Meaning in Mathematics Education (Mathematics Education Library)

262 Pages·2005·6.01 MB·English

by  Paola Valero

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
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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