DOCUMENT RESUME ,_:.. r-------ED 319 606 SE 051 416 AUTHOR Pereira-Mendoza, Lionel, Ed.; Quigley, Martyn, Ed. TITLE Canadilo Mathematics Education Study Group = Groupe Canadien d'etude en didactique des matematiques. Proceedings of the Annual Meeting (St. Catharines, Ontario, Canada, May 27-31, 1989). INSTITUTION Canadian Mathematics Education Study Group. PUB DATE 90 NOTE 149p.; For 1988 Proceedings see ED 305 235; for IAEP study, see ED 306 259. PUB TYPE Collected Works - Conference Proceedings (021) -- Reports - Research/Technical (143) EDRS PRICE MF01/PC06 Plus Postage. DESCRIPTORS Calculus; College Mathematics; *Computer Assisted Instruction; Computer Uses in Education; *Elementary School Mathematics; Elementary Secondary Education; Foreign Countries; Qeometry; *Mathematical Concepts; Mathematical Linguistics; Mathematics Curriculum; *Mathematics Education; *Mathematics Tests; *Proof (Mathematics); Secondary School Mathematics IDENTIFIERS *Canada; Constructivism ABSTRACT These conference proceedings include two invited lectures, four working group reports, five topic group reports, a list of participants, and a list )f previous proceedings. The invited lectures were: "Teaching Mathematical Proof: Relevance and Coioplexity of a Social Approach" (Nicolas Balacheff) and "Geometry Is Alive and Well:" (Doris Schattschneider). The topics of the working group reports were: (1) using LOGO-based software (Benoit Cote, Sandy Dawson); (2) using the Computer Algebra Systems (EricMuller, Stan Devitt); (3) aspects of communication and other functions oflanguage used in mathmematics (David Pim); and (4; using research methods of mathematical conceptions based on constructivism (Rafaella Borasi, Claude Janvier). The topic groups dealt with: computer uses in the elementary classroom (W. George Cathcart); constructions of fundamental mathematical cornepts and the kindergartner's construction of natural numbers (Jacques Bergeron, Nicolas Herscovics); multicultural influences (Linda Davenport); meaning of grades assigned by teachers (Rose Traub et al.); and a discussion (not presented in this document) of the results of the 1988 International Assessment of Educational Progress Study (Dennis Raphael). Appended are lists of proceedings participants and of previous proceedings available through ERIC. (YP) *********************************************************************** * Reproductions supplied by EDRS are the best that can be made * * from the original document. * **************** ***** ******************************************* ***** ** U. ORFAITIMINT OF SOUCATION Office of Educator* Reassech end improvement "PERMISSION TO REPRODUCE THIS EDllC RCES INFORMATION MATERIAL HAS BEEN GRANTED BY cENYEA (ENO document has been reproduced as Lionel Pereira -Mendosa received from the person or attenuation orientating it 0 minor changes have been roads to improve ClIC reproduction oulditv Points of view Of opinions stated in the duct, CD mod do not neateserily represent official TO THE EDUCATIONAL RESOURCES OW positionof PolICY INFORMATION CENTER (ERIC)." CD cm CANADIAN MATHEMATICS EDUCATION STUDY GROUP FA GROUPE CANADIAN D'ETUDE EN DIDACTIQUE DES MATHEMATIQUES PROCEEDINGS 1989 ANNUAL MEETING BROCK UNIVERSITY ST. CATHARINES, ONTARIO May 27-31, 1989 Edited by Lionel Pereira-Mendoza Martyn Quigley Memorial University of Newfoundland V) O BEST COPY AVAILABLE a. ( 2 CANADIAN MATHEMATICS EDUCATIONSTUDY GROUP GROUPE CANADIAN D'ETUDE ENDIDACTIQUE DES MATHEMATIQUES PROCEEDINGS 1989 ANNUAL MEETING BROCK UNIVERSITY ST. CATHARINES, ONTARIO May 27-31, 1989 Edited by Lion& Pereira-Mendoza Martyn Quigley Memorial University of Newfoundland 003-015 04/90/95 CMESG/GCEDM Annual Meeting Brock University St. Catharines May 27-31, .1989 Table of Contents Forward vii Acknowledgements ix Invited Lectures A. Teaching Mathematical Proof: 11 Relevance and Complexity of a Social Approach Nicolas Ba lathe (1RPEACS-C1VRS) B. Geometry is Alive and Well! 29 Doris Schatadmeider (Moravian College) Working Groups A. Using Computers for Investigative Work withElementary Teachers 43 Benoit COM (Universite du QuebecaMontreal) Sandy Dawson (Simon Fraser University) B. Computers in the Undergraduate Mathematics Curriculum 51 Edgar Winans (Memorial University of Newfoundland) C. Natural Language and Mathematical Language 61 David Palm (The Open University) D. Research Strategies for Pupils' Conceptions inMathematics 71 Robe& Borosi (University of Rochester) Claude !envier (CIRADE -UQAM) iv 4 Topic Groups A. Implementation of an Apple- Centre for Innovation and Year1 results....87 W. George Cathcart (University of Alberta) B. A Model to Describe the Construction of MathematicalConcepts from an Epistemological Perspective 99 Jacques Bergeron (Universite de Montreal) Mcolas Herscovics (Concordia University) The Kindergartner's Construction of Natural Numbers: An International Study 115 Mcolas Herscovics (Concordia University) Jacques Bergerac (Universite de Montreal) C. Multicultural Influences in Mathematics Education 135 Linda Ruiz Davenport (Portland State University) D. Teacher Assessment Practices in Nigh School Calculus 143 Ross E. 7'ratrb, Philip Nagy, Katherine Madbuy & Roslyn Dinners (The Ontario Institute for Studies in Education) E. The Ontario IAEP Results 161 Dennis Raphael (Ontario Ministry of Education) List of Participants 165 Previous Proceedings available through ERIC 173 V EDITORS' FORWARD We would like to thank all the contributors forsubmitting their manuscripts for inclusion in these proceedings. Without theirco-operation it would not have been possible to produce the proceedingsso quickly. Special thanks go from all ofus who attended the conference to the organizers, and particularly to Eric Muller, who workedtirelessly before and during the meeting toensure smooth sailing. I hope these proceedings will helpgenerate continued discussion on the many major issues raised during the conference. Lionel Pereira-Mendoza Martyn Quigley April 10, 1990 vii ACKNOWLEDGEMENTS The Canadian Mathematics Study Group wishes to acknowledgethe continued assistance of the Social Scienc-s and Humanities Research Council fortheir support. Without this support the 1989 meeting and proceedings would not have been possible. We would also like to thank Brock University for hosting themeeting and providing excellent facilities to conduct the meeting. Finally, we would like to thank the contributors and participants,who helped make the meeting a valuable educational experience. ix 7 Lecture One Teaching Mathematical Proof Relevance and Complexity ofa Soc:al Approach N. Balacheff IRPEACS-CNRS 8 13 Introduction What a mathematical proof consists ofseems clear to all mathematics teachers and mathematics educators. That is: "A carefulsequence of steps with each step following logically from an assumed or previously provedstatement and from previous steps" (NZTM, 1989, p.144). This description is almost thesame all over the world, and it is very close to what a logician would formulate, perhapsmore formally. Comments made on mathematical proof as a content to be taught emphasizetwo points: first they stress that it has nothing to do with empiricalor experimental verification, second they call attention to the move from concrete to abstract. Here isan example of such comments: "It is a completely new way of thinking for high school students.Their previous experience both in and out of school has taught them toaccept informal and empirical arguments as sufficient. Students should come to understand that although sucharguments are useful, they do not constitute a proof." (NCTM, 1989, p.145). We can say that the definition of mathematical proof,as an outcome of these official texts is mathematically acceptable, but there isa long way from this definition to the image built in practice along the teaching interaction.More or less, teaching mathematical proof is understood as teaching how to formulatedeductive reasoning: "Pour les professeurs, une demonstration, c'est tresnettement l'expose formel deductif d'un raisonnement logique" (Braconne, 1987, p.187). The construction of this reasoning, and its possiblerelationships with other kind of reasoning, is hidden by that over emphasison its "clear" formulation. That conception is so strong that some teachers cancome to an evaluation of a mathematical proof just considering the surface level of the discourse.For example, in her requirement for teachers comments on a sample of students formulations,Braconne reports' that: "Les professeurs ont Magi aux longueur inutiles du texts de Bertrand,au desordre dans la solution de Karina, au fait que le texts d'Elodiene suive pas le raisonnement deductif, etc. Toutefois, sect Fawners n'ont pas remsrque que, dans Is texts deBertrand, c'est la reciproque du theoreme neeessaire I la demonstrationqui &sit cit6 an premier paragraphe, et huit n'ont pas signals quo le texts de Laurentcontenait la memo erreur [...] Done pour l'eleve, et pour nous, les notesne refibteat pas le fait que le professeur se soit appergu de l'erreur ou non." (Braconne, 1987, p.99) A report on proof frames of elementary preserviceteachers shows a similar behaviour: "Many students who correctly accept a general-proof verificationdid not reject a false proof verification; they were influenced by theappearance of tea argument - the ritualistic aspects of the proof - rather than the correctness of theargument.(...] Such students 14 appear to rely on a syntactic-level deductive frame in which a verificationof a statement is evaluated according to