DOCUMENT RESUME ED 378 042 SE 055 587 AUTHOR Ernest, Paul, Ed. TITLE Mathematics, Education, and Philosophy: An International Perspecti.ve. Studies in Mathematics Education Series: 3. REPORT NO ISBN-0-7507-0290-7 PUB DATE 94 NOTE 258p.; For the companion volume, "Constructing Mathematical Knowledge," see SE 055 588. AVAILABLE FROMFalmer Press, Taylor & Francis Inc., 1900 Frost Road, Suite 101, Bristol, PA 19007. PUB TYPE Books (010) Collected Works General (020) EDRS PRICE MF01/PC11 Plus Postage. DESCRIPTORS *Cognitive Structures; *Cultural Context; Elementary Secondary Education; *Mathematics Education; Modernism; *Philosophy IDENTIFIERS *Postmodernism; *Poststructuralism ABSTRACT This book illustrates the breadth of theoretical and philosophical perspectives that can be brought to bear on mathematics and education. Part 1, "Reconceptualizing the Philosophy of Mathematics," contains the following chapters:(1) "Fresh Breezes in the Philosophy of Mathematics" (R. Hersh);(2) What Can the Sociologist of Knowledge Say About 2 2 = 4?" (D. Bloor);(3) "The Dialogical Nature of Mathematics" (P. Ernest); and (4) "Structuralism and Post-modernism in the Philosophy of Mathematics" (T. Tymoczko). Part 2, "Post-modernist and Post-structuralist Approaches," contains: (5) "Reasoning in a Post-modern Age" (V. Walkerdine); (6) "Mathematical Writing, Thinking, and Virtual Reality" (B. Rotman); (7) "Mathematics: The Problematical Notion of Closure" (A. Tsatsaroni & J. Evans);(8) "On the Ecologies of Mathematical Language and the Rhythms of the Earth" (D. W. Jardine); (9) "Discursive Saturation and SchoA. Mathematics Texts: A Strand from a Language of Description" (P. Dowling);(10) "The Dominance of Structure in 'Post-structural' Critiques of Mathematics Education" (J. Vass); ana (11) "Describing the Mathematics You Are Part Of: A Post-structuralist Account of Mathematical Learning" (T. Brown). Part 3, "The Human Face of Mathematics," contains: (12) "Mathematics and Art: Cold Calipers Against Warm Flesh?" (P. J. Davis); (13) "Skolem and Godel" (H. Wang); and (14) "Different Ways of Knowing: Contrasting Styles of Argument in Indian and Greek Mathematical Traditions" (G. G. Joseph). The final section, Part 4, "The Social Context of Mathematics and Education," contains:(15) "The Social Life of Mathematics" (S. Restivo);(16) "Research in Gender and Mathematics Education: The Production of Difference" (M. Dunne & J. Johnston); and (17) "Ethno-mathematics, the Nature of Mathematics and Mathematics . Education" (U. D'Ambrosio). Contains references with each chapter and a subject index. (MKR) ************k********************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. *********************************************************************** ...at,ieatics, w and Philosophy: An International Perspective U.S. DEPARTMENT OF EDUCATION Othce of Educational Research and Improvement OUCATIONAL RESOURCES INFORMATION CENTER tERIC) ThIS doCument has been reproduced as received horn the person or ortjantzahon ortginahng rt 0 Mrnor changes have been made to improve reproduction rsuaidy Points of view or opmions staled in thus docu. ment do not necessanty represent ottictai OE RI pOSittOn or polity "PERMISSION TO REPRODUCE THIS MP.TERIAL HAS BEEN GRANTED BY 5ve -ST2IES PrOb TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC)." The Falmer Press 2 BEST COPY AVAILABLE Mathematics, Education and Philosophy Studies in Mathematics Education Series Series Editor: Dr Paul Ernest School of Education University of Exeter Exeter, UK. 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 Paul Ernest 5 Investigating Mathematics Teaching: A Constructivist Enquiry Barbara Jaworski 6Radical Constructivisni: A W..y of Knowing and Learning Ernst von Glascrsfeld Studies in Mathematics Education Series: 3 Mathematics, Education and Philosophy: An International Perspective Edited by Paul Ernest The Falmer Press (A member of the Taylor & Francis Group) London Washington, D.C. J 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 0 Paul 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 0290 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 sp.ecified pH value on final paper manufacture of not less than 7.5 and is therefore 'acid free'. (:, Contents List of Figures and Tables vii Preface by Series Editor viii Introduction 1 Part 1:Reconceptualizing the Philosophy of Mathematics 9 Chapter 1 Fresh Breezes in the Philosophy of Mathematics 11 Reuben Hersh Chapter 2 What Can the Sociologist of Knowledge Say About 2 + 2 = 4? 21 David Bloor Chapter 3 The Dialogical Nature of Mathematics 33 Paul Ernest Chapter 4 Structuralism and Post-modernism in the Philosophy of Mathematics 49 Thomas Tymoczko Part 2:Post-modernist and Post-structuralist Approaches 57 Chapter 5 Reasoning in a Post-modern Age 61 Valerie Walkerdine Chapter 6 Mathematical Writing, Thinking, and Virtual Reaity 76 Brian Rotman Chapter 7 Mathematics: The Problematical Notion of Closure 87 Anna Tsatsaroni and Jeff Evans Chapter 8 On the Ecologies of Mathematical Language and the Rhythms of the Earth 109 David W. Jardine Chapter 9 Discursive Saturation and School Mathematics Texts: A Strand from a Language of !Description 124 Paul Dowling Chapter 10The Dominance of Structure in 'Post-structural' Critiques of Mathematics Education 143 Jeff Vass Chapter 11Describing the Mathematics You Are Part Of: A Post-structuralist Account of Mathematical Learning 154 Tony Brown Contents Part 3:The Human Face of Mathematics 163 Chapter 12Mathematics and Art: Cold Calipers Against Warm Flesh? 165 Philip J. Davis Chapter 13Skolem and Godel 184 Hao Wang Chapter 14Different Ways of Knowing: Contrasting Styles of Argument in Indian and Greek Mathematical Traditions 194 George Gheverghese Joseph Part 4:The Social Context of Mathematics and Education 205 Chapter 15The Social Life of Mathematics 209 Sal Restivo Chapter 16Research in Gender and Mathematics Education: The Production of Difference 221 Mairead Dunne and Jayne Johnston Chapter 17Ethno-mathematics, the Nature of Mathematics and Mathematics Education 230 U biratan D'Antbrosio Notes on Contributors 243 Index 245 vi List of Figures and Tables Figure 2.1 Visual Proof of 2 + 2 = 4 23 Figure 2.2 Visual Proof of 2 + 2 + 2 = 4 23 Figure 2.3 The Proof of 2 + 2 = 4 Discussed by J.L. Mackie 27 Figure 3.1 The Creative/Reproductive Cycle of Mathematics 45 Figure 6.1 The Model of Mathematical Discourse and its Users 78 Figure 6.2 Types of Inscription and Their Relations 80 Figure 7.1 Question 3 in the Interview: The London Gold Price 23 January 1980 95 Figure 9.1 Structure and Application of a Language of Description 125 Figure 9.2 SMP Book. G2, pp. 20-1 133 Figure 9.3 SMP Book Y1, p. 87 135 Figure 14.1Illustration of the Upapatti for the Avyakta Method 199 Figure 14.2Illustration of the Upapatti for the Kshetragata Method 199 Table 3.1: Examples of Dialectical Concepts in Mathematical Topics 39 Table 3.2: The Cyclic Form of Lakatos' Logic of Mathematical Discovery 42 Table 3.3: The Dialectical Form of the Generalized Logic of Mathematical Discovery 43 vii 9 Preface by Series Editor Mathematics education is established worldwide as a majorarea of study, with numerous dedicated journals and conferences serving national and international communities of scholars. Research in mathematics education is becomingmore theoretically 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 ofre- search contributions to the field based on disciplined perspectives that link theoty 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 theoretical perspectives in mathematics educa- tion as well as their critical scrutiny. It aims to havea major impact on the development of mathematics education as a field of study into the twenty-first century. Unusually for the series this book and its companion volumeare edited col- lections. Instead of the sharply focused concerns ofa research monograph the books offer a panorama of complementary and forward looking perspectives.In the