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 three-dimensional 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 geometry-workshop This workshop will facilitate a hands-on 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 non-axiomatic, non-formal 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 answers-Why?'' 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 enseigne-linéaire, algébrique, différentielle-demandent 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 pre-university mathematics curriculum. YVAN SAINT AUBIN, Département de mathématiques et de statistique, Université de Montréal, Montréal, Québec  H3C 3J7 Teaching geometry to future teachers The 4-year program for future high-school mathematics teachers at Université de Montréal contains a single 4-credit 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 pre-constructed 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 general-purpose programming and construction tools or to have them interact with pre-constructed, 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 pre-constructed 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 JavaSketches-web-based, interactive, dynamic geometry sketches-with 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 pre-constructed 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 pre-constructed 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.
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