CANADIAN MATHEMATICS EDUCATION STUDY GROUP GROUPE CANADIEN D’ETUDE´ EN DIDACTIQUE DES MATHEMATIQUES´ PROCEEDINGS / ACTES 2002 ANNUAL MEETING Queen’s University May 24 – 28, 2002 EDITED BY: Elaine Simmt, University of Alberta Brent Davis, University of Alberta i Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group / Groupe Canadien d’Étude en Didactique des Mathématiques are published by CMESG/GCEDM. They were printed in April 2003 in Edmonton, AB. ii Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group / Groupe Canadien d’Etude´ en Didactique des Mathématiques 26th Annual Meeting Queen’s University May 24 – 28, 2002 Contents / Table des matières v Acknowledgements vii Schedule MALGORZATA DUBIEL ix Introduction Plenary Lectures / Conférences plénières DEBORAH BALL & HYMAN BASS 3 Toward a Practice-Based Theory of Mathematical Knowledge for Teaching JON BORWEIN 15 The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof Working Groups / Groupes de travail SUSAN GEROFSKY, NATHALIE 33 A • Mathematics and the Arts SINCLAIR, & BRENT DAVIS RICHARD PALLASCIO 43 B • Philosophy for Children on Mathematics & ELAINE SIMMT NADINE BEDNARZ 59 C • The Arithmetic/Algebra Interface: Implications for Primary & LESLEY LEE and Secondary Mathematics / Articulation arithmétique/ algèbre : Implications pour l’enseignement des mathématiques au primaire et au secondaire DAVID PIMM & NICK JACKIW 71 D • Mathematics, the Written and the Drawn WALTER WHITELEY, PETER 81 E • Des cours de mathémathiques pour les futurs (et actuels) TAYLOR, & HARRY WHITE maîtres au secondaire 85 • Types and Characteristics Desired of Courses in Mathematics Programs for Future (and In-Service) Teachers Topic Sessions / Sessions thématiques OLIVE CHAPMAN 91 High School Mathematics Teachers’ Perspectives of Mathematical Word Problems GEORGE GADANIDIS 99 What are Critical Online Experiences for Mathematics Teachers and Students? iii New PhD Reports / Présentations de thèses de doctorat GEORGE FREMPONG 109 Socioeconomic Gradients in Mathematics Achievement: Findings for Canada from the Third International Mathematics and Science Study KATHLEEN NOLAN 117 Light and Shadows (in) Knowing Mathematics and Science MARGARET P. SINCLAIR 123 Supporting Student Efforts to Learn with Understanding: An Investigation of the Use of JavaSketchpad Sketches in the Secondary Geometry Classroom Ad hoc Sessions / Séances ad hoc ALLAN BROWN 133 Conceptualizing Limit in Calculus (Why is It Still a Problem?) GEORGE GADANIDIS 135 Amplifying Mathematical Intelligence Using Web-Based Interactive Activities 137 Grades 5-6 Teachers’ Algebra Teaching Beliefs and Practices 139 Researching the Effect of Interactive Applets in Mathematics Teaching PETER LILJEDAHL 141 Creativity and the Psychology of Mathematical Invention RALPH MASON & 143 Logical Reasoning and Mathematical Games JANELLE McFEETORS IMMACULATE NAMUKASA 145 Collective Mathematical Thinking GEOFFREY ROULET 147 Proofs and Refutations on the Web: Mathematics Environments for Grades 7 & 8 DAVID WAGNER 149 Being in a Mathematical Place: Immersion in Mathematical Investigation Special Anniversary Sessions / Séances spéciales d’anniversaire ERIC MULLER 153 Reflections on 25 years with CMESG FREDERIC´ ´ GOURDEAU 159 Réflexions d’un mathématicien sur le GCEDM CAROLYN KIERAN 165 A Historical Perspective on Mathematics Education Research in Canada: The Emergence of a Community BRENT DAVIS 187 Lessons from the Past, Questions for the Future ROBERTA MURA 197 Lessons from the past, questions for the future: méditation sur thème imposé Appendices 209 A • Working Groups at Each Annual Meeting 213 B • Plenary Lectures at Each Annual Meeting 215 C • Proceedings of Annual Meetings 217 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 2002 Annual Meeting and Confer- ence. Specifically, thank you to Peter Taylor, Marg Lambert, Geoff Roulet, Morris Orzech, Bill Higginson, Lynda Colgan, Joan McDuff, and Lionel LaCroix. On behalf of our membership, we would also like to thank the guest speakers, work- ing group leaders, topic group and ad hoc presenters, and all of the participants. You are the ones who made this meeting an intellectually stimulating and worthwhile experience. Supplementary materials to some of the contributions in these Proceedings are posted on the CMESG/GCEDM website (http://www.cmesg.math.ca), maintained by David Reid. v vi Schedule Friday Saturday Sunday Monday Tuesday May 24 May 25 May 26 May 27 May 28 9h00 - 11h00 Anniversary 09h00 - 12h15 09h00 - 12h15 09h00 - 12h15 Session 3 AM Working Working Working Groups Groups Groups 11h30 - 12h30 Closing Session Lunch 12h15 - 13h30 12h15 - 13h30 12h15 - 13h30 13h30 - 14h00 13h30 - 14h30 13h30 - 14h30 Small group Plenary 2: Questions for discussion of Borwein Borwein Plenary 1 14h00 - 15h00 14h30 - 15h30 Topic Groups Questions for 14h30 - 15h00 PM Ball & Bass Small group discussion of 16h00 - 17h00 15h30 - 16h30 15h30 - 16h30 Plenary 2 Anniversary Registration Anniversary Session 2 Session 1 16h30 - 18h30 17h00 - 18h00 Opening 16h30 - 17h30 Annual General Session New PhDs Meeting Supper 18h30 - 20h00 18h00 17h00 18h00 - 19h30 President’s Boat Trip to Dinner Gananoque 20h00 - 21h00 (Fort Henry) Theatre Plenary 1: 20h00 - 22h00 Evening Ball & Bass Displays, Ad Hocs, & 21h00 Dessert Dessert & Social vii viii Introduction Malgorzata Dubiel - President, CMESG/GCEDM Simon Fraser University It is my great pleasure to write an introduction to the CMESG/GCEDM Proceedings from the 2002 meeting, held at Queen’s University in Kingston, Ontario. A necessary part of the introduction to the CMESG/GCEDM Proceedings 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 come from mathematics departments and faculties of education, from universi- ties and colleges, and from schools and other educational institutions. What unites them is their interest in mathematics and how it is taught at every level, and their desire to make teaching more exciting, more relevant, and more meaningful. The 2002 meeting was a special one: This was our 25th meeting which was held at the very place where our first three meetings happened. Since such an anniversary is an excel- lent opportunity for both celebrations and reflections, the meeting included, in addition to all the regular components, several special sessions which followed the central theme: Les- sons from the Past, Questions for the Future. The opening session included a panel which brought back three keynote speakers from the first CMESG/GCEDM in 1997: John Coleman, Claude Gaulin, and Tom Kieren. All three were asked to reflect on the topics of their 1977 talks from today’s point of view. Tom Kieren, whose 1977 talk was titled “The state of research in mathematics educa- tion”, spoke first. He reflected on some of the work done during the part 25 years in math education in Canada, on mathematical learning, on the development of mathematical know- ing and understanding. And he also recalled the words of David Wheeler, the “spiritual father” of our organization: “Could we, as CMESG, publish something others will be inter- ested in?” Claude Gaulin, the only one to have attended all our 25 meetings, spoke in 1977 about innovations in teacher education programmes. In 2002, he remembered how he spoke then about teacher education in Quebec, where in-service education was (and still is) very strong and world renowned. He said that we need a broad picture on what is happening in teacher education—both in-service and pre-service—across Canada, and to reflect on what is needed now. John Coleman, the one mathematician in the panel, spoke in 1977 on the objectives of math education. He said that the objectives are basically the same now as they were then. Possibly the most important one is to have students learn the joy of “mathematicizing”. He cited a statement from the book “Enjoying Mathematics” by Keith Devlin, which was recently reviewed in Mathematics Monthly, to the effect that 85% of elementary school teachers in North America have math anxiety. This is something we need to change. Students pick up emotions quickly and they often “inherit” their attitudes towards mathematics from their teachers. ix CMESG/GCEDM Proceedings 2002 • Introduction Three special Anniversary Sessions continued the theme Lessons from the Past, Ques- tions for the Future throughout the conference. The first session featured two mathematicians, Frédéric Gourdeau and Eric Muller, describing what our organization means to them and how it has influenced their under- standings of the nature of mathematics, its teaching, and learning. Frédéric also shared with us some of his frustrations on being a mathematician in a group of math educators, trying to understand their view of the world and being bewildered by their language. In the second session, Carolyn Kieran gave