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Investigating Mathematics Teaching: a Constructivist Enquiry. Studies In DOCUMENT RESUME ED 381 350 SE 056 047 AUTHOR Jaworski, Barbara TITLE Investigating Mathematics Ter..ching: A Constructivist Enquiry. Studies in Mathematics Education Series: 5. REPORT. NO ISBN-0-7507-0373-3 PUB DATE 94 NOTE 250p.;. Part 5 of Studies in Mathematics Education Series AVAILABLE FROMFalmer Press, Taylor & Francis Inc., 1900 Frost Road, Suite 101, Bristol, PA 19007. PUB TYPE Books (010) Reports Research/Technical (143) Viewpoints (Opinion/Position Papers, Essays, etc.) (120) EDRS PRICE MF01/PC10 Plus Postage. DESCRIPTORS Case Studies; *Constructivism (Learning); Elementary Secondary Education; *Ethnography; Foreign Countries; *Mathematics Instruction; *Mathematics Teachers; Models; *Research Methodology IDENTIFIERS *Mathematics EducationResearch; Soci..,1 Constructivism; *Teacher Researcher Relationship ABSTRACT This book is about mathematics, teaching and learning and how these are affected by a constructivist philosophy. Chapter 1 presents background to the study of an investigative approach to mathematics teaching. Chapter 2 provides an account of constructivism as a philosophy of knowledge and learning and includes a rationale for both radical and social constructivism and relationships between constructivism and knowl-.dge communication, and the classroom. Chapter 3 presents the author's early thinking about an investigative approach to mathematics teaching. Chapter 4 examines the choice of an ethnographic approach to research through participant observation and informal interviewing and also examines issues that arise from an interpretivist analysis. Chapters 6, 7, and 9 are case studies detailing a characterization of the teaching and thinking of three teachers and the development of theory through analysis of observations. Included is a descriptive model, the Teaching Triad, which arose from the categorization of data. Chapters 5 and 8 offer interludes of research reflections. Chapter 10 presents a -ynthesis of general cnncepts in terms of characteristics of an inve.,!*igative approach and tensions that it raises for teachers in its classroom implementation. Chapter 11 offers a characterization of the teacher-researcher relationship which includes a descriptive model for reflective practice. Finally, Chapter 12 reexamines the relevance of constructivism to research outcomes in light of questions about the adequacy of radical constructivism to explain the complexities of classroom interaction. Each chapter contains notes. (Contains 221 references.) (MKR) id.********************************************************************* Reproductions supplied by EDRS are the best that can be made * from the original document. * *********************************************************************** In ti tines eM 00 t Teaching A Constructivist Enquiry Barbara Jaworski U DEPARTMENT OF EDUCATOR THIS Office ot Educations, Research ' PERMISSION TO REPRODUCE and Improvement MATERIAL HAS BEEN GRANTEDBY EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC) AD!This document has beenreproduced as eceived 'nom the person or organization originating it O Min°, changes have boonmuds to amprovo rINOOtaction c MY PointsI sew or opinions stated in the doct mint do not nocesaaMy represent&kat TO THE EDUCATIONALRESOURCES OERI poirtionOf pOliCY INFORMATION CENTER (ERIC)" to rti;The Falmer Press u.J 2 BEST CCPY AVAILABLE Investigating Mathematics Teaching Studies in I Aathematics Education Series Series Editor Paul Ernest School of Education University of Exeter Exeter 1 The Philosophy of Mathematics Education Paul Ernest 2Understanding in Mathematics Anna Sierpinska 3Mathematics, Education and Philosophy Edited by Paul Ernest 4Constructing Mathematical Knowledge: Epistemology and Mathematics Edited by Paul Ernest 5 Investigating Mathematics Teaching: A Constructivist Enquiry Barbara Jaworski 6Radical Constructivism: A Way of Knowing and Learning Ernst von Glasersfeld Studies in Mathematics Education Series: 5 Investigating Mathematics Teaching: A Constructivist Enquiry Barbara Jaworski The Falmer Press (A member of the Taylor & Francis Group) LondonWashington, D.C. UK The Falmer Press, 4 John Street, London WC1N 2ET USA The Falmer Press, Taylot & Francis Inc., 1900 Frost Road, Suite 101, Bristol, PA 19007 0 B. Jaworski, 1994 All rights resetved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mech- anical, phc:ocopying, recording or otherwise, without permission in writing from the Publisher. First published in 1994 A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data are available on request ISBN 0 7507 0373 3 cased Jacket design by Caroline Archer Typeset in 9.5/11 pt Betnbo by Graphicraft Typesetters Ltd., Hong Kong. Printed in Great Britain by Burgess Science Press, Basingstoke on paper which has a specified pH value on final paper manufacture of not less than 7.5 and is therefore 'acid free'. Tc John and to my parents Gladys and Gilbert Whitlow Th Contents Dedication Preface by Series Editor xi Preface xiii Acknowledgments xvi Epigraph xvii Chapter 1An Investigative Approach: Why and How? 1 Personally and Historically 1 The Origin_ of Investigations 2 The Purposes of Investigations 4 The Place of Investigational Work in the Mathematics Curriculum 6 Classroom Approaches 8 An Investigative Approach to Mathematics Teaching 10 Chapter 2Constructivism: A Philosophy of Knowledge and Learning 14 Introduction 14 Radical Constructivism 15 Radical Constructivism and Knowledge 18 Constructivism, Meaning and Communication 20 Social Constructivism 24 Constructivism and the Classroom 25 Piaget and Vygotsky 25 Consequences for Teaching The Social and Cultural Environment 28 Challenges to Constructivism 29 The Learning Paradox 29 Constructivism and Pedagogy 31 Constructivism and Ontology 34 Chapter 3Working with Two Teachers: Darning the Study 36 Tentative Beginnings 36 An Outline of my First Phase of Research 37 The Early Observations 38 Emergence of Field Notes 41 Observing my Own Teaching 44 vii Contents Participant Observation and Analysis 47 A Lesson on Tessellations 48 My Immediate Account of Reflections on the Pair of Tessellations Lessons 51 Later Reflections on this Account 52 Conclusion 55 Chapter 4 The Research Process 59 Introduction 59 A Very Brief Overview 60 The Choice of an Ethnographic Approach 60 Focus and Emphasis of the Researcher 62 Data Collection 63 Interpretive Enquiry 65 Data Analysis 68 Significance 69 The Place of Theory in Ethnographic Research 70 Validation and Rigour 75 A Constructivist Perspective 77 Chapter 5Interlude 1:From Phase I to Phase 2 82 Theory Validation to Theory Construction 82 The 'Fit' with Radical Constructivism 83 Relating Constructivism to Teacher development 86 Implications for Phase 2 87 Chapter 6Clare: Origins of the Teaching Triad 88 Background 88 The School and the Mathematics Department 89 Analysis of Clare's Autumn-term Lessons 90 Management of Learning 90 Categorization of Data 90 Identifying Management of Learning 90 Further Manifestations of Management of Learning 92 Sensitivity to Students 94 Mathematical Challenge 95 An Investigative Approach? 96 Analysis of the Spring-term Lessons 97 The Packaging Lesson 97 The Lines-crossing Lesson 102 Teacher's and Students' views 104 Characterizing Clare's Teaching 107 Chapter 7Mike: Significant Episodes and the Teaching Triad 112 Background 112 The Teaching Triad Related to Mike's Teaching 113 Management of Learning 113 Sensitivity to Students 117 Mathematical Challenge 123 Teacher's and Students' Views 126 viii D Contents Tensions for the Teacher 130 Balance: Management and Control, Sensitivity and Challenge 131 Chapter 8Interlude 2: From Phase 2 to Phase 3 134 Teacher and Researcher Awareness 134 Recognition of an Issue: The Teacher's Dilemma 136 The Researcher's Dilemma 138 Significance 139 Implications for Phase 3 141 Chapter 9Ben: Affirming theTeaching Triad 143 Background 143 The Teaching Triad 144 The 'Moving Squares' Lesson 146 Creating an Environmentfor Thinking and Involvement 146 The Lesson Opening 147 Working on the Task 152 An Investigative Lesson Why? 155 Didactic Versus Investigative Teaching 156 The Vectors Lesson 159 The Lesson Opening 160 Working on the Task 165 A Didactic Lesson Why? 168 Theoretical Considerations 169 Chapter 10Investigative Mathematics Teaching: Characteristics and Tensions 170 Characteristics of Classrooms Observed 170 Teaching Acts: Some Common Themes 170 Characterizing the Teaching Role: Teaching Outcomes 175 Establishing Mathematical Meaning 176 Engendering Mutual Trust and Respect 176 Encouraging Responsibility for Own Learning 177 Tensions 178 The Teacher's Dilemma 178 The Didactic-Constructivist Tension 179 The Didactic Tension 180 The Double Dialectic 181 A Counter-example 182 Chapter 11Reflection and Development Thinking and Reflection Some Starting Points 186 The Teacher-Researcher Relationship 188 Stage 1: Reflecting 190 Stage 2: Accounting for 192 Stage 3: Critical Analysis 194 Development of Teaching: Teaching Knowledge and Teaching Wisdom 196 Researcher Perceptions 197 Reflecting on the Conceptual Model 198 The Epistemology of Practice 199 ix 10 1 Contents Supporting the Reflective Teacher 200 My Own Development as a Reflective Practitioner 202 Reflective Practice Is 'Critical' and Demands 'Action' 203 Investigative Teaching 204 Chapter 12Epilogue 206 Theory and Practice 206 The Problematic Position of Classroom Knowledge 206 Constructivism and its
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