teaching mathematics within a school system possible italics added” (1986, p. 30). If the value of research rests on its generalizability, then the study of a single case might be difficult to justify, especially if the outcomes can be affirmed from existing literature. However, as Firestone (1993) argues, whereas sample to population generalization is not possible from case-study research, analytic generalization, that is, relating the findings to published work, is possible. Additionally, given a detailed and full account of the methods, data, and interpretation (i.e., “thick description,” Geertz, 1973/1993), case-to-case generalization is possible where the reader is able to apply the findings of the research reported to classes known to her/him. My contention is that the research does make a useful addition to our knowledge of classrooms through the manner in which the prolonged study enables preexisting knowledge to be woven into a coherent account of activity in one class.

REACTION BY BARBARA JAWORSKI This chapter reports the theory and methodology of a year-long ethnographic study of 14/15-year-old students learning mathematics. It begins by questioning the nature of “effective” teaching and learning, starting from the notion of “meta-concept” that, citing Howson and Mellin-Olsen (1986), “makes teaching mathematics within a school system possible.” Simon Goodchild goes on to develop the notion of metaconcept with reference to three areas of learning theory—constructivism, activity theory, and situated cognition—which “provide a set of complementary positions from which it is possible to study students’ activity.” While emphasizing that he has no intention of providing a substantive treatment of these theories, he seeks not to “diminish thoroughly worked and respected positions.” I see Simon Goodchild’s use and treatment of theory as one of the strengths of his account. First, he is honest about his usage and the context in which that use is employed. His purpose is to find some way of characterizing students’

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learning of mathematics within the naturalistic setting of their classroom. It is not to expound or contrast theoretical positions. Second, in seeking a foundation for analysis of complex data from the classroom, he justifies complementarity in his theoretical positions, recognizing that this avoids some of the finer epistemological differences between theories. This is a brave position to take given some of the fierce debates in current literature. It is also a position of some complexity, which itself deserves further consideration. From each theory, Simon Goodchild draws a construct relevant to his research, and it is these constructs with which he actually works in a cleverly woven theoretical tapestry. Thus we see interplay between individual students and the classroom arena in which they interact; students’ goals or rationales related to intrinsic, extrinsic, and practically based factors mediate the individual-in-arena setting. Within setting, we see students’ purposes in terms of creating sense or meaning from their activity, of fulfilling production targets, or of opting out of any work-related activity. Individual development is seen in terms of personal reflective activity. Within this theoretical tapestry, Simon Goodchild is the instrument of his own research. As ethnographer, he finds himself needing to tackle a range of issues and tensions: from practical concerns of his relationships with other actors in the arena to his personal perspectives and their interrelationship with the theoretical groundings of his research (which he himself has woven from established theories). From these positions he seeks to provide a deep descriptive account to characterize the learning he has observed and reach for a perception of “metaconcept.” From his position in the classroom he has to recognize that his presence will perturb the classroom, and that the classroom as constituted with him is not the same as that without. He notices, for example, that students’ goals may seem different from what they might otherwise be when students observe him to have an interest in their activity and learning. The inevitability of this position has to be rationalized and dealt with critically in research analysis. The rigor of the resulting analysis is clearly expressed and justified. This is despite protestations of a human need to “present the methodology in a positive light,” which most researchers will recognize. Steps taken to address trustworthiness of the analysis, and to include a critique of personal involvement, are convincingly described. The ethical dilemma of withholding information that might contribute to effectiveness of teaching and learning needs to be seen alongside the realization that such “information” cannot in any way be handed to a teacher. For the teacher to make sense of the information Simon Goodchild “withholds,” she would have to be personally associated with the information. This could only be achieved through a methodological approach in which teacher and researcher work together sharing perceptions and interpretations. This is a different methodology from that employed and therefore not comparable in the same arena. I felt disappointed that there was not space in this chapter to include any insights into the activity of students, their particular mathematical learning or un-

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derstandings. The reader will have to follow these up elsewhere. We are told that the researcher’s hopes for finding associations between students’ goals at different levels were not realized; that, for example, S-rationales could be associated with poor results as well as others. These seem significant findings, and the case is well made that they are related to the integration of theory and methods within the research. It is this integration that I find especially compelling as I reflect on the account in this chapter and its contribution to our wider understandings of research into mathematics learning in classroom settings. Lave’s concept of arena and its link to ethnographic methods might be seen as providing an arena of practice (of students and teachers, of learning and teaching) in which research can be conducted through an ethnographic methodology. I write it like this deliberately to draw attention to the separation of educational practice from research into educational practice. However, this separation is both a practical impossibility and a philosophical impossibility, since the arena of practice is also the arena of research where this research is concerned. This arena includes the researcher, his beliefs and motivations (including his theoretical tapestry), and the particular relationships constituted between the researcher and other actors, the teacher and students. Thus research findings are fundamentally situated within this arena. If, for example, teacher and researcher had worked together in exploring students’ learning, the arena would have been different and findings relatively so. The reported disagreement between the researcher’s findings on students’ learning, and the teacher’s perceptions of this learning might be seen in terms of the differing arenas in which teacher and researcher operated. This leaves us with the (philosophical) question of what exactly we are to make of the findings this research offers; in particular, how are we to treat these findings in terms of their contribution to a growth of knowledge about students’ learning in classrooms. Fundamental research concepts of reliability and generalizability are in question here. It is clear that no one piece of research can illuminate this complex area alone. It will be by looking at these findings alongside many others that we start to make sense of a wider picture. This calls for many further studies of this sort through which we can start to deal with the tensions and issues they individually embrace.

NOTES 1. Throughout this chapter I use the word “arena” in the sense defined by Lave, that is, “a public and durable entity. . .a physically, economically, politically, and socially organized space-in-time.” (Lave, 1988, p. 150). 2. CSMS: this research set out to define Piagetian levels attained by students. 3. Arena was defined in note 1, above. Constitutive order and setting are defined by Lave (1988) as follows: “The constitutive order is. . .the mutual entailment of culture, conceived as semiotic systems, and organized principles of the material and social universe (of political economy and social structure)” ( p. 178). “A setting is conceived here

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as a relation between acting persons and the arenas in relation with which they act” (p. 150). 4. Lave’s model also included a level above the arena, the constitutive order. This has an impact upon the activity that takes place in the classroom, and in the research I did consider the relationship and influence between the classroom and external influences such as National Curriculum, scheme of work, school ethos, and sociocultural context of the local community. However, for the present purpose, which is to consider the methodology of research within the classroom, I am omitting details of this part of the inquiry. The full account is reported in Goodchild (2001). 5. I used a small “pocket sized” tape recorder using a standard cassette, with a sensitive lapel microphone. 6. Produced for the Office for Standards in Education and made available to school inspection teams. 7. I was able to obtain this information from the audio recordings of the teacher when she took the tape recorder. 8. I was fortunate to be able to have secretarial assistance for transcribing the tapes, but I still spent many hours with each tape, refining and correcting the transcripts. This activity preceded any attempt at reconstruction or interpretation. 9. The choice was made on the basis of conversations that I considered rich in exposing students’ goals at the different levels, either through their explicit statements or that could be inferred from the full range of information available. I wanted to use the assistants to test the validity and reliability of my decisions to accept evidence, rather than evaluate my ability to identify evidence. 10. Although the main data collection period only took one year, the research from inception to reporting continued over four years.

REFERENCES Barker, R. G. (1978). Habitats, environments, and human behaviour. San Francisco: Jossey-Bass. Bateson, G. (1973). Steps to an ecology of the mind. St.Albans, UK: Paladin. Bell, A. W., Costello, J., & Küchemann, D. (1983). Research on learning and teaching, part A. A review of research in mathematical education. Windsor, UK: NFERNelson. Bishop, A. (1985). The social construction of meaning—a significant development for mathematics education? For the Learning of Mathematics, 5(1), 24–28. Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29(1), 41–62. Bruner, J. S. (1990). Acts of meaning. Cambridge: Harvard University Press. Burgess, R. G. (1989). Grey areas: Ethical dilemmas in educational ethnography. In R. G. Burgess (Ed.), The ethics of educational research (pp. 60–76). London: Falmer Press. Carr, M. (Ed.). (1996). Motivation in mathematics. Cresskill, N.J.: Hampton Press. Christiansen, B., & Walther, G. (1986). Task and activity. In B. Christiansen, A. G. Howson, & M. Otte (Eds.), Perspectives on mathematics education (pp. 243–307). Dordrecht, Netherlands: Reidel.

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Glasersfeld, E. von (1989). Learning as a constructive activity. In P. Murphy & B. Moon (Eds.), Developments in learning and assessment (pp. 5–18). London: Hodder & Stoughton. Goodchild, S. (1995). Seven dimensions of learning—a tool for the analysis of mathematical activity in the classroom. In L. Meira & D. Carraher (Eds.) Proceedings of the International Group for the Psychology of Mathematics Education: Vol. 3 (pp. 113–120). Recife, Brazil: Program Committee of the 19th PME Conference Brazil. Goodchild, S. (2001). Students’ goals: A case study of activity in a mathematics classroom. Bergen, Norway: Caspar Forlag. Hart, K. M. (Ed.). (1981). Children’s understanding of mathematics: 11–16. London: John Murray. Hart, K., Brown, M., Kerslake, D., Küchemann, D., & Ruddock, G. (1985). Chelsea diagnostic mathematics tests. Windsor, UK: NFER-Nelson. Henriques, J., Hollway, W., Urwin, C., Venn, C., & Walkerdine, V. (1984). Changing the subject: Psychology, social regulation and subjectivity. London: Methuen. Howson, A. G., & Mellin-Olsen, S. (1986). Social norms and external evaluation. In B. Christiansen, A. G. Howson, & M. Otte (Eds.). Perspectives on mathematics education (pp. 1–48). Dordrecht, Netherlands: Reidel. Hoyles, C. (1982). The pupil’s view of mathematics learning. Educational Studies in Mathematics, 13, 349–372. Hoyles, C. (1985). What is the point of group discussion in mathematics? Educational Studies in Mathematics, 16, 205–214. Jaworski, B. (1994). Investigating mathematics teaching: A constructivist enquiry. London: Falmer Press. Kelly, G. A. (1955). Psychology of personal constructs, A theory of personality (Vol. 1). New York: W. W. Norton. Kloosterman, P. (1996). Students’ beliefs about knowing and learning mathematics: Implications for motivation. In M. Carr (Ed.), Motivation in mathematics (pp. 131–156). Cresskill, NJ: Hampton Press. Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday life. Cambridge: Cambridge University Press. Leont’ev, A. N. (1979). The problem of activity in psychology. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp. 37–71). New York: M. E. Sharpe. Linchevski, L., & Williams, J. (1999). Using intuition from everyday life in “filling” the gap in children’s extension of their number concept to include the negative numbers. Educational Studies in Mathematics, 39, 131–147. Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic inquiry. London: Sage. Marshall, B., Reay, D., & Wiliam, D. (1999, September). “I found it confrontational”: Rethinking questionnaires as a large-scale research instrument. Paper presented at the meeting of the British Educational Research Association Annual Conference, University of Sussex, Brighton. Retrieved on 14 January 2002 from http:// www.leeds.ac.uk/educal/documents/00001363.doc Mason, J. (1987). Only awareness is educable. Mathematics Teaching, 120, 30–31. Mellin-Olsen, S. (1981). Instrumentalism as an educational concept. Educational Studies in Mathematics, 12, 351–367. Mellin-Olsen, S. (1987). The politics of mathematics education. Dordrecht, Netherlands: Reidel.

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Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). London: Sage. Pirie, S., & Kieren, T. (1992). Creating constructivist environments. Educational Studies in Mathematics, 23, 505–528. Q.S.R. (1994). NUD.IST. Victoria, Australia: Qualitative Solutions and Software, La Trobe University. Raeithel, A. (1990). Production of reality and construction of possibilities. Activity theoretical answers to the challenge of radical constructivism. ISCRAT Newsletter, 5/6, 30–43. Scribner, S., & Cole, M. (1981). The psychology of literacy. Cambridge: Harvard University Press. Skemp, R. R. (1971). The psychology of learning mathematics. Harmondsworth, UK: Penguin. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26. Skemp, R. R. (1979). Intelligence, learning and action: A foundation for theory and practice in education. Chichester, UK: Wiley. Skemp, R. R. (1982). Communicating mathematics: Surface structures and deep structures. Visible Language, 16(3), 281–288. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press. Walkerdine, V. (1988). The mastery of reason: Cognitive development and the production of rationality. London: Routledge. Wenger, E. (1998). Communities of practice: Learning, meaning, and identity. Cambridge: Cambridge University Press. Woods, P. (1990). The happiest days? How pupils cope with school. London: Falmer Press.

Chapter 4

Special Educational Needs in Mathematics— A Problem Developed in School? Nora Linden

INTRODUCTION This chapter describes a study that employed interviews, or more accurately, conversations to elicit evidence from informants; in these, open-ended questions were the principal research tool. The research was conducted in mainstream schools in Bergen, Norway, and focused on the curriculum provision for young children with special educational needs (SEN). Central to the study were young pupils who were diagnosed as having learning difficulties during the course of their first year in school. The problems identified included speech or language difficulties, physical disabilities, or inappropriate behavior. The research was concerned to expose how the different participants (pupils, their parents, and teachers) interpret the curriculum provision for pupils identified as being in need of special education. Mathematics was not intended to be a key issue in the study, but the informants introduced this subject as important in their verbal accounts, and its significance to the study was confirmed in the analysis of the accumulated data. I start from the premise that young pupils’ confidence is crucial when it comes to learning in school. When a child first starts school she/he is a “champion” in mathematics, as in other subjects. The young child develops recognition that mathematics is an important subject from early infancy; Fosse has revealed that even preschool children have expectations about the subject (1996). The importance of being able to count and measure is recognized in play activity, and older siblings and friends report how “difficult” it is to learn mathematics, possibly to communicate a sense of significance or awe about their school activity. These early influences give the young child joining school a motive for achieving success in mathematics. However, I conjecture that the child’s desire for success is

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often hedonistic in character because the child believes success to be evidence of being good or clever. The purpose of the study was to expose how class teachers, SEN teachers, parents, and pupils with special educational needs “understand” their participation in the arena of a pupil’s SEN provision. “Understanding” and “arena” are terms central to this study, and it is necessary to define the way they are being used. The pupil’s SEN arena is defined in broad terms that include the classroom, the small group, and the pupils’ homework; it is the physical, social, and task context within which the pupils experience SEN provision. “Understanding” is used to refer to the mental schemata, or collection of mental images on which an individual draws to interpret and make sense of events that occur in the context of the arena. Neisser (1976) refers to this alternatively as an “orienting schema.” In this context “understanding their participation” is held to be the cognitive basis for activity and the root of participants’ rationality in the SEN arena. To examine human understanding of experiences, I adopted a phenomenological approach that focuses on the interpretation of human experiences (Husserl, 1973). In this approach the researcher does not assume to know the experiences of the informants in the field but presumes to be able to interpret these experiences from informants’ accounts. My intention was to “go behind” the observable actions in the classroom. I wanted to encourage the informants to share with me their stories about, and their rationale for, their actions in the SEN arena. There was no intention to evaluate SEN teaching in the primary school. My goal was to explore, expose, and offer an interpretation of participants’ experiences. Adopting a phenomenological approach led me to eschew observation as a method for gathering evidence. I believed that through conversation it would be possible to provoke the informants to reflect on their practice and thereby gain access to their conceptual lives, and expose the way they construct their understanding of SEN education. “Understanding” and “rationale” are not simple matters to investigate; this will be made clear in the following discussion of the methodological perspectives. The approach requires care and entails compromises.

THE BACKGROUND FOR THE STUDY Before embarking on this study, I had worked for more than 15 years in teaching and researching pupils with special educational needs. From this experience emerged a desire to give a voice to the user (or beneficiary) of the SEN teaching provided in school.

Pupils’ Experience of SEN In education knowledge is sometimes regarded as an objective phenomenon in which children are intended to acquire certain knowledge through the education

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provided by school. However, the relationship between pupils and knowledge is subjective; the pupil examines new knowledge from his/her own perspective or understanding, with his/her own goals, and relates the knowledge to his/her own life. In school, pupils are sometimes perceived to be pretending not to understand and give the appearance of refusing to participate in the activity. Alternatively, pupils may pretend to join in the activity but will play a game of “pretending” rather than engaging in genuine participation. The pupil knows what is expected of her/him and acts according to this; Edwardsen (1981) describes this as “acting like a chameleon.” A widely held assumption in SEN teaching is “practice makes perfect.” Accordingly, the pupil with SEN in mathematics is presented with a continuous stream of tasks to solve and seemingly endless pages of homework to do. The activity does not seem to be important to the pupil and it does not make sense, nor does the social interaction in the learning context act as a productive learning activity. Consequently, SEN pupils live their “real life” outside school and in school many merely pretend to participate. Being able to understand the language of school, albeit in a personal and idiosyncratic manner, gives the pupil security in the situation and reliance on his/her own knowledge. From the pupil’s perspective the language of mathematics is exciting, perhaps because it is new or strange and conveys a sense of mystery. The introduction of new concepts, numbers, and rules belongs to the school context, and school mathematics has a reputation for being difficult to handle and challenges the child’s curiosity. Some children “develop” difficulties in learning school mathematics. I enclose the term “develop” between quotation marks because in some cases the child’s learning difficulties are not manifest until she starts school. Prior to starting school the child uses whatever mental “tools” that are available to him/her and any experience of failure belongs to the future (Efskin, 1998). When pupils encounter failure in their learning experiences in school, it can arouse anxiety and helplessness rather than motivation for learning. Such is evident in a study of students with special educational needs in the upper secondary school in which Lillejord (2000) revealed that the students believed themselves to be “dumb” and unable to learn. Lillejord’s study raises a crucial question: what has happened to these pupils’ self-esteem during 10 years of schooling? Much research evidence has been published detailing affective factors that influence students’ performance in mathematics. For example, Yates, in this volume, describes her study that attempts to expose evidence that confirms causal links between attainment and affect; for a wide-ranging review see McLeod (1992). This study of how four groups of participants understand SEN teaching in the mainstream classroom, which I describe here, does not take into account any future development of, or impact on, the pupils’ self-esteem. The study does not take place directly within the classroom situation, but the main issue of the study belongs to the classroom since the inquiry focuses on the informants’ experiences that take place therein. One of the assumptions of the study is that it is from their experiences within the classroom that informants construct their understanding of SEN provision.

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The Norwegian Framework for SEN Provision In Norway there are two important concepts regarding SEN education. These concepts are integration and inclusion. For more than 30 years integration of all children in mainstream schools has been seen as one of the democratic rights of all children. The notion of integration is regarded as the basis of the school syllabus. Integration in this context means that children who differ from “normal” in their behavior, appearance, or ability should attend a mainstream school with the majority of the children of their age. The system is based on a medical model in the treatment of people with disabilities. Corbett (1996) argues that this model of education, or the SEN teaching, “champions a narrow stereotype of normality, rather than fostering a celebration of differences” (p. 26). In the medical model these children need special treatment or teaching to compensate for their imperfections in order to become more like mainstream pupils. In this paradigm special education can be regarded as a means to conform a group of children to the school norm; in other words its goal is “integration” on the terms of the mainstream pupils. The other important ideology underpinning the constitution of schools in Norway, “inclusion,” is consistent with the United Nations Declaration of 1981, which proclaimed the right of all children to receive an education that enables them to develop their ability and skills to the fullest extent possible. Jenkinson (1997) argues that one of the most important issues concerning an integrated or inclusive school policy is the question of the curriculum. When SEN children were taught separately, the mainstream curriculum was regarded as out of reach for disabled pupils. They were considered to be in need of a special curriculum, possibly one that is less broad or less challenging. However, a different attitude exists when the same curriculum provision is made for all children and “specific skills and concepts are introduced and taught when the child has achieved the maturity and had the experiences necessary for their acquisition” (Jenkinson, 1997, p. 162). This view has been the foundation for the term adapted teaching, which the national curriculum of the elementary school in Norway (Core Curriculum for Primary, Secondary and Church Affairs) has as one of its cornerstones. The curriculum states that the “mode of teaching must not only be adapted to subject and content, but also to age and maturity, the individual learner and the mixed ability of the entire class” (Royal Ministry of Education, Research and Church Affairs, 1997, p. 35).

THEORETICAL FRAMEWORK A researcher’s theoretical perspective creates the basis for her/his interpretation of the world. It also constitutes the framework for the initial interpretation of the phenomenon of interest; that is, it provides a “mind-set” for exploring, exposing, noticing, and interpreting. The theoretical framework within which the

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research described here is set is based on activity theory, which is a development of Vygotsky’s (1978) sociocultural theory of cognition. It was, for me, natural to see activity theory as a useful theoretical framework for the investigation because of its consistency with the purpose of the study. In activity theory the human subject plays an active part in the development of knowledge, and my interest was in the participants’ understanding of the learning context and how they explained their actions in the classroom. Vygotsky argues that higher psychological functions appear twice, or on two planes. First they appear externally on the social plane and then reappear internally within the individual. In other words, these functions appear “first, between people (interpsychological), and then inside the child (intrapsychological)” (Vygotsky 1978, p. 57). The internalization of knowledge happens as a result of the individual’s activity or participation in the situation. Thus, from this perspective, learning is accepted as the outcome of social behavior or activity that is appropriated to become the possession of the person. Higher forms of human behavior develop from social interaction with the environment. In Vygotsky’s terminology the individual turns the interaction inwards as experience is internalized and transformed into new knowledge. Vygotsky’s intention was to create a theory about the relationship between human beings and the environment based on the dialectic of nature articulated by Engels. This model of learning includes the description of two developmental zones: the actual development zone, and the zone of proximal development. The actual zone of development is the level of development of the mental functions that has been established as a result of earlier successful learning activities; this represents what the child is able to achieve on his/her own, as a result of previous learning. Second, “the zone of proximal development which is the distance between the actual developmental level . . . and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers” (Vygotsky, 1978, p. 86, italics in original). The zone of proximal development describes the space in which reside those functions that are not yet developed but where the individual is able to act meaningfully, or learn with some assistance. Embedded in the concept of activity lie associated constructs, which are the motives and goals for the actions that arise from the activity. However, if the child does not perceive classroom activity as relevant to her/his own goals, or if the activity appears to threaten or contradict her/his understanding of the context, she/he will resist participation in the activity. If learning support is provided so that it conforms to the goal of the learner, it creates an effective context for learning. A contribution to the development of this aspect of activity theory comes from Mead’s (1934) symbolic interactionism, which focuses on social objects and the social consciousness of physical objects. Through interaction the individual develops a sense of identity, consciousness, and knowledge about him/herself and others. When the pupil lacks a motive for participating in the learning activity, she/he is asserting the right to decide whether she/he

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will learn or not. The pupil’s decision not to participate inhibits learning, and her/his withdrawal can then develop into a learning difficulty. An additional feature of activity theory that contributes to the theoretical framework for this research arises from Vygotsky’s stress on the importance of language as a “tool” for learning. Contrary to Piaget (1970), Vygotsky claims that language is crucial for the development of mental processes. When choosing conversation as a research tool, the assumption is that through language we manifest our knowledge, or understanding. This consistency between the theoretical framework and the research methods is recognized as essential to the integrity and coherence of the study.

THE RESEARCH QUESTIONS When pupils with special educational needs fail to learn in school, we seek causal explanations for their failure. However, it is not usual to ask for the teacher’s pedagogical rationale for teaching, or the pupil’s rationale for learning. Nor, normally, do we take either of these rationales as a point of departure when seeking an explanation for the pupil’s failure. However, the research outlined here is based on my recognition that learning and teaching are heterogeneous forms of activity, and I was concerned to address a variety of questions that would expose essential features of the activity. The study is predicated on my belief that knowledge of participants’ rationales will provide a valuable insight to these “essential features.” There are a number of questions implicit in the present study. For example: do young children experience problems with mathematics when they start school, or are their special needs revealed through their school experience? Do the teacher and the pupil experience the same classroom situation, or do their perceptions of what happens in the classroom create different rationales for their activity therein? Prior to his/her entry into school, a pupil will have constructed knowledge appropriate for a particular task as a “use or lose” skill. In school, where failure is explained as the result of not “being clever,” and low achievement is often confused with not being “good at,” what is the consequence of failure to the young pupil? When does a “career of failure” start, and how does the pupil experience the situation? Answers to these questions are crucial if we are to provide appropriate support for pupils with learning difficulties; the questions led me to the following key questions for the study:

.

How do young pupils with learning difficulties and their teachers understand the education provided for special education needs?

. Is it possible that differences in understanding create a less positive learning context for pupils with special educational needs, and contribute to, or reinforce, their problems?

As explained above, the purpose of the study was to explore how the participants understand the classroom context as a place for productive learning. For

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young pupils parental bonds are very strong, and I was interested to see if there was a similarity between the teachers’ and parents’ rationales for supporting the pupils. Hence, in addition to the participants in the classroom context, I wanted to explore how the parents of young children understand the education provided for their sons and daughters and their rationale for supporting them, as well as their relationship with the school. The phenomenon of understanding belongs to the micro context of a person. Each individual participant in this study is assumed to possess an understanding of the classroom from which, in turn, emerges her rationale for learning. Participants’ understanding will differ across individuals. They will have different obligations and expectations regarding the SEN provision because of the different positions they occupy in the SEN arena. In the classroom situation it is intended that the pupil acquires new knowledge. Concurrently, however, the pupil acquires knowledge about knowledge and knowledge about leaning; this is what Bateson (1972) refers to as meta-learning. In this study meta-knowledge is accepted as synonymous with a person’s understanding of the situation. If a pupil does not understand what learning is about, if the material taught in school does not belong to his/her field of interest, the pupil will not share the learning goals identified by the teacher; in this case the pupil will not possess a rationale for learning the curriculum content as interpreted by the teacher. A fundamental belief underpinning this study is that an exploration of pupils’ rationales will expose reasons for some of the learning problems developed in school.

SELECTING A RESEARCH TOOL Conversation as a Research Tool When planning this study it soon became apparent that the term interview would be inappropriate because I wanted to initiate a free exchange of reflected verbal contributions; hence in this chapter I use the term conversation rather than interview. This may need some justification before it can be accepted as a research tool. A conversation is a familiar activity for the exchange and development of consensus or otherwise, of meanings, attitudes, and experiences. The danger is that familiarity with the word can lead to a simplistic interpretation of its meaning and “conversation” can be confused with its everyday use and interpreted as mere talk. Kvale (1997) discusses the metaphors of the interviewer as a “miner” or a “traveler.” In the former, knowledge is represented as buried mineral; an interviewer uses conversation as a tool to bring the mineral to the surface. The latter designates the interviewer as someone on a journey who uses conversation as a tool to collect information along the way. Whether the researcher is seen as traveler or miner, the use of language is highly significant because it relates directly to experiences in daily life. Miles and Huberman (1994) argue that the

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strength of the qualitative research paradigm lies in its strong and close link to life as lived and experienced in daily existence. Both researcher and informant contribute to the conversation with stories from the subjective consciousness of life. When a child expresses in her/his own language how she/he feels towards school we gain access to her/his subjective experience or understanding of what is going on. I meet the informants as a researcher with my understanding, and the informants, on the other hand, meet me with their understanding, and these understandings become manifest in the language used.

Authenticity of Evidence In conducting an interview the question schedule or interview guide is often given too much weight and consequently it interferes with the flow of the conversation. The interview degenerates to little more than an interrogation, and the stories told become dry and artificial. When I planned the study, I made my questions play a subordinate role. The interviews became conversations in the true meaning of the term as illustrated by the following extract from the conversation with Terje, a SEN teacher: R (researcher): You need a lot of competence to be able to read a pupil that way. . . . T: Yes, you are right, it can be very difficult. I think we have to try and try

again. The pupils with a special interest will initiate something themselves. . . .

In the conversation I make use of my experience as a teacher and teacher educator, and this prompts Terje to confirm and explain what he means. Sometimes the conversation can take the character of an exchange between two teachers rather than between informant and researcher. If the researcher permits this to develop without interfering, the exchange of reflection can lose its implicit strength as a research tool. Thus it is important to be clear about the different roles. For example, when Birgitt, a class teacher says, “You know what it is like working with SEN pupils,” she alludes to the fact that we once taught in the same school. Here, I must be careful not to fall into the collegial role given to me by Birgitt, and to avoid this I respond, “Could you please tell me about how you understand the SEN teaching?” It is possible for the relationship between researcher and informant to become uncomfortable. For example, when one of the SEN teachers asks if he has given the “right answer” in the conversation, it can be seen as an expression of the way he experiences the researcher as a person of authority who requires a particular answer or who is evaluating the quality of the informant’s responses. This can happen even though the same informant participated in defining the conversational context at the outset. This gives an indication of the danger of overemphasizing the importance of consensus. The presence of the researcher

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is a more influential sign of power than explanations and formal agreement. Through our language the power relationship is defined, and the use of power may disturb both researcher and informant. The same evidence would not evolve from the dialogue if I, as the researcher, did not interfere the way I did. The close relationship between the researcher and the informant that is developed in the conversation provides opportunities for negotiating the outcome of the study. When one class teacher tries to lead the conversation to what she thinks I might wish to hear, I refer back to the informant’s previous statement and thereby underscore the cooperation between us. The conversation builds on the principles imbedded in the Socratic method of teaching through inquiry or dialogue. Contrary to the Socratic method, however, where the “student” pretends to know nothing about the issue of interest, the researcher does not claim ignorance; he/she openly accepts the leadership of the situation. From the outset he/she plays a dominant part in the activity by defining the opening question. Molander (1993) claims that the dialogue should be an expression of a way of life, not just a means of communication. It is an expression of an understanding, not merely a symbol of the understanding that might exist, but an expression of the actual understanding of the informant. The dialogue is a reflection, and the act of speaking is the action. The third element in this connection is the activity in the matter reflected upon. In this study the informants and I reflect on our understanding of learning and teaching through the dialogue. One of the most challenging issues I faced was that of eliciting the cooperation of young children, as well as adults, in order to gain access to their views and attitudes concerning education. Lindh-Munther (1989) analyzes the art of speaking with children as a particular research mode. She points to the fact that there is not one unique model for a conversation, and the difference in age and experience between adults and children is not the only significant variable. It can be questioned whether it is possible to analyze the child’s perspective along with the perspective of the adult. The difference between the perspectives could possibly create difficulties in the process of interpretation, both at the time of the conversation as well as in the final stages of analysis. It should not be an aim of the research to try to reduce these differences because they are of major importance for the interpretation of both the adult’s world and the child’s world and crucial to constructing an authentic model of the reality of SEN experience among the participants. In this context it can be appreciated that the research conversation is more than an everyday conversation. It can, for example, take the form of a tutorial between two individuals with a shared interest. During a conversation a new situation occurs if an informant diverts the conversation and talks about something different from what was originally agreed. The interpretation of the diversion depends upon the researcher’s preexisting ideas and sensitivity to the power relationship and the situation of the informants (Linden 1997). A reason for the diversion may be an informant’s shyness, or perhaps her/his fear of ex-

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pressing a personal view in a more explicit way. A psychoanalytic interpretation of the utterances made during a conversation can suggest that the inhibition to continue in the line initiated by the researcher might be caused unconsciously by painful emotional memories linked to the issue. As Freud proposes, there is an underlying censorship in the way views are expressed that determines the responses of the individual and her/his rationale for engaging in the course of the conversation (Freud 1992). In the conversations some of the teachers claimed that they received new insight; that is, they developed a new understanding arising from the conversation. This is clearly expected by Kvale (1997) who argues that cooperation in a qualitative interview creates new knowledge within both the informant and the researcher and that the outcome depends on the nature and quality of the interaction in the interview situation. Consensus is an important factor contributing to the motivation of the informants taking part in the research. During some of the conversations informants expressed their rationale for participating in the study. For example, Ruth, a SEN teacher, remarked, “When you asked me to participate, I reflected upon what you meant by understanding.” This was the start of a conversation about how we defined the term and the different meanings we ascribe to it. Ruth explained that although she was not sure what the study was about, she agreed to participate from curiosity, perhaps, or possibly from an interest in her work in general.

Ethical Perspective In educational research the ethical perspective must be central in the planning of the research. Vestby (1999) discusses the institutional aspect of classroom research; first he notes that mainstream school is a public arena, resourced from the general public budget, and any activity in the arena should be of public interest as well as an interesting field of research. Second, classroom research involves individuals belonging to an institution, and many of the informants are children who need protection to maintain their integrity and privacy. There are important issues that affect the portrayal and anonymity of subjects and the presentation of research data in a manner that is both trustworthy and fair to the informants. The basic ethical requirement is that the study must not create an extra burden upon, nor stigmatize or disadvantage, the informants.

Anticipating Contradictions, Discrepancies, and Divergent Goals Already in their first year in school some pupils will have had traumatic experiences of not being able to cope, and their families will share these. Earlier in this chapter the young pupil, the school-starter, was introduced as a “champion.”

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When this image of oneself is changed by the challenges of school, it can create unhappiness, fear, and disappointment. Goodchild (this volume) finds in his study of students’ goals in the mathematics classroom that many students appear ready, almost without question, to conform to the teacher’s expectations. According to Goodchild some pupils’ participation in classroom activity can be seen as a result of habit or custom because this is the accepted practice in the classroom (Goodchild, this volume). The classroom code elicits an appropriate behavior and the pupil develops a rationale for demonstrating the outward signs of the expected behavior, which is not necessarily learning activity. In some cases a more important goal is to conform to the behavior belonging to the classroom context. The young child seeks to perform as a pupil according to the signs and norms of school, and when she/he does not succeed, she/he is defined as a SEN pupil. For example, it is important for Connie, a pupil in my study, to tell about her competence and achievement in mathematics. On the other hand, Connie’s special needs teacher talks about her worries about Connie’s progress in school and specifically expresses her concern about mathematics. In this event it becomes difficult to grasp the contradiction in the two informants’ understanding of the classroom context. Perhaps Connie’s statement has to do with her wish to succeed and meet the challenges of the school or to hide or conceal her failure. The teacher, on the other hand, has the responsibility for creating a productive learning context for all pupils; she wants to observe their progress as a sign of success, and Connie’s progress in mathematics does not satisfy the criteria set by the teacher. When Connie tells me that she understands mathematics, she is telling how she experiences the subject. She knows something about mathematics; she knows something about the relationship between herself and the teacher. It could be a statement that she uses to avoid the fact that mathematics is difficult for her to handle. However, the teacher tells me that Connie does not understand mathematics; her statement is based on her knowledge of the subject and the progress of other pupils at the same age. In the teacher’s understanding, Connie has not acquired sufficient knowledge in mathematics. The reason for Connie’s lack of progress in mathematics must be that she has special educational needs in the subject. Important questions arise from this contradictory evidence: how does the teacher’s attitude toward Connie’s work influence the pupil’s possibilities for learning? Will the relationship between the pupil and the teacher create a situation in which the teacher’s assumption about Connie’s ability makes her assumption come true? That is, is it a self-fulfilling prophecy? The meeting point of the different informants’ expressed understanding constitutes the field of the child’s learning. In the conversation, there are two different languages present, the language of the informant and that of the interviewer. Are the people in conversation achieving consensus in the meanings they are constructing, or are there different discourses present in the communication? The goal and the issue of the conversation are decided and introduced by the researcher; the informant is invited

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to participate. The responsibility of the researcher is to follow up relevant issues throughout the conversation, to explore the phenomenon more deeply and systematically in order to expose authentic evidence of the informants’ understanding. The researcher brings to the conversation his/her own set of values, interests, and history, as does the informant. The qualitative research paradigm includes these issues as a crucial part of the methodology. Differences in status and ownership of the research design make the two participants unequal. To illustrate the issue about the location of authority for setting the agenda for the conversation consider the following example of a pupil, Dennis, who starts off by telling me that he is going to the swimming pool later the same day. This is an issue of great importance to him at that moment, and I have to search for a way by which he can return to the issue of the conversation without losing his self-respect in the situation. I try to connect what is going on in the pool, where he is learning to swim, to the physical education class in school. The danger in a case like this is for the researcher to silence the informant. If I had dismissed the information that Dennis wanted to give me about his plans for the day, I would be telling him that what he regards as important is not really valuable to me. In this discussion of conversation as a research tool, I have indicated the near intimate exchange of reflections and views between the informant and the researcher. The method relies on the mutual effort of researcher and informant to develop the conversation into a research situation. It is also important to take particular notice of the emphasis placed by the informants in their manner of expression. In the conversation the way the arguments are expressed, as well as the content of the utterances, is important. In the hermeneutic approach adopted in this research, evidence is a matter of what is said, how it is expressed, and the interpretation of why it is said.

THE STUDY As I have outlined at the beginning, the pupils in the study were identified as having special needs after some months into their first year at school. Some had general learning difficulties, some speech and language problems, and others had behavioral problems. The teachers in the study were the class teachers and special needs teachers of the pupils. Altogether I held 25 conversations with pupils, their teachers, and their parents. In this discussion I focus attention on the conversations with pupils and teachers because they play the most significant role in the formal setting of the SEN arena. The conversations prompt pupils and teachers to talk about their shared experiences in the classroom, their stories of social interaction are clearly told, and they offer a vivid picture of how they think about and explain their actions. In classroom research in Norway, the authority to allow researchers into the classroom rests with parents in cooperation with the school authorities. The

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young informants were selected for the study in agreement with the local school authorities. The informants were invited, in a letter, to participate in a conversation about how they understood the education provided for the pupils, and they were given the choice of time and venue for the conversations. Most of the conversations with the pupils and their parents took place in their homes in the afternoons as they considered this to be convenient. The conversations with the class teachers and SEN teachers took place in staff rooms, offices, or spare classrooms at times during the school day when they were otherwise free from commitments. The duration of the conversations varied from one to about two and a half hours. The differences in time and location are significant. The children, for instance, were tired in the afternoon; likewise the teachers, at the end of a busy day, probably would rather have left for home than engage in a research conversation. The decision to allow the informants to choose when and where the conversations were to take place arose from my belief that motivational factors were more important than the uniformity of the conversational context. I did not invite the informants to my office for the conversations because I was concerned that this would not be neutral ground for all the informants. The informants include young children, as well as adults, teachers, and parents, and it is possible that the context of a strange office could influence a pupil’s ability to tell their stories. I considered the possibility of holding the conversations with pupils during the school day. However, to require the pupils leave class to meet me, a stranger in that context, might have appeared unduly “significant” and consequently interfere with the authenticity of the feelings expressed in the conversations. It was, of course, unreasonable to ask the teachers to leave their classes. Consequently, meeting the informants in the afternoon after school seemed the most appropriate approach. For the benefit of the outcome of a study, a researcher wants motivated informants; additionally, I wanted them relaxed and confident. Thus, it is because of my recognition of the possible significance of context that I invited the informants to choose the location for the interview.

INTERPRETATION OF QUALITATIVE DATA There are many approaches open to the interpretation of the spoken and written word, just as there are a multitude of measures, tests, and programs available in the quantitative research tradition. The richness and complexity of qualitative methods and data create great problems for a standardized interpretation. However, ultimately, the outcome of a study depends upon the nature and quality of the interaction between the person who knows and the known. Thus, in a given context the specific working hypothesis is best tested and verified by the people who operate in the field. Furthermore, as Lincoln and Guba (1985) signal, serious consideration must be given to the influence of the researcher in the creation of the data.

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Crapanzo writes: The messenger is always in a precarious position. Caught between two worlds, that of the message giver and that of the message receiver he is caught between two desires. The message giver wants the messenger to convey his message and the message receiver (despite himself) wants to receive the message he desires. Each attempts to seduce the message, where does the messenger stand? He himself is not without desire. He has got some power. He lives in a world charged with value, with loyalties and with animosities. (Crapanzo, 1992, p. 3)

Who is the interpreter; who is the messenger? In many ways it can be difficult for the researcher to accept that the understanding of the world is constituted by our interpretation of the world. The interpretation of qualitative data takes place at consecutive stages throughout the study. The first and crucial stage is during the conversation itself; throughout and especially toward the end of the conversation, statements and interpretations are checked with the informants by repeating some questions or posing new questions that are intended to elicit confirmation of the researcher’s perceptions. At this stage, differences of opinion can be discussed and clarified. The researcher is confronted, on the one hand, with reasonable demands for reliability and validity and yet, on the other hand, is aware that the quality of evidence that can be exposed is highly dependent on the context and character of his/her informants. To test the validity of qualitative data, it is necessary to test the responses of the informants. To ensure this the researcher can ask, during the conversation, “Am I right when I believe you said. . . ?” In this study I was also able to send the transcripts of conversations to the informants who volunteered to read them and comment. Two informants came back with useful comments. As some of the children were not able to read for themselves, I sent them a copy of the tape of their conversation with me. The second stage of interpretation takes place when the audio recordings of the interviews are transcribed. When the researcher transcribes the audiotapes, it can serve the purpose of revitalizing the memory of the conversation; and unspoken communication, such as body language that lies behind the words, can be recalled. Lincoln and Guba (1985) assert that the researcher adopting qualitative inquiry methods should seek to establish the trustworthiness of the outcomes rather than reliability and validity, which are concepts that arise from positivist or scientific research paradigms. One method researchers use to establish the trustworthiness of their work is triangulation, in which the data, evidence, analysis, and interpretation are subject to scrutiny from different perspectives or people. Lincoln and Guba assert, “Triangulation of data is crucially important. . . . As the study unfolds and particular pieces of information come to light, steps should be taken to validate each against at least one other source” (Lincoln & Guba, 1985, p. 283). Subjectivity of the interpretation can be avoided by letting more than one person make an interpretation of the data. However, even here differences in interpretation of the same conversation might not be the result of bi-

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ased subjectivity. They could arise through different interpretations of the research questions. In this study I was able to employ a research assistant to review the transcripts and provide an independent analysis. The results of the two interpretations were compared and revealed a high measure of agreement. The text was analyzed, searching for the most used words and the way in which words were used, such as to describe and evaluate teaching and learning. Thus, an important assumption here is that the way a person understands a phenomenon and recognizes its importance is expressed in their choice of vocabulary. Connie, for example, uses the term “the red book” when she, as a pupil, describes what the SEN teaching means to her. The “red book” is one item that she recognizes as important to the SEN context. Connie does not tell about what subject the red book is used for, or if it is a textbook or a practice-book. She does not say if she uses the book every day, or seldom. These issues appear to be of little significance to her. I interpret this to mean that the red book is, to her, a signal; it organizes her school day, thinking, and understanding of the context. When the red book is present, she is participating in a SEN lesson. Practicing is another term used frequently in her story. She is practicing mathematics, reading, and writing. Practicing is what teaching and learning is about in school. Another important word used by Connie is understanding; she tells that she understands everything in math. These two terms, practice and understand, are used most frequently, in addition to “the red book,” and these may seem to Connie like synonyms for special education. It must be recognized, however, that there is difficulty in interpreting exactly what Connie means when she uses the word “understand” because the word arises in the context of school, and it is to be expected that Connie will use it, but we cannot be sure what she means by it. Anna, the teacher, stresses the concept of time when she talks about SEN teaching in school. She also emphasizes goals of learning as important in special education. In this way the analysis of informants’ language, through the frequency of certain terms in their stories as well as the choice of terms in the vocabulary, is not seen as arbitrary. Choice and frequency of use of words and expressions provide important indicators of informants’ understanding of special education.

OUTCOMES The outcome of the study depends upon the nature and quality of the interaction between the person who knows and the one who wants to learn. From the conversation with the pupils, we get an insight into how they think about their own school situation and their own ability to handle the new learning situation. The teachers’ statements give supplementary information about the activity in the classroom, their beliefs, and attitudes toward the abilities and achievements of the children. The study proved effective in that it produced answers to some of the questions outlined above (p. 72); however, some questions remained unanswered.

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I learned how the informants expressed their understanding of classroom activities, but it is not possible to draw specific conclusions in the matter of how the differences in understanding of the situation influenced the learning of the pupils. In conversation the informants reveal two different types of interpretations or understanding of activity in the classroom. First, there is the understanding expressed by both sets of teachers (class teacher and SEN teacher) with reference to the curriculum, the timetable, and the school as an institution. They often express concerns; they talk about diagnoses, about learning problems, and about more extra lessons required because of the lack of ability and achievement in mathematics among the pupils. The teacher Anna expresses this when she says, “Teaching in the SEN classroom, you have to remove the pupils from the class and teach them elsewhere. The teaching takes place on the sideline, mostly out of the ordinary classroom. Goals of the teaching of these pupils, of course, are different from those of the ordinary pupils in the class.” “Goals of teaching” and “ordinary pupils in the class” indicate her frame of reference. Such references are the language of the school and can be seen as an understanding based on the institutional references. This was common to most of the class teachers and SEN teachers. Their understanding belongs to the institutional context, the school language. I needed to develop a classification system to organize my findings. What was common in their statements about their understanding, and what characterized the differences? I started out by interrogating the data to answer the following question: what do the conversations tell me about the participants’ understanding? The teachers used the institution, that is, the school, as their frame of reference. I classify this understanding as an I-understanding as it refers to the school as an institution. When outlining activity theory as the basis for the theoretical framework of the study, I indicated the centrality of the motive and goal of the activity. Behind the stated motive for action lies an understanding of the context. The motive for action arises from the person’s rationale for participating in the activity. When a teacher refers to the institutional rules, limits, and possibilities as signs of her/his understanding, it is possible to interpret these as a rationale for her/his actions in the classroom. The pupils referred to mathematics as an important subject (Linden, 1997). They enjoyed mathematics, it was exciting, they understood mathematics, and they were good at it. These are some of the expressions they used. School is fun. They work in small groups because the other children in the small group are nice. They learn certain subjects because the teacher wants it, and they work in a “red book.” Their statements are couched in the strong language of experience. Their language belongs partly to the private sphere, partly to their shared experiences in the classroom. Taking the common reference as their experience, I classify their understanding as an E-understanding, where E stands for experience. Consequently, I find that their rationale for learning must rely on their experiences in the learning situation, their learning about how to learn, or their meta-learning. I call this an E-rationale. Bauersfeld (1976) stresses the differ-

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ence between the matter intended, the matter taught, and the matter learned. The matter intended is the content of the subject, the syllabus in mathematics in school’s first year, while the matter taught relates to how the teacher presents the subject, and the matter learned refers to how the pupil handles the subject content. Bauersfeld’s distinction between intended and received lessons implies acceptance of the fact that the pupil does not always recognize the subject the way the teacher does, or the way the teacher meant to explain it to the class (Bauersfeld, 1976). The analysis of the discrepancies between the matter taught and the matter learned can serve as one explanation for the I- and E-understandings revealed by this study.

FINAL CONSIDERATIONS The close relationship between the research methods and the properties of the phenomenon of interest of a study provides a strong reason for the choice of a qualitative approach. Qualitative data cannot be analyzed to give a complete view of the researched field. In studying people’s “lived-in” world we need to get access to their own personal views or understanding. We need to open up what they want to bring forth in their own words. In the conversation the participants will manifest the importance that they ascribe to a certain idea or event. This idea will, in the course of the conversation, develop from a general communication to a conversation about some common issues of interest, in a formal way. In the course of the study, data constitutes new knowledge about the actual phenomenon. If we use the data to illustrate our arguments, they represent our personal interpretation of the situation. However, if they are presented as proof or generalizations, they are false because there is no evidence in the data as such. The data might be used to bring some phenomenon into light and derive new insights about the problem statement. The close relationship between sociology and education in qualitative studies may seem obvious. At the same time as the study is approaching an educational field, the sociological perspectives of the study are important. The informants of the study participate as subjects in the conversational activity, not as objects of the researcher’s interest. The process entails taking another person’s perspective as important data. The crucial stage of classroom research, as in all research, is when the methodological approach is defined; methodology comes from the Greek word methodos, meaning the road to a goal. My intention in this chapter has been to give an outline of the methodological issues concerning one qualitative method in classroom research. My choice of research tool was based on my beliefs in the close relationship that exists between the informants and myself as researcher, and my respect for the informants as both subject and object in their own lives. It is possible that alternative approaches would have led to the goal, but I had to choose from my own individual perspectives. My choice was to use conversa-

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tion as a tool for getting to know more about the participants’ understanding of an important part of their lives. In this study the informants showed me that the tool was appropriate in their enthusiastic participation and their willingness to explore some important questions in the field of special needs education. In this chapter my aim has been to discuss a particular study, as well as one methodological approach to educational research. The consideration of the subjective nature of the conversation as a research tool has led me to a deeper understanding of the significance of the researcher as an influencing factor in the process of generating the data, and this influence is a key factor in qualitative research. For decades research in education has asserted the ideal of a qualitative paradigm in which the researcher aims at neutralizing her/his influence on the data production. This ideal of the researcher “stepping back,” as an important feature of a study, is not consistent with the methodological principles of qualitative research. When planning the study, the participation of me as the researcher was of particular importance. My own experience as a former practitioner, a SEN teacher, gives me a mental schema of the field that an inexperienced researcher, or one unfamiliar with SEN provision, would lack. In choosing conversation as a research tool, experience in the field was one of the determining issues. Another influencing aspect was the adoption of activity theory as the basic theoretical perspectives of the study. In this the foundation for a theoretical influence was laid. Earlier in this chapter I stressed the necessity of a close relationship between the researcher and the informants in the process of the conversation. Human relationships are impossible to standardize or operationalize in the scientific meaning of these terms, that is, when objectivity is the goal. The research tool, or scientific method, is chosen according to the aim of the study. The method is constituted throughout the study itself through the interaction between the researcher and the informants. The data is a product of this interaction as the researcher is her/his own instrument in the research process. There are some psychological qualities influencing the process of eliciting the data. The vocabulary chosen during the conversation, the interpretation of the questions and answers, and the degree of experienced empathy from both sides are examples of such qualities. When I ask the informants “By what you said just now, do you mean. . . ?” I can give the impression that I want the informants to confirm my interpretation, and the informants could interpret me as wanting “the right answer.” Some of these qualities are invisible and unconsciously manifest. Therefore, the effect on the study is unknown. This effect should, however, be explored as an issue of interest in the qualitative research paradigm.

REACTION BY CYRIL JULIE Nora Linden sets herself the task of giving “an outline of the methodological issues concerning one qualitative method in classroom research.” These issues center on the data collection, data transcription, and data interpretation. The

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main research endeavor is about various participants’ understanding of the SEN experience. In critiquing the methodology I developed descriptors while reading and rereading the chapter. These descriptors are minimization of contamination and surfacing and redirection. Minimization of contamination refers to the procedures taken to ensure that data were collected, transcribed, and interpreted in a responsible and accountable manner. The author accomplishes this in various ways during the different phases of the research. First, during the data collection care was taken to leave the choice of site for conversation to the interviewees. The author views context uniformity, her office in this case, as a possible contaminating factor. This need not necessarily be so. My own experience is that one can engage in a fairly motivated and meaningful conversation about a topic of interest in unexpected environments where research participants are met. Of course, in such cases the data collection tools might not be readily available. This can, for example, be done by the production of a report of the conversation immediately after the incidental meeting. The report can then be presented to the research participant for verification. The gathering of data through spontaneous conversation at sites not originally catered for in the research plan and its verification by the “living” should not be summarily dismissed. The minimization of contamination is, in the second instance, striven for during the transcription of data. Of particular interest here is the apparent insertion of the nuances, the “body language,” into the data by the researcher. There is, however, no indication of whether the transcribed data, with the inclusion of the nuances, were subjected to some form of verification. Third, minimization of contamination is sought during interpretation. The procedures used here are participant checking and independent interpretation as a triangulation mechanism. In asserting that useful comments were obtained from some participants after they had read the transcripts or listened to the audio recordings, no indication is given of what happened to these comments or whether they were fused or not with the data. Another issue with the triangulation exercise is the resolution of differences of interpretation between the independent interpreter and the researcher. Although there were differences (albeit characterized as a small measure of disagreement) between the outcomes of the analysis of the two parties, there is silence on how these differences were resolved. Consensus is mentioned but not how consensus was reached. Given that the author refers to the differences that might arise because of differing interpretations of the research questions, one is left wondering whether these small measures of disagreement resulted from research question, interpretation, or something else. Depending on which is which, different resolution strategies would have been followed. It might, perhaps, be read between the lines that the judicial procedure of competitive argumentation was used to reach consensus. This does not absolve the author from addressing the power relations since in many instances the inequality of power is already inherent in the classification of researcher and research assistant.

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Surfacing and redirection is a composite singular descriptor and refers to the centrality of concentrating the conversation on the issue of pursuit. Within this form of data collection it easily happens that informants want to, at times, bring to the attention of the researcher an issue which, according to data being sought, is not really of immediate interest. Judgments of this nature must be made on the spot by the researcher, and she or he must then find ways to redirect the conversation in an appropriate direction. Nora Linden uses a variety of techniques to address surfacing and redirection. At the level of getting the conversation started “invitation by provocation” is utilized. The researcher ostensibly engineers this provocation. In doing such the proclaimed “open-endedness” is, in a sense, compromised, and there is a need for qualifying what “open-endedness” means in this study. A second instance of surfacing the researcher refers to is that of collegial familiarity between the researcher and informant and the informant steering the conversation in a seemingly unwarranted direction. In referring to “the danger of overemphasising the importance of consensus,” and “the presence of the researcher is a more influential sign of power than explanations and formal agreement,” the redirection strategy suggested for these kinds of situations appears to be subtle assertion of researcher power. Personally, I would have preferred that in these cases consensus be renegotiated since the assertion of researcher power is, in a sense, a reneging on the tenets that underpin research of this genre. A question in this case is whether the authenticity of the data is compromised. This is a judgment that must be made by readers and enough information is to be supplied for readers to indeed make these judgments. A third instance of surfacing and redirection can occur when informants steer the conversation in an unwanted direction because they operate under the perception that they can deviate and talk about their interest of the moment. The author refers to this in her relating of the case of a boy who wanted to speak about his swimming activity. The suggested redirection strategy that was employed is that of connecting the informant’s interest to the research topic. This requires quite some insight from the researcher since the topic of interest for the informant might be so distant from the issue of pursuit that finding linkages might be extremely difficult. Nevertheless, this is an issue on which more light will be shed as more studies of this nature appear. Notwithstanding the above comments on the data collection, it can be concluded that sufficient mechanisms were set in place to ensure that the data were authentic in the sense of being related to the issue that is being pursued. The analysis procedure followed was basically a compare and contrast process driven by the question, “What do the conversations tell me about the participants’ understanding?” By analyzing the text in the search for the most used words and the way words were used, the usage of language by the informants was confirmed as an important tool for the interpretation of the data. This was separately done for the teachers and the students. Construction of categories was effected through a search for the frequency of word and phrase occurrences motivated by the view that “the frequency of cer-

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tain terms in their stories as well as the choice of terms in the vocabulary is not seen as arbitrary. Choice and frequency of use of words and expressions provide important indicators of informants’ understanding of special education.” Whether this process was wholly inductive, or moved from an initial inductive one to a later deductive one, is not clear. I assume it was, as in most research of this nature, the latter, but only the researcher can verify this. Although frequency of occurrence was used, the particular words or phrases or a collective name for a phrase was not seen as the categories. The researcher moves beyond this initial coding-like analysis to a more abstract level, driven by both the literature under scrutiny and the question at hand, to develop two categories: one for the learners’ understanding—the E-rationale, and one for the teachers—the I-rationale. On the surface it appears that the two rationales are similar, both embedded within the experiences of both parties in the institution of special education needs. The data that are provided and the techniques and procedures used for their collection and analysis within the chosen theoretical framework are, however, such that this distinction can legitimately be made for this study.

RESPONSE TO CYRIL JULIE I want to thank Professor Julie for his thorough and helpful comments. The remarks on the minimization of contamination and the interpretation of the data offer a welcome reminder of how careful one should be in explaining and controlling one’s handling of the data. The comments elicit an important critique as far as the clarity in the presentation of such studies is concerned. I want to comment on two issues raised by Cyril Julie regarding the minimization of contamination of the data. First I will address the issue of the informants’ choice of time and place for the interview. I left the choice to the informants mainly due to the fact that the group of informants comprised children and adults. Children with special needs will have a history of several visits to “the office” such as to doctors, speech therapists, and psychologists. This history could make them vulnerable to fear and anxiety in a similar context. It was important for me to meet them in a good and secure environment for the data collection, and I regarded the home environment better for this purpose than, for instance, my office. Security and willingness to participate was a significant issue, and I refer to Matre (2000), who stresses the importance of making the participation pleasurable, natural, and secure. As for the parents some of the reasons stressed above are equally relevant. In addition, it has to do with the practicability of meeting the parents who are, mostly, at work during the day, and for some it could be a problem to arrange meetings in the afternoon outside their home. Similarly, in the course of a school day it is difficult for the teachers to meet for an interview. They all chose school as the most convenient place to meet. My argument for leaving the choice of where to meet with the informants has to do with seeking the best outcome of

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the data collection and to secure this by seeking the most favorable and motivated participation of the informants. I assume it could have been done by other means; these were what I chose for the present study. My second comment regards Cyril Julie’s remarks about how the informants’ vocabulary could indicate traits of their understanding, and how the categories of terms were developed: I will refer to LeCompte and Preissle (1993). As Cyril Julie points out, in the qualitative research paradigm both a deductive and an inductive dimension are commonly represented. The categories were developed during the course of the study, that is, as a consequence of the data gathered. Since the interview as a research method depends on the use of language, one of the basic theoretical emphases was on the importance of language for learning. I want to add that the above critique of the methodological reflections and interpretation of the data I presented in this chapter has evoked a fresh understanding for me about how careful one should be when planning and performing a study such as I have described. An example of this is when Cyril Julie points to my emphasis on the significance of the uniformity of context in my study. This issue seemed to me to be the best way, and in fact, crucial to the process of collecting authentic data when the study was initiated. When reflecting on this issue from the view expressed by Cyril Julie, I clearly understand that I could have regarded different means to secure the motivated cooperation of the informants. Cyril Julie’s suggestion about making a report immediately after the interview, which could be presented to the informants for verification, seems like a good alternative. I want to thank Professor Cyril Julie for an insightful and constructive critique, for giving me a new perspective on the study, and for the opportunity to “discuss” my research methods in this dialectical structure.

REFERENCES Bateson, G. (1972). Steps to an ecology of mind. New York: Ballantine Books. Bauersfeld, H. (1976). Research related to the mathematical learning process. In New trends in mathematics teaching. IV (pp. 199–213). Paris: UNESCO. Corbett, J. (1996). Special educational needs in the twentieth century. London: Cassell. Crapanzo, V. (1992). A critique of the conservative defense of culture. Chicago: University of Chicago Press. Edwardsen, E. (1981). Skolen som parantes i samfunnet. Norsk Pedagogisk Tidskrift, 7, 283–291. Efskin, R. (1998). Matematikkutvikling gjennom bruk av matematikkverksted i førskole og småskole. In B. Selvik, et al., Klasselæreren som matematikklærer (pp. 141–148). Bergen, Norway: Caspar Forlag. Fosse, T. (1996). Hva venter de seg av skolens matematikk? In M. Johnsen Høines (Ed)., De små teller også (pp. 137–143). Bergen, Norway: Caspar Forlag. Freud, S. (1992). Psykoanalyse. Oslo, Norway: Gyldendal Norsk Forlag. Husserl, E. (1973). Fænomenologiens Ide. København, Denmark: Reitzels Forlag.

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Jenkinson, J. (1997). Mainstream or special: Educating students with disabilities. London: Routledge. Kvale, S. (1997). Interviews. Oslo, Norway: Gyldendal Norsk Forlag. LeCompte, M. D., & Preissle, J. (1993). Ethnography and qualitative design in educational research. San Diego, CA: Academic Press. Lillejord, S. (2000). Handlingsrasjonalitet og spesialundervisning. Doktoravhandling: Universitetet i Bergen. Lincoln, Y. S., & Guba, E. G. (1985). Naturlistic inquiry. Beverly Hills, CA: Sage. Linden, N. (1997). Education and educability. Nordiske Udkast, 2, 31–43. Lindh-Munther, H. (1989). Barnintervju som forskningsmetod. Uppsala, Sweden: Universitetet i Uppsala. Matre, S. (2000). Samtaler mellom barn. Oslo, Norway: Det Norske Samlaget. McLeod, D. B. (1992). Research on affect in mathematics education: A reconceptualisation. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 575–596). New York: Macmillan. Mead, G. H. (1934). Mind, self and society. Chicago: University of Chicago Press. Miles, M. B., & Huberman, A.M. (1994). Qualitative data analysis. London: Sage. Molander, B. (1993). Kunnskap i handling. Gøteborg, Sweden: Bokforlaget Daidalos. Neisser, U. (1976). Cognition and reality: Principles and implications of cognitive psychology. San Francisco: W. H. Freeman. Piaget, J. (1970). Barnets imitasjon, leg og symbolspråk. København, Denmark: Munksgaard og Reitzel. Royal Ministry of Education, Research and Church Affairs. (1997). Core curriculum for primary, secondary and church affairs. Oslo, Norway: National Centre for Educational Resources. Vestby, G. (1999). Samfunnsvitenskapelig forskning med barn—status presens for forskningsetikken. In R. Bugge (Ed.), Etikk og forskning med barn. Skriftserie nr.13. Oslo, Norway: De Nasjonale Forskningsetiske Komiteer. Vygotsky, L. S. (1978). Mind in society. Cambridge: Harvard University Press.

Chapter 5

A Methodology of Classroom Teaching Experiments Kay McClain

The research reported in this paper was supported by the National Science Foundation under grant no. REC-9814898 and by the Office of Educational Research and Improvement through the National Center under grant no. R305A60007. The opinions expressed do not necessarily reflect the views of either the foundation or the center.

INTRODUCTION The purpose of this chapter is to describe a methodology for conducting classroom-based research known as the classroom teaching experiment (cf. Cobb, 2000; Cobb & Steffe, 1983; Confrey & Lachance, 2000; Simon, 2000; Steffe & Thompson, 2000; Yackel, 1995). The particular type of teaching experiment discussed in this chapter is conducted in collaboration with a classroom teacher who participates as a member of the research and development team. The primary goal of the classroom teaching experiment is to test and revise an instructional sequence that is designed to support students’ mathematical development in a particular content domain. Concurrent with those efforts, the research team is also focused on the means of support necessary to achieve that development, such as norms for argumentation, the role of the teacher in deliberately facilitated whole-class discussions, and the use of tools. Classroom teaching experiments vary in duration from a few weeks to an entire school year. During this period of time, the concerns and issues discussed by the research team are frequently highly pragmatic since the teacher and researchers are together responsible for the quality of the students’ mathematics

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education. As a consequence, conjectures about how best to support students’ mathematical development in a particular content domain are continually being modified against the background of informal daily analyses of students’ ongoing mathematical activity. These analyses inform microlevel decisions about the appropriateness of particular mathematical activities from the instructional sequence. Detailed, retrospective analysis of data sources1 provides the basis for large-scale modifications and revisions to the instructional sequence. The particular methodology that my colleagues and I2 employ falls under the general heading of design research. Design research involves cycles of research and development in which ongoing classroom-based research feeds back to inform instructional design decisions in a cyclic manner as shown in figure 5.1. The initial instructional design efforts result in a conjectured sequence of instructional tasks. This sequence is subject to modification and revision based on results of daily analysis of the students’ mathematical activity in the classroom as they engage in these tasks.Therefore, daily analyses are used to guide the microlevel decisions about appropriate tasks for the following day. These decisions are also informed by the initial design efforts that provide a meta-level frame in which to work. In this way, while the intent of the sequence is outlined prior to entering the classroom, decisions about specific mathematical tasks are made on a daily basis. The instructional design efforts are guided by a domain-specific instructional theory. In particular, the theory that we use to guide our efforts is taken from Realistic Mathematics Education (RME), which was developed at the Freudenthal Institute in the Netherlands. (This theory will be elaborated in the next section of the chapter.) An interpretive framework known as the emergent perspective guides the classroom-based research efforts. The emergent perspective involves coordinating constructivist analyses of individual students’ activities and meanings with an analysis of the communal mathematical practices in which they occur (cf. Cobb & Yackel, 1996). This framework was developed out of attempts to coordinate individual students’ mathematical development with social processes in order to account for learning in the social context of the classroom. It, therefore, places the students’ and teacher’s activity in social context by explicitly coordinating sociological and psychological perspectives. In the following sections of this chapter I begin by elaborating the two theoretical positions that form the basis of the design research cycle, the emergent perspective and RME. Next, the methods used in conducting a design experiment are clarified.A discussion of the trustworthiness, generalizability, strengths, and weaknesses of the methodology follows. I conclude with a summary of the significance of the classroom teaching experiment with respect to its value and usefulness to other mathematics educators and classroom teachers.

THEORETICAL FRAMEWORKS The reader will recall that the design research cycle involves the interplay of (1) the development of instructional materials and (2) the ongoing research of

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Figure 5.1 The Design Research Cycle

the effectiveness of those materials in the context of a classroom. The emergent perspective, explicitly coordinating sociological and psychological perspectives, guides analyses conducted during the classroom-based research phase of the cycle. The psychological perspective is constructivist and treats mathematical development as a process of self-organization in which the learner reorganizes his/her activity in an attempt to achieve purposes or goals (Glasersfeld, 1995). The sociological perspective is interactionist and views communication as a process of mutual adaptation wherein individuals negotiate mathematical meaning (Bauersfeld, Krummheuer, & Voigt, 1988). From this perspective, learning is characterized as the personal reconstruction of social means and models through negotiation in interaction. Together, the two perspectives treat mathematical learning as both a process of active individual construction and a process of enculturation into the mathematical practices of wider society. Individual and collective processes are viewed as reflexively related in that one does not exist without the other (Cobb, Perlwitz, & Underwood, 1994). This coordination of the social and psychological perspectives within the emergent perspective is highlighted by the two columns shown in figure 5.2 (cf. Cobb & Yackel, 1996). The entries in the column under “Social” indicate three aspects of the classroom microculture that we have found useful when con-

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Figure 5.2 The Interpretive Framework

ducting our analyses of the classroom community. The three entries in the column under “Psychological” clarify the three related aspects of individual students’ activity as they participate in the classroom community. To further clarify, social norms can be characterized as general norms that are necessary for engaging in classroom discussions. Such norms include explaining and justifying solutions, attempting to make sense of explanations given by others, and challenging others’ thinking. These types of norms, however, are general classroom norms that may apply to any subject matter (Cobb & Yackel, 1996; Yackel & Cobb, 1996). The psychological correlate of social norms is the teacher’s and students’ beliefs about their role, others’ roles, and the general nature of mathematical activity in school. In describing the reflexive relationship between these two correlates, Paul Cobb and Erna Yackel maintain that, on the one hand, social norms are seen to evolve as students reorganize their beliefs (1996). On the other hand, the reorganization of these beliefs is seen to be enabled and constrained by the evolving social norms. Sociomathematical norms focus on the normative aspects of mathematical discussions that are specific to students’ mathematical activity. These norms include what counts as a different mathematical solution, a sophisticated mathematical solution, an efficient mathematical solution, and an acceptable mathematical explanation and justification. These norms all deal with a taken-as-shared sense of when it is appropriate to contribute to a discussion. The term taken-as-shared is

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used to describe these norms since ensuring that all members of the classroom community have a “shared” interpretation would entail conducting individual interviews. As a result, analyses of the classroom participation structure are used to determine normative ways of participating that are then characterized as takenas-shared. Students’ personal beliefs and values about engaging in mathematical discussions make up the psychological correlate of sociomathematical norms. These two aspects are also related reflexively. Students develop their own beliefs about judging if solutions are different, sophisticated, efficient, or acceptable. As students then participate in discussions, new sociomathematical norms are continually negotiated and redefined which, in turn, influence students’ individual beliefs (McClain & Cobb, 2001; Yackel & Cobb, 1996). Classroom mathematical practices are the taken-as-shared ways of reasoning, arguing, and symbolizing established by the classroom community while discussing particular mathematical ideas. Just as sociomathematical norms are specific to mathematical activity, classroom mathematical practices are specific to particular mathematical ideas (Cobb, 2000). An indication that a classroom mathematical practice has been established is that explanations about that practice are no longer necessary; it is beyond justification. The reflexive nature of this framework is again noted in the relationship between classroom mathematical practices and their psychological correlate, individual students’ mathematical conceptions. Individual students’ mathematical interpretations and actions contribute to the constitution of the classroom mathematical practices. At the same time, the classroom mathematical practices influence individual students’ activity. Therefore, students are seen to contribute to the evolution of the classroom mathematical practices as they reorganize their activity while contributing to their establishment (Cobb & Yackel, 1996). The theoretical framework outlined above serves to guide analyses of the classroom by providing a way to account for the learning of individual students and the classroom community. In addition, it situates the analyses of individual students in the social context of the classroom or of the interview situation, thus providing another lens on the learning process. It could be argued that enclosing the framework within larger boxes that account for the social context of the school and community might better represent the framework (see figure 5.3). However, for the basis of analyses to be discussed in this chapter, the framework as configured in figure 5.2 is sufficient. The reader will recall that the instructional design theory that guides the development of our instructional sequences during a classroom teaching experiment is taken from Realistic Mathematics Education. It is important to note that RME did not seek a basis in general instructional theory. Neither did it allow specific instructional objectives to serve as the points of departure. Instead, a philosophy of education was adopted in which learning mathematics was seen as an “active process and the teaching of mathematics as a process of (guided) reinvention” (Gravemeijer, 1994, p. 21). By taking the solution procedures that students use as starting points for instructional design, opportunities present them-

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Figure 5.3 Elaborated Interpretive Framework

selves for strategies to develop naturally out of the students’ investigations. The fundamental challenge is then to support students’ transition from informal mathematical activity to more formal yet personally meaningful activity. One of the central tenets of RME is that the starting points of instructional sequences should be experientially real to students in the sense that they can immediately engage in personally meaningful mathematical activity (Gravemeijer, 1990; Streefland, 1991). As a point of clarification, it should be stressed that the term “experientially real” means only that the starting points should be experienced as real by the students, not that they should necessarily involve situations from their personal experience. In this way, students’ imagery of the situation provides support for their mathematical investigations. The goal is that initially the students’ solutions will be informal and grounded in their conceptions or imagery of the situation. This initial grounding can then support mathematical shifts toward more abstract ways of reasoning about the situations. A second tenet of RME is that in addition to taking account of students’ current mathematical ways of knowing, the starting points should also be justifiable in terms of the potential end points of the learning sequence. This implies that students’ initially informal mathematical activity should constitute a basis

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from which they can abstract and construct increasingly sophisticated mathematical conceptions as they participate in classroom mathematical practices. At the same time, the situations that serve as starting points should continue to function as paradigm cases that involve rich imagery and thus anchor students’ increasingly abstract mathematical activity. A third tenet argues that instructional sequences should involve activities in which students create and elaborate symbolic models of their informal mathematical activity. This is accomplished by encouraging students to develop symbolic models of their informal mathematical activity and then guiding their evolution into models for reasoning mathematically about problem situations (Gravemeijer, 1994). This transition in modeling activity might initially involve making drawings, diagrams, or tables, or it could involve developing informal notations or using conventional mathematical notations. As an example, when students are introduced to addition, they might be asked to make drawings to represent the combining of some number of items, say two pencils and four pencils. As their understandings of the process develop, they would be encouraged to develop symbolic notations to communicate their mathematical activity. This might entail the introduction of conventional symbols such as the sign for addition. These more sophisticated models (in comparison to the pictures) could then become tools for reasoning mathematically about a diverse range of problem situations. This third tenet is based on the psychological conjecture that, with the teacher’s guidance, students’ models of their informal mathematical activity can evolve into models for increasingly abstract mathematical reasoning (Gravemeijer, 1994). Instructional approaches in most textbooks used in the United States also stress that mathematical learning should involve the use of models. However, the starting points typically consist of manipulatives and notational schemes that are assumed to embody the mathematical structures and relationships to be learned. In this approach, acquisition precedes the application, and the instructional sequence typically consists of structural models, formal mathematical activity, and applications. In contrast, the instructional sequences we develop in the course of a classroom teaching experiment reflect a bottom-up approach that emerges from students’ informal mathematical activity and supports their developing concepts and strategies (Cognition and Technology Group, 1992; Lesh & Akerstrom, 1982; Thompson, 1992). In this approach, the instructional sequence involves (1) informal, situated problem solving, (2) the development of models of informal mathematical activity, (3) the development of models for mathematical reasoning, and (4) more formal mathematical activity in a wide range of situations in which the manipulation of conventional symbols signifies conceptual actions on abstract yet personally real mathematical objects. The two theoretical frameworks, the emergent perspective and RME, provide a basis for our work and a “frame” for our methodology. In the next section of the chapter, I will articulate the tenets of design research that serve as an over-

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arching guide for our work. I will then use these tenets to organize a discussion of the different methodological components that constitute a classroom teaching experiment.

DESIGN RESEARCH METHODOLOGY Developing Instructional Sequences The design research cycle that we use while conducting a classroom teaching experiment is framed around three tenets that guide our research and development efforts:

. . .

The development of a conjectured learning trajectory which guides the development of the instructional sequence The development of an instructional sequence guided by a domain-specific instructional theory Daily minicycles of revision based on the ongoing analysis of students’ participation in classroom activities

The first of these tenets involves envisioning a conjectured learning trajectory for the students’ mathematical development. This would entail a proposed route through the mathematics that would support the students’ mathematical development. Our first step in developing a conjectured learning trajectory is to define the “big ideas” in mathematics that we hope to address. A thorough reading of the research literature helps identify these. This then provides us with a mathematical basis for planning our instructional sequence. The approach we take follows the heuristics of RME in that it takes students’ initial understandings as starting points and attempts to build in a bottom-up manner toward more sophisticated mathematical understandings. Against this background, we envision a conjectured learning trajectory that defines our instructional sequence. This trajectory would merely identify significant mathematical benchmarks on the route to a mathematical endpoint. As an example, in thinking about how one might support young children’s understanding of addition facts to twenty, it would be important to help them develop strategies for operating on single digit numbers such as the use of going-through-ten and doubles. These would be benchmarks that could be used to frame the development of tasks in the instructional sequence. The particular tasks and the frequency with which they are assigned would depend on analysis of the students’ activity in the class sessions. For that reason it is important to note that during the classroom teaching experiment the instructional sequence is constantly being changed and modified based on information gained from informal, ongoing analyses of students’ current understandings.3 In this way, the instructional sequence serves as a means of framing the general intent while revisions are typically necessary on a day-to-day basis. Simon (1995) has addressed this distinction by clarifying the difference between a conjectured learning trajectory and a realized learning trajectory.

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As an example, in a recent teaching experiment conducted in a third-grade classroom, our overarching goal was to develop an instructional sequence designed to provide the means for supporting students’ ability to add and subtract three-digit numbers so that their activity had quantitative significance and was not merely composed of manipulating symbols in meaningless algorithms. In order to support the students’ development, we acknowledged the importance of ensuring that they had a strong understanding of place value to ground their development of a conceptual understanding of the algorithms for three-digit addition and subtraction. Based on analyses of prior classroom teaching experiments and a review of the research literature, we acknowledged that it would be important for students to view 100 and 10 as composite units that are both individual units that can be counted and composites of 10 and smaller units. Another step along the conjectured learning trajectory was to support students’ understanding of different ways to configure a collection. In other words, they might come to think of 63 as 63 ones, or 6 tens and 3 ones, or 5 tens and 13 ones, or 4 tens and 23 ones, and so on. Further, they need to understand that a transformation of 63 ones into 4 tens and 23 ones preserves the quantity. This is important when they begin working on problems where they have to “borrow” or “carry.” The envisioned endpoint of the learning trajectory involved students using the algorithm for subtraction in a meaningful way. For instance, when students solved a task such as 265–194, they viewed the problem as composed of two quantities, the latter to be subtracted from the former, not merely symbols to be manipulated. In addition, the quantities could be transformed to facilitate the students’ problem-solving activity. As an example, viewing 265 as 1 hundred, 16 tens, and 5 ones simplifies the task at hand and counteracts the use of the “buggy” algorithm. The learning trajectory was then created based on these conjectures about how to support students’ development, which are grounded in the research literature (see figure 5.4). From figure 5.4, the reader will notice that while the main mathematical benchmarks have been identified a priori, the transitions between the benchmarks are developed based on analyses of the students’ understandings. The route between the benchmarks can, at times, be quite circuitous and involve the introduction of numerous additional concepts. After mapping out a conjectured learning trajectory, the next step is to develop an instructional sequence designed to support the evolution of the trajectory. With the aforementioned overarching mathematical goals in mind, the instructional sequence is “sketched out” prior to beginning the classroom teaching experiment. However, it is important to note that the particular activity sheets that will be used each day are not created. Instead the research team devises a more general outline of the type of activity and how the sequence will flow. Within this loose framing, the ideas that will guide the development of activities serve as a general outline of the sequence. As an example, in the third-grade classroom teaching experiment, the research team planned to begin the instructional sequence with estimating and quantifying tasks. The next phase of

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Figure 5.4 A Conjectured Learning Trajectory for Three-Digit Addition and Subtraction

the sequence involved students determining the different ways a collection could be configured. However, the actual activity sheets and the particular number of class sessions for each type of activity were not predetermined. These decisions were made on a day-to-day basis, based on ongoing analyses of the students’ current mathematical understandings. The reader will recall that the tenets of RME highlight the importance of starting points of instructional sequences that are experientially real to students in the sense that they can immediately engage in personally meaningful mathematical activity. As a result, our efforts at developing an instructional sequence began with an experientially real context. The sequence we developed in the course of the third-grade classroom teaching experiment centered on the scenario of a candy factory4 where students were asked to help the factory man-

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Figure 5.5 Symbolizing Unpacking a Roll

ager keep a record of the activity that was taking place. In doing so, students were asked to solve tasks that initially involved Unifix cubes as models of the candies and later involved the development of ways of recording transactions in the factory (cf. Bowers, 1995; Cobb, Yackel, & Wood, 1992). During initial wholeclass discussions, the students and teacher negotiated the convention that single pieces of candy were packed into rolls of 10, and 10 rolls were packed into boxes of 100. Ensuing activities included estimating and quantifying tasks designed to support the development of enumeration strategies. These activities involved showing students drawings of rolls and pieces using an overhead projector and asking them to determine how many candies there were in all. In addition, students were shown rectangular arrays of individual candies and asked to estimate how many rolls could be made from the candies shown. Instructional activities developed later in the sequence included situations in which the students “packed” and “unpacked” Unifix cubes into bars/rolls of 10. To support students’ development of a rationale for these activities, the teacher explained that the factory manager, Mr. Strawberry, liked his candies packed so that he could quickly tell how many candies were in the factory storeroom. In order to record their packing and unpacking activity, the students developed drawings and other means of symbolizing as models of their mathematical reasoning (Gravemeijer, in press). The goal of subsequent instructional activities was then to support the students’ efforts to mathematize their actual and recorded packing and unpacking activity so that they could interpret it in terms of the composition and decomposition of arithmetical units. To encourage this process, the teacher capitalized on the students’ contributions by describing purely numerical explanations in terms of packing activity and vice versa. For example, a student who describes subtracting 29 from 43 as “unpacking a roll” might find this symbolized as shown in figure 5.5. In contrast, a student whose activity was reported in purely symbolic terms would have his/her explanation recast in terms of packing and unpacking rolls of candies. This, in turn, served to support the students’ development of situation-specific imagery of transactions in the factory that would provide grounding throughout the sequence. In a subsequent phase of the sequence, students were asked to make drawings to show different ways that a given amount of candies might be found in the storeroom if the workers were in the process of packing them. For example, 143 candies might be completely packed up into 1 box, 4 rolls, and 3 pieces, or they

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might be found as 12 rolls and 23 pieces. When students first described their different ways, many drew separate pictures for each. Later, the teacher encouraged students to use numerals to make a record of their drawing. To this end, she introduced an inventory form that was used in the factory to keep track of transactions in the storeroom. The form consisted of three columns that were headed from left to right, “Boxes,” “Rolls,” and “Pieces” (see figure 5.6). The issue of how to symbolize these different ways, and thus the composition and decomposition of arithmetical units, became an explicit topic of discussion and focus of activity. For example, a typical explanation when verifying that 143 candies could be symbolized as 12 rolls and 23 pieces was to suggest packing 20 pieces into two rolls and then packing 10 rolls into a box, as shown in figure 5.6. As a final phase of the sequence, the inventory form was used to present addition and subtraction tasks in what was, for us, standard vertical column format (see figure 5.7). These problems were posed in the context of Mr. Strawberry filling orders by taking candies from the storeroom and sending them to shops or by increasing his inventory as workers made more candies. The different ways in which students conceptualized and symbolized these transactions gave rise to discussions that focused on their emerging addition and subtraction algorithms. Throughout the sequence, students’ use of drawings, diagrams, and symbols evolved from merely representing the candies in the candy factory to becoming a tool for reasoning in problem situations. In this way their drawings, diagrams, and symbols initially served as a “model of” the candy in the storeroom and became a “model for” mathematical reasoning. The evolution of the symbolizations can be traced along with the evolution of students’ mathematical development. Initially, many students had to actually manipulate Unifix cubes in order to solve the tasks. As they worked through activities in the sequence, they began to devise ways of symbolizing and recording their ways of reasoning that fit with their current understandings. Later, the use of the inventory form

Figure 5.6 The Inventory Form for the Candy Factory

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Figure 5.7 An Addition Problem Posed on the Inventory Form

emerged as a means for recording their activity in the storeroom. (For a detailed analysis of the candy factory instructional sequence, see Bowers, 1996; Bowers, Cobb, & McClain, 1999).

DATA COLLECTION Student Interviews Our work in the classroom is preceded by individual clinical interviews conducted with each student, which allow us to document their current understandings. The interviews also serve to inform our starting points for instruction. Although our work at outlining an instructional sequence has already begun prior to the interviews, it is important to verify our conjectures against the results of informal analysis of the interviews. In this way, the starting points of the instructional sequence can be justifiable in terms of both the potential end points of the learning sequence and the students’ current understandings. As a result, the interview tasks are designed to investigate the students’ current understandings about the concepts we intend to address through the classroom teaching experiment. Continuing the example of the candy factory, the reader will recognize a need for the researchers to investigate students’ current understandings of (1) place value and (2) algorithms for multidigit addition and subtraction. As a result, the interviews conducted with the students in the third-grade class were intended to provide us with information about how they reasoned on tasks that involved understanding in these two areas. As an example, one task given to the students involved a transparent bag containing 360 crayons. Students were asked to tell the interviewer how many collections of 10 crayons could be formed from the 360 crayons in the bag. The fact that only 7 of the 23 students could success-

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fully complete the task supported our belief in the need to begin the instructional sequence with activities designed to build understanding of place value. As a result, the pre-interviews serve a dual purpose. First of all, they provide information that can ensure that the instructional sequence “fits” with the students’ current understandings. In this way, the sequence can build in a bottomup manner and support the students’ development of more increasingly sophisticated ways to reason. Second, they provide baseline data on each student so that the researchers can determine the mathematical development of individual students through the course of the classroom teaching experiment. In order for this documentation to be possible, similar interviews must be conducted with each student at the end of the classroom teaching experiment. It is important to note that while many of the tasks in the post-interview are the same as those in the pre-interview, situations often arise that require the addition or deletion of tasks. In the candy factory teaching experiment, for instance, students worked with three computer microworlds during the course of the classroom teaching experiment. It was important for the research team to analyze the effectiveness of these tools in supporting students’ mathematical development. To do so, we needed post-interview data that documented how the students reasoned with the tools in problem situations. However, it would have been inappropriate to pose pre-interview tasks that involved the computer microworlds since the students would have had no means of determining their purpose or function. The data collection techniques used during the interviews involve videotaping each interview. The interviewer, using a protocol sheet as a guide as he/she makes decisions about the tasks to be posed, characterizes the process of the interview. These decisions are made against the background of later being able to use the interview to determine the depth of the student’s understanding of a particular concept. As a result, the interviewer may choose a sequence of increasingly difficult tasks to tease out the nature of the student’s ways of reasoning. In like manner, the researcher may skip certain tasks that would be judged to be beyond the scope of the student’s ability during the interview. In the early stages of the classroom teaching experiment, a preliminary analysis of the pre-interviews is also used to identify “target” students who will be followed more closely throughout the classroom teaching experiment. Four to six students are typically identified and are selected based on their varying abilities. The research team is interested in monitoring the activity of students at all levels throughout the process. The target students are watched more closely during classroom interactions and typically are the subject of a more in-depth study by one or more members of the research team. The focus on these students also provides additional data for the project team as they make daily decisions about appropriate instructional activities.

Classroom Episodes During the classroom episodes, all members of the research team are collecting data. Two members videotape the classroom events. During whole-class dis-

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cussions, one camera is focused on the teacher and students who come to the board to share their solutions. The second camera is focused on the class to capture contributions of other class members. During individual or small-group work, the two cameras are focused on the target students. All members of the research team take field notes during the class period. In addition, the teacher typically keeps a reflective journal which she updates daily. These records are then used as data for the project team as they make decisions about appropriate next steps in the instructional sequence and later for longitudinal analyses of classroom events. The format of a class typically begins with the teacher giving a task to the students in a whole-class setting. This would involve the teacher introducing the scenario in which the task is situated through “story telling.” Students would then engage in a discussion with the teacher about features of the scenario that are important to the problem-solving activity. For example, in the candy factory scenario, it was important for the teacher and students to discuss the fact that 10 candies were packed in each roll. This was only introduced after the students had an opportunity to offer suggestions as to how the manager might organize his inventory. The convention was then introduced as the manager’s solution to the problem the students had just discussed. Afterwards, students work individually or in small groups to solve the task. While the students are working, members of the project team circulate around the room to observe the students’ activity. It is important to note that during this time the team, other than the teacher, does not intervene with the students. The goal is not to “correct” the students’ ways of reasoning, but to simply gather information about the varied ways in which the students are solving the task in order to better orchestrate the upcoming whole-class discussion. In this way the teacher can make informed decisions about which student solutions to highlight in order to further the mathematical agenda. After the students have completed the assigned tasks, they return to a wholeclass setting to discuss their solutions. It is during this discussion that the importance of both the teacher’s role and the norms for argumentation are highlighted. These discussions need to be orchestrated in such a way as to sequence the shared solutions toward more sophisticated ways of reasoning. In doing so, the teacher works to build from the students’ contributions to support shifts in their ways of reasoning. As an example, the teacher might begin with solutions to the problem: There are 34 candies in the storeroom, and the workers make 17 more. How many candies are now in the storeroom? This involves drawing both collections and then counting by ones to determine the total number of candies. The teacher might then move on to a solution that involved drawing the candies but also entailed the use of a notational scheme to symbolize the activity in the storeroom. The third solution in the sequence would highlight a student who was able to work from modified drawings that showed rolls instead of each individual candy and was linked to a notational scheme. In this way, the teacher is able to build from the students’ activity as she works to achieve her mathematical agenda. However, this can only be achieved if the students un-

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derstand and accept the expectation that they are to explain and justify their ways of reasoning. In other words, the social and sociomathematical norms need to become constituted in the classroom in such a way as to create a participation structure that supports students’ becoming active participants in their mathematics. In this setting the explanations then become the focus of discussion, not the solution. This approach to instruction values students’ construction of nonstandard algorithms. However, it also emphasizes the critical role of the teacher and of instructional activities in supporting the development of students’ numerical reasoning. In addition, this approach highlights the importance of discussions in which students justify their algorithms. It, therefore, treats students’ development of increasingly sophisticated algorithms as a site for conceptual learning. In our view, an approach of this type is consistent with reform recommendations in the United States that stress the need for students to develop what Skemp (1976) calls a relational understanding rather than to merely memorize the steps of standard procedures (National Council of Teachers of Mathematics NCTM, 1989, 1991). This contrast is captured by the difference between “How can I figure this out?” and “What was I told I was supposed to do?” (Yackel, Cobb, Wood, Wheately, & Merkel, 1990). Relational understanding occurs as teachers systematically support students’ construction of personally meaningful algorithms. These algorithms emerge when students engage in sequences of problem-solving activities designed to provide opportunities for them to make sense of their mathematical activity. An immediate consequence of this focus on algorithms as expressions of numerical relationships is that it precludes the unreasonable answers that often occur from misapplication of little understood standard algorithms.

Daily Project Meetings At the end of each class session the project team meets to discuss their individual analyses of what happened. These discussions are grounded in the students’ activity and framed against the mathematical agenda as defined by the learning trajectory. It is during these meetings that major revisions or small modifications are made to the instructional sequence. The reader will recall that these daily minicycles are highlighted in the tenets of design research. The reader will also recall that the instructional sequence is developed prior to entering the project classroom. However, the sequence as initially defined is subject to change depending upon the students’ ways of participating. As a result, it is offered as a way to support the development of a conjectured learning trajectory. The revisions and modifications that result from daily analysis from interacting with the students result in the realized or actual learning trajectory (Simon, 1995). The actual learning trajectory would then inform future design efforts. While these daily project meetings result in minicycles of revision, the larger cycle

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would involve revisions to the initial trajectory that would guide the modification of the overall sequence. Decisions made during these daily debriefing sessions are documented either through audio recording or field notes. Tracking the decision-making process of the project team is critical in revising the sequence. As a result, this aspect of the classroom teaching experiment is also viewed as important with respect to data collection.

DATA ANALYSIS Throughout the classroom teaching experiment, data are being informally analyzed in order to provide a basis for the numerous decisions that are being made with respect to the appropriateness of instructional tasks for the students in the project classroom. The decisions are based on an assessment of the students’ mathematical development and conjectures about how best to support their further development. These decisions are framed against the background of the conjectured learning trajectory and its ongoing viability. These analyses serve to guide the decision-making process in the classroom during the classroom teaching experiment. For that reason it is important to acknowledge the role that the conjectured learning trajectory and the sketched out instructional sequence play in that decision-making process. Because of the time constraints imposed by daily class sessions, it is not possible to conduct detailed analyses of students’ activity or to completely revise the instructional sequence. As a result, the decisions take the form of modifications to an instructional sequence that is itself the focus of the research. The approach we take to analyzing data at the conclusion of a teaching experiment is much more extensive and involves a method described by Cobb and Whitenack (1996) that was developed for longitudinal analyses of large sets of data. This method follows from Glaser and Strauss’s (1967) constant comparative method. It involves the researcher tracking regularities and patterns in the data to substantiate claims about such issues as ways to support the students’ development or the evolution of classroom mathematical practices. The process begins with the researcher making initial conjectures based on the ongoing analyses of data that have been occurring throughout the classroom teaching experiment. In the beginning these conjectures are grounded only in the preliminary analysis that was conducted as the project team was engaged in the classroom teaching experiment. As a result the strongest statement that can be made is that they are “conjectures.” However, once made, they are continually tested against the data as the researcher works through the data in a systematic and typically chronological manner. This constant comparison then supports the development of theories about how the classroom community developed mathematically and the means of supporting that development. Cobb (2000) describes these analyses as “the result of a complex, purposeful problem-solving process.”

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The analysis then becomes a problem-solving activity wherein the researcher poses conjectures that she/he verifies or refutes through a process of longitudinal data analysis. The learning of the classroom community is documented by a detailed analysis of the classroom mathematical practices that emerged. The reader will recall that classroom mathematical practices are the taken-as-shared ways of reasoning, arguing, and symbolizing established by the classroom community while discussing particular mathematical ideas. A detailed analysis of the project classroom can, therefore, document the evolution of classroom mathematical practices, thereby documenting the realized learning trajectory of the classroom community. (An analysis of classroom mathematical practices is beyond the scope of this chapter. For a detailed analysis of the classroom mathematical practices that emerged during the candy factory instructional sequence, see Bowers, Cobb, & McClain, 1999).

TRUSTWORTHINESS AND GENERALIZABILITY The trustworthiness of analyses conducted within the design research paradigm is focused on the credibility of the analyses. The most important consideration in this regard is the extent to which the analysis of the data set is both systematic and thorough. This is an extremely important issue given that large sets of video recordings and transcripts are generated in the course of a teaching experiment. What is required is a systematic analytical approach in which provisional claims and conjectures are continually open to modification. It is, therefore, important to document all phases of the analytic process including the refining and refuting of initial conjectures. Final claims and assertions can then be justified by backtracking through the various levels of the analysis, if necessary to the video recordings and transcripts. There is then a strong distinction between systematic analyses in which sample episodes are used to illustrate general assertions and questionable analyses in which a few possibly atypical episodes are used to support unsubstantiated claims. Additional criteria that enhance the trustworthiness of an analysis include both the extent to which it has been critiqued by other researchers and the extent to which it derives from a prolonged engagement with students and teachers (Taylor & Bogdan, 1984). This latter criterion is typically satisfied in the case of classroom teaching experiments and constitutes a strength of the methodology (cf. Cobb, 2000). The issue of generalizability with respect to analyses resulting from classroom teaching experiments can be addressed by noting the importance of viewing classroom events as “paradigmatic of broader phenomena” (Cobb, 2000). Treating classroom activities and events as exemplars or prototypes gives rise to generalizability. This, of course, is not generalization in the traditional sense such that the characteristics of particular cases are either ignored or treated as interchangeable with other similar situations. Instead, the post hoc theoretical

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analysis conducted after conducting a classroom teaching experiment can be deemed to be relevant when interpreting other cases. Thus, what is generalized is a way of interpreting and acting that preserves the specific characteristics of individual cases. For example, my colleagues and I conjecture that much of what we learned when analyzing third graders’ understanding of place value can inform analyses of other students’ mathematical learning in a wide range of classroom situations. The distinction is further highlighted by noting the difference between analyses whose primary goal is to assess a particular instructional innovation and those analyses whose goal is the development of theory that can feed forward to guide future research and the development of activities.

CRITICAL REFLECTION In reflecting back over the classroom teaching experiment methodology as outlined in this chapter, the reader will notice that it is highly interventionist. The researcher interacts with both the students and the teacher over an extended period of time. In addition, a productive collaboration with the classroom teacher is critical to the effectiveness of the methodology. By accepting that underlying premise, the researcher acknowledges the importance of an ongoing collaboration in order to develop a mutually acceptable professional relationship. The overriding concern should, therefore, be that of establishing an effective basis for communication so that the teacher and researchers are united by a common purpose. A primary goal then becomes to support the teacher’s development while simultaneously learning from him/her rather than to ensure that he/she acts in accord with the researchers’ personal interpretations and judgments. This stems from the fact that unless negotiated otherwise, the collaborating teacher might assume that his/her role is merely to translate the researchers’ decisions and judgments into practice. The highest priority should, therefore, be given to establishing relationships with the collaborating teacher that are based on mutual respect and trust. It is, therefore, imperative that the teacher and researchers develop a shared understanding of the mathematical intent of the instructional sequence before the teaching experiment begins. A second important point to note is that the instructional materials used in the course of a classroom teaching experiment are potentially revisable and are themselves the focus of research. As a result, the researcher is involved in the development of instructional materials as an integral aspect of the classroom teaching experiment. While this might range from developing an instructional sequence for use for the first time, to working from materials that have been researched in similar settings, the day-to-day decisions about what tasks to pose to the students constitute an important aspect of the decision-making process involved in a classroom teaching experiment. It is an equally important aspect of the design research cycle. The daily decision-making requires a firm grounding in both research literature and design theory. This differs markedly from ex-

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periments designed to assess a particular instructional innovation. In this latter work, the instructional innovation is already developed and is not subject to modification. The research is intended to judge whether the innovation proved effective when compared to other forms of instruction. In the classroom teaching experiment, the instructional sequences are continually subject to revision and the focus of the research is on testing the conjectured learning trajectory and the means of supporting its development.

IMPLICATIONS In reading this chapter, it becomes clear that the type of classroom research described requires a substantial investment of both time and resources. However, I would be remiss if I did not point out that the methodology outlined here is characteristic of the classroom teaching experiment at its most intense level. I have participated in classroom teaching experiments where the research team was composed of three members, two of whom were graduate research assistants, and where the research team was composed of nine members, five of whom were graduate research assistants. The point to be made is that what is presented in this chapter is offered as a methodological tool. It is intended as a resource for other researchers who are interested in conducting classroom-based research. In that way, aspects of this methodology that are applicable can be utilized as they best fit with the goals and purposes of other researchers. For example, researchers who are interested in developing innovative instructional sequences and accompanying tools might employ all aspects of the methodology in an attempt to enhance their development efforts. Individuals concerned with particular aspects of an instructional sequence might use the design research cycle as a way of refining a portion of the sequence and omit the use of student interviews. A rationale for engaging in the classroom-teaching experiment should then be grounded in the questions to be answered and the issues to be addressed. In this way, the question drives the decision for the choice of method. An example of the applicability of the classroom-teaching experiment can be found in thinking about its utility for classroom teachers. Teachers are faced with the same decisions that the research team makes while conducting a classroomteaching experiment. In particular, teachers have a set of instructional materials intended to provide them with the means of support in guiding their students’ mathematical development. While the classroom teacher’s goal is not to design an instructional sequence, the teacher is focused on appropriate tasks for use with students. These decisions are made not only on a daily basis, but also for larger periods of time, such as units or grading blocks. The instructional materials can, therefore, provide the initial conjectured learning trajectory. The task of the classroom teacher then becomes that of testing and refining this conjecture based on students’ daily activity.

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Teachers make professional decisions each day about the appropriateness of instructional materials. They choose to modify, insert, and/or omit tasks based on their ongoing assessments of their students. This activity parallels that of the teaching experiment—it just has not been formalized as such. The process of making this process explicit can then become a forum for professional growth and development for the classroom teacher. Instead of route-marching through a series of instructional materials, the teacher views him/herself as a professional who makes informed decisions about appropriate next steps for the students. This involves “analyses” of the students’ daily activity in the form of ongoing assessment. In this way, the process of analyzing students’ activity with an eye toward selecting appropriate instructional tasks places the students’ current understandings at the forefront of the decision-making process instead of the a priori decisions of the materials designer. Such practices as building from students’ emerging understandings and modifying instructional materials are already instantiated in many classrooms; the links to the design research cycle merely need to be made explicit. In doing so, the classroom-teaching experiment methodology can then be a tool that assists the classroom teacher in the tasks that are central to the profession.

REACTION BY MARJA VAN DEN HEUVEL-PANHUIZEN Toward Scientific Research in Classrooms? Mathematics education can be investigated in a variety of research settings, but the place of all places for carrying out educational research is the spot where it all happens: the classroom. At least, these are our present thoughts about educational research, which are also expressed by Kelly and Lesh (2001) who, in their handbook on research designs in mathematics and science education, conclude that the site for educational research moves out from the laboratory. However, it should not be overlooked that classroom-based research and teaching experiments have a long history in educational research. Recall, for instance, reformers like Pestalozzi and Dewey who opened their experimental schools in 1809 and 1889, respectively. Bearing this in mind, it cannot be denied that we have moved on from the time when answers to educational questions were sought in laboratory settings. Currently a substantial proportion of educational investigations are done in situ, in regular classrooms. Irrespective of the important reasons that have led to present practices in education research, it is clear that the move into the complexity of the learningteaching environment of classrooms can entail some serious threats to the quality of research activity. It is necessary to consider how to conduct research in such a way that it leads to reliable and valid evidence. Indeed, whatever new interpretations are given to these classic issues of reliability and validity, they can still be considered as central to the integrity of research activity. This is also why

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it is necessary to have a methodology: a system of methods, procedures, and guiding principles for carrying out educational research in classrooms. Such a methodology is a necessary requirement for the scientific status of educational research. McClain’s chapter focuses on one important approach to classroom-based research: the classroom-teaching experiment. An overview of this approach given by Kelly and Lesh (2001) reveals a diversity of approaches that are referred to as teaching experiments ranging from studies aimed at the development of students or teachers through to developing instructional activities, curriculum materials, or—in case these are more broadly conceptualized—instructional environments. McClain’s chapter belongs to the third category. The chapter describes an example of a classroom-teaching experiment that develops an instructional sequence for adding and subtracting three-digit numbers in a third-grade classroom. The primary goal of the research is to test and revise the instructional sequence that is designed to support students’ mathematical development in this particular content domain; thus it is properly described as design research. In the classroom-teaching experiment the instructional sequence is constantly modified on a day-to-day basis following concurrent analyses of classroom experiences. The main purpose of McClain’s chapter is, however, to describe a methodology for conducting classroom-teaching experiments. McClain starts her chapter by sketching the dual theoretical basis for the teaching experiments that she and her colleagues employed. Put simply, the instructional design efforts are guided by the Dutch approach to mathematics education called Realistic Mathematics Education (RME), and the classroom-based research efforts are guided by what is called the Emergent Perspective (EP). The basic tenets of RME—which, among other things, include taking students’ initial understanding as a starting point, providing them with problem situations, which they can imagine, and scaffolding the learning process via models—were adopted as a philosophy of mathematics education that eventually led to the instructional sequence developed for the experiment. The basis of the EP includes the social and the psychological perspectives including social norms, sociomathematical norms, and classroom mathematical practices. These perspectives are used to construct an integrated analysis of both the individual student’s mathematical development and the social context of learning in the classroom. Together with the conjectured learning trajectory, which was provided by the initial design efforts and functioned as a meta-level frame, these perspectives provide the interpretive framework for the analysis of the students’ understanding. It is the results of this analysis on which the daily minicycles of revision of the instructional sequence are based. In addition to this, a retrospective analysis of the teaching experiment was conducted—based on a wide variety of classroom-connected data sources—to provide the basis for large-scale modifications of the instructional sequence. The chapter reveals the different stages of the experiment and provides an overview of the research activities:

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Defining the “big ideas” Mapping out a conjectured learning trajectory Sketching the instructional sequence Collecting data (including interviewing students to inform the starting points for instruction, taking field notes, and video recording the classroom events) Discussing the classroom events in daily project meetings Analyzing the data both informally on a daily basis and more systematically on a post hoc longitudinal basis using the constant comparative method of Glaser and Strauss

Apart from this concrete direction given for carrying out the retrospective analysis, McClain offers few practical guidelines. She explains what research activities were done, but pays little attention to how they were carried out or why they were done in the way they were done. Similarly, the section devoted to the issues of trustworthiness and generalizability is too brief to be of much value in informing the reader. This section raises many questions, for instance, about the conjecture that much of what was learned when analyzing the third graders’ understanding of place value can inform analyses of other student’s mathematical learning in a wide range of classroom situations. It is not clear how this conjecture emerged and how the gains from analyzing the students’ understanding in one classroom can inform other students’ mathematical learning in different classroom contexts. Although it is unreasonable to require that a complete methodology be described in one relatively short chapter, I have to say that merely portraying what is possible when carrying out a classroom-teaching experiment is not sufficient to define a methodological tool for other researchers to follow. A methodological tool must tell how data must be collected and analyzed and must give the rationale for why it has to be done in that way. Furthermore, I miss what Freudenthal (1991) refers to as “a posteriori reflecting on one’s methods.” Rather than a methodology I see the chapter as a paradigmatical example of a classroomteaching experiment. To be clear, this does not mean that I do not consider the chapter as valuable—nor is it the reason for putting a question mark in the title of my reaction. McClain’s description provides us with a clear view of the ingredients of a well thought out and carefully executed classroom-teaching experiment. As such, this chapter will certainly inspire other researchers. The chapter demonstrates that a classroom-teaching experiment can hold a magnifying glass up to the complex process of teaching and learning and can open the “black box” of the classroom. In so doing the experiment provides us with knowledge about instructional environments in classrooms and with means to create or improve them. As such, classroom-teaching experiments clearly fall under the heading of design research. The issue for me concerns the potential of classroom-teaching experiments to answer questions about the macrodidactics of classrooms. In McClain’s par-

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adigmatic example, the longitudinal perspective—the view that is mostly connected to the content and goals of mathematics education—is almost beyond the scope of the experiment and its methodology. The chapter focuses particularly on microlevel decisions and is not very explicit about both the initial and concluding design efforts. For me, this part cannot be left out in design research. Within the Dutch equivalent of this type of research—called “developmental research”—it is essential that long-term learning processes be covered (Freudenthal, 1991). Compared to the microperspective, which is more focused on the how of teaching, the macroperspective brings us more to what should be taught. It is here where my question mark comes in view. It has all to do with decisions made about the content of mathematics education. The content of the instructional sequence described in McClain’s chapter is more or less determined by the initial design efforts that were based on the RME tenets. Apart from the fact that the chapter does not reveal how the instructional sequence on adding and subtracting three-digit numbers was designed, the result is quite remarkable. Although the sequence reflects some important RME principles, I would not call it a RME teaching unit. The narrow focus on algorithms in conjunction with heavy emphasis on place value supported by Unifix cubes is quite different from an RME approach in which written and mental calculations are integrated, and in which an orientation toward numbers supported by activities on the number line form the basis for developing strategies to solve problems like 265 –194. Without saying that this latter approach is better than the approach that was chosen in the classroom-teaching experiment described by McClain, I am left wondering about the evidence base that motivated this different choice in content. An explanation for this might be that the initial design efforts were only deduced from the RME teaching tenets. These tenets, however, do not completely encapsulate the domain-specific education theory of RME. In addition to these teaching principles, RME includes choices for particular goals and content—in its broadest sense—which resulted from didactical phenomenological analyses, knowledge of the history of mathematics, evidence from students’ learning, and experience from collaboration with teachers, teacher educators, teacher counselors, and textbook writers. In other words, the RME tenets alone cannot lead to an instructional sequence, not to mention a longitudinal learning-teaching trajectory. Taking this into account, it is understandable that the content of the instructional sequence is different from an RME approach. Nevertheless, I am still puzzled about the rationale for the choice of content. Clearly the choice was made, deliberately, before the classroom experiment started, and it is apparent that the experiment, once underway, did not bring new insights that led to a fundamental revision of the conjectured trajectory. Indeed, it appears that the revisions that were made in the instructional sequence and the conjectured learning trajectory related to the microdidactical how questions (how do the students learn and how to teach or how to organize an instructional environment to optimize the students’ learning). It might then be concluded that the macrodidactical what questions (what should be the goals? And what content should

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constitute the program?) cannot be covered within the scope of classroomteaching experiments. However, I do not believe this is a correct conclusion. Although the answers to macrodidactical what questions are significantly informed by didactical phenomenological analyses combined with experiences and opinions of experts in and outside the school, evidence from classrooms can play an important role in providing answers to these what questions. To achieve this, however, it is necessary to include research activities that can inform our macrodecisions. To conclude my reaction I want to draw attention to an example from the Dutch TAL-project in which I am involved; this is aimed at the development of learning-teaching trajectories for primary school mathematics (one trajectory on whole-number arithmetic has been finished recently, Van den Heuvel-Panhuizen, 2001). This longitudinal trajectory describes broad “stepping-stones” that students will take as they pass from K1 to grade 6. The description is intended to offer the teachers an overview of the long-term process and provide them with a mental educational map that can facilitate their didactical decision making. Given my remarks above, it is perhaps unnecessary to emphasize that the development of these trajectory descriptions cannot be restricted to how questions but that they also need the answers to many what questions. I will not deal now with all the research activities we applied to come to answers to these what questions but will only focus on how we collected empirical evidence from classroom teaching. Crucial in this respect is Treffers’s argument for using problems that can be solved in a variety of ways—preferably student-generated problems, which we refer to as “own productions.” Students’ work on these problems can bring to light the levels of understanding and the arithmetic skills of the students at one particular moment. Apart from the fact that this information is important for taking microdecisions, it guides also the macrodecisions. The cross section of the class that is produced in this way shows at the same time a longitudinal section of a trajectory or a part of it. The solution strategies of individual students reveal collectively essential elements of the long-term path students need to traverse. What is found in the classroom in the present anticipates what is on the horizon and coming later. Returning to the purpose of McClain’s chapter, I think that research activities by which we can expose students’ future learning must have a place within the methodology of classroom-teaching experiments in the context of developmental research. They can bring us closer to the heart of the design process where instructional trajectories and sequences come into being.

NOTES 1. Data sources typically include video recordings of classroom sessions, videotaped individual clinical interviews conducted with each of the students before and after the

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teaching experiment, copies of the students’ written work, daily field notes, records of project meetings and debriefing sessions, and the teacher’s daily journal. 2. During the past seven years, I have been involved in collaborative research efforts involving classroom teaching experiments with Paul Cobb, Koeno Gravemeijer, Cliff Konold, Erna Yackel, Janet Bowers, Jose Cortina, Lynn Hodge, Maggie McGatha, and Joy Whitenack. 3. These analyses are conducted on a daily basis from field notes and conversations among the project team. 4. This was the third time a version of this sequence had been used in a classroomteaching experiment. For detailed analyses of the candy factory instructional sequence see Cobb, Yackel, and Wood, 1992; Bowers, 1995; and Bowers, Cobb, and McClain, 1999.

REFERENCES Bauersfeld, H., Krummheuer, G., & Voigt, J. (1988). Interactional theory of learning and teaching mathematics and related microethnographical studies. In H.G. Steiner & A. Vermandel (Eds.), Foundations and methodology of the discipline of mathematics education (pp. 174–188). Antwerp, Belgium: Proceedings of the Theory of Mathematics Education Conference. Bowers, J. (1995). Designing computer learning environments based on the theory of realistic mathematics education. In L. Miera & D. Carraher (Eds.), Proceedings of the Nineteenth Conference of the International Group for the Psychology of Mathematics Education (pp. 202–210). Recife, Brazil: Program Committee of the 19th PME Conference. Bowers, J. (1996). Children’s emerging conceptions of place value in a technologyenriched classroom. (Unpublished doctoral dissertation, Peabody College of Vanderbilt University, 1996.) Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical practices: A case study. Cognition and Instruction, 17(1), 25–64. Cobb, P. (2000). Conducting teaching experiments. In A. Kelly & R. Lesh (Eds.), Research design in mathematics and science education (pp. 307–334). Mahwah, New Jersey: Erlbaum. Cobb, P., Perlwitz, M., & Underwood, D. (1994). Construction individuelle, acculturation mathématique et communauté scolaire. Revue des Sciences de l’Éducation, 20, 41–62. Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14, 83–94. Cobb, P., & Whitenack, J. (1996). A method for conducting longitudinal analyses of classroom videorecordings and transcripts. Educational Studies in Mathematics, 30, 213–238. Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175–190. Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2–33.

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Cognition and Technology Group of Vanderbilt (1992). The Jasper experiment: An exploration of issues in learning and instructional design. Educational Technology Research and Development, 40(1), 65–80. Confrey, J., & Lachance, A. (2000). Transformative teaching experiments through conjecture-driven research design. In A. Kelly & R. Lesh (Eds.), Research design in mathematics and science education (pp. 231–266). Mahwah, New Jersey: Erlbaum. Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Dordrecht, Netherlands: Kluwer. Glaser, B., & Strauss, A. (1967). The discovery of grounded theory: Strategies for qualitative research. New York: Aldine. Glasersfeld, E. von. (1995). Radical constructivism: A way of knowing and learning. Washington, DC: Falmer Press. Gravemeijer, K. (1990). Context problems and realistic mathematics instruction. In K. Gravemeijer, M. Van den Heuvel, & L. Streefland (Eds.), Contexts, free productions, tests, and geometry in realistic mathematics education (pp. 10–32). Utrecht, The Netherlands: OW & OC Research Group. Gravemeijer, K. (1994). Educational development and developmental research in mathematics education. Journal for Research in Mathematics Education, 25(5), 443–471. Gravemeijer, K. (in press). Mediating between concrete and abstract. In T. Nunes & P. Bryant (Eds.), How do children learn mathematics? Hillsdale, NJ: Erlbaum. Kelly, A. E., & Lesh, R. A. (2001). Handbook of research design in mathematics and science education. Mahwah, NJ/London: Erlbaum. Lesh, R., & Akerstrom, M. (1982). Applied problem solving: Priorities for mathematics education research. In F. Lester & J. Garofalo (Eds.), Mathematical problem solving: Issues in research (pp. 117–129). Philadelphia: Franklin Institute Press. McClain, K., & Cobb, P. (2001). The development of sociomathematical norms in one first-grade classroom. Journal for Research in Mathematics Education, 32(3), 236–266. National Council of Teachers of Mathematics NCTM. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics NCTM. (1991). Professional standards for teaching mathematics. Reston, VA: Author. Simon, M. (2000). Research on the development of mathematics teachers: The teacher development experiment. In A. Kelly & R. Lesh (Eds.), Research design in mathematics and science education (pp. 335–360). Mahwah, NJ: Erlbaum. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114–145. Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267–307). Hillsdale, NJ: Erlbaum. Streefland, L. (1991). Fractions in realistic mathematics education. Dordrecht, Netherlands: Kluwer. Taylor, S., & Bogdan, R. (1984). Introduction to qualitative research methods: The search for meanings (2nd ed.). New York: Wiley.

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Thompson, P.W. (1992). Notations, principles, and constraints: Contributions to the effective use of concrete manipulatives in elementary education. Journal for Research in Mathematics Education, 23(2), 123–147. Van den Heuvel-Panhuizen, M. (Ed.). (2001). Children learn mathematics. A learning teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school. Utrecht, Netherlands: Freudenthal Institute, Utrecht University. Yackel, E. (1995). The classroom teaching experiment. Unpublished manuscript, Department of Mathematical Sciences, Purdue University at Calumet, IN. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentations and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477. Yackel, E., Cobb, P., Wood, T., Wheatley, G., & Merkel, G. (1990). The importance of social interaction in children’s construction of mathematical knowledge. In T. Cooney & C. Hirsch (Eds.), Teaching and learning mathematics in the 1990s. Reston, VA: National Council of Teachers of Mathematics.

Chapter 6

Context and Content:What Are Student Teachers Learning about Teaching Mathematics? Ruth Shane

INTRODUCTION The student-teaching practicum is a significant feature of elementary preservice teacher education. Mathematics educators often suspect that this practicum serves to reinforce a narrower, computation-oriented approach to mathematics and does not contribute to reform efforts that put more emphasis on problem solving and higher-order thinking strategies. My own experience as a mathematics educator, teaching courses for prospective elementary school mathematics teachers and supervising their practicum, led me to research the nature of the mathematical activity in the classroom as a background for understanding the knowledge development of the student teacher in that classroom. The questions that provoked my research were: What do student teachers learn from the practicum about teaching mathematics? Does the particular mathematical environment of the practicum classroom play a role in what the student teachers learn? Is there an interrelationship between how the children in a classroom are learning about mathematics and what the student teachers are learning about teaching mathematics? The research was designed to capture the day-to-day details of the mathematics classroom environment and to undertake a microanalysis of the parallel learning of the student teachers in those classrooms. It took the form of a case study of four student teachers, two of whom taught mathematics in secondgrade classrooms with a traditional approach, and two of whom taught in second-grade classrooms with a reform-oriented, alternative approach. This research plan reflected a holistic view of the distinctive culture of the mathematics classroom as being initiated, maintained, and developed by the teacher. It is acted out, extended, clarified, and ”dialogized” by the teacher and

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pupils, and continuously affects what the pupils and the teacher and student teacher learn about mathematics and teaching mathematics. Classroom culture in this research is the sum of those norms that have been “taken-as-shared,” that is, accepted by all parties as common knowledge without need for explanation, around the business of dealing with mathematics. The research plan reflected a social-constructivist perspective of teachers and students learning together in the classroom about mathematics and about teaching mathematics. The research plan needed to include documentation of the classroom culture in conventional and alternative mathematics classrooms in order to define and establish the difference between the two cultures. There needed to be documentation of the nature of the children’s understanding of the mathematics in order to validate the premise that the classroom culture was reflected in the children’s learning of mathematics. Furthermore, there needed to be an assessment of the student teachers’ knowledge of the mathematics they were teaching on an ongoing basis, and it was necessary to expose their beliefs about mathematics and teaching mathematics. Each component presented its own methodological challenges. One challenge involved capturing the culture of the mathematics classroom by designing an observational approach that would focus on the process of instruction, in particular on the verbal interactions between student teacher and pupils that take place in the context of mathematical tasks. This led to the decision to record on video a sequence of lessons in order to provide data that would hold evidence of classroom discourse arising from as near to the regular classroom routine as possible. A second challenge in this research was to adopt an assessment tool for knowledge about teaching mathematics. Particularly for student teachers teaching second-grade classes, there would be no value in a written test of mathematical content because the difficulty of the mathematics taught should not present the students with challenges that might expose the type of evidence sought. It was believed that interviewing the student teachers would provide them with a way to articulate their thinking about the mathematical ideas of the lesson and about teaching and learning mathematics. Another major issue was that the research design would have to take into account the fact that the student teachers were teaching mathematics on only two days a week while the mentoring teachers were teaching mathematics to the same children for the other three days of the week. While this proceeded naturally in the traditional classrooms without any intervention by the researcher, the alternative classrooms were the product of team meetings between the student teachers, the mentoring teachers, and the researcher, in order to ensure the consistency of the alternative approach. This clearly offered the student teachers extra opportunities for building their knowledge about teaching mathematics during these sessions for reflective discussion, evaluation, and planning. Furthermore, the scheme of work and approach in the alternative classrooms were constructed on the researcher’s beliefs about a desirable environment for mathematics teaching and learning. Thus the researcher had a significant influ-

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ence in the alternative classrooms. The use of the video camera, while not neutralizing the bias, allowed for “observation” and interpretation by others of the same lesson. Likewise, the use of a standard, semistructured interview protocol with common lines of enquiry pursued with all the student teachers, reviewing their lessons or summarizing their experience, was meant to establish a common basis on which the variety of responses could be compared.

THEORETICAL BACKGROUND The research reported here is based on the theoretical perspectives that address the following questions:

. . .

What is considered (in this research) to be knowledge of mathematics and knowledge for teaching mathematics? How is this knowledge acquired? How is the culture of the mathematics classroom connected to this learning process? What are the particular issues involved in the student teachers’ learning about teaching mathematics in the classroom-based practicum?

Knowledge of Mathematics and Knowledge for Teaching Mathematics Researchers have defined the different components of what is considered mathematical knowledge and the relationship between these components. Stieg Mellin-Olsen and Richard Skemp (1976/1989) make a significant contribution to our understanding with the notions of two kinds of mathematical understanding, “relational” and “instrumental,” each of which can be associated with a distinctive approach to teaching and learning. Skemp argues that an instrumental approach based on sequences of instructions for specific mathematical exercises restricts the learner to instrumental mathematical knowledge that is useful only in the routine activities of the classroom tasks. Relational understanding, on the other hand, provides the genuine conceptual framework for mathematics that includes the necessary computational proficiency and is available for problem solving, nonroutine tasks, and further development (Skemp, 1976/1989). Hiebert and Lefevre (1986) define the terms “procedural” and “conceptual” knowledge as two potentially complementary goals and present a similar model. Procedural knowledge includes all the standard algorithms, rules, and procedures for solving problems; on the other hand conceptual knowledge is rich in the connections that give meaning and power to mathematical concepts. Research by Hiebert and Wearne (1994) supports the claim that procedural knowledge alone is not a strong base for conceptual understanding, but conceptual knowledge contributes to computational proficiency. While Skemp asserts that

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the two kinds of understanding he describes are different and incompatible, Hiebert and Wearne suggest a complementarity, albeit based on a onedirectional relationship between procedural and conceptual knowledge. Ephraim Fischbein (1993) presents an alternative model of knowing mathematics that fuses the intuitive component, the algorithmic, and the formal. Knowing mathematics in this framework arises through building the right “fit” among the three, drawing on each component effectively and efficiently, settling any apparent differences. The algorithmic component is similar to Hiebert’s procedural knowledge of mathematics; the formal component is the understanding of the underlying structures, patterns, and laws of mathematics; and the intuitive component is the kind of mathematical knowledge that comes from common-sense, everyday experiences that precede the formal instruction of the same topic. (For instance, addition is intuitively commutative.) Fischbein (1987) posits the importance of the intuitive knowledge in mathematics as a critical component of mathematical activity. However, intuitive knowledge has not yet found its way explicitly into the elementary school mathematics classroom. There is a hint of this “intuitive” inclination in the examples, metaphors, and analogies that Skemp chooses from music, mapping, and nature to clarify the deeper meanings of a relational understanding of mathematics. Skemp also explicitly describes relational understanding as including “the separate rules. . . knowing also how they are inter-related. . .” Fischbein would probably argue, therefore, that Skemp’s presentation of a relational understanding of mathematics as “organic in quality” includes the integration of the algorithmic, the formal, and the intuitive understandings of any mathematical idea. These formulations of the nature of mathematical knowledge offer similar frameworks for analyzing the mathematical content of any curriculum as it emphasizes standard versus student-invented algorithms, drill-and-practice versus problem solving, applying procedures versus choosing and justifying strategies. They provide the researcher with descriptive terms for defining the classroom culture, for assessing both children’s knowledge and student teachers’ knowledge of mathematics. Another perspective presented by Deborah Ball adds a further dimension to the identification of knowledge in the mathematics classroom. Ball (1990) uses the language of Schwab (1961/1978) to describe the knowledge of a discipline as substantive and syntactic. The substantive knowledge of mathematics includes all the components of the standard mathematics curriculum as described above. The syntactic knowledge of mathematics includes “the rules of the game,” what counts as evidence, what is the nature of the discourse. This is quite similar to what Paul Cobb (1994) introduces as sociomathematical norms. In the context of this research, it is clearly part of the knowledge of mathematics for the children and for the student teachers (see chapter 5 by Kay McClain in this volume). In this research, knowledge of mathematics was interpreted from these theoretical perspectives. The data were explored for evidence of the construction of substantive and syntactic knowledge, procedural and conceptual knowledge, and

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the interplay of intuitive and formal knowledge. While this was sufficient for analyzing the children’s knowledge of mathematics, in order to analyze the student teacher’s knowledge, attention had to be focused on a higher level. The research needed to consider the knowledge that student teachers must have in order for their pupils to develop their understanding of mathematics. In order to collect and interpret data on student teachers’ knowledge base for teaching mathematics, there needed to be an understanding of what an appropriate knowledge base for teaching mathematics might be. Knowledge for teaching mathematics was traditionally seen as the combination of subject-matter knowledge and general pedagogic knowledge. Lee Shulman posits a third component, pedagogical content knowledge. Shulman (1986) explains that neither generic knowledge about teaching nor subject-matter knowledge alone is sufficient without the particular repertoire of understandings, representations, examples, and insight into the specific subject matter which will connect the mathematical ideas with the particular classroom of children. This was one of the major premises of this research; namely, student teachers need to learn how to choose the most mathematically rich tasks, to engage pupils in a mathematically significant classroom discussion, to choose the most appropriate examples, and to anticipate the potential misunderstandings of the arithmetic topics arising in the elementary grades. The assumption was that this is a significant and challenging component of what is to be learned for teaching mathematics. These theoretical constructs were the basis for planning and interpreting the research about the nature of the knowledge of mathematics being constructed in these second-grade classrooms and about the process of constructing that knowledge. These constructs provided the framework for the design of the interview protocol for the student teachers where the intention was to expose their knowledge for teaching mathematics. The constructs were at the root of the choice of focus for the video cameras and the categories of analysis of the discourse.

Knowledge Acquisition and the Relationship between Classroom Culture and the Learning Process Designing a research project on the knowledge growth about mathematics and teaching mathematics was inseparably bound to a theory of learning. While research has brought evidence about how much learning of mathematics goes on outside of school, the nature of that knowledge tends to be contextually bound and intuitive. In order to integrate an efficient and appropriate use of procedures, and certainly to build a formal structure of mathematics as a basis for advanced mathematical thinking, it is generally accepted that mathematics has to be learned in a formal setting. Except for the rare “natural” mathematician, mathematics is not an intuitive discipline but one that is dependent on the learning that goes on in mathematics classes. Since mathematics requires abstract cognitive effort rather than behavioral skill, interest is focused on how the mind

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can develop more and more complex schema to “hold” new understandings of mathematics. Jean Piaget’s genetic epistemology has been the basis of the constructivist theories of learning that have prevailed in mathematics education at the end of the twentieth century. Constructivism places the emphasis on the individual’s learning as a growing organism, responsible for his or her own assimilation and accommodation of experiences as they interact with his or her personal maturation (Davis, Maher, & Noddings, 1990). By contrast, L. S. Vygotsky’s sociocultural theory of learning accounts for the recognition that mathematics, like other knowledge, is learned through interaction with other people. For Vygotsky (1978), learning takes place in a social context through interaction with peers and adults; in this, language has a role of particular importance together with other socially mediated tools. The emphasis within a Vygotskian framework is on being enculturated into the mathematical community. Paul Cobb analyzes these two contrasting models of cognition and finds them complementary. Rather than one theory contradicting the other, they account for different planes of the learning process. The social-constructivist theory such as that argued by Cobb (1994) was a key plank of my research since it describes a mathematics classroom where the children are learning mathematics from/with each other, and the student teachers are learning about teaching mathematics from their interaction with the children’s learning. In this paradigm the classroom is perceived metaphorically as a “learning web” rather than a “ping-pong table.”

Central Issues Relating to Student Teachers’ Learning about Teaching Mathematics during the Practicum Where do teachers develop their knowledge for teaching mathematics? Several researchers have cited practical experience within the classroom as one of the significant arenas that may stimulate the development of teacher knowledge. Donald Schön considers the teacher as reflective practitioner where naturally the classroom provides the stimulus for reflection-in-action and reflection-on-action (1983). Fennema et al. (1996) present evidence from long-term research data where teachers learn about teaching mathematics from their own teaching. The classrooms that were studied were classrooms where the mathematics teaching was based on the children’s mathematical thinking. Teachers participated in the in-service workshop and learned to plan their instruction from encouraging the children’s strategies to be the focus of discussion in the classroom. From this “uncovering” of children’s thinking, the teachers’ knowledge developed. My research design was intrinsically related to this view, that knowledge is being constructed by the student teacher from teaching in the mathematics classroom. The issue that confronted me was to articulate the nature of that knowledge and relate it to the particular instructional approach. I wanted to explore whether an instructional approach that, by definition, encourages more reflec-

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tion by the children on their mathematical activity (like Fennema et al., 1996) also promoted more reflection by the student teachers on the mathematics taught and the mathematics pedagogy. I wanted to expose what the student teachers were learning in a classroom where children were doing more computation and recall rather than sharing their thinking strategies. I wanted to investigate how student teachers were building on their intuitive understandings in a classroom where pupils were extending their intuitive knowledge. Additionally, I wanted to expose how the student teacher’s knowledge was expanding when the classroom discourse is formal and not based on “number sense.” Student teachers, however, usually occupy a different position in the classroom from the regular teacher. The students are teaching in classrooms where the mentoring teachers, from whom they are supposed to be learning about teaching, determine the nature of the culture in the mathematics classroom. The expected process is one of being an apprentice to the teacher and learning through that relationship. Eisenhart et al. (1993) document the potential tensions around this traditional field placement of student teachers who are educated for a reform agenda but whose mentoring teachers focus almost exclusively on procedural knowledge. They recommend creating special contexts for learning to teach which reflect the reform-oriented mathematics curriculum. Denise Mewborn operationalized this stance and designed a program for increasing the value of the practicum for learning to teach mathematics. Mewborn (1999) articulates five features of this practicum design: 1. An inquiry perspective 2. A community of learners (comprising mentoring teachers, student teachers, and college supervisor) 3. A nonevaluative stance by the mentoring teacher and college supervisor 4. Release of the student teachers from responsibilities that are not directly related to instruction 5. A focus on mathematics instruction

Her conclusion is that given particular but realistic conditions for the field-based practicum, the student teachers could develop their reflective thinking skills in the context of teaching mathematics. My research design included these features identified by Mewborn, with particular emphasis on (1), (2), and (5). The significant premise motivating the research was that student teachers could learn from the practicum about the mathematics they are teaching, about how children learn mathematics, and about the implications of both for their teaching. The logical extension of the social constructivist position as articulated above is that the student teachers are also constructing their knowledge from their interactions in the mathematics classroom with the pupils and each other, and therefore broadening their web of understandings for teaching mathematics.

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SUMMARY All these issues raised methodological dilemmas. The nature of the knowledge of mathematics is more complex than any one tool can measure, both in respect to the children’s mathematics and the student teachers’ knowledge. Even the formulation of a test or interview protocol for the children represented particular biases about what knowledge I value. Moreover, since I was more involved in the alternative classrooms there was clearly the possibility that the student teachers and children in those classrooms were already familiar with my frame of reference, vocabulary, intentions, and values. Therefore the sources of evidence needed to provide a variety of means for understanding the mathematical knowledge being constructed from the range of assessment tools available. From standard criterion-referenced objective tests to more open-ended assessment of mathematical reasoning, the measures of students’ knowledge needed to look at procedural and conceptual knowledge, the attitude towards new problems, the flexibility of strategies, the comfort with varieties of representations, and the links between formal and intuitive understandings. A particular challenge was to document how children are learning mathematics. From the social-constructivist perspective the evidence could focus primarily on the small-group interactions between the children or on their interactions with the teacher in whole-class or smaller group discussion. I chose to focus on the interactions between the teacher and the children in whole-class discussions. This choice reflected my research interest in simultaneously documenting the student teachers’ beliefs about teaching mathematics as they were reflected in their classroom activity. This choice would significantly highlight the social-constructivist dimension of the children’s learning, capturing the child’s individual remarks as she or he constructs a response to the tasks on hand, as well as the social dimension of the interaction in the discourse between teacher and children. The more interactions that would be captured, the richer the evidence of the nature of the learning process in the classroom. However, since more observers, cameras, or other intrusions in the classroom also take space, both literally and sociologically, I needed to balance the concern to collect a wider range of evidence with the concern to maintain, as far as possible, a natural or regular context. The documentation of mathematics lessons needed to provide both evidence of particular features of the instructional-learning process and a sense of the classroom culture in which the process occurs. The third issue concerns what the student teachers were learning about teaching mathematics from the practicum and how this learning occurs. The evidence was dependent on the student teachers’ self-reporting both at the level of explaining particular classroom events and of generalizing their beliefs and understandings about teaching mathematics. It was only from their articulated understanding of their own classroom behavior, their rationales, their reflection, their conclusions, their generalizations, their beliefs, their frames of reference,

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their choice of examples, and so forth that I was able to find evidence of their construction of knowledge for teaching mathematics.

RESEARCH DESIGN The research was designed to provide meaningful evidence of the mathematics classroom culture and, concomitantly, children’s learning and student teachers’ learning. Any one of these three domains—classroom culture, children’s mathematics, student-teachers’ knowledge for teaching mathematics—would have been interesting in itself, but the novelty of the research was to hold up the three to concurrent examination to expose the interrelationships among them. This holistic nature of the classroom organism and the particular nature of each component required an integration of research tools. In order to understand the culture, to identify its salient features, and to capture these in all their richness to provide a thick description, the research was planned as a case study of a small number of mathematics classrooms. In order to assess the student teachers’ knowledge for teaching mathematics from their own perspectives, without coming to them with predetermined questions and answers, the qualitative paradigm was particularly relevant. Since the mathematics classroom was approached as a culture to be understood, an ethnographic inquiry including nonparticipant observation and semistructured interviews were the most appropriate research instruments for describing the instructional-learning process of the mathematics classrooms and the teachers’ knowledge about this process. For assessing the children’s knowledge, assessment procedures included two different instruments, tests and interviews—allowing for some insight into the range of understanding mathematics in second grade. The methodological dilemma that confronted me at the outset was the need to choose either to only follow and describe the student teachers in the classroomteaching experiment, which is identified as the alternative classrooms, or to follow at the same time two student teachers in “conventional” classrooms. While case studies are not inherently comparative in the quantitative sense, there is a sharpening of the sense-making lens that arises from looking simultaneously at an experimental group and a nonexperimental group. Therefore, despite the small number of classrooms in the study, it seemed important to include two conventional and two experimental classrooms, all of which were ostensibly implementing the same second-grade mathematics curriculum, imposed by Israeli state law. The four classes were all working on two-digit addition and subtraction, with all the necessary knowledge supporting that topic, and, therefore, the distinctiveness of the classroom culture would stand out in the foreground. The research, therefore, was a case study of four urban, heterogeneous, second-grade mathematics classrooms in Beersheva, Israel. The four student teachers were third-year students at the regional college of education, majoring in early childhood education. All four were in the same mathematics methods

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course I had taught for two years. All four had experience of conventional teaching in local first-grade classrooms teaching reading and mathematics lessons during the previous academic year. In their third year, students are placed in second-grade classrooms with responsibility for teaching two days a week. In all of the practicum classrooms, the mathematics lessons were conventional and fairly uniform, so I needed to “create” two classrooms for the research that would reflect an “alternative” approach to the teaching of mathematics. Two of the student teachers who had a mathematics minor in their college preparation volunteered to do their practicum in the alternative classrooms. Two other student teachers agreed to participate in the research in conventional classrooms, agreeing to their lessons being videotaped and to giving the time for an audiotaped interview after each recorded lesson. “Creating” the alternative classrooms clearly required the willing participation of the mentoring teachers in order to provide a consistent culture in the mathematics classroom. This methodology was influenced considerably by the large research studies conducted in the United States where “classroom-teaching experiments” provide the alternative mathematics classrooms for exploring children’s mathematics.

The “Teacher Development Experiment” The “teacher development experiment” is an extension of the “classroomteaching experiment” associated with studying children’s learning in classes with alternative instructional approaches (see chapter 5 by McClain in this volume). Researchers alternatively train classroom teachers in a novel approach using inservice workshops and meetings or teach the classes themselves. The “teacher development experiment” as described by Simon and Tzur (1999) was used for research on teachers’ professional development in the course of their teaching. Such experiments clearly include a high level of personal involvement by the researcher with the treatment group. In order to study student teachers’ knowledge growth in mathematics classrooms with an alternative instructional approach, the following plan guided the mathematics instruction in the two alternative classrooms:

. .

The emphasis in the instruction would be on problem solving and building upon children’s intuitive knowledge, with an emphasis on thinking strategies. The mentoring teachers and student teachers would participate in a five-day summer workshop to plan the structure of the mathematics program for the year within the theoretical framework of the project. Specific planning for the first two months of the year would be part of the workshop as well as preparation of classroom materials. The summer workshop would provide an introduction for the mentoring teachers into a different approach for teaching mathematics and provide practical reinforcement for the student teachers who had learned about alternative approaches as part of their college course.

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.The mentoring teachers and student teachers would meet every week with the researcher for a 90-minute session to discuss, reflect, and plan. This would be a creditcourse for the mentoring teachers and an essential feature of the project.

While Simon and Tzur’s (1999) interest in the “teacher development experiment” is in the professional development of practicing teachers, they include prospective teachers as candidates for participation. The model includes a role for the mathematics teacher educator: as well as being the researcher, she/he is promoting the professional development of the teachers in a process of interaction and reflection. Concurrent with the work in the experimental classrooms, the two student teachers in the conventional classrooms participated in the standard studentteaching practicum framework. This included meeting with the mentoring teachers a week before the start of the school year for general orientation to the classrooms and school and lesson planning during the year by consulting with the mentoring teacher on an ad hoc basis.

Data Collection and Analysis Three aspects of the mathematics classroom were examined, each with its own procedure for data collection and analysis:

. . .

the classroom culture: videotaped observations children’s construction of knowledge: written tests, individual interviews student teachers’ construction of knowledge: reflective interviews

The Classroom Culture: Videotaped Observations—Data Collection Six mathematics lessons were videotaped in the alternative-experimental classrooms and six lessons were videotaped in the conventional classrooms. For each group, two lessons were in October, two in December and two in January. The videotapes focused on interactions between the student teacher and the class, which is remarkably easy to videotape with a high degree of visual and audio clarity. Each lesson was 45–50 minutes long. The purpose of the observation was to capture the activity in the mathematics lesson as a natural unit of organization. Attention was on the teacher-directed activity around the mathematical tasks, whether with the whole class or small group. This attention to tasks as organizing subunits of classroom mathematical activity was influenced by the research of Hiebert and Wearne (1993) and the earlier work of Doyle (1988). All activity of the student teacher, including

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her verbal communication, demonstration, and writing at the board, was considered relevant; the children’s verbal communication was also relevant as it connected with this activity. There was no attempt to document the personal conversations between children as they discussed the tasks among themselves, as this would have involved a whole new set of considerations for gathering, analyzing, and interpreting the data, which was beyond the scope of this already complex research plan. I was well known to the children and student teachers from many visits; however, during the lessons when events were videotaped, there was almost no interaction between the subjects and me. The children and student teachers were accustomed to supervisors from the college who make regular visits and observe, take notes, and ask questions. This familiarity minimized distraction by the researcher while the nonparticipation allowed for uninterrupted documentation. The documentation of the observations was done by me, using a video camera, one small Sony Handycam™. The equipment was nonintrusive and highly mobile, the closest approximation to a notebook and pencil. Only one of the observations, the last, was filmed by a professional cameraman. The videotape was chosen for documentation for two main reasons: first, it allowed for an extended period of time to analyze the data without losing any details. This helped to eliminate any premature categorization scheme that would bias the observer in later observations (Erickson, 1986). Second, given the interpretive nature of the research, it allowed for sharing the total experience of the classroom lesson with other researchers who were not present. The four student teachers were told that the videotapes would be analyzed for my research but not used in any other way without their consent. I had been in each of their classrooms on several occasions during the previous year, and they all felt that they would “learn something” from my visits. One copy only of the videotapes was made, and this was for the assistant working on the transcription. The district schools’ supervisor, responding to my request and an abstract of my research proposal, authorized the school principals to allow my videotaping in the classrooms. My relationship with the principals was ongoing, with them offering full cooperation. The transcription of the videotapes, including what was written on the blackboard, was done by a research assistant who had not been present at the observations and who had no familiarity with the classes. I edited the transcription by reviewing the tapes. Four months later I checked the transcription again (against the videotapes), which resulted in some minor corrections. In the course of analyzing the data on the classroom observations, it became evident that the accuracy of the transcriptions was highly significant. Every word of the children and the teacher was relevant for the analysis. For that reason the third round of checking and correcting, which was not part of the original plan, became necessary.

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Data Analysis The analysis of the classroom observations closely followed the structure proposed by Erickson (1986) and Marshall and Rossman (1995) for data analysis and reporting in qualitative research. The first stage after the videotaping was to watch the videotapes and to read the transcriptions many times, jotting down associations, and noting other points of interest. This is the process of turning documentation into data, during which one becomes “intimately” familiar with the data. At the same time, the material was being read from a holistic perspective and dissected, categorized, marked, and reassembled according to categories. The purpose of the first stage was to stimulate associations and insights from the raw data and then to start clustering these thoughts into broader themes, such as “attention to children’s strategies” and “directed versus open-ended use of materials.” These themes would later be supported from the two points of view proposed by Erickson (1986): the “particular description” and the “general description.” The particular description is based on the narrative vignette taken from an observation or interview. For Erickson, this is the foundation of the qualitative research report. Strengthening the evidence of the narrative vignette is the general description: General descriptive data are reported synoptically; that is, they are represented so that they can be seen together at one time. One kind of synoptic reporting medium is the simple frequency table showing raw frequencies whose patterns of distribution are apparent by inspection. In the analysis of fieldwork data, pattern discovery is done qualitatively. The frequency tables presented to the reader are usually a tabulation of qualitative judgments on nominal scales. (p. 151)

In this study I first developed the general description and then organized the particular narratives. This was because there is a certain confidence in proposing categories of the general description for which frequencies can be counted and a measure of significance calculated. This “buys a certain amount of time” with the data while the particular descriptions are uncovering themselves based on some intuitive leaps of faith.

The General Description The first stage of analysis, therefore, was to generate specific, tentative categories (Marshall & Rossman, 1995). This initial list of categories was influenced by the analysis of tasks in Hiebert and Wearne (1993), with some of the categories identical, some modified, and some new. This was formatted roughly into a table of characteristics of the tasks in which the student teachers engaged the pupils. The transcriptions were then reexamined, the tasks were assigned identification codes and ascribed to categories, and the table was modified for greater

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accuracy. There was initial difficulty in identifying tasks as discrete units and the following “rules” were developed: Each question presented by the student teacher as a new line of thinking for the student was counted as a task. This could be, for example, an addition exercise (2  9), or a word problem, or one part of a complex problem situation (If two darts together scored 10 points, what might be the value of each dart?). If there was a further probing of the specific task, not an extension or a simplification, but a clarification, this was not considered a new task. The 22 task categories organized themselves into 6 groups: tasks of computation, recall, semantic analysis, using procedures, explanation, and integrating manipulatives. While tasks might fall in several categories (labeled A–T), the last two categories were mutually exclusive: Category Y: the task was meant to elicit more than one correct response Category Z: the task was meant to elicit exactly one correct response

While interpreting the results on the tasks, a picture began to emerge of classrooms that were very different. In order to check the nature of the tasks as reflected in the children’s responses, an additional analysis of the observations was done focusing on the children’s statements around the mathematical tasks. The same three stages shaped the analysis: first, a draft list of characteristics of the children’s responses; second, a reexamination of the transcripts, numbering the responses and assigning categories, followed by a refinement of the list of categories as a more accurate analysis of the data. The categories of children’s responses were based on the categories used by Hiebert and Wearne (1993) in describing the nature of discourse in the mathematics classroom. The children’s responses were first analyzed for the length of the response: while almost the same number of responses were monitored in the conventional and alternative classrooms, their distribution as to length of response was clearly different. The children in the alternative classrooms expressed many more responses that were sentences of explanation rather than short numerical answers to exercises. The nature of the responses also varied significantly between the conventional and alternative classrooms, as seen in table 6.1. Taking together the data from the videotapes, whether looking at the data analysis of the mathematical tasks or the analysis of the nature of the children’s responses, a similar picture became clear; that of two distinctly different cultures of the mathematics classroom. In the conventional classrooms, the mathematical activity is computation-focused, with the tasks eliciting a single correct answer, recall of a fact, or identification of the formal structure of the problem. In the alternative classrooms, there was significant attention to the children’s explanations of their solution strategies. This general description from an analysis of each task and each response exposed features of the culture of the mathematics classroom. This would be expanded by turning to the holistic analysis of the vignettes, the significant little stories in the observational data. Together,

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this was the necessary background for examining what mathematics the children were learning and what the student teachers were learning, in very different mathematics cultures.

The Particular Description The reading and rereading of the data in the first stage of analysis suggested a number of important issues for the culture of the mathematics classroom around each of which at least four vignettes were relevant evidence. Eleven issues emerged, which were organized into two themes: 1. The value attributed to children’s thinking: subthemes

. . . . .

Context Attention to children’s mistakes Group work Use of blackboard Problem solving, or what’s the question

2. The teacher’s presentation of subject matter: subthemes

. . . . . .

Big questions/little questions Connections or exclusions Number sense Symbols Use of physical materials Diversity of learners

This stage of drawing out issues and organizing themes from videotaped observations is the heart of qualitative analysis and clearly subjective. Cobb and Whitenack (1996) address this particular methodological challenge from their experience in mathematics classroom research and propose that the issue of trustworthiness be connected to how reasonable and justifiable are the findings. Table 6.1 Example of Categories of the Nature of Children’s Responses

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Other researchers, with other agendas, may be able to organize the data differently and interpret the same mathematical activity from other perspectives. The challenge for my research was to make one plausible claim from these observations that was supported by the data. The transcriptions were reexamined to locate representative vignettes for each of the issues, now identified as subthemes. These vignettes, identified by student teacher and date, are included in the research report (Shane, 1996). An example of the analysis is the following discussion of the first subtheme (context) of the first theme (the value attributed to children’s thinking). The vignettes are taken from the videotaped observations. The student teachers are referred to by pseudonym: Alternative class: Rachel (R) Conventional class: Sara (S)

Vignette 1 (Oct. 30) Rachel is sitting with the children in front of a number line. R: Now I’m going to tell you a different story. Dilugi went for a walk, and his

mother was very worried. She called his friends to see where he had been. I’ll write down the series of numbers where he had been, ok? Watch—(she writes) 3,6,9,12—those are the numbers Dilugi’s mother wrote down. What instruction did Dilugi follow? Child: He skipped by three. R: How do you know? Explain. Child: 9,10,11,12

Vignette 2 (Jan. 8) S: In the last lesson with the teacher, you learned about equalities and number

relationships and today we’ll practice that. I wrote on the board an equality which is missing a number: 24  10? I am not asking what is the missing number. I’m asking what is the sum in this equality? Child: 10 S: Does someone think differently? Child: 24 S: Why do you say the sum is 10? Child: Because 10 plus something equals 24.

One of the important techniques for uncovering children’s intuitive understandings and encouraging their personal attachment to building more formal

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understandings is to situate the learning in a known context. In the alternative classrooms, every lesson was built around a unique context such as a game, a manipulative, or a problem situation. The tasks were presented as part of a “story.” The knowledge for addressing the tasks was therefore “taken-as-shared” by the class with the teacher. The teacher did not explain how to pursue a task even when this was the first time she posed this kind of problem. In the conventional classrooms, there was no comparable shared context in any of the lessons. There was a topic from the curriculum that the student teacher was addressing; this was her organizing principle. The connection between tasks was structural with no situated context: exercises with the same outcome such as “What is an addition exercise with the sum of 18?” or several examples of the same procedure such as the exercises: 24–17, 35–19, 56–28. The choice of context for a lesson can be situational, which all children can share, or it can be formal-mathematical. This was a distinguishing feature of the different cultures of the mathematics classrooms explored. My interpretation is that it is not only a choice of didactic strategy but also a reflection of sociomathematical norms and of beliefs about how mathematics is learned.The issue of context, therefore, became an example of the larger theme concerning the children’s intuitive knowledge and the attention accorded to that knowledge. It appeared to be a significant discriminating factor if the mathematics in the classroom started from the pupils’ existing knowledge and that the student teacher’s choice of tasks provided opportunities for extending pupils’ knowledge forward and sideways. In order to identify this aspect of the culture of the mathematics classroom, the holistic analysis of the classroom documentation is singularly successful. While the frequency distributions of the nature of the tasks and discourse could uncover other qualities of the instructional approach, this interest in issues such as the grounding in familiar contexts was reinforced in the vignettes. The general and the particular descriptions of the videotape data both reinforced each other and extended the analysis more deeply. The data from the classroom observations, as analyzed from the general and the particular descriptions, provided a rich source of evidence for articulating the distinguishing elements of the culture of the alternative and conventional mathematics classrooms. Having established this base, that in the alternative and conventional classrooms the features of the instructional approach for teaching mathematics were significantly different, this had a most significant implication. It provided the base for comparing results from the assessment of what the children learned and what the student teachers learned, in the conventional and alternative classrooms as two significantly different environments.

Children’s Construction of Knowledge: Written Tests; Individual Interviews The methodology for identifying the children’s knowledge of mathematics was the most straightforward aspect of the research plan. Many large-scale stud-

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ies have been carried out in first and second grade that include measurement of children’s knowledge of the relevant arithmetic topics.

Written Tests of Children’s Knowledge At the beginning of the school year a standard, written test was administered to all the children in the four classrooms (p  140) to establish the relative achievement level of the children on arithmetic items from the first-grade curriculum. At the end of the year, a semi-standard, written test of second-grade arithmetic material was administered to all the children (p  139). This test included items which measure computation skills, conceptual understanding, and problem solving (Wood & Cobb, 1989).

Test I. I administered the first test on the second day of the school year in each of the four classrooms. The items in the test were taken from the first-grade curriculum, adapted from the test suggested by the standard textbook series used by the classes. The tests were marked immediately by the student teachers. This first test involved little reading and no writing of text since it was only the beginning of second grade. The items were one-step computation (addition, subtraction, multiplication), sequences, and two word problems.

Test II. I administered Test II in all four classrooms in the spring. Most of the secondgrade arithmetic material had been covered by this time. There were 25 items. Twenty items were taken from the standard textbook test for second grade or from the bank of test items for second grade recommended by the Ministry of Education (1988). Two items were translated from the Indiana Cognitive Test (Wood & Cobb, 1989), and I generated three items. The test had several items that could be answered with more than one correct response. The test was designed to measure the standard achievement in mathematics at the second-grade level. There were one-step computational exercises as well as items that had a stronger conceptual component. Given the rationale of the research, that knowledge of mathematics means a relational understanding of number, a flexible use of number operations and properties, and the ability to choose strategies with viable explanations, it was necessary to include a further means for collecting data on the children’s knowledge.

Task-Based Interviews In recent years many researchers in mathematics education have adapted Piaget’s clinical interview technique to investigate children’s understanding of a

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wide variety of mathematics concepts. Ed Labinowicz (1985) presents the rationale and technique for this adapted clinical interview method with children in the early elementary grades. The basis of these interviews is a verbally presented task, usually in the context of physical objects, which the children are asked to perform and to explain their response. The children’s responses are recorded and analyzed. This task-based interview method has been used in research on second-grade arithmetic concepts, specifically as an alternative assessment of understanding the base-ten structure and two-digit addition and subtraction. The interview tasks in this research were adapted from Labinowicz (1985), Bednarz and Janvier (1988), and Hiebert and Wearne (1994).

Pilot Interview A month into the school year, an interview was designed. Six tasks were framed, four of which focused on different concrete representations of the baseten structure and two of which were questions of two-digit mental computation. Twenty children were randomly chosen to be interviewed. The pilot interview had three purposes: 1. Identifying the relative level of difficulty of a variety of tasks (One task that was identified as being too easy was eliminated from the later version of the interview.) 2. Settling the technicalities of how and where to conduct the interviews 3. Developing a scoring system based on the results

The scoring system (0,1,2) that developed from the nature of the children’s responses was a modified version of Charles and Lester’s (1984) scoring for problem solving: Score of 0: Child was silent. Child using inappropriate strategy. Score of 1: Partial answer. Solution strategy logical but solution incorrect. One part of the task done correctly and one part incorrectly. Score of 2: Appropriate strategy; correct answer.

The Task-Based Interview, End of Second Grade Since the class size was quite large (about 35 per class) and I wanted to personally conduct all the interviews in a period of approximately two weeks, I decided to interview about one third of the children. The scores on Test II were the basis for selecting the children from each class who would be interviewed about a month before the end of the school year. Scores for each class individually were arranged from highest to lowest, and every third child was selected to be interviewed, leaving out the lowest-attaining pupils in each class. Forty-four inter-

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views were held, 11 from each class, comprising 25 boys and 19 girls evenly distributed across the classrooms. Interviews lasted about 20 minutes. On the table in front of the child were pencil and paper, which they were told they could use whenever they chose. I conducted all the interviews using a written protocol of tasks and a small bag of equipment. Children’s responses were noted with careful attention to detail at the time of each interview in a small notebook. The interviews were not audiotaped since the children also wrote and manipulated objects. It did not seem appropriate to videotape these personal encounters, which had their own privacy.

Data Analysis of Children’s Knowledge: Written Tests, I and II The results from both written tests were processed using standard statistical procedures. The mean for each test item was compared between alternative and conventional classrooms, between the two classrooms in the same treatment group, and across all four classes. Tendencies toward significance were noted when the statistical measures were not conclusive.

The Task-Based Interview Once all the interviews were completed, the tasks were analyzed for the range of responses. This analysis was inherently qualitative since the interest in an interview response is not merely whether it is right or wrong. Therefore, there were two levels of analysis: a quantitative assessment of the child’s performance on each task and a holistic assessment of the range of strategies used in approaching the tasks. From my experience in the pilot interviews earlier in the year, the responses were coded using a system of 0,1,2. The range of responses for each task and the corresponding score were organized for reference. A sample scoring is presented in table 6.2. While the mathematical topics of the interview tasks were all taken from the standard second-grade curriculum, the children in the alternative classrooms had much more experience in verbalizing the explanations for their responses. Beyond the verbalizing, however, the tasks presented an opportunity to expose flexibility of mathematical thinking that would be harder if not impossible from responses to a written test. Therefore, the integration of data from the second test and from the interviews should give complementary pictures of how the children understand the arithmetic at the end of second grade. It was believed that the results would reveal the effect of certain mathematical habits and/or norms on the children’s use of their knowledge. In fact, the results were quite significant from both the tests and the interviews. The results of the first test (on the first day of school) had shown that al-

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Table 6.2 Children’s Interview, Sample Scoring

though the two schools were five minutes apart and I had thought that the achievement levels would be the same, the one with the conventional classrooms started the year with a significantly higher score. On the test at the end of the second grade, the scores of all four classrooms were virtually the same. That is, the children in the alternative classrooms closed the gap with the children in the conventional classrooms. Given that the same children were all continuing with the same teachers from the first grade and that the second-grade arithmetic is based on skills from the first grade, one might have expected the gap to widen. The results of the interview showed the children in the alternative classes to have a higher mean score on six of the eight interview tasks. On four of the five tasks that drew heavily on understanding the base-ten structure, the children in the alternative classrooms scored significantly higher. For example, on a word

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problem that was based on subtracting 11 from 100, the children in the conventional classrooms tended to make errors using the traditional paper-andpencil algorithm while the children in the alternative classrooms succeeded with mental computation strategies such as 100–10–1. The children in the alternative classrooms generally tended to “talk themselves through” problems that were unfamiliar while the children in the conventional classrooms had no backup strategies when they could not recall a procedure.

Student Teachers’ Construction of Knowledge: Reflective Interviews While the data on children’s knowledge of mathematics was the easiest to collect and analyze, the knowledge of the student teachers for teaching mathematics is most difficult to expose and interpret. The model developed in Michigan State (Kennedy, Ball, & McDiarmid, 1993) for interviewing teachers about their lessons seemed to offer the most promising approach for uncovering their teacherthinking about mathematical ideas, about children’s mathematical activity, and about their teaching of the mathematics.

Data Collection The purpose of the reflective interviews, therefore, was to uncover the explicit and implicit knowledge of the student teachers for teaching mathematics. The interviews were held, in most cases, immediately after the classroom observations. In addition, a summary interview was held with each student teacher at the end of the school year. All the interviews were recorded on audiotape and transcribed by a research assistant.

Lesson-Specific Interviews There were eleven interviews with the student teachers, each held about an hour after they taught a mathematics lesson. The six interviews with the student teachers in the conventional classrooms were individual, since each taught a different lesson according to the directions of the mentoring teacher. The five interviews with the student teachers in the alternative classrooms were part of a general debriefing that went on each week; these included the student teachers, the mentoring teachers, and the researcher. While these were clearly different contexts for the interviews, each context was consistent with the way the student teacher was “doing” her practicum. The student teachers in the conventional classrooms were not planning together since their mentoring teachers did not plan together, and their trajectory of in-

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struction was essentially a one-way process from the mentoring teacher to the student teacher to the children. In contrast, the natural pattern in the alternative classrooms was “circles of communication,” the mentoring teachers with the two student teachers, the student teachers with the second graders. Therefore, the natural place of the reflective interview with the students in the conventional classrooms was an individual interview alone in the teacher’s room, while the reflection on the lessons in the alternative classroom took place at their weekly meeting. This has significant implications for interpreting the process around the student teachers’ learning from their reflection. The sample questions in table 6.3 formed the basis of the interviews. The interviews were designed as semistructured, with the questions providing an entry into the teacher’s thinking. These questions led to other questions; not all these questions were asked at any given interview. These questions were of the most general pedagogic nature unconnected to the nature of the instructional practice. I saw them as unbiased in terms of beliefs about the nature of mathematics or the teaching of mathematics. Since the four student teachers had also studied with me for two years at the college, they all had heard my approach to appropriate mathematical activity in the classroom. I had reason to claim that if their responses to these questions were significantly different, then this was the effect of the context of their practicum experience. Table 6.3 Sample Questions from the Lesson-Specific Interview Protocol

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There is a certain similarity between this reflective interview and the stimulated recall interview technique. In the stimulated recall interview, most often a video of the lesson is shown to promote reflection and insight into the teacher’s thinking. In this case, although the video was not shown, both the student teacher and I had the shared experience of participating in the lesson just before the interview. This provided the same kind of stimulation for reflection on teaching in the context of an authentic specific experience. Therefore, since the interviews had an immediacy in their relation to particular lessons, it seemed important to include a summary interview with general questions of how they looked back at the total practicum experience.

The Summary Interviews The concluding interview with each of the four student teachers took place as the school year was ending. The interviews were held at the teachers college and each lasted about 50 minutes. The core interview questions (table 6.4) were the same for all four student teachers, while questions of clarification or expansion were added when appropriate. Some of the questions were adapted from the Michigan State research on tracking teachers’ knowledge (Kennedy, Ball, & McDiarmid, 1993); some I have used previously with student teachers, others were designed as the result of reflection on features of the actual classroom activity. The interviews were audiotaped and then transcribed by a research assistant during the summer after the school year. I personally checked the transcriptions and made final corrections. These questions also seemed to me to be unbiased toward any particular instructional approach and appropriate for asking any student teacher at the end of her practicum experience. My only hesitation was that I thought I might hear the same generalizations, “politically correct” educational slogans from all the student teachers. I knew that the strength of the interview data would be in the particular examples the student teachers might choose and their way of explaining these choices.

Data Analysis: Lesson-Specific Interviews Two lines of analysis suggested themselves to me as I read and reread the interview transcripts:

. .

Looking at all the responses from the 11 interviews question by question Rereading the interviews to identify issues which recurred, unconnected to the particular questions

Looking at each question individually and grouping the responses by treatment group uncovered significant differences between the student teachers in

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Table 6.4 Sample Questions from the Summary Interview Protocol

the conventional classrooms and those in the alternative classrooms. The questions in their very neutrality were able to expose elements of the knowledge of the student teachers. The following example is typical of the use of vignettes from the responses to particular interview questions and my interpretation of the responses. “What was new in today’s lesson? How is the subject of today’s lesson connected to other topics?”

Sample Response from the Conventional Classrooms S: The equals sign the children already know, so it was to learn the unequals

sign. . . . ” Interviewer: Why did you choose not to accept the boy’s answer who suggested the

sign for “more than”? S: Because we were talking about equals and not equals, it’s either equals or

not equal; it doesn’t matter if it’s larger or smaller. What matters right now is if the two sides are equal or not equal. . . . I’m not asking right now if it’s bigger or smaller, even though that’s a kind of inequality. Here I wanted to focus, the point is the symbol. So I can’t let them use greater or smaller, only the inequality sign. Even though you can’t really say they are wrong, one side really is bigger than the other.

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Interviewer: Was there a moment when you considered accepting it, or were you sure

you didn’t want to? S: At the beginning I saw it in one of the children’s notebooks and I thought

it was correct, they used the sign for greater than or smaller than. Then after that, first I said correct, but immediately afterwards I decided it was inappropriate. We want the unequals sign.

Sample Response from the Alternative Classrooms R: In the last lesson we talked about sequences, patterns. I gave the rule, the

number of jumps, and the children continued. This time they got sequences where numbers were missing at the beginning or the middle or the end and they had to figure out the rule. Because what I wanted, after all what is a number sequence? . . . you have to know the rule to continue. . . . As soon as they figured out the rule they had no problem continuing to higher numbers, because they know the numbers, they know the order of the numbers and the value of the numbers. . . . As soon as they figured out the rule, it was like a trick, . . . they were very excited.

Discussion In the conventional classroom, the student teachers did not choose to link the math lesson with previous or upcoming lessons. When the child in the first vignette makes an appropriate connection to earlier mathematics knowledge, the student teacher rules out the answer because it was not her intention to recall that particular topic. She sees the correctness of the response and purposely rejects it because it interferes with her objective. In the alternative classroom, the topic of the lesson is the natural development of the previous lesson. For Rachel, the organizing principle is her mathematical question, “After all, what is a number sequence?” Clearly the choice of the vignettes and their accompanying interpretation is a subjective filtering of the responses to the interview questions. On the other hand, the intention of the interview was just that, to elicit the beliefs and dispositions, the knowledge that the student teacher has constructed for teaching mathematics. The next and more general stage was reading over the interview transcriptions, looking for the issues of each interview, noting particularly “loaded” passages that had not surfaced in the analysis of responses to a particular question. Some of these issues that became identified as subthemes were present only in the interviews of the student teachers in the alternative classrooms and absent from the responses of the student teachers in the conventional classrooms. For example, instances of children’s personal computational strategies were only evident in responses by the student teachers in the alternative classrooms. For

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other subthemes there were vignettes supporting different approaches in the alternative and conventional classrooms. The following examples are from such situations of contrast. They are examples from the subtheme: extrinsic or intrinsic motivation:

Vignette from the Conventional Classrooms Maybe I should have insisted more that they solve it. . . . if I had left a space on the side, they would have to solve it. . . . it would have forced them to solve the exercise.

Vignette from the Alternative Classrooms They were supposed to decide what happens to the price on Friday, if it goes up or down and by how much. And Simon saw the price of the hat was 10 shekel. He wanted to make it more interesting so he decided that the price would go down by 14 shekel, and he said to me—“See, now the price is minus 4. You said I could do what I wanted.”

Discussion The student teachers in the conventional classrooms presumed that the motivation for doing the mathematical tasks is extrinsic—the teacher assigns work and the children produce. By formulating the task differently from a technical point of view, the children will have to use the strategy that the teacher has chosen. The implied conception of the learner is one who would “cut corners” if the teacher is not watchful. In the alternative classrooms, the student teachers described their observation that children relate to intrinsic challenges in the tasks and can be motivated by providing tasks which are flexible. Their conception of the child was of a learner looking to “stretch” his or her knowledge, willing to challenge him/ herself with more difficult tasks. The evidence of the student teachers’ knowledge from these lesson-specific interviews organized itself into three themes: Knowledge of mathematics Knowledge of how children learn mathematics Knowledge of how to teach mathematics

The Concluding Interviews The concluding interview data were clearly structured around the interview questions with less scope for variation than the lesson-specific interviews. The

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analysis had two stages: first, identification of the four responses to each interview question. Second, qualitative comparison of the responses between treatment groups, that is, two student teachers in conventional classrooms and two in alternative classrooms. In general, there was an internal check between the lesson-specific interviews where the knowledge is implicit in the student teachers’ verbal responses and the summary interviews where the knowledge is explicit in the direct responses to summarizing questions. The responses to the summary interview were considerably simpler than the lesson-specific interviews. The richness of example and detail that followed the immediate experience of the mathematics lesson was not matched in the generalizations of the summary interview. What was surprising to me were not the responses of the student teachers in the alternative classrooms, since the summary of their experience was very similar to the sum of the parts of their comments throughout the year. I did not anticipate the responses from the student teachers who had taught in the conventional classrooms. Although the lesson-specific interviews had focused on rather matter-of-fact pedagogic analysis during the year, their honest assessment in this summary interview was very powerful in its “innocence.” The following is a sample of the responses in this summary interview: A: response of student teacher in alternative classroom C: response of student teacher in conventional classroom

Were Games Used in the Class? How? Why? A: One teaches the other, chance to hear how kids think, pick up mistakes, integral

part of the lesson. . . . To practice what they learned, connected to the topics C: no

Tell about a Lesson Which Was Particularly Successful A: Lessons on strategies for addition and subtraction were living course in Math

Methods—how kids think. C: Class with riddles—who am I?— was very successful, worked out each solution,

but I didn’t really teach them anything.

Did You Hear Kids’ Thinking? A: Yes and it was important for me and for them. C: I don’t know, I don’t remember. I had to cover from this page to that. I am a guest in the class.

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I had been concerned that the student-teachers would all answer the way they thought was the “correct” answer, but the two who had been in the conventional classrooms described their experience with all its problems. In fact, they summarized the experience in their own words in a quite similar fashion to the features observed by the researcher from other evidence. They revealed little or no attention to children’s thinking; a successful lesson was criticized because it had not taught anything from their perspective—because there was no teachertelling; they identified children’s knowledge as expressed in answers to exercises in the book; they explained that the mathematics content had to be “broken into little pieces and spoon-fed.”

DISCUSSION OF RESULTS Results are reported in Shane (1996) for each of the three sections, and an attempt is made to connect these sections in order to address the issue of the interrelationships between the context of the classroom and the content of the knowledge being constructed for learning and teaching mathematics. Expressions of this interrelationship include the following: When the repertoire of the instructional approach included tasks that were more open-ended, which required a problem-solving approach and connections between concepts and procedures, the student teachers in those classrooms demonstrated knowledge of the complexity of the mathematical ideas. When the emphasis in the instructional approach was on one-step computation and recall of symbols, then the student teachers in those classrooms focused on issues of “interesting didactic packaging” without deliberating about the mathematics. When the instructional approach supported a broad view of the range of potential mathematics within a lesson, then the student teachers in those classrooms verbalized their new knowledge of how much more mathematics the children are familiar with than they had imagined. When the instructional approach supported a narrow view of the topics of the lesson, then the student teachers in those classrooms verbalized a preference even for incorrect answers that would not overstep the topics of the lesson. In the classrooms where the content of much of the lesson was how the children arrived at their solutions, the student teachers verbalized the importance of uncovering children’s thinking strategies for that child’s learning, for the learning of the other children, and for the learning of the student teacher herself. In the classroom where the instructional approach did not encourage the invention and sharing of solution strategies, the student teachers did not demonstrate any knowledge of children’s thinking and it was not on their agenda.

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METHODOLOGICAL ISSUES AND IMPLICATIONS The integration of quantitative and qualitative methods in this research was naturally appropriate because different aspects of the mathematics classroom lend themselves to different modes of analysis. In fact, this strengthened the claims of the connections between the features of the instructional approach, the knowledge of the children, and the knowledge of the student teachers for teaching mathematics.

Limitations of the Study and Issues of Validity In a case study of four classrooms, it is necessary to exercise caution and not to be overambitious in generalizing. Furthermore, classroom research with such a small sample runs the risk of collapse when, say, a student teacher becomes pregnant, or her other children are sick, or the school takes on a project that changes its schedule. Therefore, the time framework for the intensive classroom visits was shortened to five months while the testing and interviewing took place later in the year. This schedule turned out to be more appropriate for the pacing of everyone’s school year, children and student teachers alike. The choice of student teachers for the research was not random. The two student teachers in the alternative classrooms were mathematics specialists, and they volunteered for a special program, both not insignificant factors. On the other hand, mathematics specialists have been placed every other year in conventional classrooms, and there has been no informal evidence of the kind of knowledge leap that occurred in the course of this research. Issues of validity are particularly controversial in the data analysis stage of qualitative research. Two doctoral students checked the coding of the classroom tasks and responses. Categories were linked into subthemes and subthemes into themes, all based on my own insight, connecting events from the observations or interviews. Erickson (1986), in discussing the subjective nature of these generalizations, reminds the reader that the aim of this research is not proof, but “plausibility.” Kagan (1990), in her survey of research on teacher cognition, indicates one of teachers’ weaknesses to be in the area of ecological validity: whether the thinking that is attributed to teachers is consistent with their classroom behavior. She answers the charge by quoting other researchers who claim that this aspect of validity is irrelevant to research that is constructivist by nature and designed to generate insights and explain events, not to prove a theory. In this research there actually was the opportunity to correlate the teacher interviews with the classroom observations.

CONCLUSION For myself, as a mathematics teacher educator, there was a lot of knowledge construction that took place while analyzing the mathematics lessons in the alter-

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native classrooms. Children’s thinking and student teachers’ negotiating with the children’s thinking dominated these mathematics lessons. There was a considerable amount of intrinsic motivation that encouraged my involvement. On the other hand, the conventional classrooms provided little opportunity for me to learn about the children’s mathematics or the student teachers’ consideration of their teaching. I kept thinking that if I find it difficult to stay “mathematically awake” in these lessons, how are the children being engaged? How are the student teachers learning about teaching mathematics? For an elementary teacher turned mathematics educator and researcher, classroom research is both alluring and awesome. It looks so intuitively “within reach” and so appropriate, yet formally full of obstacles because of its complexity and fuzzy edges. The qualitative methodology presents the opportunity to document and manage a rich description that may provide the necessary insight into the dynamics of teaching and learning elementary school mathematics.

REACTION BY GILAH C. LEDER For those involved in educational research, the ERIC database offers a wealth of information on a multitude of topics. Surveying its contents offers a useful, although somewhat crude, indication of the areas attracting most attention from the research community. In an attempt to identify topics of particular interest in mathematics education, Lubienski and Bowen (2000) searched the contents of 48 major national and international educational research journals accessible through ERIC and likely to include at least some mathematics education–related research. Their enterprise revealed that articles concerned with elementary education were most prevalent (37 percent of the articles), that relatively little attention was given to early childhood education (2 percent), and even less to what they classed as adult basic education (0.2 percent). An examination of more general topics related to teaching and learning revealed that cognition was the most popular of the categories considered, relating to 49% of the 3,011 articles surveyed . . . . Student achievement was also a popular topic . . . (23%). Teacher actions (20% of the articles), curriculum (17%), technology (15%), student characteristics (15%), and student affect (12%) also received significant attention. Teacher education (6%), student assessment (5%), educational environment (5%), and students in classrooms (4%) received the least attention. (Lubienski & Bowen, 2000, p. 630, emphasis added)

Teacher education, it seems, has to date attracted only moderate research attention. The chapter by Ruth Shane, with its emphasis on (preservice) teacher education, represents a thoughtful addition to an area of research arguably in need of additional work. It is a nicely organized and written chapter, which offers much useful and indeed provocative information for those of us engaged in teacher training. Rather than dwell on the undoubted strengths of the study described, I will use the (limited) space allocated for this commentary to raise some

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issues—particularly ones concerned with methodology—stimulated by Ruth’s account of her work. Ruth gives a comprehensive yet lucid description of the key theoretical assumptions that shaped the research design and instruments selected. These do not need to be repeated here except to note the acknowledged tension between the wish to gather a rich pool of data and yet minimize the level of intrusion foisted upon the class. This challenge is all too familiar to those involved in exploring complex phenomena in a real, practical setting. Yet considerable intervention did occur. In order to compare classrooms with distinct instructional cultures, two classrooms were “created.” To achieve this, the mentoring teachers as well as the preservice teachers participated in a special training program and regular meetings during the year. (The latter also took place for those working in the traditional settings.) It is not clear whether the instructional strategies advanced by Ruth for the “alternative” classrooms were well accepted among Israeli teachers generally and readily fostered by the “second-grade mathematics curriculum, imposed by Israeli state law.” I would have liked to hear more about the methods used to counter the difficulties of maintaining an “alternative” classroom culture for one subject area in a climate otherwise attuned to “conventional” methods. It would also be useful to know how possible flow-on effects were monitored (e.g., whether the alternate approaches Ruth shared with the mentoring teachers were discussed by them with their colleagues, whether the participating teachers believed that the “alternative” approaches flowed onto their instruction in other subject areas, and if not, what tensions were experienced by them, and their students, in coping with quite different expectations in different subject areas). Similarly, it would have been good to hear whether the preservice teachers introduced the “alternative” strategies they employed in mathematics lessons during their two teaching days per week into their teaching of other subject areas, and whether this caused friction with other supervising teachers or confusion among the students they taught. It is intriguing that the third year experience in the “alternative” classroom should have been more powerful than the students’ earlier experiences of conventional teaching. What aspect of the design of her study, I would have liked Ruth to reflect, might be particularly responsible for this? The multiple data-gathering methods employed, nonparticipant observation, videotaping of lessons, semistructured interviews of the participating student teachers, and written tests and interviews of the students being taught were all varied and extensive, as might be expected given the complexity of the issues being explored. Ruth is clearly familiar with the work of Simon and Tzur (1999). Would she describe her study as an attempt to explain “the teacher’s perspective from the researcher(s)’ perspective” (p. 254)? With hindsight, which methods proved most informative in capturing the perspectives of her participants, which measures seemed redundant, and what would she modify should she embark on a similar research project? It is worth noting that the methods selected by Ruth were similar to ones employed in comparable research. For example, in another study which attempted

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to trace the interaction between course and field work for preservice teachers, Ebby (2000) relied on interviews, audio recorded pre- and postobservational conferences, lesson observations supported by videotaped records, and review of documents written by the participants. Ebby noted considerable variations in the quality and rate of student growth among the three participants in her study. Was Ruth able to trace individual differences among her main participants? The preservice teachers prepared to work in the “alternative” classrooms seemed relatively well qualified in mathematics. Might this have influenced the findings obtained? Which of the data gathering approaches might furnish a tentative answer to this question? It is clear that the study on which this chapter is based took place some years ago. Did Ruth have the opportunity to explore, formally or informally, the longer-term effects on the mentoring teachers, the preservice teachers, and the grade 2 students of working in the “alternative” classroom environment for one teaching year? Given the overriding theme of the book, a critical examination of methodology, I would have liked to read more in the concluding section of the chapter about Ruth’s reflections on the methodological issues raised by this piece of research.

REFERENCES Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90(4), 449–466. Bednarz, N., & Janvier, B. (1988). A constructivist approach to numeration in primary school: Results of a three year intervention with the same group of children. Educational Studies in Mathematics, 19, 291–331. Charles, R. I., & Lester, F. K. (1984). An evaluation of a process-oriented instructional program in mathematical problem solving in grades 5 and 7. Journal for Research in Mathematics Education 15(1), 15–34. Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13–20. Cobb, P., & Whitenack, J. (1996). A method for conducting longitudinal analyses of classroom videorecordings and transcripts. Educational Studies in Mathematics, 30(3), 213–228. Davis, R. B., Maher, C. A., & Noddings, N. (1990). Constructivist views on the teaching and learning of mathematics. Reston, VA: National Council of Teachers of Mathematics NCTM. Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psychologist, 23 (2), 167–180. Ebby, C. B. (2000). Learning to teach mathematics differently: The interaction between coursework and fieldwork for preservice teachers. Journal of Mathematics Teacher Education, 3(1), 69–97. Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D., & Agard, P. (1993). Conceptual knowledge falls through the cracks: Complexities of learning to teach

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mathematics for understanding. Journal for Research in Mathematics Education, 24(1), 8–40. Erickson, F. E. (1986). Qualitative methods in research on teaching. In M. C. Wittrock (Ed.), Handbook of research on teaching (3rd ed.) (pp. 119–161). New York: Macmillan. Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996) A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27(4), 403–434. Fischbein, E. (1987). Intuition in science and mathematics. Boston: Reidel. Fischbein, E. (1993). The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity. In R. Biehler, R. W. Scholz, R. Straßer, & B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 231–245). Dordrecht, Netherlands: 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: Erlbaum. Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic. American Educational Research Journal, 30, 393–425. Hiebert, J., & Wearne, D. (1994, April). Instruction, understanding and skill in multidigit addition and subtraction. Paper presented at the annual meeting of the AERA, New Orleans, LA. Kagan, D. M. (1990). Ways of evaluating teacher cognition: Inferences concerning the Goldilocks principle. Review of Educational Research, 60(3), 419–469. Kennedy, M., Ball, D., & McDiarmid, G. W. (1993). A study package for examining and tracking changes in teachers’ knowledge. East Lansing, MI: The National Center for Research on Teacher Education, Michigan State University. Labinowicz, E. (1985). Learning from children: New beginnings for teaching numerical thinking. Menlo Park, NJ: Addison-Wesley. Lubienski, S. T., & Bowen, A. (2000). Who’s counting? A survey of mathematics education research 1982–1998. Journal for Research in Mathematics Education, 31(5), 626–633. Marshall, C., & Rossman, G. B. (1995). Designing qualitative research (2nd ed.). Thousand Oaks, CA: Sage. Mewborn, D. S. (1999). Reflective thinking among preservice mathematics teachers. Journal for Research in Mathematics Education, 30(3), 316–341. Ministry of Education, Israel. (1988). Mathematics Curriculum, K–6. Jerusalem: Department of Education. Schön, D. A. (1983). The reflective practitioner. New York: Basic Books. Schwab, J. J. (1978). Education and the structure of the disciplines. In I. Westbury & N. Wilkof (Eds.), Science, curriculum, and liberal education (pp. 229–272). Chicago: University of Chicago Press. (Original work published 1961). Shane, R. (1996). Constructing knowledge in the mathematics classroom. Unpublished doctoral dissertation, Ben-Gurion University of the Negev, Israel. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

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Simon, M. A., & Tzur, R. (1999). Exploring the teacher’s perspective from the researchers’ perspectives: Generating accounts of mathematics teachers’ practice. Journal for Research in Mathematics Education, 30(3), 252–264. Skemp, R. (1989). Relational understanding and instrumental understanding. In R. Skemp, Mathematics in the primary school (pp. 152–163). London: Routledge. (Original work published 1976). Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press. Wood, T., & Cobb, P. (1989). The development of a cognitively-based elementary school mathematics test. Calumet, IN: Purdue University School Mathematics and Science Center.

Chapter 7

Longitudinal Measurement of Student Motivation and Achievement in Mathematics Shirley M.Yates

INTRODUCTION The central focus of this longitudinal study was the relationships among students’ characteristic optimistic or pessimistic explanations for the causes of everyday events in their lives, their motivational goals, and achievement in mathematics. The impetus for the research came from a South Australian primary school principal who had noted differences in patterns of students’ achievement in mathematics, which persisted over time. Each year the school administered a calibrated test of mathematics achievement to all classes. While some students continued to exhibit a sound level of achievement as they progressed through the primary school, others fared less well. The teachers reported that they found the students who were doing well in mathematics to be a pleasure to teach. They had positive attitudes toward mathematics, applied themselves during classroom lessons, tried hard, persisted in the face of difficulties, and generally worked well. However, students with lower levels of achievement in mathematics posed greater challenges for the teachers. These students often expressed dislike for the subject, were less motivated during mathematics lessons, less willing to try, more easily discouraged and distracted, and were more likely to give up in response to difficulties. Over time, these differences between the students became accentuated, so that by the end of the primary school years some students appeared to have “given up” on mathematics. In light of these trends, the school was interested in gaining a better understanding of factors that influence student achievement in mathematics and, in particular, investigating why some students were motivated while others, clearly, were not. Such information would be of value in informing the school’s educational decisions and enhancing the learning of mathematics for all students.

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The problems pertaining to individual differences in motivation and achievement in mathematics noted by the principal and teachers are by no means unique to this school. They are critical concerns for teachers and students in every classroom and pose important questions for research. Numerous studies have investigated relationships between achievement in mathematics and several attitudinal and intrapersonal factors (McLeod, 1992). Attitudes, beliefs, and emotions about mathematics and about mathematics learning and teaching have been examined in both teachers and students (for reviews see, Leder, 1987; McLeod, 1992; Reyes, 1984). Individual differences in students’ self-confidence, self-efficacy, self-concept, mathematics anxiety, learned helplessness, and attributions for the causes of success and failure have also been considered (Kloosterman, 1990; McLeod, 1992). In light of this research, it was decided to investigate students’ optimism, pessimism, and goal orientations in mathematics as these are significant motivational constructs that had not been examined previously in relation to achievement in mathematics. Goal orientation theory proposes that the task or ego goals that students hold have an influence on their achievement (Dweck, 2000). Students who espouse task-oriented goals strive to achieve understanding and gain mastery in their learning, while ego-oriented students focus on normatively referenced standards (Duda & Nicholls, 1992). Academic achievement has also been shown to be influenced by the causal attributions that students make for general events, particularly when these events are negative (Nolen-Hoeksema, Girgus, & Seligman, 1986, 1992). Explanatory-style theory hypothesizes that people characteristically explain the causes of events from either optimistic or pessimistic frameworks (Seligman, 1990). By contrast with optimism, pessimism carries with it the expectation that negative events are permanent, personal, and pervasive; and positive events temporary, external, and isolated (Peterson & Bossio, 1991). It was, therefore, of interest to consider whether the customary explanatory styles and task involvement and ego orientation goals developed during childhood were related to student achievement in mathematics. From the outset, the design of the study took into account the desire of the school staff to explore classroom motivation and to improve the learning outcomes of all students. Learning and development in education occur across a span of several years. The project therefore needed to involve intact classes of students across different grade levels, with the measurement to be repeated across time so as to capture students’ developmental progression through the grades and to facilitate the examination of patterns of relative stability and change in their motivation and achievement. Once the relevant theories of motivation had been identified, it was necessary to select the most appropriate methods of investigation. It was decided to collect information from the students in their classrooms with an objective test of achievement and pencil and paper questionnaires since these measures were relatively efficient to administer and score, caused minimal disruption to the classes, and, at the same time, provided useful information from which relationships

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among the students’ explanatory style, motivation, and achievement in mathematics could be explored and explicated. Teacher contribution to the study was also sought since the teachers were best placed to comment on students’ behavior during mathematics lessons in their classrooms. In the first year of the study, students’ task and ego goal orientations toward mathematics were canvassed together with information about their optimistic or pessimistic causal frames of reference and their achievement in mathematics. Comparable data were collected at the same time in a second primary school where the principal had expressed a similar interest in student achievement in mathematics. Students from these two schools were then followed over three years during which time approximately half of them moved to a number of lower secondary schools. The measures of achievement, motivational goals, and optimism and pessimism administered to students in the first year were repeated in the third year together with an additional index of self-reported depression. In the intervening year, classroom teachers rated the students’ behavior in the mathematics classroom and achievement in mathematics. Specific measurement challenges were presented by the longitudinal nature of this study. Students had to be traced across three measurement points over almost three years. The tests of achievement and questionnaires needed to be selected carefully as the study involved students across the primary school years and into the first two years of secondary school. It was also necessary to choose methods of analysis that took into account the sample characteristics yet which allowed for a detailed investigation of relationships between the variables. In addition to the use of conventional parametric statistics, the data were scaled with the Rasch measurement procedure and proposed causal models were tested by path analysis with latent variables. The latter two procedures were significant innovations in this type of research. The use of path analysis enabled the causal interrelationships between the students’ explanatory style, depression, motivational attitude toward mathematics, and achievement in mathematics and teachers’ ratings to be examined over time. Exposure of these causal relationships enabled the schools to gain a better appreciation of factors that influence student motivation and achievement in mathematics.

THEORETICAL FRAMEWORK There are a variety of research methods available for classroom-based studies in mathematics. Quantitative and qualitative modes of inquiry have been developed to examine a broad range of research questions encompassing the three different worlds of human inquiry proposed by Popper and Eccles (1977) (Keeves, 1997a). The real world is represented in World 1 and includes physical objects as well as structures such as schools and universities created by society. World 2, the mind of the learner, encompasses students’ subjective experiences, conscious and unconscious thoughts, and psychological dispositions. World 3

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comprises the objective body of knowledge generated by and through World 2. Objects in World 3 acquire a reality of their own and can influence World 1 (Keeves, 1997a). For example, propositional knowledge gained through the physical and biological sciences acts through technology to produce changes in the real world (Keeves, 1997a). Investigations may be directed toward each world separately or to the interactions between them. Studies of relationships between World 1 and 2 are referred to as heurism while theorism is concerned with the processes operating between World 2 and World 3 (Keeves, 1997a). The theoretical framework adopted by educational researcher(s) should, therefore, take into consideration the realm of inquiry within which their research questions most properly fit. It was clear from the outset that this study fell within the aegis of Worlds 1 and 2, as the focus was clearly on the psychological dispositions of the students, viewed as part of the real world. The theoretical orientations were adopted from World 3 with the outcomes of the study expected to add to the body of knowledge in that domain. The discipline of psychology has a long tradition of theoretical and empirical research that indicates people develop beliefs that organize their world and give meaning to their experiences (Dweck, 2000). Information collected from thousands of individuals across numerous studies shows that students’ inherent belief systems or “meaning systems” differ such that under identical circumstances they think, feel, and act differently. In this tradition access to the unique psychological world inhabited by each student is commonly gained through pencil and paper questionnaires with results aggregated to the group level. The responses made by the participants to the items in the questionnaires are used as an index of their psychological disposition. For this study students’ beliefs were assessed with questionnaires measuring their goal orientation toward mathematics and their explanatory style. Information on their internal feelings of depression was also obtained. The decision to gather information with questionnaires clearly indicated that this study would be conducted within a quantitative framework. Furthermore, pragmatic considerations including availability of funding, access to schools and teachers, the inclusion of entire classes of students, and demands upon researcher time indicated that this approach would be most suitable. Quantitative studies serve to measure qualitative differences objectively, since “quantities are of qualities, and a measured quality has just the magnitude expressed in its measure” (Kaplan, 1964, p. 207). All measurement involves a degree of abstraction. Within a quantitative framework observable characteristics and relationships are clearly defined, specified, measured, and analyzed with appropriate mathematical and statistical procedures (Keeves, 1997b). These constructs, characteristics, or relations are represented by numerical values and are specified accurately and precisely prior to their measurement. The assigned numerical values do not of themselves have an inherent worth but provide an objective means of exploring the nature and degree of the constructs and of their interrelatedness. Judgments of the characteristics or constructs generally occur across multiple ob-

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servations, so as to ensure their adequate representation, with trustworthiness determined through unidimensionality (Keeves, 1997b). In primary and lower secondary schools, mathematics is a compulsory core subject in which all students receive varying weekly allocations of instructional time. In the recent international study of mathematics achievement in the middle school years conducted in 45 countries, a clear positive relationship between a strong liking of mathematics and higher achievement was observed within nearly every country (Beaton et al., 1996). However, many studies have indicated that some students hold unfavorable attitudes toward mathematics that influence their approach to and participation in mathematics lessons (McLeod, 1992). These negative attitudes arise in part from the nature of the subject matter with success and failure more salient in mathematics than in other subjects (McLeod, 1992). Some students believe that mathematics is governed by rules and that problems should be able to be solved relatively quickly (McLeod, 1992). In addition, mathematics is often considered to be a subject only for the very able (McLeod, 1992), with many students believing that success is dependent upon ability rather than hard work (Dweck, 2000; McLeod, 1992). These attitudes and beliefs influence students’ expectations toward mathematics, particularly in relation to failure, and are likely to be reflected in their behaviors within the mathematics classroom. This study examined students’ expressed goal orientations in mathematics, to determine whether they were related to their characteristic optimistic or pessimistic styles of causal explanations and to consider the relationship of both to the students’ concurrent and subsequent achievement in mathematics.

Motivation In developing a suitable instrument to measure motivation toward mathematics in this study, it was relevant to consider why students wanted to achieve academically. It has been found that the goals that students espouse affect the quality of their motivation (Dweck, 1986), which in turn influences behavioral, cognitive, and affective outcomes (Urdan, 1997). In achievement motivation research, students’ task involvement and ego orientation goals have been identified as important attitudinal constructs, as they reflect students’ reasons either to achieve mastery of the subject matter, to be competitive with their fellow students, or to do both (Nicholls, Cheung, Lauer, & Patashnick, 1989; Nicholls, Cobb, Wood, Yackel, & Patashnick, 1990; Nicholls, Patashnick, & Nolen, 1985; Thorkildsen, 1988). Students who are task involved hold learning goals which motivate them to learn, improve, seek challenges, persist in the face of difficulty, and focus on mastery of the topic or task (Nicholls & Miller, 1984). They are also more likely to believe that ability is incremental (Dweck, 2000; Schunk, 1996) and to seek appropriate assistance (Butler & Neuman, 1995). Students who espouse ego ori-

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entation goals are focused on their performance goals relative to others. They are motivated by the need to appear to be successful, to be better than others, and to avoid failure. They are not likely to expend effort on tasks especially when they are difficult as the very fact of having to make an effort is tantamount to an admission of a lack of ability (Covington & Omelich, 1979). They are also more likely to believe that ability is fixed (Dweck, 1986, 2000; Pintrich & De Groot, 1990). Such students are motivated in the classroom only when their performance is being evaluated so that they choose tasks and expend effort accordingly (Stipek, 1996). In this study students’ task involvement and ego orientation goals in mathematics were measured with Your Feelings in Mathematics: A Questionnaire (Yates, 1997a, 2000b; Yates, Yates, & Lippett, 1993, 1995).

Explanatory Style It was hypothesized in this study that students’ characteristic optimistic or pessimistic explanations for the events in their lives, referred to as their explanatory style (Peterson & Seligman, 1984), might be related to their achievement in mathematics. Explanatory-style theory proposes that when people are faced with uncontrollable events, they characteristically advance positive or negative explanations for the causes of those events (Peterson & Seligman, 1984). Those students who view the world from an optimistic framework are likely to perceive positive events as being permanent rather than transient, personal rather than due to circumstances beyond their control, and pervasive rather than limited to the immediate context. For optimistic students negative events are viewed as temporary, external, and specific. By contrast, those students who construct causal explanations from a pessimistic framework see negative events as long lasting, due to their own ineptitudes, and global and positive events as short-term, unrelated to their actions, and context-specific. Since 1980, research has supported the theory that explanatory style is related to achievement in various domains including education, work, and sports (Schulman, 1995; Shatte, Reivich, Gillham, & Seligman, 1999). In the workplace people with an optimistic explanatory style have greater productivity than pessimists (Seligman & Schulman, 1986). The deleterious effects of a pessimistic style have been implicated also in studies of athletic performance and illness (see Seligman, 1990), academic performance (Peterson & Barrett, 1987), and final grade in algebra (Pierce & Henry, 1993). With school-aged students relationships have been found between their explanatory style and their general achievement (Nolen-Hoeksema, Girgus, & Seligman, 1986, 1992) as measured by the standardized California Achievement Test (California Test Bureau, 1982). Success and failure are highly salient in learning mathematics (Dweck & Licht, 1980). Students bring to the classroom a myriad of experiences that influence how they account for their successes and failures. Seligman (1990) has asserted

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that students’ optimistic or pessimistic explanatory frameworks are likely to exert an influence on their explanations for their successes and failures in school. Explanatory style interacts with achievement to create self-fulfilling prophecies, which either enhance or undermine performance (Schulman, 1995; Seligman, 1990). The explanations that individuals habitually make for their successes and failures lead to expectations that affect their reactions to future successes and failures. In turn, these expectations affect performance through a variety of behaviors. Individuals with a more optimistic explanatory style are more likely to take the initiative, persist under adversity, take risks, be decisive, engage in quality problem-solving strategies, and be more assertive (Schulman, 1995). Students with pessimistic cognitive frameworks are at risk for doing less well academically (Seligman, 1995). When they encounter negative events, they are likely to exhibit a constellation of helpless behaviors including cognitive deficits, passivity, sadness, lowered self-esteem, and lowered assertiveness and competitiveness (Nolen-Hoeksema, Girgus, & Seligman, 1986). While students’ attributions for success and failure in mathematics have been studied in relation to specific factors such as luck, ability, effort, and task difficulty (McLeod, 1992), relationships between students’ characteristic positive or negative causal attributions for events in their lives and their motivations toward, and achievement in, mathematics have not been examined empirically. Students have been found to have developed a characteristic manner or style of explaining the causes of everyday events in their lives by the age of eight or nine years (Nolen-Hoeksema & Girgus, 1995; Seligman, 1990). In general, explanatory style is seen to be fairly stable over the Piagetian period of concrete operations, although as students enter the fourth Piagetian stage of formal operations, explanatory style for positive events is seen to become somewhat more optimistic (Nolen-Hoeksema & Girgus, 1995). These developmental trends, established from research involving children in the United States, would suggest that any study involving the examination of the explanatory-style construct over time would need to include both children and adolescents. Furthermore, it would be important to consider the stability of explanatory style over time. Negative explanatory style has been related to students’ self-reported depression (Nolen-Hoeksema, Girgus, & Seligman, 1991). Several studies have indicated that children with more pessimistic explanatory styles give more depressed responses on self-report depression scales (Kaslow, Rehm, & Siegel, 1984; Nolen-Hoeksema, Girgus, & Seligman, 1986, 1992; Seligman et al., 1984) with the strength of the relationship between explanatory style and depression increasing with age (Nolen-Hoeksema, Girgus, & Seligman, 1992). Children with diagnosed depressive disorders have been found to have more pessimistic explanatory styles in comparison with nondepressed children (Asarnow, Carlson, & Guthrie, 1987; Kaslow, Rehm, & Siegel, 1984; McCauley, Mitchell, Burke, & Moss, 1988). When they recovered from an episode of depression, these children’s pessimistic outlook remained (Nolen-Hoeksema, Girgus, & Seligman, 1992). This depression was associated with lower achievement and chronic

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deficits in social skills, leading Nolen-Hoeksema, Girgus, and Seligman (1986, 1992) to conclude that school failure and poor peer interactions could convince children that bad events were indeed stable, global, and internally caused. Licht and Kistner (1986) found that students with learning difficulties entered a vicious cycle in which negative beliefs reciprocally interacted with academic failures. With respect to gender differences in depression, most studies of preadolescent children either reported no difference, or a tendency for boys to be somewhat more depressed than girls (Anderson, Williams, McGee, & Silva, 1987; Kashanie, Cantwell, Shekim, & Reid, 1982; Nolen-Hoeksema, Girgus, & Seligman, 1992). It was, therefore, essential to examine students’ self-reported depression in this study, particularly as the students entered adolescence.

Teacher Perceptions As students’ experiences in school are likely to be important determinants of both their motivation and achievement in mathematics, it was necessary to take some measure of their behavior in the classroom in this study. In their everyday interactions in the classroom, teachers are in a position to perceive students’ overt reactions to success and failure in mathematics. It was not practicable in this study to observe directly such a large number of students in their respective classrooms over a long period of time. Teachers were asked, therefore, to rate students on dimensions of learned helplessness and mastery orientation in their mathematics classroom with the Student Behavior Checklist (Fincham, Hokoda, & Sanders, 1989) and to give a single rating of achievement in mathematics. These ratings were taken in the second year of the study when some of the students had entered their first year of secondary school, when many more had changed to a different primary school, and almost all were taught mathematics by different teachers. In developing the Student Behavior Checklist that was used in this study, Fincham, Hodoka, and Sanders (1989) generated items that reflected the range of behaviors associated with learned helplessness and mastery orientation in previous research studies. The checklist contained 12 items measuring learned helplessness and 12 items measuring mastery orientation. In their study, Fincham et al. (1989) found that teachers were not only able to detect learned helplessness as rated by the Student Behavior Checklist, but that their ratings predicted achievement two years later. The same checklist was used by Nolen-Hoeksema, Girgus, and Seligman (1986, 1992), although in the 1992 study they used only the 12 learned helplessness items, which they renamed “achievement helplessness.” In this study it was also considered worthwhile to obtain an estimate of teachers’ perceptions of students’ achievement in mathematics. In a recent review of research, Hoge and Coladarci (1989) located 16 studies in which teachers’ judgments of their students’ academic performance were compared against actual

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scores on objective test measures. Across the studies the median correlation was 0.66, suggesting a strong correspondence between teacher judgments and student achievement. When the judgments of teachers were compared, Hoge and Coladarci (1989) noted that a number of studies indicated large variations among individual teachers. Moreover, they reported that the accuracy of teacher judgments appeared to be relatively higher in the case of judgments made on average to above average ability students. Teacher ratings of academic brightness have been found to be significantly correlated with examination success five years later (Kenealy, Frude, & Shaw, 1991). The data from several studies suggested that teachers achieved a success rate of around 70 percent accuracy when asked to assess whether individual students were able to succeed on specific test items. In a review of 42 studies, Follman (1990) found that although the correlations ranged from about 0.10 to 0.90, the best estimate of the correlation between teachers’ estimates of students’ achievement and their actual scores on standardized achievement tests was 0.50.

Summary In light of the available research literature, and taking into account the concerns of the primary school principal about factors influencing students’ achievement in mathematics, it was of considerable interest to investigate possible relationships between students’ explanatory style or characteristic optimistic or pessimistic mental framework and their goal orientations and achievement in mathematics. These relationships needed to be explored longitudinally as students’ beliefs and attitudes develop over a relatively long period of time (McLeod, 1989, 1992). It was hypothesized that students with more optimistic explanatory styles would be more likely to be task oriented and to have higher achievement in mathematics. By contrast, pessimism was more likely to be associated with ego-oriented goals. Students who had developed a pessimistic cognitive framework would not perceive failure as part of the fabric of learning mathematics but were likely to recast it as permanent, personal, and pervasive, leading them to expect further negative outcomes and setting up a vicious circle. As many studies had indicated a strong relationship between pessimism and depression, it was necessary to include a measure of students’ self-reported depression. Grade level and gender differences also needed to be taken into account. Teachers’ observations of the students in the mathematics classroom and their ratings of the student achievement in mathematics were also essential data in this study.

RESEARCH DESIGN Considerations of relationships between students’ explanatory style, depression, motivation, and achievement in mathematics and the impact of these factors on

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their classroom behavior and achievement in mathematics as perceived by teachers have guided the design of this study. The longitudinal design involving children and adolescents facilitated a thorough investigation of the following:

. The relationship of explanatory style to motivational goals . The relationship of explanatory style to achievement in mathematics . Teacher perceptions of students’ achievement and behavior in the classroom . The interrelationships among teacher perceptions and explanatory style, depression, goal orientations, and achievement in mathematics

In this study motivation was measured in terms of task involvement and ego orientation goal constructs.

Research Plan It was necessary to devise a study in which the interrelatedness among explanatory style, motivational goal orientation, and mathematics achievement could be explored in a large group of school children over a period of time. Quantitative research methods were most appropriate for these investigations, particularly in view of the relatively long time span of the study and the size of the student sample. The use of a longitudinal design facilitated an examination of relative gain in mathematics achievement over time and the factors that were hypothesized to influence this gain as an index of change over time. Such a longitudinal design also enabled cross-sectional comparisons, both across age levels as well as between the genders, to be undertaken. Ratings from classroom teachers were planned for the second year of the study. This research took place over three years beginning in term 1 in the first year and finishing in term 4 at the end of the third year. The factors measured and the timing of these factors within the study are presented in figure 7.1. The factors have been shown in temporal order from left to right, after the listing of the antecedent variables of gender, the students’ grade level, and the primary school attended at the commencement of the study. The initial measurements of students’ explanatory style, goal orientations, and achievement in mathematics on the left (referred to as Time 1) are mirrored on the right with the same measurements taken at the end of the third year (referred to as Time 3), with the additional factor of depression considered at Time 3. The teacher ratings taken in the intervening year (referred to as Time 2) are in the center of the diagram. By its very nature longitudinal research entails the follow up of a sufficiently large group of students over a considerable period of time (Keeves, 1994). At the commencement of the study all students who participated in the study were in grades 3 to 7 in two metropolitan primary schools in South Australia, but by the end of the time period, almost half had moved to secondary schools. Therefore,

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Figure 7.1 An Overview of the Longitudinal Plan of the Study

in setting up this plan it was essential at the outset not only to select measures that would be applicable to such a wide age range of students and which could be repeated on separate occasions, but also to consider the methods of analysis that would be used. An Australian calibrated test, the Progressive Achievement Tests in Mathematics (The Australian Council for Educational Research ACER, 1984), suitable for all students from grades 3 to 9, with different forms of the test for different age and grade groups, had been administered to all students in the first school in term 1. As this test had been Rasch scaled, the performance of students across grade levels and over time could be located on a single scale of achievement. The Rasch scaling procedure was also applied to all of the measures employed in the study so as to provide the properties of an interval scale for each instrument. This was a particularly important aspect of this study as it addressed one of the major difficulties associated with measurement of change over time in longitudinal studies. Furthermore, with Rasch scaling it is possible for items to be omitted, included, or excluded across the grade levels so that all students are compared on the same scale even if they have not answered the same items. The Rasch measurement procedure is discussed later in greater detail. In order to investigate relative change over time and to determine which factors were responsible for shifts in performance between the two occasions, it was also necessary to develop and test a causal model using the PLSPATH procedure (Sellin & Keeves, 1997). The essential feature of path analysis is that it provides the researcher with a vehicle for exploring or testing theories about causal relationships between sets of variables. A model is developed between the theoretical constructs of interest with the resultant path diagram presenting these hypotheses as a model that can be rigorously tested and the magnitude of the causal relationships estimated (Klem, 1995). In a longitudinal study, the advantage of using a causal model is that it can be tested against the observed data,

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the parameters of the model estimated, and the adequacy of the model assessed. With the PLSPATH procedure the size of the causal paths could be estimated so that the magnitudes of both the direct and indirect effects of explanatory style on motivation toward, and achievement in, mathematics could be obtained. The causal relationships of depression to both explanatory style and goal orientation and achievement were also considered. Furthermore, the use of path analysis enabled the relationships between teachers’ ratings and students’ explanatory style, depression, goal orientation and achievement in mathematics to be examined over time (Yates, 1998a, 2000a). Students’ grade level, gender, and school were also taken into account in these analyses. The causal model is presented in figure 7.2 on page 180 and discussed more fully later.

Factors in the Design of the Study Longitudinal research is complex and time consuming as the same students need to be followed over time with at least two but preferably more data collection points (Keeves, 1997c). The conduct of this study was shaped by a number of factors including the nature and size of the student sample, the research instruments selected, ethical considerations, available funding, and the methods of data analysis employed.

The Nature and Size of the Sample The study commenced with students in grades 4, 6, and 7 in one government primary school and was replicated in a second government primary school with all students in grades 3 to 7, giving a total sample of 335. In the second year, 258 of these students in grades 4 to 8 were traced to 31 schools where they were rated by 58 teachers. By the third year of the study, the 243 students in the final sample were located in 26 primary and 24 lower secondary schools in both the government and nongovernment sectors. The gender and grade levels of the final sample are presented in table 7.1. While every attempt was made to keep track of the students at Time 2 and Time 3, natural attrition occurred through student movement, nonparticipation, and parental failure to return consent forms. Furthermore, some teachers did not complete the rating scale in term 4 (Time 2). Although the retention rate of the final sample was less than ideal, the 243 students for whom complete data were available was sufficient for all statistical analyses. The Rasch scaling of the questionnaires was undertaken with all the students for whom data were available while the relational and causal analyses were conducted only on the 243 students.

Selection of the Research Instruments In the selection of the research instruments, it was necessary to take into account the longitudinal design of the study, the factors to be studied, the nature, range, and locations of the student sample, and the psychometric properties of

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Table 7.1 Numbers of Students by Grade Level and Gender at Time 1 and Time 3

each instrument. While the achievement test was administered under timed, standardized conditions, the amount of time that students and teachers respectively would need to complete the questionnaires was also a deciding factor. The following questionnaires and tests were administered over the three years of the study: Time 1

. . .

Children’s Attributional Style Questionnaire (Seligman et al., 1984) Your Feelings in Mathematics: A Questionnaire (Yates, Yates, & Lippett, 1995) The Progressive Achievement Tests in Mathematics Form A, Tests 1, 2, or 3. (ACER, 1984)

Time 2

.

The Student Behavior Checklist (Fincham, Hodoka, & Sanders, 1989)

Time 3

. . . .

Children’s Attributional Style Questionnaire (Seligman et al., 1984) Children’s Depression Inventory (Kovacs, 1992) Your Feelings in Mathematics: A Questionnaire (Yates, Yates, & Lippett, 1995) The Progressive Achievement Tests in Mathematics Form A, Tests 1, 2 or 3. (ACER, 1984)

These instruments are considered now in detail.

Achievement in Mathematics The problem of selecting a suitable mathematics achievement scale that could be used for such a wide age range of students yet which yielded scores that could be compared meaningfully across time as well as grade levels was resolved when it was evident that the Progressive Achievement Tests in Mathematics (PAT-

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Maths) (ACER, 1984) had been administered already to all students in the first school in term 1, Time 1. This instrument has three different tests suitable for the range of grade levels of the student sample. All three tests contain items measuring number, computation, measurement, money, statistics, spatial relations, and graphs. Fractions are included in Tests 1 and 2, logic and sets are introduced in Test 2, and relations and functions are added to Test 3. Within each test, items are arranged in content areas and in order of increasing difficulty as determined by the Rasch analysis of the responses from the Australian calibration sample tested in November 1983 (Teachers Handbook, ACER, 1984). The Rasch scaled scores published in the Teachers Handbook allow for all student raw scores to be placed on a single scale, irrespective of the level of the test and the time at which the test was administered. The PATMaths were adapted and calibrated for Australian schools by the Australian Council for Educational Research (ACER, 1984). Reliability coefficients, determined by a Kuder-Richardson reliability coefficient (KR 20), were reported as relatively high (see Teachers Handbook, 1984). These KR 20 coefficients were also cited as evidence that test scores were satisfactorily stable. Validity was established from professional opinion, the Rasch item calibration procedures, and from the regular and marked increase in achievement from one grade level to the next. These increments indicated that the tests measured abilities that were sequential and that developed from both years of instruction in mathematics and exposure to an increasing range of mathematical ideas and processes. A decision was therefore made to use the PATMaths in the second school at Time 1 and to continue with the same tests at Time 3.

Explanatory Style It was evident from the published research that the Children’s Attributional Style Questionnaire (CASQ) (Seligman et al., 1984), a forced-choice, penciland-paper instrument, was the most appropriate measure of optimism and pessimism for the student sample. The psychometric properties of the CASQ have been investigated with classical test theory, with adequate measures of validity and indices of internal consistency and test-retest reliability reported (for details see, Nolen-Hoeksema, Girgus, & Seligman, 1986, 1992; Panak & Garber, 1992; Seligman et al., 1984). The questionnaire was readily available (Seligman, 1990) and as a nonstandardized measure could be administered either to individuals or groups of students on both occasions. While the CASQ had been designed for children between the ages of 8 years and 14 years, no upper age limit had been established for its use. The CASQ consists of 24 positive and 24 negative items of hypothetical events involving the child, followed by two possible explanations. Typical items include

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(2) You play a game with some friends, and you win. (a) The people I played with did not play the game well. (b) I play that game well. (21) You do a project with a group of kids, and it turns out badly. (a) I don’t work well with the people in the group. (b) I never work well with a group.

It is scored by the assignment of 1 to each internal or stable or global response, and a 0 to each external or unstable or specific response. Scales are commonly formed for composite positive (CP) events and composite negative (CN) events separately (Peterson, Maier, & Seligman, 1993) as well as a composite total score (CPCN) formed by taking the composite negative from the composite positive score (Nolen-Hoeksema, Girgus, & Seligman, 1986).

Motivation Your Feelings in Mathematics: A Questionnaire, a 25-item variant of the Motivation Orientation Scales (Duda & Nicholls, 1992; Nicholls, Cobb, Wood, Yackel, & Patashnick, 1990), was designed specifically for this study to assess students’ dispositions toward task involvement and ego orientation goals in mathematics. These two attitudinal goal orientations were determined by factor analytic studies as independent dimensions of both personal academic goals and beliefs about the causes of school success (Nicholls et al., 1989; Nicholls et al., 1990). In this questionnaire 15 of the 25 items measured task involvement, 6 items measured ego orientation, with the remaining 4 designated as filler items. Each item commenced with the stem Do you really feel pleased in maths when . . . followed by a statement that related to student mathematics behavior. Typical task involvement items were (1) you get really busy with the work. (15) something you learn makes you want to find out more.

Ego orientation in mathematics items included (4) you know more than the others. (23) you score better on a test than others.

Students circled the rating that most closely approximated their feeling about the situation presented in the item on a five-point Likert-type scale ranging from a strong yes to a strong no. Items were coded from 1 to 5, with a 5 being allocated for a strong yes through to a 1 for a strong no.

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Depression The Children’s Depression Inventory (CDI) (Kovacs, 1992) was chosen as the preferred index of depression for the third year of the study since it was appropriate for the age range of the student sample, had satisfactory psychometric qualities (see, Kovacs, 1992, The Children’s Depression Inventory Manual), and had been used most often in conjunction with the explanatory-style measure. It was also considered to be sensitive to changes in depression over time and has been used extensively as a routine screening device with both normal and clinical child groups (Kovacs, 1992). Since its initial development in 1977, the psychometric properties of the CDI have been examined extensively with classical test theory. While the validity of the instrument was not investigated with the normative standardization sample (Finch, Saylor, & Edwards, 1985), a voluminous literature has attested to the strong explanatory and predictive utility of the instrument (Kovacs, 1992). The Manual for the CDI also reported an acceptable level of stability (Saylor et al., 1984; Saylor, Finch, Cassel, Saylor, & Penberthy, 1984; Smucker, Craighead, Craighead, & Green, 1986). This inventory, suitable for administration in either individual or group settings, was developed as a self-rating, symptom-orientated scale for school age children and adolescents aged 7 to 17 years (Kovacs, 1992). It consists of 27 items covering a range of depression symptoms. For this study the questionnaire comprised 26 items, as item 9 concerning suicide ideation, deemed inappropriate for the student sample, was omitted. It was administered to students as an Attitude Survey since this title was thought to be less anxiety provoking than the original title. For each of the 26 items, students rated the one of three statements that described them best for the past two weeks. The statements presented contexts with which students were likely to be familiar, with the ratings ranging from an absence of the symptom, through a mild symptom, to a definite symptom. About half the items started with a choice that represented the greatest symptom severity while in the remainder of the items the sequence of choices was reversed. The items were scored as 0 for the absence of symptom, 1 for a mild symptom, and 2 a definite symptom.

Teacher Rating Scale Teachers rated students’ behavior in their mathematics classrooms with the Student Behavior Checklist (Fincham, Hodoka, & Sanders, 1989). This instrument, designed to measure learned helplessness and mastery orientation in the classroom, was selected because it provided a means of sampling the behaviors of interest in the study in the students’ own classrooms, had been used previously in related research (Nolen-Hoeksema, Girgus, & Seligman, 1986, 1992), and was not too time consuming for teachers to complete. In developing the checklist, Fincham, Hodaka, and Sanders (1989) generated items that reflected

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the range of behaviors associated with learned helplessness and mastery orientation in previous research studies. Thus, by their very nature, the items reflected student characteristics that were directly observable by teachers, rather than being inferred from an internal state as measured in student self-reports. Fincham, Hodoka, and Sanders, (1989) reported that although the learned helplessness and mastery orientation subscales were highly correlated (r  –0.81), the psychometric robustness of the instrument had yet to be established. Furthermore, Fincham et al. (1989) raised the issue as to whether the scales specifically measured learned helplessness and mastery orientation or whether they reflected academic competence. The written instructions for the completion of the checklist requested the teacher to consider the nominated child over the last two or three months and, for each of the 24 items, circle the number that indicated how true that description was of that child. Sample statements included (2) Expresses enthusiasm about his/her work. (23) When s/he encounters an obstacle in schoolwork s/he gets discouraged and stops trying. S/he is easily frustrated.

The ratings were made on a five-point scale with 1 designated not true, 3 described as somewhat or sometimes true, and 5 as very true. Teachers were asked to read the items carefully as they were directed toward several different aspects of the child’s behavior. Teachers also gave a single rating of the students’ achievement in mathematics on a five-point scale that ranged from 1 (excellent) through 3 (average) to 5 (poor). Completed questionnaires were returned by post.

Ethical Considerations Ethical principles are an essential consideration in any study involving human participants. It is incumbent upon researchers to design a study in which the principles of integrity, a respect for persons, beneficence, and justice are exemplified (National Health and Medical Research Council, 1999). There needs to be relevant consultation, informed consent from the participants (or their parents or guardians), a maintenance of confidentiality and anonymity, harm minimization, and provisions made for appropriate retention and long-term storage of information and publication of the information gathered and methods used. Researchers must ensure that the study is not only conducted in accordance with these principles but that the personnel involved in the data collection are appropriately qualified and supervised. Approval for this study was obtained from both Flinders University and the South Australian Department of Education, Training and Employment Ethics Committees. Permission for students to take part in the study was obtained in writing from the parents or guardians. Teachers also obtained consent from

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parents or guardians prior to completing the Student Behavior Checklist for each student. Letters were sent home to parents or guardians outlining the nature and purposes of the study and specifying the voluntary nature of the student’s participation. The aims and objectives of the study were also explained verbally to the students by the university researcher prior to their participation. At all times the students were informed that they were free to withdraw from the study or not to answer any questions if they wished. Students were assured of the confidentiality and anonymity of their answers and, in particular, that the information they provided for the research would not be divulged to their school at any time. These procedures were followed for all aspects of the study except for the measure of achievement in mathematics in the primary school in which the study originated. At the commencement of the first term 1, as part of their schoolwide assessment procedures, the school administered annually the PATMaths tests to all primary school students. Scores were, therefore, obtained for Time 1 and Time 3 for each student involved in the study from school records. In order to follow the students over the three-year period, it was necessary to keep confidential records of their names, gender, date of birth, grade level, and current school. All students were given an ID number, which was used for data entry and matching on an SPSS file (Norusis, 1993). Procedures were put in place to ensure the confidentiality and security of the data file, test record forms, and questionnaires. While the students were followed individually, the data were analyzed only at the group level. At the conclusion of the study, permission to publish the results was obtained in writing from the Department of Education, Training and Employment.

Funding Longitudinal studies are time consuming and relatively costly as each student must be tracked individually. While some funding was available from the Flinders University Research Board, the resources were not sufficient for students to be followed up once they moved to other Australian states or overseas. As the students were located in 50 different schools by the third year of the study, and as the instruments were administered to them in their own schools, time was also a major consideration. It was not feasible to observe or interview such a sizeable group of students spread over a large number of schools.

Methods of Data Collection: Student Data Longitudinal data were collected on 243 students initially drawn from two primary schools but spread over 26 primary and 24 secondary schools almost

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three years later. The achievement tests and questionnaires were administered to students in their schools during normal school hours by a university researcher except in the first school where the test of mathematics achievement was administered by teachers. In both the first and third year the achievement test was administered first, followed by the Children’s Attributional Style Questionnaire, Your Feelings in Mathematics: A Questionnaire, and in the third year only, the Children’s Depression Inventory. Together these measures took no more than 90 minutes to administer.

Teacher Data Teachers rated the students’ classroom behavior in the intervening year with a checklist and gave a single rating of their achievement in mathematics. While some of the students were located in grade 8 in secondary schools, many had moved to different primary schools. The principals of these 31 schools were contacted initially by telephone. The checklist was then posted to the school together with separate letters to the principal and teacher. The letters confirmed the invitation to take part in the study and outlined the purpose of the study and the intent of the Student Behavior Checklist. The voluntary nature of participation was stressed and confidentiality of the information assured. Students were spread across a number of schools at Time 2, and as the funds available covered only the costs of printing and postage, it was not possible to provide direct training to teachers in the administration of the questionnaire. Clear instructions were printed on the checklist, and the teachers were aware of the purposes of the study through the initial telephone call to the school and the accompanying letter. Teachers also needed to obtain parental permission in writing before completing the questionnaire.

METHODOLOGICAL ISSUES Representativeness of the Student Sample As the study arose in response to the request from a primary school principal, the sample of students was neither randomly selected nor representative of the population. It would have been preferable to use a “probability proportional to size” sample of schools at the first stage of sampling and a simple random sample of 25 students at the second stage from a defined target population (Ross & Rust, 1997). However, the use of the two schools in the design of this study did allow for replication. Allowances were made for differences between the two schools in the path analysis.

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Test-Retest Effects The study commenced at Time 1 in the first term and concluded at Time 3 in term 4, with the administrations of the test of mathematics achievement, goal orientation, and explanatory style measures separated by at least 2 years 8 months. With the exception of the Children’s Depression Inventory, which was administered at Time 3 only, the same attitudinal instruments were employed on the two occasions. The length of time between these administrations should have minimized any practice effects from the first administration, particularly as the students were not provided with any results from the study. However, it was also necessary to consider the stability of the measures over such a long time span with interclass and intraclass correlation coefficients.

Scoring Differences between the Achievement and Questionnaire Variables Dichotomous scoring was used for the Children’s Attributional Style Questionnaire. The Feelings in Mathematics: A Questionnaire and the Children’s Depression Inventory employed Likert-type scales, while the Progressive Achievement Tests in Mathematics contained either four- or five-option multiple-choice items. Since the scales associated with these instruments had no predetermined metric, it was necessary to form common scales over time in order to avoid the interpretation of the different arbitrary metrics (Sellin, 1990). This was achieved in this study through the use of the one-parameter model of item response theory referred to as the Rasch model (Rasch, 1966, 1960/1980). Each of the scales was calibrated with the Rasch procedure in order to bring them to common scales with interval properties and a natural logistic metric, with the exception of the Progressive Achievement Tests in Mathematics that had been already Rasch calibrated (ACER, 1984).

Trustworthiness of the Findings The validity and trustworthiness of the findings of the study ultimately rest in their meaningfulness and their consistency with previous research that was reviewed before the study was undertaken. Each instrument contributes to the reliability of the scores obtained, and the scores are interrelated through the causal model developed. The strengths of the relationships within the causal model are, as a consequence, the major findings that must be assessed at the end of the study, not only for trustworthiness but also for utility, coherence, simplicity, and fruitfulness in giving rise to further investigation (Quine & Ullian, 1978).

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DATA ANALYSIS In addition to the use of conventional parametric statistics, the data were scaled with the Rasch measurement procedure (Hambleton, 1989; Kline, 1993; Wright, 1988). In this study, students were administered instruments that were designed to measure specific attitudinal and achievement variables. Wright and Masters (1981) have set out seven criteria by which any quantitative measurement scale should be evaluated. First, each of the items within a scale must function as intended. Second, the relative position or difficulty of each item within the scale that is the same for all participants must be estimated. Third, the responses of each participant must be checked to ensure that they form a valid response pattern. Fourth, each participant’s relative score must be estimated. Fifth, the participant scores and the item scores must fit together on a common scale defined by the items, and they must share a constant interval throughout the scale such that they mark off the scale in a linear fashion. Sixth, standard errors should accompany the numerical values as an indication of the precision of the measurements in the scale. Lastly, items should remain relatively similar in their function and meaning across individual participants and groups of participants so that they are seen as stable and useful measures. To meet these criteria, the Rasch measurement model was employed for each of the instruments used in this study.

Rasch Analysis A more detailed discussion of the Rasch model has been provided in the appendix. The use of the Rasch model as the method of determining the scalability of the Children’s Attributional Style Questionnaire (Yates & Afrassa, 1994; Yates, Keeves, & Afrassa, in press), the Children’s Depression Inventory (Yates, 1998c), Your Feelings in Mathematics: A Questionnaire (Yates & Yates, 1996), and the Student Behavior Checklist (Yates & Afrassa, 1995) allowed for the calibration of the items and the scales independently of the sample of students and for the measurement of motivation and performance in mathematics independently of the sample of items employed (Wright & Stone, 1979). With the exception of Your Feelings in Mathematics: A Questionnaire, the psychometric properties of the questionnaires used in this study had been investigated in previous studies with classical test theory methods. While the reported and published indices of the validity and reliability of these instruments were useful in determining their suitability for inclusion in the study, they were limited as their estimation was dependent upon the samples of children who took the questionnaires (Hambleton, 1989; Hambleton & Swaminathan, 1985; Osterlind, 1983; Weiss & Yoes, 1991; Wright, 1988). Similarly, information on items within the questionnaires was not sample free, with the various composite scores being calculated solely from the number of correct items answered by subjects. With classical test theory, estimates of item difficulty, item discrimination, item qual-

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ity, and the spread of the subjects’ ability levels associated with raw scores are mathematically confounded (Snyder & Sheehan, 1992). The use of Rasch scaling procedures, based on item response theory, addresses the shortcomings of classical test theory at the same time as bringing each of the instruments employed in the study on different occasions to common interval scales. Student scores and the items within each of the instruments were analyzed together on the same scale, but independently of each other, and the data compared across the instruments and over time. In any longitudinal study in which multiple indicators are measured across time, it is also necessary to choose a method in which missing data and changes in the instruments can be handled effectively.

Scoring of Student Responses Data quality has been found to be related to the educational level of subjects (Alwin & Krosnick, 1991; Craig & McCann, 1978; Downs & Kerr, 1986; Ferber, 1966–67), age (Downs & Kerr, 1986; Kaldenberg, Koenig, & Becker, 1994), and gender (Downs & Kerr, 1986; Taylor, 1976). Quality has also been found to be a function of item sensitivity (Marquis, Marquis, & Polich, 1986), item order (Converse & Presser, 1986), item wording and ambiguity (see Clark & Schober, 1992), with Davis and Jowell (1989) cautioning against single-item measures. Rasch modeling has an advantage over both these subject- and item-related problems of classical test theory as not only could item characteristics be examined independently of the subjects, but the individual response patterns of students that varied substantially from the mathematical model could also be detected (Snyder & Sheehan, 1992). Essentially, the Rasch scaling procedure provides an estimate of student performance or attitude based on the probability of responses to the items within each instrument. The Rasch model assumes that guessing should not be a factor in students’ responses to items (Hambleton & Swaminathan, 1985; Lord, 1980; Scheuneman, 1979) as guessing is a characteristic of individuals rather than that of the items (Hambleton & Swaminathan, 1985; Skaggs & Lissitz, 1986). As this study commenced with students in two government primary schools, it was possible that some students had had little exposure to either formal achievement testing or self-report questionnaires. Furthermore, as students voluntarily took part in the study, the low-stakes nature of the data collection should have ensured that guessing would not be a major problem. After the items have been reviewed, and the items that fitted the Rasch model retained, estimates of student fit to the Rasch model were determined. This analysis would detect if students had provided incorrect but consistent responses, and the pattern of student responses would not fit the model if they had lied, cheated, did not take the task seriously, acted with a response set, or were inconsistent in their responding (Green, 1996). Because of the long period of time

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between the two administrations of the instruments to the participants in this study, it was considered that a test-retest or response set was unlikely to be a factor. The low-stakes nature of the testing and completion of the questionnaires was likely to have reduced the propensity toward untruthful responses. Thus, analyses of the variability in participant response patterns could be undertaken, both across time for each student and across all students, with any aberrant patterns readily discernible. It should be noted that students who responded with all items correct or favorable and with all items incorrect or unfavorable were automatically excluded from the calibration of the scales. However, scores for such students were calculated by extrapolation from a logit table. With Rasch modeling, scores from parallel forms of a test or from the same measure administered on different occasions can be placed on a single scale thus facilitating comparisons of performance across time (Stocking, 1994). As the students in this study had taken the same measures of explanatory style, achievement in, and goal orientation toward mathematics in both the first and third year of the study, it was also necessary to equate their performance across time. In this study concurrent equating was employed for the scoring of the responses of the 243 students to the CASQ and the Feelings in Mathematics: A Questionnaire, as complete data were available for these students from the administrations of these instruments at both Time 1 and Time 3. Concurrent equating involved pooling of the data as this method has been found to yield stronger case estimates than equating based on anchor item equating methods (Mahondas, 1996; Morrison & Fitzpatrick, 1992). All of the statistical analyses of the data were based on the Rasch case estimate scores.

PLSPATH Analysis with Latent Variables In quantitative studies, complex sets of interrelationships between variables of interest to the researcher can be represented mathematically and in the form of a diagram or model which is then subjected to rigorous statistical analyses (Tuijnman & Keeves, 1997). As was the case with this study, the causal model is usually drawn prior to the data collection phase of a study and represents the variables to be observed or measured, their underlying theoretical perspective(s), and the hypothesized interrelationships between them. The process of determining a model is particularly important in longitudinal studies where variables are measured across time and where explanations of change are central to the study. Relationships between the variables in a model can be explored statistically, both directly and indirectly, allowing the researcher to take into account the influence of more than one variable upon another and of their possible mediating effect(s). The analytical techniques used to explore these relationships allow for a more thorough and comprehensive analysis of the interrelationships of the variables than would be possible with correlational and multiple regression analyses alone.

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Causal modeling, sometimes referred to as path analysis (Falk, 1987), was ideal for this study. It allowed for the testing of the hypothesized relationships between the variables that had been measured over the three years and the theoretical constructs on which they were based. Furthermore, estimation of the magnitude and direction of these causal relationships served to advance the theoretical understandings of the development of explanatory style and goal orientation in children and of their relationships to each other and to achievement in mathematics. As the initial selection of the sample from the two primary schools was not random, it was necessary to identify particular statistical modeling methods that would control for variables that might confound the patterns of covariation observed between variables (Tuijnman & Keeves, 1997). The size of the final sample was also a consideration. Path analysis with latent variables was chosen as the most appropriate technique for this study as it is flexible (Falk, 1987), simple, quick in computation, not dependent upon large samples, makes no assumptions about the shape and underlying distributions of the observed and latent variables, and does not employ significance tests that have strong assumptions of normality in the distributions (Sellin, 1986, 1990; Sellin & Keeves, 1997). The flexibility inherent in path analysis means that both dichotomous and continuous variables can be considered within the same model (Tuijnman & Keeves, 1997). It is also efficient, as empirical or manifest variables are clustered and linked in terms of their commonalities to underlying latent variables, with the extent to which each contributes to that higher-order theoretical construct (Falk, 1987) tested thoroughly. Not only are the number of manifest variables reduced in the analysis, but the magnitude of relationships between the actual theoretical or latent variables that were hypothesized to underlie the observed measures can be estimated rigorously (Sellin & Keeves, 1997). For this study, the path analysis was carried out with PLSPATH (Sellin, 1990), with the PLS referring to the partial least squares regression analyses (Wold, 1982) used by the computer program to explain the variance and maximize prediction in the model. The fact that path analysis with latent variables does not rely on the assumption of the multivariate normal distribution of the variables (Tuijnman & Keeves, 1997) made it an ideal choice of method for this study. However, the use of conventional tests of statistical significance to estimate the magnitude of effects is necessarily precluded in the path analysis because this assumption is not made. As an alternative, the PLSPATH procedure utilizes a jackknife technique (Tukey, 1977) in which each case is progressively left out in the computation of the coefficients for the group. The use of the jackknife technique has the advantage of giving a more realistic estimate of the relationships of the predictors to the endogenous variables as it takes into account the bias that might have accrued from any one case (Tabachnick & Fidell, 1996). In addition, the jackknife procedure enables the standard errors of the path coefficient to be estimated (Sellin & Keeves, 1997). As this study was longitudinal in nature, the time sequence was of particular relevance, both in terms of the temporal sequence of the data and the predictive

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nature of the relationships to be investigated. Not only was it important to investigate the more precise nature of the causal relationships between the measures of explanatory style over time, but also to consider whether depression was predicted by either the proximal or distal measures of explanatory style or both, as well as to look at any gender differences prior to and during the onset of adolescence. Furthermore, path analysis enabled the explorations of any associations between explanatory style and depression with academic achievement in the area of mathematics, as well as taking into account any influences from student motivation toward mathematics and teacher perceptions of classroom academic behavior and achievement. Thus this study brought together information regarding the short- and long-term impact of motivational goal orientations, as measured through task involvement and ego orientation, on both explanatory style and achievement in mathematics by clarifying the causal relationships between teacher ratings and student explanatory style, depression, and goals toward, and achievement in, mathematics over time. In the path diagram presented in figure 7.2 the observed or manifest variables, which form the outer model, are represented diagrammatically by rectangles, while the inner-model latent variables are represented as ellipses. The small circles with arrows leading into each endogenous latent variable are the residuals or amount of unexplained variance. The latent variables were determined on the basis of a study of the research literature and on the central hypotheses to be explored. Within this study, 11 latent variables, presented in table 7.2, were specified and numbered in the order in which they would enter the analysis. The temporal sequence of the data was also taken into account in the ordering of the variables with the Time 1 (T1) variables preceding the Teacher Judgments variable at Time 2 (T2), which in turn preceded the Time 3 (T3) variables. The model in figure 7.2 is recursive with the latent variables in an ordered sequence and single-headed arrows indicating that there were no reciprocal relationships between the latent variables. By convention, the causal flow in the latent variables in a path model diagram is from left to right, with the causal relationships being shown by single-headed arrows (Klem, 1995). It is the task of the researcher to determine whether the latent variables are endogenous or exogenous. In figure 7.2 latent variables 4 to 11 are shown with incoming arrow heads indicating that as endogenous variables, they could be affected by one or more other variables in the model. The Gender (1), T1 Student Grade (2) and T1 School Attended (3) variables, however, were specified as exogenous variables, with their lack of incoming arrow heads indicating that they functioned as antecedent causes and not as effects. Exogenous variables have values that may be influenced by variables that are not in the model (Klem, 1995). Once the manifest and latent variables to be considered in the model were determined and specified, relationships between all possible paths in the model were estimated by the PLSPATH computer program. The model was then progressively trimmed, with trivial paths deleted and only the significant paths retained.

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Figure 7.2 Significant Paths in the PLSPATH Model

Figure 7.2 presents the final model developed from this process of refinement, with the significant relationships between the observed variables and the latent variables in the outer model, and between the 11 latent variables within the inner model depicted. Paths in the outer model were considered significant and were from the latent-value ellipse to the observed variable in the rectangle, were at least .30, or more stringently .55 (Sellin & Keeves, 1997). Inner model paths were retained if their coefficients, written along each arrow line in figure 7.2 between latent variables, were less than .1. These values are accompanied by a standard error score, which appears in parentheses alongside the coefficient in each case. Each coefficient should be at least twice its respective standard error for the retention of the path. As a recursive model, the path coefficients in the inner model are standardized regression coefficients that are related to the amount of variance explained by a construct when other significant constructs have been taken into account. The values on each factor loading in the outer model and each path coefficient in the inner model, respectively, were then used to describe and discuss the significant relationships between the variables. However, not all information about the model can be gleaned by visual inspection of the path coefficients presented in figure 7.2 alone. In addition to the direct effects presented by the values along the significant paths in the inner

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Table 7.2 Latent Variable Order and Acronyms for the PLSPATH Analysis

model, PLSPATH provides the researcher with information about indirect effects that can be added to the direct effects to give an overall total effect, sometimes referred to as an effect coefficient (Tuijnman & Keeves, 1997). The computer program essentially derives the values of these indirect effects by tracing them through the significant pathways that lie between the variables of interest. In figure 7.2, for example, there is a direct path of medium magnitude (.34) between T1 Explanatory Style and T3 Explanatory Style. However, there is also a significant direct path (–.15) between T1 Explanatory Style and T1 Maths Motivation, which in turn is linked directly (–.15) to T3 Explanatory Style. Thus T1 Maths Motivation serves to mediate the effects of Explanatory Style indirectly from Time 1 to Time 3. This indirect effect of T1 Maths Motivation, with a value calculated by PLSPATH as (.02), strengthened the direct path presented in the model, giving a total effect coefficient of .37. T1 Explanatory Style was influenced directly by the Gender (–.21) and T1 Student Grade (.16) of the students with boys (coded as 1) being more pessimistic than girls (coded as 2). Moreover, students in later year levels had more negative explanatory styles although these influences were weak. Using a similar process of tracing through all relevant significant paths, the PLSPATH program indicated that the Gender and T1 Student Grade

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variables exerted only indirect effects (.06 and –.08, respectively) on T3 Explanatory Style, operating through T1 Explanatory Style and T1 Maths Motivation. In this manner, direct, indirect, and total effects can be explored between all of the latent variables in the inner model. The numerical values for the direct, indirect, and total effects are presented to the researcher as part of the PLSPATH output (see Yates, 1998a) but have not been discussed in detail in this chapter because of limitations of space. The total variance in achievement in mathematics explained by the model, as presented by the R2 value in figure 7.2, was 0.62, indicating that the model fitted the data well. Clearly the variables measured did account for much of the variance associated with achievement in mathematics, enabling the relative contribution of explanatory style to be discerned. The Q2 value of 0.60 also demonstrated that the model was stable with respect to the data, taking into account the variability associated with individual cases. The strength and stability of the model was further supported by the use of the jackknife procedure in the model estimation process, which reduced the need to replicate the analysis through the use of the half cross-sample validation procedure (Sellin & Keeves, 1997).

DISCUSSION This study set out to examine, using a quantitative framework, relationships between the optimistic or pessimistic explanatory style of a sample of South Australian students in primary and lower secondary schools and their motivation toward, and achievement in, mathematics over time. Students’ achievement in mathematics was measured with a calibrated test of achievement, while data on their explanatory style and goal orientations toward mathematics were gained from pencil-and-paper questionnaires on two occasions separated by almost three years. On the second occasion a measure of self-reported depression was also obtained. In the intervening year teachers rated the behavior of the students in their mathematics classrooms as well as their achievement. These students’ attitudinal self-reports, the calibrated achievement measure, and teachers’ ratings accounted for a substantial proportion of the variance associated with achievement in mathematics. The principal focus of this chapter has been on the use of specific quantitative research methods rather than a detailed discussion of the results obtained. Through the use of path analysis, it was clearly established that explanatory style influenced and was influenced by students’ motivational attitude toward mathematics and that explanatory style and goal orientation were implicated in students’ achievement in mathematics. Students’ predispositions to view the world from predominantly optimistic or pessimistic outlooks were evident in the early years of their schooling. Their habitual explanations for the causes of events impacted on and interacted with their motivation toward mathematics, with these goal orientations in turn related to their achievement in mathematics.

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While a previous longitudinal study had found a link between explanatory style and students’ general achievement at school (Nolen-Hoeksema, Girgus, & Seligman, 1986, 1992), this study clearly established that once other variables were taken into account, the relationship between explanatory style and achievement was indirect rather than direct (Yates, 1998a, 2000a). It was also more evident in older students where explanatory style was mediated by their goal orientations toward mathematics and their self-reported depression. Over time, optimistic students relative to pessimistic students were more likely to be task oriented toward mathematics and as a consequence to perform better on achievement outcomes in mathematics. Male students were more likely to be pessimistic, to be less motivated, and to have lower levels of achievement in mathematics. While there was no direct relationship between teachers’ ratings of classroom behavior and achievement in mathematics (Yates, 1997b) and students’ explanatory style (Yates, 1998c), teachers were both influenced by and influenced students’ espoused goals toward mathematics, as well as their achievement in mathematics (Yates, 1998a). It would be advantageous for future studies to be conducted on more than two occasions, with depression measured at each data collection point, estimates of the relationships of the change scores calculated, and estimates of absolute change rather than relative change used in the analyses. Not only would multiple measurements be more likely to capture students prone to depressive episodes (Nolen-Hoeksema, Girgus, & Seligman, 1992), but it would also be possible to estimate the stability of depression with interclass and intraclass correlations. Finally, interviews with both teachers and students would assist in the identification of factors not examined with the achievement tests and the questionnaires.

Generalizability and Usefulness of the Study This study has demonstrated the usefulness of gathering quantitative information from both students and teachers in their classrooms to examine common educational problems of attitude and achievement. Performance in mathematics is likely to demand a continuing level of high motivation and persistence. Explanatory style is a motivational characteristic that might conceivably impact upon the disposition to maintain effortful responding over extended time sequences. Failure is an inevitable part of learning (Ames & Archer, 1988), but it is the manner in which students explain the causes of their failure to themselves that is crucial (Pressley & McCormick, 1995). If students can explain the causes of their failures as temporary, specific, and due to factors other than their lack of ability, then they may be more likely to persist in the face of adversity, to maintain high levels of effortful responding, and to focus on task involvement goals (Dweck, 2000). In this study it was the students who espoused task involvement goals who gave evidence of higher achievements in mathematics over time. Negative attitudes toward mathematics have been repeatedly cited as affecting students’ achievement in mathematics (McLeod, 1992). This study not only

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confirmed that students’ motivational goal orientations were predictive of their achievement in mathematics, but also demonstrated that these goals are at least in part influenced by students’ characteristic, habitual attributions for the causes of events in their lives. The commonly held belief that an optimistic outlook on life is generally beneficial was borne out, as students with a positive explanatory style were predisposed to have more favorable goal orientations, to be less vulnerable to depression, and, as a consequence, to make better achievement gains in mathematics. Such students were also more likely to receive more positive ratings from teachers (Yates, 1998b). Children develop their optimistic or pessimistic beliefs by the time that they are eight or nine years old (Seligman, 1990; Yates, 1998a), and these beliefs have an indirect impact on their achievement as well as their daily lives. Therefore, it would be advantageous for teachers to be cognizant of students’ explanatory style. All teachers, particularly those at the primary school level, need to be sensitive to the attributions that students make, particularly in relation to their failures. Intervention programs have been developed for adolescents (Jaycox, Reivich, Gillham, & Seligman, 1994; Peterson, 1988) and college students (DeRubeis & Hollon, 1995). This study would suggest that such interventions should begin when students are in primary schools and should target attributions in specific subject areas, particularly as attributions have been found to be subject specific (Marsh, 1986). Students’ levels of task involvement are also important considerations. Furthermore, teachers need to be cognizant of the attributions that they make about students’ work, particularly in relation to failures. Research focused on World 2 (Popper & Eccles, 1977), the mind of the learner, is imperative in education. In the conduct of any study into factors influencing learning and development, researchers are faced with a number of theoretical, methodological, and practical decisions. As this project was conducted over such a long period of time, it was important to consider carefully the theoretical framework as well as the design, implementation, and interpretation of the study. There were distinct advantages in the adoption of pencil-and-paper instruments to measure the variables in this study, since they were an efficient and effective means of gathering data on a large number of students in a variety of settings over time. Not only could the instruments be administered with minimum disruption to students either individually or in groups within their own schools, but also teachers were able to rate the students’ classroom behavior and achievement in mathematics proficiently. Furthermore, it was possible to ascertain the extent to which the measures accurately reflected the theoretical constructs of interest to the study. The use of the Rasch and path analytic techniques were significant innovations in this attitudinal research as they not only provided information that could not be ascertained from classical test theory and conventional parametric statistical approaches, but they also facilitated the investigation and explanation of the causal relationships both between the variables and over time.

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REACTION BY WENDY KEYS Introduction In this chapter, Shirley M. Yates describes very clearly how she planned and carried out a longitudinal quantitative study focusing on the relationships between students’ attitudes and achievement in mathematics. She provides wouldbe quantitative researchers with a step-by-step account of how to carry out a longitudinal study and gives them an insight into the logistical challenges inherent in this type of research. This chapter will be useful reading for anyone wishing to replicate, extend, or adapt this study.

Main Strengths Shirley Yates sets out very clearly the aims of the study and explains that the research was designed to throw light on a problem relating to mathematics achievement identified by teachers in one of the primary schools taking part in the study: Why did some children appear to “give up” on mathematics by the time they reached the top of the primary school? It is pleasing to note that the impetus for the research came from a real-life problem. The chapter contains an excellent review of the salient findings of previous research. Shirley Yates clearly knows the area well, and the causal model that she developed is thoroughly grounded in previous research. She gives a good justification of her choice of measures and how each relates to her causal model. Each of the measures used—the test of mathematics, the attitude inventories, and the teachers’ checklists—is described clearly with example items given for illustration. A few additional examples would, perhaps, have been useful. The research design is set out clearly in figure 7.1 and described in the associated text. Figure 7.1 is especially helpful as it gives the reader an idea of the timescale of the research. The section on data collection is clear and gives useful information on how to carry out a research study, including ethical considerations such as obtaining state or local authority permission and parental consent and assuring respondents of confidentiality, administering tests and attitudinal inventories, and gaining teachers’ cooperation in completing the checklists. Carrying out a research study is, as Shirley Yates makes clear, much more challenging than it appears from a cursory look at the original research design. A particular strength of this chapter is the fact that Shirley Yates highlights some of the challenges she met and describes how she dealt with them. She explains, for example, that she used pencil-and-paper tests and questionnaires, rather than individual interviews, not only for reasons of cost effectiveness and larger sample size but also because it minimized the burden on schools (a very important consideration). She describes how she was able to make use of the results of mathematics achievement tests administered by the school for monitoring purposes

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in one school and acknowledges that this influenced the choice of mathematics test for the study. She highlights the difficulties in following up a sample from two primary to some 50 primary and secondary schools and the resulting attrition (about a quarter of the original sample was lost), and acknowledges that her sample (drawn initially from two primary schools) was not randomly selected or nationally representative. In the real world it is not possible to carry out a perfect research study. It is important, though, to be aware—as Shirley Yates clearly is—of the challenges in any research study and to devise strategies to cope with them. Anyone wishing to carry out a similar study would do well to read the sections on research design and data collection very carefully. The analytical techniques used in this study are somewhat complex and may be unfamiliar to many readers. The method of analysis used—PLSPATH analysis with latent variables (which is essentially a series of multiple regression analyses presented in diagrammatic form in which correlations are adjusted to take account of other variables in the analysis)—and Shirley Yates’s rationale for using it to test hypothesized relationships between variables is described simply and clearly for the lay reader. She then goes on to describe the results of the study in terms of the path model (figure 7.2). Again, the reader is led gently toward the main conclusions of the research that are set out in the discussion section. It would, perhaps, have been useful to include more discussion of the implications of the findings for schools.

Possible Alternative Approaches The research design described in this chapter was quite complex and would be costly to replicate, especially in terms of researchers’ time: the study was longitudinal (over three school years); data were collected on three different occasions with children, in some cases, traced from primary to secondary school; and the 243 children in the achieved sample, drawn initially from two primary schools, were selected from five grades (school years). It might be simpler, for anyone wishing to carry out a similar study, to select the sample from a single year group (surveyed in three consecutive years) in a larger number of primary schools. Furthermore, since the research question arose from concerns within a primary school, would it have been possible to confine the study to the primary school years? This would avoid much time-consuming tracing of pupils and, possibly, have allowed Shirley Yates to sample from more primary schools. Increasing the number of schools would, of course, have increased the amount of administration required (contacting schools, persuading them to take part, chasing up returns, etc.), which might not have been possible within the resources available. In her chapter, Shirley Yates highlighted some innovative analytical procedures, Rasch scaling and PATH analyses. It is possible that less statistically minded researchers would be put off by these complex statistical analyses. It should be remembered, however, that path analysis is only one of the many tools available

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to quantitative researchers. There are many other ways in which this type can be analyzed and reported, such as parametric and nonparametric statistics, and techniques developed for exploratory data analysis (Tukey, 1977). Shirley Yates, for example, also analyzed her data using traditional statistics. This type of analysis can provide interesting descriptive statistics about what children thought about mathematics and other issues. Furthermore, simple comparisons between the attitudes of children who do well and not so well in mathematics could illustrate, and possibly clarify, the conclusions set out in this chapter.

Concluding Comments Shirley Yates’s chapter focuses on a number of important issues, many of which are relevant for qualitative as well as quantitative research. Her account of the way in which she tackled her research provides a useful tool for anyone wishing to carry out a similar study.

APPENDIX The Rasch Model The basic principles of Rasch scaling are really quite simple. The oneparameter item response model, or Rasch model, assumes that the relationship between an item and the student taking the item is a conjoint function of both the performance of the student and the difficulty level of an item on the same latent-trait dimension (Snyder & Sheehan, 1992). This relationship is expressed as a mathematical function (Lawley, 1942), referred to as an item characteristic function and portrayed by an item response curve (Fischer & Molenaar, 1995; Hambleton, 1989; Hambleton & Swaminathan, 1985; Weiss & Yoes, 1991). In the calculation of item difficulty, the Rasch model takes into account the performances of the students in the calibration sample and then frees the item difficulty estimates from these performances. Likewise, student performance is estimated by freeing it from the estimates of item difficulty (Snyder & Sheehan, 1992). With scales measuring attitudes, the term performance is taken to mean the level of the students’ attitude, with response possibilities reflecting the level of the items on the underlying attitude scale (Green, 1996). In this study, explanatory style, depression, and goal orientation are attitudinal constructs. The probability of a student answering an item correctly is defined as a function of the student’s performance level and the difficulty level of the item. Provided the items in the scale fulfill the requirements of unidimensionality, neither the item discrimination parameter nor a guessing factor associated with each item is taken into consideration. With attitudinal scales, the likelihood of

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any particular response is determined jointly by the student’s attitudes and the level of the item on the underlying attitudinal scale, thus providing for interaction between the student and the content and format of the item (Green, 1996). Estimates of both student performance or attitude and item difficulty are compared on a logistic scale in which the item difficulty and student performance or attitude may hold any value from – to  (Snyder & Sheehan, 1992). As these natural logistic scales are not bounded and are interval in nature, they more adequately serve linear regression estimation of developmental change across time (Snyder & Sheehan, 1992). The major advantage of the Rasch model is that students’ estimated performance or attitude is independent of the sample of items while at the same time the difficulty level of the items is not dependent on the sample of students who take the items (Hambleton, 1989; Wright, 1977; Wright & Stone, 1979). As items that have been calibrated with the Rasch model are sample free (Green, 1996), they yield equivalent performance or attitude estimates for any group or individual. Similarly, any sample from the specified population yields equivalent item difficulty estimates (Snyder & Sheehan, 1992) provided that in both cases the items and the students satisfy the requirements of unidimensionality. This specific objectivity (Rasch, 1960/1980) overcomes the dependence on a standardization group in classical test theory where the interdependence of items and students confounds independent estimates of students’ performance and the examination of the psychometric properties of a test or instrument (Snyder & Sheehan, 1992). Furthermore, as the measures of students’ performance or attitude are algebraically freed from the calibrations of the items, commensurate estimates for items and students can be made over time (Green, 1996). Missing items and missing persons are not a problem as not all students need to answer all items (Green, 1996), provided that at least 80 percent of the items in the particular set administered are answered by each student (Anderson, 1994). The performance of students who take different items from the same test battery can then be compared, provided that the items and students have been calibrated on a common scale (Green, 1996).

Unidimensionality of the Instruments The item response model employs the notion of a single specified construct (Snyder & Sheehan, 1992) or an inherent latent-trait dimension (Hambleton, 1989; Weiss & Yoes, 1991), referred to as the requirement for unidimensionality (Wolf, 1994). While this concept needs to take into account the fact that in any measurement situation items and persons are multifaceted, measures need to be thought of, and behave, as if the different facets act in unison (Green, 1996). In classical test theory, scores are created by summing responses. By contrast, scores on Rasch-calibrated tests or instruments represent the probabilistic estimation of the performance or attitude level of the respondent based on the pro-

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portion of correct responses and the mean difficulty level of the items attempted. This has a distinct advantage over classical test theory procedures, as the scale from which such scores have been obtained is built with items that satisfy the requirements of unidimensionality. Prior to the use of the Rasch model in this study, it was essential to determine whether each instrument met the requirement of an inherent latent-trait dimension (Lord, 1980; Weiss & Yoes, 1991). In the case of the Children’s Attributional Style Questionnaire (CASQ), the items had been designed to measure the construct of a single trait of explanatory style. The issue of unidimensionality had been specifically examined in the construction of the Children’s Depression Inventory (CDI) through the use of the oblimin rotation factor analytic procedure (Kovacs, 1992). Thus, it was only necessary to examine the factor structure of the Your Feelings in Mathematics: A Questionnaire and the Student Behavior Checklist. Principal-components analysis and the oblimin rotation procedure indicated that Your Feelings in Mathematics: A Questionnaire contained two separate scales of Task Involvement in Mathematics and Ego Orientation in Mathematics, each of which independently met the criterion of unidimensionality for the application of the Rasch procedure (Yates, 1998b, 2000b). Confirmatory factor analysis of the Student Behavior Checklist with the LISREL8W (Joreskög & Sorböm, 1993) computer program indicated that the checklist contained a single factor of academic behavior (Yates & Afrassa, 1995).

Calibration of the Instruments A further requirement of the Rasch model is that students’ answers to each item should be independent of their answer to any other item, except for the influence of the latent trait (Green, 1996). Misfitting items are therefore deleted before student estimates are made. The requirement of classical test theory that all items in a scale to be calibrated should have very strong discriminating power is not appropriate, as some items, particularly those in achievement tests with a high level of discrimination, differ in the extent to which they relate to an underlying trait (Hambleton & Swaminathan, 1985; Lord, 1980; Scheuneman, 1979). In practice, the Rasch model allows for the predetermination of a range of variability in the slope of the ideal item characteristic curve, through the use of the infit mean square range. Adams and Khoo (1993) indicate that an infit mean square range of 0.77 to 1.30 is satisfactory, but the more stringent conditions of 0.83 to 1.20 were chosen for this study to ensure that all items fitted well the requirement of unidimensionality and conformed to the Rasch model. Items with an infit mean square of less than 0.83 have insufficient band width and provide redundant information. They discriminate too sharply between students who are high on the attitude scale in comparison with those who are not. Those items with an infit mean square greater than 1.20 are considered to be inappropriate, as the probability of the students responding to the item in the same direction expected from the pattern of their other responses is dependent upon

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factors other than those measured by the scale. With such items, students who are relatively high on a scale respond in the opposite direction, while those who are relatively low on a measure respond as if they are higher on the scale. With the exception of the Progressive Achievement Tests in Mathematics (ACER, 1984), which had been Rasch analyzed with the BICAL3 program, the QUEST program (Adams & Khoo, 1993) was used for Rasch scaling of the instruments. Item characteristic curves used in the examination of the relationship between a student’s observed performance on an item and the underlying unobserved trait or ability it measured are dependent upon a large number of participants taking the item. Therefore the item analyses for the CASQ, the Task Involvement in Mathematics, and the Ego Orientation in Mathematics were carried out with all students who took part in the study at Time 1 (N  293), (N  328), and (N  328) respectively. The small variability between these numbers occurred because the questionnaires were not necessarily administered on the same occasion to all participants. The Student Behavior Checklist was analyzed with the responses for the 258 students who were rated by the teachers at Time 2. The CDI was administered only at Time 3 so the analysis was based on the total group of subjects (N  335) who participated in the final phase of the study. In all analyses the probability level for student responses to an item was set at 0.50 (Adams & Khoo, 1993). Thus the threshold or difficulty level of any item reflected the relationship between students’ attitude and the difficulty level of the item such that any student had a 50 percent chance of responding favorably to that item. For the analyses of the CASQ, the Task Involvement in Mathematics, the Ego Orientation in Mathematics, and the CDI, the infit mean squares of each item were inspected to determine whether they fell within the predetermined ranges of 0.83 and 1.20. Items that fell outside these ranges were progressively deleted so that the final scales were composed of those items that met the requirements of the Rasch model. While the item infit mean squares for the CP, CN and CPCN scales of the CASQ were within the specified range, some item deletion was necessary for the Task Involvement in Mathematics, Ego Orientation in Mathematics, and the CDI. The final scales for the Task Involvement in Mathematics, and Ego Orientation in Mathematics were composed of 12 and 5 items respectively, while the CDI was composed of 20 items (Yates, 1998a). More detailed information on the principles and uses of Rasch scaling can be found in the recent book by Masters and Keeves (1999).

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ACKNOWLEDGEMENTS Deep-felt thanks are extended to Professor John P. Keeves for his invaluable advice and guidance throughout the study. This study was supported by a Flinders University Research Board Establishment Grant.

Afterword

Applying the Theory to the “Practice” in This Book Leone Burton

With the discussion in chapter 1 in mind, and your own readings of the chapters to guide you, I now review the chapters in this book particularly from the perspectives of how the author(s) deals with methodology and method but noting, too, the style of writing, and the ways in which the author is inserted into that writing. This review was written on the basis of a prepublication draft so some of the comments might well not seem to apply to the printed text. However, because the comments were applicable, and because they most frequently are applicable to much writing, I decided, together with the editors, to leave the comments as written. The chapter by Maria Manuela David and Maria da Penha Lopes reports on a study on flexibility of thinking by lower secondary pupils in three classes. They used “classroom observation to expose evidence of teachers’ and students’ mathematical thinking and explore the influence that the teacher may have on developing mathematical thinking”(p. 11). To support their work they drew on Gray and Tall, on Schoenfeld, and on Luria and Vygotsky, translating these theoretical stances, as appropriate, into application to the secondary mathematics classrooms in which they were interested. They underline their belief that school has a dual function, providing access to so-called “scientific” knowledge while, at the same time, valuing “everyday” knowledge. This poses a recognizable dilemma that they do not explore. However, they do appear to accord great respect to the body of knowledge, mathematics, and those who practice it, mathematicians, and to have expectations that they would find some of the practices of mathematicians “being carried out, at least in some schools”(p. 18). They do not explain from where or what these expectations arise. Another belief, which they express, is in “the potential of the teacher’s influence” (p. 18) that led them

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to focus upon the procedures used by the teachers with whom they worked. Throughout this extensive exploration of the theoretical background to their research and despite their claim to outline their values as well as their beliefs, we get no insights into the values that drove their research choices. Indeed, the research is cited in the phenomenological tradition and is described as being “qualitative research of an interpretative character” (p. 19, 20), but the authors go on to describe “subjects” and an “inevitable risk of subjectivity”(p. 20). It is possible, here, that not being first language English speakers constrains these authors to using words that a first language speaker might avoid because of their cultural baggage. Nonetheless, the authors are careful to assert the expected permeation by their personal views and life experiences of their theoretical conclusions. However, such a caveat does not stop them from claiming that they “construct a model that can be used in the analysis of mathematics classes” (p. 20)—a not inconsiderable claim. I am drawing attention, here, to contradictions between the methodology, which lies behind many of the methods that they choose to use, and the language that they use to describe what they did. The authors position themselves as holding an “observational and interpretative methodology” (p. 32) and recognize and adjust to the constraints such a position carries with it. I think that observation and interpretation are methods, and associated with them are difficulties of the kind that they outline, such as absence of evidence, not meaning absence of object of interest. At the same time, their language, for example of “subjects,” of “ability,” of “instruments,” springs from a much more traditional position that separates knowledge from the knower and makes assumptions about the possibility of positioning the “objects” of study in such a way that the researcher can come to general conclusions. The language of “ability” is used not only to describe cognition but also metacognition, but we have no way of knowing what meaning to ascribe to it. Is it “ability” in the psycholinguistic sense of something innate? Is it “ability” meaning “attainment” as in the language of so many teachers? Is it some combination of these two or some other meaning altogether? The confusion leaves the reader with no guidance as to the covert meanings that are carried by the language. But, furthermore, “ability” carries further covert meanings associated with the power of mathematics as a discipline to be learned which, to me, seem to be contradicted by the theoretical positioning adopted by these authors. In a later section of the chapter (see p. 25), the authors write about the classification of “productive” and ”nonproductive” dialogues and of “correct” mathematical language without giving us any methodological insight into how they can make these judgments and under what conditions. A further example provides evidence of some confusion between what is meant and what is said. On page 22, in reference to the claim that their data expose “the importance of the role of the teacher in the development of a productive dialogue” and “those dialogues that lead a student to reelaborate the argumentation and the mathematical thinking from those which do not,” they say,

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“This offers us the affirmation of the appropriateness of the methodology used.” Editors’ comment: here is an example of the text being revised in the process of final drafting to which Leone Burton refers above. I could see how the collection of such data might reinforce the appropriateness of the methods used, but cannot imagine how the methodology would, or could, be so affirmed, and the statement does not provide convincing evidence of either. The authors are very clear about the analytical structure that they have used on their data, but in the examples that they provide, this structure is not used to inform the reader about the conclusions. Like Anne Watson, I am very conscious of the imperialism of my first language, English, and also of the many impositions this creates for researchers working outside the English language community. Nonetheless, in this chapter, I am asking to know “how the researchers would have made decisions, and what beliefs they would have brought to bear on the work” (Anne Watson’s reaction, p. 34). Indeed, I am asking for more, in calling for researchers to articulate not only their beliefs, but also the attitudes and values that have influenced them. If these are offered to the reader, I think there is just a possibility that the differences between research communities might be made meaningful and a subject of learning for the members of each. Without them, I fear that the more powerful community remains in a position to impose its values on others. In his chapter, Simon Goodchild is punctilious about informing us of who he was when he came to the research which he describes. As he says, “The preceding paragraphs not only provide my rationale for engaging in classroom research but also expose the underlying threats and opportunities of the inquiry” (p. 41). He articulates clearly his values: to cause minimal disturbance in the classroom, to mitigate any threats that he might pose to teacher/pupils while making best use of the opportunities to understand his observations in the mathematics classroom. Additionally, he deliberately chooses to position himself theoretically within three different perspectives in an attempt “to overcome the problem of the research being saturated by a personal belief system” (p. 44). He explains how he derived four levels of analysis, the constitutive order, the arena, the setting, and the individual, which helped to frame his research questions. This substantial section of the chapter devoted to methodology leads naturally into an explanation both of why he adopted an ethnographic approach and why he rejected other alternatives. He is clear as to how the study was designed and what data were collected. He indicates how consistency was sought through data gathered from different sources. The approach to analysis of the data is also explained with, again, an emphasis on consistency, authenticity, and trustworthiness. The chapter ends with a personal reflection that reinforces the person of the researcher who has been present throughout the writing. From the point of view of asking, in reading a report of a piece of research, that the methodology of the researcher should be made apparent and the consistency of that methodology with the methods of research and analysis be maintained, this chapter provides the reader with an excellent exemplar.

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In her chapter, Nora Linden explains her purposes in undertaking a study focused on young children with special educational needs, their parents, and their teachers. In this study, mathematics became an important issue because of its introduction by her informants. Her intentions were to share her informants’ stories about, and their rationale for, their actions and their understandings of their education. She did this through conversation. Her values are implicit in the desire to “give a voice” to these pupils and also to explore issues relating to their self-esteem. The official values about SEN education are made explicit in her references to integration and inclusion. In her explanation for the theoretical positioning of her study within activity theory, Nora Linden gives further indications of her methodological stance. She highlights the importance to her of what I would call the “authorship” of the learner, the need for the learner to be the principal agent of her/his own learning, and emphasizes, for her, the centrality of language to learning, both in how it helps her frame her research questions, but also in how it contributes to the methods she chooses for collection of her data. In this way, she draws links between her methodological stance and the methods of research and analysis that together constitute her study. She articulates her intentions for the chapter as being “to give an outline of the methodological issues concerning one qualitative method in classroom research” (p. 83). In this I think she has been successful, not only in how she contextualizes her research, but also in the description and subsequent reflection that she makes upon her data and its analysis. If I were to appeal for any changes, I think it would be to make more overt the methodological issues instead of leaving the reader to unpick the values and beliefs that drove her and are certainly there, but not overtly addressed. The chapter by Kay McClain is called “A Methodology of Classroom Teaching Experiments,” so it was with interest that I sought to find out what she means by this. The classroom teaching experiment described in the chapter is not, I believe, an example of a methodology but, again, of a method. The method is situated within design research which, as Kay McClain explains, “involves cycles of research and development where ongoing classroom-based research feeds back to inform instructional-design decisions in a cyclic manner” (p. 92). Realistic Mathematics Education is the instructional theory on which design decisions are based and the “emergent perspective” (p. 92) guides an interpretative approach to the classroom-based research. “The emergent perspective involves coordinating constructivist analyses of individual students’ activities and meanings with an analysis of the communal mathematical practices in which they occur. . .in order to account for learning in the social context of the classroom. (p. 92). Kay McClain outlines these theoretical frameworks in detail, making clear what are the classroom expectations (called “norms”) that drive the classroom activity. However, from where these expectations come remains unstated, as do why they are chosen, what values and beliefs they reflect, and how they define the research that is being described. In her discussion of Realistic Mathematics Education, Kay McClain highlights three beliefs that underlie this approach. These are, first, that instructional start-

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ing points should be experientially real to students so that the mathematical activity in which they engage carries personal meaning. Second, the instructional starting points should be linked to appropriate mathematical learning. Third, students should create and elaborate symbolic models of the mathematical activity upon which they are engaged. These beliefs are embedded, for those who work within a Realistic Mathematics Education framework, in a philosophical and empirical context although such context is not explored in this chapter. However, as Kay McClain explains, the instructional sequence she is describing is of a different order, involving informal, situated problem solving out of which models of this mathematical activity are developed followed by models of mathematical reasoning and, finally, more formal mathematical activity. This structure is neither explained nor justified, nor are the differences explored other than citing country-based differences as a reason. Like Marja Van den HeuvelPanhuizen, I found the chapter “provides us with a clear view of the ingredients of a well-thought-out and carefully executed classroom teaching experiment” (p. 113). But in order to help me evaluate the contribution that the chapter might make to my thinking, I need to know more than what is done. I need to know why the choices that were made were seen by the researcher to be the best for her purposes. Ruth Shane, in her chapter, poses three research questions: What do student teachers learn from the practicum about teaching mathematics? Does the particular mathematical environment of the practicum classroom play a role in what the student teachers learn? Is there an interrelationship between how the children in a classroom are learning about mathematics and what the student teachers are learning about teaching mathematics? (p. 119)

To try to answer her questions, Ruth Shane did a case study of four student teachers, two teaching mathematics in a traditional classroom and two in a reform-oriented classroom, all classes being second grade. (In using these terms, Ruth Shane is relying upon a “consensus of understanding” without specifically explaining her meanings.) The research was designed as a case study “to capture the day-to-day details of the mathematics classroom environment and to undertake a microanalysis of the parallel learning of the student teachers in those classrooms” (p. 119). Ruth Shane expresses a holistic view of the distinctive culture of the mathematics classroom as being initiated, maintained, and developed by the teacher. It is acted out, extended, clarified, and “dialogized” by the teacher and pupils, and continuously affects what the pupils and the teacher and student teacher learn about mathematics and teaching mathematics. Classroom culture in this research is the sum of those norms which have been “taken-as-shared.” (p. 119, 120)

Her adoption of a social-constructivist perspective, her sensible use of the frameworks extracted from the theory in order to help describe the classroom culture, and her expressed need to try to capture the culture of the mathematics class-

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room give an indication to the reader of her methodology. This outline of her methodological stance places the teacher as central but does not explain what she means by a “holistic view” nor why she views the culture of the mathematics classroom as distinctive and how this belief has influenced what she did and why she did it. Interestingly, as part of the design, she felt that she needed to expose the beliefs of the student teachers but her own beliefs about the research exercise remain hidden. We are told about the kinds of data which she hoped to “expose” but not why she sought that particular data as the best means of answering her research questions. She provides an extensive review of the appropriate literature in order to root the research into a theoretical context. But although her chosen questions (p. 121) are interesting, they are not the only ones possible and we do not find out why these are the questions that drove her theoretical stance. I would say that there is a heavy emphasis on knowledge, both of mathematics and its learning/teaching, which might be seen to be most appropriate to a study focusing upon the learning of student teachers from their practicum. However, I would suggest that knowledge is neither the only thing nor, perhaps, the most important that they learn, and that knowledge itself does not inhabit an undisputed terrain. Will this focus constrain the research such that its outcomes are less helpful than they might have been? Although I do not find a direct presentation of the researcher’s methodology, there are some words, together with the knowledge focus, which help me to interpret her position. For example, “the intention was to expose their knowledge for teaching mathematics” (p. 123) feels positivist to me, carrying as it does the implication that there is a recognizable knowledge base and that it is reasonable for a researcher to “expose” it. Later, she says “except for the rare ‘natural’ mathematician, mathematics is not an intuitive discipline but one that is dependent on the learning that goes on in mathematics classes” (p. 123) making clear the epistemology that she holds about mathematics and its power. but also refuting what the majority of mathematicians, and indeed Ephraim Fischbein (1987) on whom she earlier leans, have to say about the role of intuition in mathematics (see Burton, 1999). While it would be foolish to disagree with Ruth Shane that learning mathematics is, at least to some extent, dependent upon the learning that happens in mathematics classes, it would be reasonable to challenge her to support a statement that mathematics itself is dependent upon such learning. Many authors have written of the damage done to the discipline in mathematics classrooms, so her position can be critiqued. Most of all, however, it reflects her values and her epistemology, even though these are not addressed directly. But it also poses, for me, a contradiction with the position she takes on p. 124, that she perceives the classroom “metaphorically as a ‘learning web’ rather than a ‘ping-pong table.’” This contradiction is maintained in the description of her choice of methods as consisting of “the most appropriate research instruments” (p. 127) and, later, of the children and student teachers as her “subjects.” In her summary (pp. 126–127), Ruth Shane acknowledges methodological dilemmas, the most overwhelming of which is the complexity of the field she

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wishes to research. To deal with this, she explains why she designed the study in the ways in which she did and follows this with a rich description of what was done and how the data were collected and analyzed. The chapter concludes with a section called “Methodological Issues and Implications,” which focuses upon the methods that were used, any limitations that she found with them, and the ways in which she tried to deal with these. Like Gilah Leder, in her reaction, I, too, “would have liked to read more. . .about Ruth’s reflections on the methodological issues raised by this piece of research” (p. 151). Shirley Yates offers the final chapter, a report on a longitudinal study, originating in two primary schools, of factors influencing pupil achievement in mathematics, particularly with respect to their motivations to learn. The three motivational factors relating to achievement in mathematics, pupil optimism, pessimism, and goal orientations, were chosen as the focus for the study. While the choice of these factors is related to the motivational literature, Shirley Yates does not tell us how her values, beliefs, and attitudes about undertaking this study contributed to this choice. The style of writing is third person, which, of course, hides the motivations of the researcher although, no doubt she hopes, makes clearer the motivations to be researched! She does indicate the value of the school staff associated with this study as being to improve pupil learning outcomes. Shirley Yates chose to gather her data through the use of “objective test of achievement and paper-and-pencil questionnaires” (p. 156). Reasons given for choice are efficiency, minimal disruption, and the gaining of information relating “pupils’ explanatory style, motivation, and achievement” (p. 156, 157). Teacher ratings of pupil behavior in class and mathematics achievement were also gathered. The study was made over three years. It seems to me that motivational information is among the more difficult to gather in this way, so I would have appreciated knowing why questionnaires were thought to be the best method to use. Indeed, in this chapter, although considerable space is given to explaining how particular statistical techniques were applied to the data and a justification for their use is offered in the “better appreciation of factors which influence student motivation and achievement in mathematics” (p. 157), the major methodological claim appears to be one of analytical innovation but not of why the application of such innovations provides the reader with more convincing evidence. The statement, for example, that “access to the unique psychological world inhabited by each student is commonly gained through pencil-and-paper questionnaires, with results aggregated to the group level” (p. 158) does not provide me with convincing reasons as to why I should have confidence in such data and their analysis. My expectation is, therefore, that at the end of the study I will be able to agree that these pupils had these dispositions expressed, on questionnaires, in these ways on these days. That, it seems to me, does not enable me to come to generalizations about the links between the pupils’ motivations to learn and their achievement! The choice of quantitative methods, as I said earlier, may be the most legitimate and sensible. In her justification of her choice, Shirley Yates cites pragma-

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tism and “objectivity” (my quote marks, to indicate that objectivity is not without its challenges). She uses the language of such studies without ever indicating any need to question it, or deal with hidden meanings. One example is that teachers were asked to rate students on dimensions of learned helplessness and mastery orientation as well as to give a single rating of achievement. Despite recognizing that these ratings were made by teachers, some of whom were not well known to the pupils, in schools where more than one teacher taught mathematics, the third person writing hides these problematics which are simply stated and not explored at all. A further example is provided by the presentation of a correlation found between teacher judgments and student achievement, a correlation that has been challenged by studies such as Rosenthal and Jacobson’s (1968) Pygmalion in the Classroom, updated in Rosenthal and Jacobson (1992). These results are stated without discussion, whereas it could be said that teachers who judge a student as a high achiever have set up the conditions for a selffulfilling prophecy. Whether this is the case, or not, it does seem to me that the results need unpicking, discussing, and problematizing in the context of the research and the questions it asked. In discussing the measurement of achievement on multiple-choice tests, Shirley Yates states that “these increments indicated that the tests measured abilities that were sequential” (p. 168). I think that what is measured, here, is achievements, and the use of the word “abilities” I believe perpetuates a psychologistic confusion that permeates the thinking of teachers and pupils. Also, queries have been raised in the literature about the application of teacher rating scales and their potentiality for misplaced sexism and racism, but, again, these questionnaires are not discussed as in any way requiring justification. In her final discussion, Shirley Yates indicates that there were factors that her study could not examine and that other methods of inquiry might be beneficial to broaden the scope. For me, it is not an issue as to how the research was done. As I indicated in chapter 1, as a researcher one chooses the best methods to do the job that one wants to do. Much more important is to provide the reader of the study with confidence that the problematics have been identified and attempts have been made either to deal with them or to be open with the reader about the constraints necessarily set or incurred in doing the study. If such a reflection had been included in this chapter at every stage, I would have been able to feel more confidence in the choice of method of research and analysis as justifiable. The chapters in this volume present a range of research methods for seeking the data demanded by their questions. Many of them, however, do not explore openly and problematically the attitudes, beliefs, and values that led them to embark upon the study in the manner they describe, and to analyze their data in the ways outlined. That does not make their chapters nor, of course, their research less worthwhile. It has led me, however, to raise the questions in my mind about why the study question, the foci, the design, and the analysis were done in the manner reported and what influence other choices might have had on the

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outcomes and, indeed, to point to issues about the reporting style which inevitably influenced this reader. We should, of course, celebrate the growing basis of knowledge that the results of research in mathematics education offer. Nonetheless, it is my expectation that were we, as editors of journals and books, as authors of papers and chapters, to be more transparent in our treatment of our research subjectivities, we would offer our readers more robust grounds for having confidence in our results and treating them with respect.

REFERENCES Burton, L. (1999). Why is intuition so important to mathematicians but missing from mathematics education? For the Learning of Mathematics, 19(3) 27–32. Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht, Netherlands: Reidel. Rosenthal, R., & Jacobson, L. (1968). Pygmalion in the classroom: Teacher expectations and pupils’ intellectual development. New York: Holt, Rinehart and Winston. Rosenthal, R., & Jacobson, L. (1992). Pygmalion in the classroom: Teacher expectations and pupils’ intellectual development. New York: Irvington.

Index

Note: References in bold refer to main contributions to the text. ability, 200, 206 abstraction: definition, 14 ACER. See Australian Council for Educational Research achievement. See Progressive Achievement Tests in Mathematics (PATMaths); student motivation and achievement activity theory, 42–43, 44, 45–46, 71–72 Adams, R. J., 189 algorithms, 12–13, 58, 106 arena, 41, 43, 45, 61, 61 n1, 68 attitudes towards mathematics, 67–68, 159, 183–84. See also Children’s Depression Inventory; Student Behavior Checklist Australia. See student motivation and achievement Australian Council for Educational Research (ACER), 165, 168 autonomous thinking: definition, 14 awareness, 58–59 Ball, D., 122 Barker, R. G., 45 Bassey, M., 6 Bateson, G., 58–59, 73 Bauersfeld, H., 83 Biklen, S., 19 Bishop, A., 40 “blind activity,” 45–46 Boaler, J., 47, 58 Bogdan, R., 19 Borko, H., 48 Bowen, A., 149

Brazil: education system, 25; mathematics teaching, 17–18, 25–30, 36 Brown, M., 40 Bruner, J. S., 45 Burgess, R. G., 55–56 Burton, L., 1–10, 199–207 CASQ. See Children’s Attributional Style Questionnaire CDI. See Children’s Depression Inventory Children’s Attributional Style Questionnaire (CASQ), 167, 168, 174, 175, 189, 190 Children’s Depression Inventory (CDI), 167, 170, 174, 175, 189, 190 classroom culture. See student-teaching practicum, classroom culture classroom environment, 23 classroom mathematical practices, 95, 108 classroom teaching experiments, 91–116 theoretical frameworks, 92–98; emergent perspective, 92, 93–95, 94f, 96f; Realistic Mathematics Education (RME), 92, 95–96, 114, 202–3 design research methodology, 92, 93f, 113–14; conjectured learning trajectory, 98–99, 100f; development of instructional sequence, 99, 100–102, 101f, 102f, 109–10 data collection: classroom episodes, 104–6; daily project meetings, 106–7; realized/actual learning trajectory, 106; student interviews, 103–4

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classroom teaching experiments (continued) data analysis, 107–8; generalizability, 108–9, 113; trustworthiness, 108 critical reflection, 109–10 implications, 110–11 reaction, 111–15 review, 202–3 Cobb, P., 42, 94, 107, 108, 122, 124, 133 cognition: cognitive skills, 14, 18–19, 31; metacognition, 15; situated cognition, 43, 46; social nature of, 42–43 cognitive psychology, 43–44, 48 Colardarci, T., 162–63 Coles, M. J., 16 Concepts of Secondary Mathematics and Science research, 41 constitutive order, 43, 61 n3, 62 n4 constructivism, 42, 43, 44, 45, 124 conversation as research tool, 49, 50, 54–55, 73–78, 83; language, 73–75, 77–78, 81, 86, 88; surfacing and redirection, 86; transcription of data, 52, 85 Convery, A., 48–49 Cooper, P., 47, 49, 56 Corbett, J., 70 Crapanzo, V., 80 cultural relevance, 34 David, M. et al., 5 David, M. M., 11–32, 36–37, 199–201 Davis, J. A., 176 Denvir, B., 40 depression, 161–62, 170 design research, 92, 93f, 109–10, 113–14. See also classroom teaching experiments Doyle, W., 45, 129 Duda, J. L., 169 Ebby, C. B., 151 Eccles, J. C., 157–58 education for research, 35 Edwardsen, E., 69 Eisenhart, M., 48 Eisenhart, M. A., 47

Index

Eisenhart, M. et al., 125 emergent perspective, 92, 93–95, 94f, 96f E-rationale, 82–83, 87 Ericcson, K. A., 48 Erickson, F. E., 131, 148 ethical perspectives, 55–56, 60, 76, 171–72 ethnographic style, 44, 46–47. See also students’ goals in classroom activity explanatory style, 156, 160–62, 168–69, 183–84 failure, 69, 72, 160–61, 162, 183, 184 Fennema, E. et al., 124 Fincham, F. D. et al., 162, 170 Fischbein, E., 122, 204 flexible thinking, 11, 12, 14, 22 Forgasz, H. J. et al., 48 Fosse, T., 67 Freud, S., 76 Freudenthal, H., 113 Gallagher, D. J., 51, 56 gender bias, 2, 8 generalizability/usefulness of research, 33; in studies reported, 33, 36, 37, 108–9, 113, 183–84 generalization in mathematical thinking, 14, 27, 29 Glaser, B. G., 52, 107 goal orientation theory, 44–46, 47, 156, 159–60. See also students’ goals in classroom activity Goodchild, S., 39–59, 77, 201 Gravemeijer, K., 95 Gray, E. M., 11, 12, 13, 14 Guba, E., 4, 79, 80 Harding, S., 1, 2 Harré, R., 2 Hart, K. M. (Ed.), 41 Herbert, C., 6–7 Hextall, I., 3 Hiebert, J., 121, 129, 131, 132 Hoge, R. D., 162–63 Howson, A. G., 40, 41 Huberman, A. M., 53, 73–74

Index

inference: definition, 14 initial teacher education. See studentteaching practicum instructional sequences, 99, 100–102, 109–10 instructional theory, 95–96, 97 instrumental understanding, 121, 122 interviews, 73, 103–4, 136–38, 140–42. See also language of interviews/conversations I-rationales, 45, 57, 82, 83, 87 Israel. See student-teaching practicum Jacobson, L., 206 Jaworski, B., 23, 24, 48, 59–61 Jayaratne, T., 8 Jenkinson, J., 70 Jowell, R., 176 Julie, C., 84–87, 88 Kaplan, A., 158 Keeves, J. P., 157–58 Keller, E. F., 2 Kelly, A. E., 111, 112 Keys, W., 185–87 Khoo, S. K., 189 Kistner, J. A., 162 knowledge: acquisition, 68–69, 123–24, 204; children’s construction of, 135–40; conceptual, 121–22; Fischbein’s model, 122; internalization of, 71; intuitive, 122, 123, 204; meta-knowledge, 73; procedural, 121–22; student teachers’ construction of, 140–47; substantive, 122; syntactic, 122; for teaching mathematics, 123. See also mathematical thinking; understanding Kovacs, M., 170 Kvale, S., 73, 76 Labinowicz, E., 137 language: of interviews/conversations, 73–75, 77–78, 81, 86, 88; of mathematics, 15, 23–24, 32–33, 69; of researchers, 32–33, 200; as tool for learning, 72 Lather, P., 2, 4, 5

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Lave, J., 43, 44, 45, 46, 57, 61 n1, 61–62 n3 learning theories, 41–44; activity theory, 42–43, 44, 45–46, 71–72; constructivism, 42, 43, 44, 45, 124; emergent perspective, 92, 93–95, 94f; situated cognition, 43, 46; social-constructivist theory, 124; sociocultural theory, 41, 42–43, 124; zone of proximal development, 42, 71. See also learning trajectories; teaching and learning learning trajectories: conjectured, 98–99, 100f; realized/actual, 106 LeCompte, M.D., 88 Leder, G. C., 149–51, 205 Lefevre, P., 121 Leont’ev, A. N., 41, 42 Lesh, R. A., 111, 112 Licht, B. G., 162 Lillejord, S., 69 Lincoln, Y., 4, 79, 80 Linden, N., 67–84, 87–88, 202 Lindh-Munther, H., 75 longitudinal study. See student motivation and achievement Lopes, M. P., 11–32, 36–37, 199–201 Lubienski, S. T., 149 Lynch, K., 1 Mahony, P., 3 management of learning, 23 Marshall, B. et al., 48 Mason, J., 36 Masters, G., 175 mathematical challenge, 23 mathematical thinking, 11–17; abstraction, 14; awareness, 58–59; cognitive skills, 14, 18–19, 31; development of, 16–17; flexible thinking, 11, 12, 14, 22; generalization, 14, 27, 29; investigation, 23; language, 15, 23–24, 32–33, 69; mathematical discourse, 23; metacognitive aspects, 15; metaconcepts, 40, 41, 52, 58–59; procepts, 12; reflection, 42, 45–46, 58, 124; sense making, 13, 14, 18, 68; socialization, 15;

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mathematical thinking (continued) sociomathematical norms, 94–95, 122. See also knowledge; learning theories; student-teacher interactions; understanding mathematics education research, 149, 150–51 Matre, S., 87 McClain, K., 91–111, 202–3 McClintock, B., 2 McIntyre, D., 47, 49, 56 Mead, G. H., 71 Mellin-Olsen, S., 40, 41, 44–45, 57, 121 member checking, 51 methodology, 1–10; choice of, 1; definitions, 1, 83; design, methods and analysis, 6–9; epistemological issues, 1, 6, 8; objectivity, 2, 3, 6, 8–9; openness and reflection, 9; operationalization, 4; a position on, 3–6; research design, 4–5, 6–7, 8; research reporting, 5–6, 206–7; values and beliefs, 1–3, 4, 6, 7–8, 206–7 methods, 1, 202, 206 Mewborn, D. S., 125 Miles, M. B., 53, 73–74 modeling: definition, 14 Molander, B., 75 motivation, 159–60, 169, 205. See also student motivation and achievement National Research Council, 14 Neisser, U., 68 Netherlands: Realistic Mathematics Education (RME), 92, 95–96, 114, 202–3; TAL-project, 115 Nicholls, J. G. et al., 169 Nolen-Hoeksema, S. et al., 161–62 Norway: classroom research, 78–79; SEN provision, 70 objectivity, 2, 3, 6, 8–9 O’Connor, M. C., 18–19 orienting schema, 68

Index

parallelism, 25–27 path analysis, 165–66, 178, 186–87. See also PLSPATH analysis with latent variables PATMaths. See Progressive Achievement Tests in Mathematics Piaget, J., 124, 136 PLSPATH analysis with latent variables, 165-66, 178–82, 180f Popper, K. R., 157–58 P-rationale, 45 Preissle, J., 88 procepts, 12 Progressive Achievement Tests in Mathematics (PATMaths), 165, 167, 168, 174, 190 proofs and demonstrations, 14 quadrilaterals, 27–29, 28f, 29f qualitative studies: data interpretation, 79–81, 83, 85; particular description and general description, 131; validity, 80, 133, 148. See also special educational needs; student-teacher interactions; student-teaching practicum quantitative studies, 158–59, 175, 205–6. See also student motivation and achievement questionnaires: on achievement, 167–68, 206; analysis, 175–76; on explanatory style, 168; on motivation, 169, 205; on psychological disposition,158, 167, 170; scoring, 174; topaz effect, 48 Raeithel, A., 42 Rasch analysis, 165, 175–76, 187–90 rationales for learning: E-rationale, 82–83, 87; I-rationales, 45, 57, 82, 83, 87; P-rationale, 45; and SEN, 72, 73; S-rationale, 44–45, 57 Realistic Mathematics Education (RME), 92, 95–96, 114, 202–3 reflection, 9, 42, 45–46, 58, 124 relational understanding, 106, 121, 122

Index

research, mathematics-education related, 149, 150–51 research design: dialectical theorybuilding, 5; empowerment, 4–5, 8; informed consent, 5; sensitive issues, 6–7; validity, 5, 7 research reporting, 5–6, 206–7 research tools. See conversation as research tool; interviews; questionnaires Rittenhouse, P. S., 23–24 RME. See Realistic Mathematics Education Rosenthal, R., 206 Royal Ministry of Education, Research and Church Affairs (Norway), 70 rules and algorithms, 12–13, 58, 106 Schoenfeld, A. H., 11, 13, 16 Schön, D. A., 124 Seligman, M. E. P., 160–61 Seligman, M. E. P. et al., 167, 168 SEN. See special educational needs setting, 43, 45, 61–62 n3 Shane, R., 119–49, 203–5 Shulman, L. S., 123 Simon, H. A., 48 Simon, M. A., 95, 106, 128, 129 Skemp, R. R., 41, 44, 106, 121–22 social-constructivist theory, 124 sociocultural theory, 41, 42–43, 124 sociomathematical norms, 94–95, 122 special educational needs (SEN), 67–88 background for study, 68–70; Norwegian framework, 70, 78–79; pupils’ experience of SEN, 68–69, 72, 77 theoretical framework, 70–72 research questions, 72–73 conversation as research tool, 73–78; authenticity of evidence, 74–76; ethical perspective, 76; anticipating contradictions, 76–78 the study, 78–79 interpretation of qualitative data, 79–81, 83, 85 outcomes, 81–83 final considerations, 83–84

213

reaction, 84–87; minimization of contamination, 85, 87–88; surfacing and redirection, 86; response to, 87–88 review, 202 S-rationale, 44–45, 57 Stewart, A., 8 Strauss, A. L., 52, 107 Student Behavior Checklist, 162, 167, 170–71, 172, 175, 189, 190 student motivation and achievement, 155–90 theoretical framework, 157–63; earlier research, 156, 159, 183; explanatory style, 156, 160–62; goals, optimism and pessimism, 156, 159–60; motivation, 159–60; psychological disposition of students, 158; quantitative studies, 158–59, 205–6; summary, 163; teacher perceptions, 162–63; worlds of inquiry, 157–58 research design, 156–57, 163–73; achievement scale, 167–68; depression, 170; ethical considerations, 171–72; explanatory style, 168–69; funding, 172; motivation, 169; nature and size of sample, 166, 167t; research instruments, 166, 167, 184, 205; research plan, 164–66, 165f; teacher rating scale, 170–71 data collection methods, 172–73 methodological issues, 173–74 data analysis, 175–82; Rasch analysis, 175–76; scoring of student responses, 176–77; PLSPATH analysis with latent variables, 177–82, 180f, 181t discussion, 182–84; generalizability and usefulness, 183–84 appendix, 187–90; Rasch model, 187–90; calibration of instruments, 189–90; unidimensionality of instruments, 188–89 reaction, 185–87; main strengths, 185–86; possible alternative approaches, 186–87 review, 205–6

214

students’ goals in classroom activity, 39–62 background of researcher, 40–41, 46 theoretical framework, 41–46; theories of learning, 41–44; goals, 44–46, 47 method: ethnographic style, 44, 46–47; implementation, 49–52; research instrument, 47–49 analysis: data searching, 52–53; validation, 53 personal critique, 53–56; disturbance, 54; ethical dimension, 55–56, 60; lack of structure, 54–55; researcher fatigue, 55 summary, 56–59 reaction, 59–61 review, 201 student-teacher interactions, 11–37 theoretical framework: earlier research, 16–17; mathematical thinking, 12–17 research design, 17–22; beliefs and values, 17–19; classroom research, 19–22, 19f, 35, 37; limitations, 32 analysis of data, 22–25; examples, 25–30; positive/negative interventions, 24–25; productive/nonproductive dialogues, 25, 30, 36, 200; sensitivity to students, 23, 24; stepping in/stepping out, 24 final considerations, 30–32; assessment instruments, 31–32 reaction, 32–36; common experience, 36; generality/usefulness, 33, 36, 37; humility, 32–33; interpretation of data, 34, 37; problems, 34–36; response to, 36–37; validity/credibility, 34, 36 review, 199–201 student-teaching practicum: context and content, 119–51 theoretical background, 121–27; knowledge acquisition, 123–24, 204; knowledge of mathematics, 121–23; knowledge for teaching mathematics, 123; student teachers’ learning, 124–25; summary, 126–27

Index

research design, 120, 127–29; data collection and analysis, 129; “teacher development experiment,” 128–29 classroom culture, 120, 129–35; context, 134–35; data analysis, 131–35; general description, 131–33, 133t; particular description, 133–35; videotape data collection, 129–30 children’s construction of knowledge, 135–40; data analysis, 138–40, 139t; task-based interviews, 136–38; written tests, 136 student teachers’ construction of knowledge, 140–47; reflective interviews, 140–42, 141t, 143t; data analysis, 142, 143–44; discussion, 144–47 results: conclusion, 148–49; discussion, 147; methodological issues and implications, 148 reaction, 149–51 review, 203–5 symbolism: definition, 14 Tall, D. O., 11, 12, 13, 14 TAL-project, 115 “teacher development experiment,” 128–29 teacher education. See student-teaching practicum teachers: perceptions, 162–63; role of, 23, 31. See also special educational needs; student motivation and achievement; student-teacher interactions; student-teaching practicum teaching and learning: existing research, 40; instructional sequences, 99, 100–102, 109–10; instructional theory, 95–96, 97; intended and received lessons, 83; learning support, 71; learning trajectories, 98–99, 100f, 106; management of learning, 23; meta-learning, 58–59, 73; participation, 71–72; rationales, 44–45, 57, 72, 73, 82–83, 87; reflection, 9, 42, 45–46, 58, 124;

Index

teaching and learning: existing research (continued) traditional vs. innovative, 17, 120. See also activity theory; special educational needs; student-teacher interactions; student-teaching practicum thinking skills: autonomous thinking, 14; flexible thinking, 11, 12, 14, 22. See also mathematical thinking; reflection Torbert, W. R., 7 trustworthiness. See validity/credibility Tzur, R., 128, 129 understanding: instrumental, 121, 122; pretending and, 69; and rationale for learning, 73; relational, 106, 121, 122; sense making, 13, 14, 18, 68. See also knowledge; mathematical thinking United States: instructional approaches, 97; National Academy of Sciences, 2; reform recommendations, 106 validity/credibility: qualitative studies, 80, 133, 148; research design,

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validity/credibility (continued) 5, 7; in studies reported, 34, 36, 53, 108 values and beliefs: and methodology, 1–3, 4, 6, 7–8, 206–7; in studies reported, 17–19, 78, 201, 204 Van den Heuvel-Panhuizen, M., 111–15, 203 Vestby, G., 76 videotape data collection, 129–30 Vygotsky, L. S., 15, 18–19, 41, 42, 71, 72, 124 Watson, A., 32–36, 37, 201 Wearne, D., 121, 129, 131, 132 Whitenack, J., 107, 133 worlds of inquiry, 157–58, 184 Wright, B. D., 175 Yackel, E., 94 Yackel, E. et al., 106 Yates, S. M., 155–84, 187–90, 205–6 Your Feelings in Mathematics: A Questionnaire, 167, 169, 174, 175, 189 zone of proximal development, 42, 71

About the Contributors

LEONE BURTON is Emeritus Professor of Mathematics Education at the University of Birmingham, and currently, Visiting Professor at King’s College, London. MARIA MANUELA DAVID is Professora Adjunto da Faculdade de Educação at the Universidade Federal de Minas Gerais, Brasil. LYN ENGLISH is Professor of Mathematics Education at Queensland University of Technology, Brisbane, Australia. Lyn English is also Editor of the international journal Mathematical Thinking and Learning. SIMON GOODCHILD is Senior Lecturer in Mathematics Education at The College of St. Mark and St. John, Plymouth, UK. MARJA VAN DEN HEUVEL-PANHUIZEN is Senior Researcher in Mathematics Education at The Freudenthal Institute, Utrecht University, Netherlands. BARBARA JAWORSKI is Reader in Mathematics Education at the University of Oxford, UK. Barbara Jaworski is also Editor in Chief of the international Journal of Mathematics Teacher Education. CYRIL JULIE is Professor of Mathematics Education at the University of the Western Cape, Cape Town, South Africa. WENDY KEYS was, until her recent retirement, a Principal Research Officer and Deputy Head of Professional and Curriculum Studies at the National Foundation for Educational Research (NFER), UK. GILAH C. LEDER is Director of the Institute for Advanced Study and Professor at the Institute for Education, La Trobe University Bundoora, Victoria, Australia. NORA LINDEN is Senior Lecturer at the Bergen University College, Norway.

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About the Contributors

MARIA DA PENHA LOPES is Professora do Departamento de Matemática in the Faculdade de Ciências Humanas de Pedro Leopoldo, Minas Gerais, Brasil. KAY MCCLAIN is Assistant Professor of Mathematics Education at Vanderbilt University, Nashville, Tennessee. RUTH SHANE is Chair of Elementary Mathematics Education at Kaye College of Education, Beersheva, Israel. ANNE WATSON is Lecturer in Mathematics Education in the Department of Educational Studies, University of Oxford, UK. SHIRLEY M. YATES is Senior Lecturer in Education at Flinders University, Adelaide, Australia.