




Mathematical Education: The teaching and learning of geometry: why, what, how. / Éducation mathématique : L'enseignement
et l'apprentissage de la géométrie : pourquoi, quoi, comment. (F. Gourdeau and B. R. Hodgson, Organizers)
 JANOS BARACS, Université de Montréal
The teaching of spatial geometry: the 9 basic problems

We would like to promote a pedagogical change. The teaching of geometry
at any level should begin with spatial geometry without the use of
projections in order to liberate the young minds from the confines of
``Flatland''. Plane geometry should than follow as a special case and
as a means to reproduce space. A real obstacle of this program is the
weakness of some teachers and students alike in a basic aptitude: the
perception of the threedimensional space.
We recognized in our previous research and experience the crucial phase
of spatial perception: the creation of an imagery, which is an
invisible mental image of an abstract geometrical scene. The teaching
of the 9 basic spatial geometry problems proved to be a true didactical
laboratory for the creation of such an imagery in the classroom.
The presentation starts with a short introduction of the vocabulary,
the grammar and the syntax of points, lines and planes, their unions
and intersections.It is followed by five theorems of spatial geometry
which are then applied in the solutions of the 9 problems of increasing
complexity.
 DAVID W. HENDERSON AND DAINA TAIMINA, Cornell University
Experiencing geometryworkshop

This workshop will facilitate a handson cooperative experience of the
geometries of various surfaces, including cones, cylinders, spheres,
and possibly the hyperbolic plane. Studying the intrinsic geometry (as
a crawling bug would experience it) of these surfaces enhances our
understanding of plane geometry. In addition, these explorations help
to demonstrate a nonaxiomatic, nonformal view of mathematics and
mathematics learning. Students at all levels (including elementary
school) should and can understand mathematics and can communicate this
understanding with others and we believe that this involves proofs but
we define ``proof'' as: ``A convincing communication that
answersWhy?'' This notion of proof is appropriate for all levels of
mathematics classrooms. If we listen to the students' proofs, we can
often learning new things in mathematics from them.
 WILLIAM HIGGINSON, Queen's University, McArthur Hall, Kingston, Ontario L7L 3N6
Beyond monocular mathematics

Much time is spent in educational circles in bemoaning the numerous and
diverse weaknesses of mathematics teachers and learners. From an
external perspective it might, however, be argued that, on a more
general level, mathematics educators have been exceptionally successful
insofar as much of the contemporary world is dominated by the various
manifestations of the concept of number.
This `wrap up' session will claim that this localized success has
brought with it many grave problems for both individuals and societies.
It will be argued that a partial solution for some of these excesses
might be realized through the implementation of a much greater role for
geometry (of a certain sort) in curricula. In particular this attempt
to move toward a more balanced conception of mathematics and its
potential contributions to 21st century education will emphasize the
cultural and aesthetic aspects of geometry.
 KLAUS HOECHSMANN, Pacific Institute for the Mathematical Sciences, 1933 West Mall
at UBC, Vancouver, British Columbia V6T 1Z2
A la recherche du plan perdu

Le manque de géométrie dans nos écoles secondaires correspond à
une lacune dans les programmes universitaires: les géometries que
l'on y enseignelinéaire, algébrique,
différentielledemandent une étude préalable (souvent
avancée) d'analyse et d'algèbre; le peu de géométrie
synthétique, qu'on y trouve, sert presque uniquement à véhiculer
la méthode axiomatique. Ce ne sont pas des modèles prêts à
porter pour les futurs enseignants de lycée. Il faudrait songer à
une résurrection de la géométrie euclidienne, au niveau
universitaire, comme elle existait il y a moins de cent ans.
Aujourd'hui elle sera sans doute moins formelle (sans perte de
rigueur), plus axée sur les idées, le débat, et le genre de
problèmes que ce discours signalera.
 JACQUES HURTUBISE, Centre de Recherches Mathématiques, Montréal,
Québec H3C 3J7
Invariance and global properties

This lecture will be a rather discursive and anecdotal examination of
two essential features of geometry that permeate research in the area:
the first is the importance of invariant descriptions of geometric
objects, reflecting their intrinsic properties and their stability
under the action of a relevant group, and the other is the importance
of global properties. Examples are taken from Euclidean geometry,
differential geometry, and complex geometry. It is hoped that these
reflections will stimulate some discussion as to how these two features
can be reflected in the preuniversity mathematics curriculum.
 YVAN SAINT AUBIN, Département de mathématiques et de statistique, Université
de Montréal, Montréal, Québec H3C 3J7
Teaching geometry to future teachers

The 4year program for future highschool mathematics teachers at
Université de Montréal contains a single 4credit compulsory course
on geometry. In a new (elective) course devoted to the applications of
mathematics (mostly to technology), five hours were devoted to friezes
and mosaics. During these five hours, affine coordinates were
introduced, the seven frieze groups classified and the equivalent
problem for mosaics glossed over. I shall describe two exercises done
with the students.
 MARGARET SINCLAIR, York University, 4700 Keele St., Toronto, Ontario M3J 1P3
Using JavaSketchpad to enhance geometry learning

Dynamic geometry is the exploration of geometric relationships by
observing geometric configurations in motion. Although, in most
classrooms, these configurations are constructed by the students
themselves, sketches preconstructed by the teacher or downloaded from
a website can also be used. There is a continuing discussion in the
educational technology community about whether it is better to give
students powerful generalpurpose programming and construction tools or
to have them interact with preconstructed, interactive models. Some
strongly support student constructions because they believe that
students develop a deeper understanding of the object by explicitly
connecting the parts. Others believe that preconstructed models are
valuable as learning tools because ability to recognise connections
between geometric objects is a necessary stage before students can
effectively carry out many constructions. In an attempt to inform this
debate my research investigated the benefits and limitations of
using JavaSketcheswebbased, interactive, dynamic geometry
sketcheswith senior high school students in geometric activities
related to proof. In this seminar I will present some of my findings,
and demonstrate the features of preconstructed sketches that helped
students focus attention on mathematically meaningful details. By
reflecting on my results in relation to the extensive research on
Geometer's Sketchpad and Cabri, I will attempt to characterise
situations in which preconstructed sketches can play a purposeful role
in the geometry program.
 WALTER WHITELEY, York University, 4700 Keele Street, Toronto Ontario M3J 1P3
Geometry with eye and hand

Following the lead of recent writings on mathematical cognition, we
focus on the role of visual (and associated kinesthetic) experience and
reasoning in the practice of geometry. This understanding matches some
rereadings of the van Hiele model of geometry learning, and we will
offer a diagrammatic model for a modified version of the connections
among visual and verbal processes in the development of understanding
and reasoning in geometry education.
We will illustrate these larger points with examples and conceptual
approaches based on symmetries and transformations as the central core
of geometry, following the path of Klein's Erlanger program. These
will be applied to simple school tasks, such as definitions and
classifications of quadrilaterals (a source of confusion and debate in
the school curriculum). The examples will include Geometers SketchPad
sketches, origami, and other tools, as well as connections to areas
such as stereochemistry.

