PERSPECTIVES ON MATHEMATICAL PRACTICES

LOGIC, EPISTEMOLOGY, AND THE UNITY OF SCIENCE VOLUME 5

Editors Shahid Rahman, University of Lille III, France John Symons, University of Texas at El Paso, U.S.A. Editorial Board Jean Paul van Bendegem, Free University of Brussels, Belgium Johan van Benthem, University of Amsterdam, the Netherlands Jacques Dubucs, University of Paris I-Sorbonne, France Anne Fagot-Largeault Coll`ege de France, France Bas van Fraassen, Princeton University, U.S.A. Dov Gabbay, King’s College London, U.K. Jaakko Hintikka, Boston University, U.S.A. Karel Lambert, University of California, Irvine, U.S.A. Graham Priest, University of Melbourne, Australia Gabriel Sandu, University of Helsinki, Finland Heinrich Wansing, Technical University Dresden, Germany Timothy Williamson, Oxford University, U.K. Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic. At present, no single logical, semantical or methodological framework dominates the philosophy of science. However, the editors of this series believe that formal techniques like, for example, independence friendly logic, dialogical logics, multimodal logics, game theoretic semantics and linear logics, have the potential to cast new light no basic issues in the discussion of the unity of science. This series provides a venue where philosophers and logicians can apply specific technical insights to fundamental philosophical problems. While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and the philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity.

Perspectives on Mathematical Practices Bringing Together Philosophy of Mathematics, Sociology of Mathematics, and Mathematics Education Edited by

Bart Van Kerkhove and Jean Paul van Bendegem Vrije Universiteit Brussel, Belgium

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-5033-X (HB) 978-1-4020-5033-6 (HB) 1-4020-5034-8 (e-book) 978-1-4020-5033-6 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.



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I am grateful to Miguel Gimenez of the University of Edinburgh for translating this paper. For further detail of the proof see [2], [25] or [12].

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  The Learning to Teach Secondary Mathematics project (LTSM) is a multi-year

research project funded in part by the National Science Foundation (NSF Grant #REC-9605030 and REC-0087653). We would like to acknowledge Hilda Borko and Lew Romagnano for their work on this aspect of the paper as they are Co-PIs with Dominic Peressini on the LTSM project.



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2.

EDUCATION THEORY: A SITUATIVE PERSPECTIVE

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2.1

Proof and justification

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3.

EDUCATIONAL PRACTICE: WHAT IS A PROOF?

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  The names of participants and their schools are pseudonyms.



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s = 0.999… 10s = 9.999… 9s = 9.0 s = 1.0 0.999… = 1.0

| | | | |

definition multiplying both sides by 10 subtracting s from both sides and (1) dividing both sides by 9 substitution and (1)

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