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CANADIAN MATHEMATICS EDUCATION STUDY GROUP ´GROUPE CANADIEN D’ETUDE EN DIDACTIQUE ´DES MATHEMATIQUES PROCEEDINGS 2000 ANNUAL MEETING Université du Québec à Montréal May 26 – 30, 2000 EDITED BY: Elaine Simmt, University of Alberta Brent Davis, John Grant McLoughlin, Memorial University i ii th24 Annual Meeting Canadian Mathematics Education Study Group / ´Groupe Canadien d’Etude en Didactique des Mathématiques Université du Québec à Montréal, 2000 PROCEEDINGS Table des matières / Contents v Acknowledgements vii Schedule MALGORZATA DUBIEL ix Introduction PLENARY LECTURES GILBERT LABELLE 3 Manipulating Combinatorial Structures MARIA BARTOLINI BUSSI 21 The Theoretical Dimension of Mathematics: A Challenge for Didacticians WORKING GROUPS CAROLINE LAJOIE 35 A • Des cours de mathématiques pour les futurs enseignants et & ED BARBEAU enseignantes du primaire 43 Mathematics Courses for Prospective Elementary Teachers LOUIS CHARBONNEAU 47 B • Crafting an Algebraic Mind: Intersections from History and & LUIS RADFORD the Contemporary Mathematics Classroom TOM KIEREN & 61 C • Mathematics education et didactique des mathématiques : y a- ANNA SIERPINSKA t-il une raison pour vivre des vies séparées? Mathematics education et didactique des mathématiques: Is there a reason for living separate lives? DAVID A. REID & 81 D • Teachers, Technologies, and Productive Pedagogy ROSAMUND SUTHERLAND SYLVIE DESJARDINS & 87 E • Calculus Reform: A Critical Assessment HARALD PROPPE TOPIC SESSIONS VICKI ZACK & 95 A Proof Ought to Explain: A Classroom Teacher-Researcher, a DAVID A. REID Mathematics Educator, and Three Cohorts of Fifth Graders Seek to Make Meaning of a Non-Obvious Algebraic Equation LEIGH N. WOOD & 103 Assessment for All GEOFFREY H. SMITH iii RALPH T. MASON 109 Mathematics—By Invitation MHAIRI (VI) MAEERS 115 Math Central: An Internet Service for Teachers and Students & HARLEY WESTON HD REPORTSNEW P LYNN GORDON CALVERT 123 Mathematical Conversations within the Practice of Mathematics SUSAN GEROFSKY 129 The Word Problem as Genre in Mathematics Education 135 How Visual Perception Justifies Mathematical ThoughtDENNIS LOMAS LYNDON MARTIN 141 Folding Back and Growing Mathematical Understanding La place et les fonctions de la validation chez les futurs enseignantsCLAUDINE MARY 147 des mathématiques au secondaires ELAINE SIMMT 153 Mathematics Knowing in Action: A Fully Embodied Interpretation PANEL DISCUSSION BERNARD R. HODGSON 163 Pourquoi enseigner les mathématiques à tous? SANDY DAWSON 165 Why teach mathematics to all students? 167BRENT DAVIS NADINE BEDNARZ 173 181 Why teach mathematics to everyone? APPENDICES 191 A • Working Groups at Each Annual Meeting 195 B • Plenary Lectures at Each Annual Meeting 197 C • Proceedings of Annual Meetings 199 D • List of Participants iv Acknowledgements On behalf of the members, the CMESG/GCEDM executive would like to take this opportu- nity to thank our local hosts for their contributions to the 2000 Annual Meeting and Confer- ence. Specifically, thank you to: Lesley Lee, chair of local organizing committee; Caroline Lajoie, electronic equipment; Bernadette Janvier and Carolyn Kieran, meals; Nadine Bednarz, coffee; Nathalie Prévost and Jeanne Laporte-Jobin, registrations; Asuman Oktaç and Astrid Defence, walking tours; Hassane Squalli and Rina Herscovics, email. Supplementary materials to some of the contributions in these Proceedings are posted on the CMESG website (, maintained by David Reid. The editors would like to express their appreciation to Darren Stanley, who provided assis- tance in proofreading, and to Dave Wagner, who reproduced the CMESG logo. v vi Schedule Friday Saturday Sunday Monday Tuesday May 26 May 27 May 28 May 29 May 30 8h30 - 9h00 9h00 - 12h15 9h00 - 12h15 9h00 - 9h30 Registration WG A WG A Ad hoc sessions WG B WG B9h00 - 12h15 9h30 - 11h00 WG C WG CWG A Special Panel WG D WG DWG B DiscussionAM WG E WG EWG C 11h00 - 11h15WG D Coffee BreakWG E 10h30 - 11h00 10h30 - 11h00 11h15 - 12h1510h30 - 11h00 Coffee Break Coffee Break Coffee Break Closing Session 12h15 - 13h30 12h15 - 13h30 12h15 - 13h30Lunch 15h45 13h30 - 14h20 14h00 - 15h00 13h30 - 14h00 Friends Topic Groups Plenary II: Small group of FLM Bartolini Bussi discussion of Plenary II 14h30 - 15h00 14h10 - 15h1015h10 - 16h00Small group16h00 - 17h00 Questions forTopic Groupsdiscussion ofRegistration Bartolini BussiPlenary I 15h10 - 15h30PM 16h00 - 16h30 Coffee BreakCoffee Break17h00-18h00 15h00 - 15h15 15h30 - 15h55Opening Coffee Break New PhDsSession 16h30 - 16h55 New PhDs 16h00 - 16h30 Ad hoc sessions15h15 - 16h15 Questions for 17h00 - 17h25 16h45 - 17h45 New PhDsLabelle General Meeting 19h00 19h00 Supper Restaurant18h00 - 19h15 Walking Tours Restaurant Le Caveau La Rotonde 19h30 - 20h30 Plenary I: Evening Labelle 20h25 Social vii viii Introduction Malgorzata Dubiel - President, CMESG/GCEDM Simon Fraser University It is a great pleasure to write an introduction to the CMESG/GCEDM Proceedings from the 2000 meeting held at the Université du Québec à Montréal (UQAM). A necessary part of the introceedings is an attempt to explain to readers, some of whom may be newcomers to our organization, that the vol- ume in their hands cannot possibly convey the spirit of the meeting it reports on. It can merely describe the content of activities without giving much of the flavour of the process. To understand this, one needs to understand the uniqueness of both our organization and our annual meetings. CMESG is an organization unlike other professional organizations. One belongs to it not because of who one is professionally, but because of one’s interests. And that is why our members are members of mathematics and education departments at Canadian and other universities and colleges, and school teachers, united by their interest in mathematics and how it is taught at every level, by the desire to make teaching more exciting, more relevant, more meaningful. Our meetings are unique, too. One does not simply attend a CMESG meeting the way one attends other professional meetings, by coming to listen to a few chosen talks. You are immediately part of it; you live and breathe it. Working Groups form the core of each CMESG meeting. Participants choose one of several possible topics, and, for three days, become members of a community which meets three hours a day to exchange ideas and knowledge, and, through discussions which often continue beyond the allotted time, create fresh knowledge and insights. Throughout the three days, the group becomes much more than a sum of its parts—often in ways totally unexpected to its leaders. The leaders, after working for months prior to the meeting, may see their carefully prepared plan ignored or put aside by the group, and a completely new picture emerging in its stead. Two plenary talks are traditionally part of the conference, at least one of which is given by a speaker invited from outside Canada, who brings a non-Canadian perspective. These speakers participate in the whole meeting; some of them afterwards become part of the group. And, in the spirit of CMESG meetings, a plenary talk is not just a talk, but a mere beginning: it is followed by discussions in small groups, which prepare questions for the speaker. After the small group discussions, in a renewed plenary session, the speaker fields the questions generated by the groups. Topic Groups and Ad Hoc presentations provide more possibilities for exchange of ideas and reflections. Shorter in duration than the Working Groups, Topic Groups are ses- sions where individual members present work in progress and often find inspiration and new insight from their colleagues’ comments. Ad Hoc sessions are opportunities to share ideas, which are often not even “half- baked”—sometimes born during the very meeting at which they are presented. A traditional part of each meeting is the recognition of new PhDs. Those who com- pleted their dissertations in the last year are invited to speak on their work. This gives the ix CMESG/GCEDM Proceedings 2000 • Introduction group a wonderful opportunity to observe the changing face of mathematics education in Canada. In 2000, the Tuesday morning panel discussion on curriculum reform was a new fea- ture. Lively and exciting, it created a lot of interest, and so we invited the panelists to submit brief summaries of their presentations for the proceedings, to preserve some of the flavour of the discussion. Our annual meetings are traditionally set on university campuses with participants staying in dormitories rather than hotels, both to make the meetings more affordable and to allow for discussions to continue far beyond the scheduled hours. The 2000 Annual Meet- ing was no exception. It was hosted on the campus of the Université du Québec à Montréal, and the participants stayed in the UQAM residence. A successful, working meeting in the middle of Canada’s most exciting city, in early Summer—is it possible? Can you keep people in? The executive had been worrying about this before the meeting. But the exciting pro- gram, excellent organization and great food (probably the best of all our meetings!) all con- tributed to one of the largest and most successful meetings ever. Thanks to Lesley Lee and her team for their hard work! Editing the Proceedings of the meeting is a formidable task, one we tend to take for granted. I would like therefore to extend our gratitude to the editors of this volume: John Grant McLoughlin, Brent Davis and Elaine Simmt. Special thanks to Elaine and Brent, who started their work earlier than expected to give a hand in difficult circumstances. x