DOCUMENT RESUME ED 378 043 SE 055 588 AUTHOR Ernest, Paul, Ed. TITLE Constructing Mathematical Knowledge: Epistemology and Mathematics Education. Studies in Mathematics Education Series: 4. REPORT NO ISBN-0-7507-0354-7 PUB DATE 94 NOTE 297p.; For the companion volume, "Mathematics, Education, and Philosophy: An International Perspective," see SE 055 587. AVAILABLE FROMFalmer Press, Taylor & Francis Inc., 1900 Frost Road, Suite 101, Bristol, PA 19007. PUB TYPE Books (010) Collected Works General (020) EDRS PRICE MFO1 /PC12 Plus Postage. DESCRIPTORS *Constructivism (Learning); Elementary Secondary Education; *Epistemology; *Hermeneutics; *History; *Mathematics Education; *Psychology ABSTRACT This book illustrates the breadth of theoretical and philosophical perspectives that can be brought to bear on mathematics and education. Part 1, "Constructivism and the Learning of Mathematics," contains the following chapters:(1) "A Radical Constructivist View of Basic Mathematical Concepts" (E. von Glasersfeld);(2) "Interaction and Children's Mathematics" (L. P. Steffe & R. Tzur);(3) "Radical Constructive Criticisms of von Glasersfeld's Radical Constructivism" (R. S. D. Thomas);(4) "Articulating Theories of Mathematics Learning" (S. Lerman);(5) "Is Radical Constructivism Coherent?" (M. Otte);(6) "Social Constructivism and the Psychology of Mathematics Education" (P. Ernest); (7) "Mathematics, Computers and People: Individual and Social Perspectives" (E. Smith); and (8) "The Context of Cognition: The Challenge of Technology" (K. Crawford). Part 2, "Psychology, Epistemology and Hermeneutics," contains:(9) "Another Psychology of Mathematics Education" (D. Pimm);(10) "On Interpretation" (D. Tahta);(11) "Potent'ial Space and Mathematical Reality" (P. Maher); (12) "Towards a Hermeneutical Understanding of Mathematics and Mathematical Learning" (T. Brown); and (13) "The Myth of Mathematics" (F. Seeger & H. Steinbring). Part 3, "Enquiry in Mathematics Education," contains: (14) "The Problem of the Problem and Curriculum Fallacies" (S.I. Brown); (15) "Enquiry in Mathematics and in Mathematics Education" (J. Mason);(16) "Demystifying Mathematics Education through Inquiry" (M. Siegel & R. Borasi); and (17) "Reading to Learn Mathematics in the Primary Age Range" (C. W. Desforges & S. Bristow). The final section, Part 4, "History, Mathematics and Education," contains: (18) "The Idea of 'Revolution' As an Instrument for the Study of the Development of Mathematics and Its Application to Education" (F. Speranza); and (19) "Mathematical Practices, Anomalies and Classroom Communication Problems" (A. Sfard). Contains references with each chapter and a subject index. (MKR) nstru tin Mati.ematical Knowledge: Epistemology and Mathematical Education U.S DtPARTMENT OF EDUCATION Office of Educational Research and Improvement DUCATIONAL RESOURCES INFORMATIOh CENTER IERIC) The document haS en reproduced as received from the person or organization originating A C Minor changes have been made to improve reproductIon quahlY "PERMISSION TO REPRODUCE THIS Points& new or opinions staled in thiocu. went do not necessarily representofsdfal MATERIAL HAS BEEN GRANTED BY OERI positron or policy a U RI TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC)." Edited by AM Paul Ernest BEST COPYAVAILABLE The Fahner Press I* Constructing Mathematical Knowledge 1 Studies in Mathematics 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: An International Perspective Edited by Paul Ernest 4Constructing Mathematical Knowledge: Epistemology and Mathematics Education Edited by Pau! Ernest 5 Investigating Mathematics Teaching: A Constructivist Enquiry Barbara Jaworski 6Radical Constructivism: A Way of Knowing and Learning Ernst von Glasersfeld 4 Studies in Mathematics Education Series: 4 Constructing Mathematical Knowledge: Epistemology and Mathematics Education Edited by Paul Ernest The Faimer Press (A member of the Taylor & Francis Group) LondonWashington, D.C. r1; UK The Falmer Press, 4 John Street, London WC1N 2ET USA The Falmer Press, Taylor & Francis Inc., 1900 Frost Road, Suite 101, Bristol, PA 19007 P. Ernest 1994 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, 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 0354 7 cased. Jacket design by Caroline Archer Typeset in 9.5/11pt Bembo 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 thertftre 'acid free'. Contents Series Editor's Preface Introduction Part 1Constructivism and the Learning of Mathematics 1 Chapter 1 A Radical Constructivist View of Basic Mathematical Concepts 5 Ernst von Glasersfeld Chapter 2 Interaction and Children's Mathematics 8 Leslie P. Steffe and Ron Tzur Chapter 3 Radical Constructive Criticisms of von Glasersfeld's Radical Constructivism 33 Robert S.D. Thomas Chapter 4 Articulating Theories of Mathematics Learning 41 Stephen Lerman Chapter 5 Is Radical Constructivism Coherent? 50 Michael Otte Chapter 6 Social Constructivism and the Psychology of Mathematics Education 62 Paul Ernest Chapter 7 Mathematics, Computers and People: Individual and Social Perspectives 73 Erick Smith Chapter 8 The Context of Cognition: The Challenge of Technology 92 Kathryn Crawford Part 2Psychology, Epistemology and Hermeneutics 107 Chapter 9 Another Psychology of Mathematics Education 111 David Pimm Chapter 10On Interpretation 125 Dick Tahta Chapter 11Potential Space and Mathematical Reality 134 Philip Maher Chapter 12Towards a Hermeneutical Understanding of Mathematics and Mathematical Learning 141 Tony Brown V sit Constructing Mathematical Knowledge Chapter 13The Myth of Mathematics 151 Falk Seeger and Heinz Steinbring Part 3Enquiry in Mathematics Education 171 Chapter 14The Problem of the Problem and Curriculum Fallacies 175 Stephen I. Brown Chapter 15Enquiry in Mathematics and in Mathematics Education 190 John Mason Chapter 16Demystifying Mathematics Education through Inquiry 201 Marjorie Siegel and Raffaella Borasi Chapter 17Reading to Learn Mathematics in the Primary Age Range 215 Charles W. Desforges and Stephen Bristow Part 4History, Mathematics and Education 237 Chapter 18The Idea of 'Revolution' As an Instrument for the Study of the Development of Mathematics and Its Application to Education 241 Francesco Speranza Chapter 19Mathematical Practices, Anomalies and Classroom Communication Problems 248 Anna Sfard Notes on Contributors 274 Index 276 List of Tables and Figures Table 17.1 Incidences of Different Purposes Expressed as Pupils Engaged with the Maths Packs 228 Table 17.2 Levels of Constructive Activity in Learning from Text 231 Table 17.3 Incidence of Levels of Constructive Activity in Response to Text in Science and in Area 232 Figure 2.1 The Computer Screen of Sticks 14 Figure 2.2 The Screen after Arthur's Attempt 17 Figure 2.3 The Screen during Nathan's Attempt 18 Figure 2.4 The Srreen after Arthur Had Used 'Pull Parts' 22 Figure 2.5 Computer Screen after Nathan's and Arthur's Work 23 Figure 2.6 Social Interaction, Learning and Development Figure 2.7 Types of Interaction 29 Figure 7.1 Individual Constructivist Perspective 78 Figure 7.2 Social Constructivist Perspective 79 Figure 7.3 Computer from Social Constructivist Perspective 86 Figure 7.4 Computer from an Individual Constructivist Perspective 87 Figure 13.1Measuring the Height of a Tree 154 Figure 14.1The Standard Problem-solving Paradigm 177 Figure 14.2 A Reversal of the Standard Problem-solving Paradigm 179 Figure 14.3A Summary of Some Non-standard 'Problem' paradigms 182 Figure 17.1A Sheet front the Averages Pack 218 Figure 17.2 A Sheet from the Fractions Pack 219 Figure 17.3Full Learning Pack for Area 220 Figure 17.4An Extract from the Fractions Learning Pack 224 Figure 17.5An Extract from the Fractions Learning Pack 224 Figure 17.6Examples of the Science Text Presented to the Pupils 229 Vii t.) Preface by Series Editor Mathematics education is established worldwide as a major area of study, with numerous dedicated journals and conferences serving national and international com- munities of scholars. Research in mathematics education is becoming more theoret- ically orientated. Vigorous new perspectives are pervading it from disciplines and fields as diverse as psychology, philosophy, logic, sociology, anthropology, history, feminism, cognitive science, semiotics, hermeneutics, post-structuralism and post- modernism. The series Studies in Mathematics Education consists of research contri- butions to the field based on disciplined perspectives that link theory with practice. It is founded on the philosophy that theory is the practitioner's most powerful tool in understanding and changing practice. Whether the practice is mathematics teaching, teacher education, or educational research, the series intends to offer new perspectives to assist in clarifying and posing problems and to stimulate debate. The series Studies in Mathematics Education will encourage the development and dissemination of