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Meaning in Mathematics Education (Mathematics Education Library) PDF

262 Pages·2005·6.01 MB·English
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Meaning in Mathematics Education a - Springer Mathematics Education Library VOLUME 37 Managing Editor A.J. Bishop, Monash University, Melbourne, Australia Editorial Board J.P. Becker, Illinois, U.S.A. C. Keitel, Berlin, Germany G. Leder, Melbourne, Australia A. Sfard, Ha@, Israel 0. Skovsmose, Aalborg, Denmark S. Turnau, Rzeszbw, Poland The titles published in this series are listed at the end of this volume. Jeremy Kilpatrick Celia Hoyles Ole Skovsmose (Editors, in collaboration with Paola Valero) Meaning in Mathematics Education Q- Springer Jeremy Kilpatrick, University of Georgia, Georgia, USA Celia Hoyles, University of London, London, United Kingdom Ole Skovsmose, Aalborg University, Aalborg, Denmark Paola Valero, Aalborg University, Aalborg, Denmark Library of Congress Cataloging-in-PublicationD ata A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 0-387-24039-X e-ISBN 0-387-24040-3 Printed on acid-free paper. O 2005 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 SPIN 11323051 CONTENTS vii 1 9 "Meaning" and School Mathematics 17 Geoffrey Howson The Meaning of Conics: Historical and Didactical Dimensions Mariolina Bartolini-Bussi Reconstruction of Meaning as a Didactical Task: The Concept of Function as an Example Rolf Biehler Meaning in Mathematics Education Ole Skovsmose COLLECTIVMEE ANINGAN D COMMONSE NSE Mathematics Education and Common Sense Christine Keitel and Jeremy Kilpatrick COMMUNICATIAONND CONSTRUCTIOONF M EANING Making Mathematics and Sharing Mathematics: Two Paths to Co-Constructing Meaning? Celia Hoyles The Hidden Role of Diagrams in Pupils' Construction of Meaning in Geometry Colette Laborde What's a Best Fit? Construction of Meaning in a Linear Algebra Session Joel Hillel and Tommy Dreyfus VI CONTENTS Discoursing Mathematics Away Anna Sierpinska Meaning and Mathematics Michael Otte PREFACE This book is a product of the BACOMET group, a group of educators-mainly educators of prospective teachers of mathematics-who first came together in 1980 to engage in study, discussion, and mutual reflection on issues in mathematics education. BACOMET is an acronym for BAsic Components of Mathematics Education for Teachers. The group was formed after a series of meetings in 1978-1979 between Geoffrey Howson, Michael Otte, and the late Bent Christiansen. In the ensuing years, BACOMET initiated several projects that resulted in published works. The present book is the main product of the BACOMET project entitled Meaning and Communication in Mathematics Education. This theme was chosen because of the growing recognition internationally that teachers of mathematics must deal with questions of meaning, sense making, and communication if their students are to be proficient learners and users of mathematics. The participants in this project were the following: Nicolas Balacheff (Grenoble, France) Maria Bartolini Bussi (Modena, Italy) Rolf Biehler (Bielefeld, Germany) Robert Davis (New Brunswick, NJ, USA) Willibald Dorfler (Klagenfurt, Austria) Tommy Dreyfus (Holon, Israel) Joel Hillel (Montreal, Canada) Geoffrey Howson (Southampton, England) Celia Hoyles-Director (London, England) Jeremy Kilpatrick-Director (Athens, GA, USA) Christine Keitel (Berlin, Germany) Colette Laborde (Grenoble, France) Michael Otte (Bielefeld, Germany) Kenneth Ruthven (Cambridge, England) Anna Sierpinska (Montreal, Canada) Ole Skovsmose-Director (Aalborg, Denmark) Conversations about directions the project might take began in May 1993 at a NATO Advanced Research Workshop of the previous BACOMET project in VIII PREFACE Villard-de-Lans, France. The main themes and topics for papers began to emerge at a meeting in June 1994 in Melle, Germany, and were put in final form at a conference in Athens, Georgia, USA, in September 1995, where participants' papers were critiqued and revised, the structure of the book was created, and interludes were drafted to introduce the sections of the book. Subsequently, several project members either withdrew because of the press of other responsibilities (Davis & Dorfler), decided to publish their work elsewhere (Balacheff), or decided that their papers no longer represented their current thinking (Ruthven). The chapters in this book were written by the remaining project members. We would like to thank the Scientific Affairs Division of the North Atlantic Treaty Organisation for funding the 1993 workshop and the Office of the Vice President for Academic Affairs of the University of Georgia for funding the 1995 conference. We are grateful to Brian Lawler of the University of Georgia for helping to polish the English of the text. The book would never have seen its way into print without the collaboration of Paola Valero of Aalborg University. Her many contributions were invaluable, and we acknowledge them with pleasure and gratitude. Bob Davis, charter member of BACOMET, founding editor of the Journal of Mathematical Behavior, and a pioneer in studying mathematics education from the perspective of cognitive science, died in 1997. This book is dedicated to his memory. Jeremy Kilpatrick Celia Hoyles Ole Skovsmose Mathematics Education Library Managing Editor: A.J. Bishop, Melbourne, Australia 1. H. Freudenthal: Didactical Phenomenology of Mathematical Structures. 1983 ISBN 90-277-1535-1 HB; 90-277-2261-7 PB 2. B. Christiansen, A. G. Howson and M. Otte (eds.): Perspectives on Mathematics Education. Papers submitted by Members of the Bacomet Group. 1986 ISBN 90-277-1929-2 HB; 90-277-21 18-1 PB 3. A. Treffers: Three Dimensions. A Model of Goal and Theory Description in Mathematics Instruction TheWiskobas Project. 1987 ISBN 90-277-2165-3 4. S. Mellin-Olsen: The Politics of Mathematics Education. 1987 ISBN 90-277-2350-8 5. E. Fischbein: Intuition in Science and Mathematics. An Educational Approach. 1987 ISBN 90-277-2506-3 6. A.J. Bishop: Mathematical Enculturation. A Cultural Perspective on Mathematics Education. 1988 ISBN 90-277-2646-9 HB; 1991 0-7923-1270-8 PB 7. E. von Glasersfeld (ed.): Radical Constructivism in Mathematics Education. 1991 ISBN 0-7923-1257-0 8. L. Streefland: Fractions in Realistic Mathematics Education. 'A Paradigm of Developmental Research. 1991 ISBN 0-7923- 1282-1 9. H. Freudenthal: Revisiting Mathematics Education. China Lectures. 1991 ISBN 0-7923-1299-6 10. A.J. Bishop, S. Mellin-Olsen and J. van Dormolen (eds.): Mathematical Knowledge: Its Growth Through Teaching. 1991 ISBN 0-7923-13 44-5 11. D. Tall (ed.): Advanced Mathematical Thinking. 1991 ISBN 0-7923-1456-5 12. R. Kapadia and M. Borovcnik (eds.): Chance Encounters: Probability in Education. 1991 ISBN 0-7923-1474-3 13. R. Biehler, R.W. Scholz, R. StraBer and B. Winkelmann (eds.): Didactics of Mathematics as a Scientific Discipline. 1994 ISBN 0-7923-2613-X 14. S. Lerman (ed.): Cultural Perspectives on the Mathematics Classroom. 1994 ISBN 0-7923-293 1-7 15. 0. Skovsmose: Towards a Philosophy of Critical Mathematics Education. 1994 ISBN 0-7923-2932-5 16. H. Mansfield, N.A. Pateman and N. Bednarz (eds.): Mathematics for Tomorrow's Young Children. International Perspectives on Curriculum. 1996 ISBN 0-7923-3998-3 17. R. Noss and C. Hoy1es:Windows on Mathematical Meanings. Learning Cultures and Computers. 1996 ISBN 0-7923-4073-6 HB; 0-7923-4074-4 PB 18. N. Bednarz, C. Kieran and L. Lee (eds.): Approaches to Algebra. Perspectives for Research and Teaching.1996 ISBN 0-7923-4145-7 HB; 0-7923-4168-6, PB 19. G. Brousseau: Theory of Didactical Situations in Mathematics. Didactique des Mathkmatiques 1970-1990. Edited and translated by N. Balacheff, M. Cooper, R. Sutherland and V. Warfield. 1997 ISBN 0-7923-4526-6 20. T. Brown: Mathematics Education and Language. Interpreting Hermeneutics and Post-Structuralism. 1997 ISBN 0-7923-4554-1 HB Second Revised Edition. 2001 ISBN 0-7923-6969-6 PB

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What does it mean to know mathematics? How does meaning in mathematics education connect to common sense or to the meaning of mathematics itself? How are meanings constructed and communicated and what are the dilemmas related to these processes? There are many answers to these questions, some of whi
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