Elaine Simmt and I are engaged in a 4-year study of the sorts of mathematical competencies that are necessary for effective teaching where "mathematical competencies" are understood to include both awarenesses of how subject matter knowledge is developed (on individual and collective levels) and competence with already-generated subject matter knowledge. In this session, I would like to report on some of the emphases in this research, our rationales for these emphases, some preliminary interpretations, and their possible implications for undergraduate mathematics courses for teachers.

Chaired by:

Malgorzata Dubiel, Department of Mathematics, Simon Fraser University, and

Peter Liljedahl, Faculty of Education, Simon Fraser University

Panelists:

Brent Davis, Faculty of Education, University of Alberta,

Tom O'Shea, Faculty of Education, Simon Fraser University,

Peter Taylor, Department of Mathematics, Queens University,

Walter Whiteley, Department of Mathematics, York University

Abstract:

At the 2003 CMS Forum on School Mathematics, held last May in Montreal, one of the key issues discussed was the problem of overloaded school curriculum, the result of pressures felt across Canada to include more and more topics, and cover them in less time. The rationale often given for maintaining current programs of study is: "The kids will need this if they're going to be successful at university math". But students' learning does not increase proportionally to the increase in concepts covered, and so, the end product of such curriculum is frequently a student with a toolbox full of disconnected knowledge, but not necessarily well prepared for university courses.

In preparation for the second CMS Forum, which will be held in 2005 in Toronto, the discussions on curriculum were continued by one of the Working Groups at the 2003 CMESG meeting at Acadia University. One of the outcomes of this work was the Position Statement on School Mathematics Curriculum . This Statement will be a starting point of this panel discussion.

The title of the panel was inspired by a session held ten years ago, at the joint AMS-CMS-MAA meeting in Vancouver, which was titled "Less is More". The mathematicians present there expressed an idea that, if we exercise some restraint and provide more freedom, teachers can be more creative and students are much more likely to come away with the skills we want them to have. Everyone at the session agreed. But, in the past ten years matters grew even worse. So, is there any realistic hope for a less-is-more curriculum? A new geometrical formulation of the hydrodynamics equations is given for point-vortices on surfaces with costant curvature. More specifically, we consider regular polygonal configurations and the transition in their stability properties when passing from spherical to hyperbolic geometry. The problem has been formulated in a unifying geometrical setting. (work in collaboration with Tadashi Tokieda)

In 1943 Jacques Hadamard gave a series of lectures on mathematical invention at the Ecole Libre des Hautes Etudes in New York City. These talks were subsequently published as The Psychology of Mathematical Invention in the Mathematical Field (Princeton University Press). Hadamard's seminal work outlines the beliefs of contemporary mathematicians as to the mechanism by which they come to create new mathematics. In 2002 I resurrected this classic questionnaire and used it to survey 25 of the most prestigious mathematicians in the world. In this presentations I share some of the results of this survey as they pertain to, among other things, the learning of mathematics.

In mathematics, as in many fields, there are often several ways to solve a problem. Does it matter which method a student is taught? What happens when the student meets a different method at a later stage? Is the only goal to get the right answer?