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Working Group 13: Rethinking Math Thinking in Secondary Maths Classes / Repenser les mathématiques au secondaire
(E. Barbeau, D. Tanguay, P. Taylor)

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EDWARD BARBEAU, DENIS TANGUAY AND PETER TAYLOR, Department of Mathematics, University of Toronto; Département de mathématiques, UQAM; Department of Mathematics and Statistics, Queen’s University
Rethinking Math Thinking in Secondary Maths Classes / Repenser les mathématiques au secondaire [PDF]

Rethinking math thinking in secondary maths classes

Secondary mathematics courses face the dilemma of providing an active and lively classroom experience for all students, while at the same time preparing those whose vocation demands a more technical and comprehensive background. Can our teaching go beyond the "basics" (whatever these are) to provide an intellectual foundation that includes sense of organization, structure, synthesis, strictness and clarity, as well as astute judgment in selecting among competing approaches, and the ability to think intuitively and rigorously? Can we foster this in the face of an ideology that is narrowly utilitarian? How can we encourage in the education of teachers a more holistic view embracing these broader goals, and help them to transfer these to their own classrooms? How can we train pre-service teachers to a less procedural way of tackling and teaching mathematics? Participants in this workshop should reflect on these issues, bring their own experiences and perspectives and consider what changes in curricula, local and provincial systems and even in social values and attitudes might address them.

Repenser les mathématiques au secondaire

En classe de mathématiques au secondaire, l’on est confronté au dilemme de procurer à tous les élèves une expérience d’apprentissage vivante et stimulante, tout en assurant une formation plus complète et technique à ceux dont la vocation l’exige. Cet enseignement peut-il aller au-delà des << bases >> (quelqu’elles soient), et transmettre une véritable << formation de l’esprit >> qui incluerait : le sens de l’organisation, la synthèse, la structure, la rigueur et la clarté, aussi bien que le jugement qui permet un choix judicieux entre deux approches divergentes, ou que l’habileté à diriger sa pensée par l’intuition et la déduction ? Comment le faire à contre-courant d’une idéologie étroitement utilitaire ? Comment ouvrir les maîtres en formation à de tels objectifs, plus larges ? Comment favoriser chez eux une approche moins procédurale, une appréhension plus holistique des mathématiques ? Comment les aider à mettre tout cela en œuvre en classe ? Les participants sont conviés à réfléchir à ces questions, à faire part de leurs expériences et point de vue, à évaluer quels changements dans les curriculums, les systèmes éducatifs locaux et provinciaux, et mêmes les attitudes et valeurs sociales, sont susceptibles d’apporter des solutions.

Session 1: Edward Barbeau, Department of Mathematics, Toronto University

Title: Understanding, Power and Applicability

Modern society requires people with a broad range of mathematical skills, from the citizen who must negotiate the demands of daily life to the scientist on the frontlines of research. This cannot be achieved by a `toolchest´ approach to education, where the emphasis is merely on providing a lot of technical procedures. The modern student needs a more holistic view of mathematics that involves a feeling for structure, strategic thinking, astute judgment in selecting among competing approaches, and the ability to think intuitively or rigorously as required. Only in this way can the power of mathematics be tapped. How can the curriculum meet these requirements, while being accessible, meaningful and interesting to all students? How does this bear on the expectations we have of teachers and developers of the syllabus?

Session 2: Peter Taylor, Department of Mathematics and Statistics, Queen’s University

For many years now we have all written passionately and eloquently about mathematics education in the schools and why it most often seems to fail. But little changes and one wonders if we are wasting our time. A common view is that the biggest culprit is the overstuffed curriculum. It is clearly not enough simply to recognize this problem and resolve to do something about it. The group who recently wrote the policy document for the new Ontario curriculum were all agreed on this, and yet somehow it (or someone else) produced a list of topics that is far too heavy for all but the most able students, at least if understanding and independence are to be thought of as reasonable goals. So we have to dig more deeply and understand why this keeps happening. I believe that the problem ultimately has to do with our conviction that mathematics can only be taught in a systematic manner, that if we are teaching calculus, we of course have to do this and this and this... It is true that mathematics is a systematic and hierarchical discipline and much of its beauty and power derives from this structure, but do most students learn in this manner? In being systematic, we might serve the subject, but do we serve the student? And if not, can we really serve the subject? In this working group we will pursue these questions.

Session 3: Denis Tanguay: Département de mathématiques, UQAM

Titre : Déployer un raisonnement en mathématiques : comment développer cette compétence chez les enseignants du secondaire en formation

Le nouveau programme d’études secondaires du Ministère de l’éducation du Québec fait de la compétence << déployer un raisonnement en mathématiques >> une parmi trois compétences fondamentales. Cela renvoie aux difficultés qu’éprouvent les enseignants en formation face à la preuve, entre autres les difficultés d’ordres logique (départager la thèse des hypothèses, ne pas utiliser la thèse comme argument, différencier une implication de sa réciproque, comprendre le statut de l’exemple et du contre-exemple, etc.) et les difficultés d’ordre sémantico-langagier (distinguer les définitions formelles des descriptions informelles, des conceptions intuitives, des significations méta ou extra-mathématiques véhiculées par le terme défini, comprendre et repérer les quantifications implicites, etc.). Des questions se posent alors à l’égard de la démonstration (la preuve formelle) : peut-elle ou doit-elle faire l’objet d’un enseignement spécifique ? Est-il possible de donner accès aux savoirs de logique formelle sous-jacents sans discréditer d’autres formes de discours comme l’argumentation ou l’explication, sans inhiber chez l’étudiant ses capacités à recourir à l’intuition, aux associations, aux métaphores, sans faire de la démonstration l’unique but à atteindre en mathématiques, plutôt qu’un outil privilégié de validation permettant une meilleure appréhension du sens?