PERSPECTIVES ON MATHEMATICAL PRACTICES
LOGIC, EPISTEMOLOGY, AND THE UNITY OF SCIENCE VOLUME 5
Editors Shahid Rahman, ...
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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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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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EDUCATION THEORY: A SITUATIVE PERSPECTIVE
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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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$QWKRQ\3HUHVVLQLDQG'RPLQLF3HUHVVLQL 182 important connection for her students. In this activity, students cut paper circles into sectors and rearranged the sectors to form a rectangle-like figure. Ms. Savant then posed a series of questions about the figure, to help students connect the formula for the area of a rectangle to the formula for the area of a circle. She concluded by telling her students that they had just proved the formula for the area of a circle: “You cut up the area, you rearranged it, and you proved that the area [of a circle] is ‘pi-r-squared.’ That’s great!” We were interested in Ms. Savant’s use of the word “proved” in this context. In the interview conducted later that day, the we asked Ms. Savant whether she thought the sectors-of-a-circle activity was a proof. In her response, Ms. Savant described proof as an informal sense-making process that shows why something is true. WKLQNWKLVLVDSURRI@,W VQRWDIRUPDOSURRILQWKDWZHVLWGRZQ DQG ZH ZULWH LW DOO RXW %XW FHUWDLQO\ LW¶V D ZD\ WR SURYH WR \RXUVHOI ³+H\ORRNWKLVDFWXDOO\UHDOO\GRHVZRUN,FDQXVHWKLQJVWKDW,NQRZ DQGWKLVUHDOO\GRHVZRUN´ She commented further, ³0\LGHDRIDIRUPDOSURRILVZKHQ\RXDFWXDOO\VLWGRZQDQG\RXXVH V\PEROLF ODQJXDJH ZLWK (QJOLVK ODQJXDJH DQG \RX JLYH HLWKHU D SDUDJUDSKRUDVWHSE\VWHSSURRIWKDW¶VYHU\ORJLFDO´ Ms. Savant contrasted this notion of a formal proof with that of an informal proof, or convincing argument, which is acceptable and appropriate in a high school classroom. 0V6DYDQWKDGPDGHWKLVGLVWLQFWLRQEHWZHHQIRUPDODQGLQIRUPDOSURRIV RIWHQ LQ LQWHUYLHZV SULRU WR WKLV LQFLGHQW 6KH KDG UHSHDWHGO\ GHVFULEHG IRUPDO SURRI DV WKH VWUXFWXUHG V\PEROLF SUHVHQWDWLRQV DFFHSWDEOH WR PDWKHPDWLFLDQV³$SURRILVDORJLFDODQGFOHDUVHULHVRIVWHSVWKDWWDNH\RX IURP SRLQW $ WR SRLQW %´ ,Q FRQWUDVW IRU 0V 6DYDQW LQIRUPDO SURRIV DUH ZD\V WR H[SODLQ ZK\ VRPHWKLQJ LV WUXH SDUWLFXODUO\ WR VWXGHQWV ³, WKLQN FRQYLQFLQJ >LQIRUPDO SURRI@ LV GLIIHUHQW IURP IRUPDO SURRI %XW LQ WKH YHUQDFXODU(QJOLVKLWZRXOGSURYHWRWKHNLGVWKDWLWZRUNV´ 7KURXJK RXU VLWXDWLYH OHQV ZH VHH WKDW $XGUH\ 6DYDQW GHYHORSHG WZR VRPHZKDW GLIIHUHQW FRQFHSWLRQV RI SURRI DV VKH SURJUHVVHG WKURXJK KHU WHDFKHU HGXFDWLRQ SURJUDP²IRUPDO SURRIV WKDW SURYH DQG LQIRUPDO SURRIV WKDW H[SODLQ 6KH OHDUQHG KRZ WR GUDZ XSRQ WKHVH FRQFHSWLRQV LQ RUGHU WR SDUWLFLSDWHVXFFHVVIXOO\LQWZRGLIIHUHQWW\SHVRIPDWKHPDWLFDOFRPPXQLWLHV ,Q KHU FRQWHQW FRXUVHV DQG RQ RXU UHVHDUFK WDVNV ZKHUH WKH VWDQGDUGV RI SURRIZHUHVHHQWREHPRUHUHODWHGWRVWUXFWXUHDQGIRUPWKDQWRH[SODQDWRU\ SRZHU²SURRIVWKDWSURYHQRWQHFHVVDULO\H[SODLQ²0V6DYDQWGUHZXSRQD FRQFHSWLRQRISURRIWKDWVXSSRUWHGKHUVXFFHVVIXOSDUWLFLSDWLRQDVDVWXGHQW
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4.
PHILOSOPHICAL DIFFERENCES AND RECONCILIATION
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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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