MATH5450.06 Course Outline:
Geometry for Teachers.
Course Director: Walter Whiteley
Office: South 616 Ross
Telephone: 736-5250 Extension 33971
E-mail:
whiteley@mathstat.yorku.ca
WWW: http://www.math.yorku.ca/Who/Faculty/Whiteley/menu.html
Office hours: TBA (currently by
appointment)
Lectures: Monday 6:00 - 9:00, Thursday 6:00-9:00 Ross S525
Introductory Remarks:
Geometry has an important classical side: Euclidean Geometry from the Greeks moving, in the last two centuries, to non-Euclidean geometries (which differ by their assumptions about parallel lines), including spherical, hyperbolic and projective geometries. This transition is one of the critical `paradigm shifts' in the history of mathematics. The hierarchy of geometries (organized by their transformations) will be one
theme of the course.
In modern geometry, the interplay of abstraction, axiomatics, synthetic geometry, analytic methods, and groups of transformations presents a rich mix of mathematical methods and problems to be explored. We will explore simple plane and spherical geometry from
several of points of view, beginning with synthetic, but expanding to include
analytic, and axiomatic
Geometry also has important modern applications, to such areas as Computer Aided Geometric Design, computer graphics, computational geometry, robotics, modern physics and engineering. Even how we practice Euclidean geometry (and teach geometry) is being changed by computer programs, both symbolic (such as Maple) and visual (such as Geometer's Sketchpad).
In this course we will introduce these plane geometries in their classical and their modern settings. This course is designed to further reflection on the teaching and learning or geometry.
Geometry is also a good subject to explore the role of
the visual in the practice of mathematics: generating
insights, problem solving, communication, remembering,
etc. An underlying goal of the course is to 'change the way we see'. How we see is key to many of the ways
in which we work in geometry (and in other parts of
mathematics).
Prerequisites: The formal prerequisites are minimal: I will assume familiarity with linear algebra (vector spaces, matrices, linear transformations, eigen vectors) and some mathematical maturity. All other background will be developed as needed. I will expect you
- to join in group work regularly in class and some group work outside of class;
- to work with and build physical
models in class (such as plastic spheres for spherical geometry, plastic
"polydron" for polyhedral models, kaleidescopes);
- to work with a dynamic plane geometry
program called Geometer's Sketchpad (which is installed in the Steacie Labs and also accessible in S525). All high schools or school boards already have copies available.
" Student Versions" of this program (full copies with limited documentation) can be purchased, for Windows or the Macintosh, through the instructor for about $65). Meanwhile, you can practice (but not save or print) with a downloaded demonstration copy for either Mac or PC.
- to explore the use of visual material in the
learning and communication of mathematics (including geometry);
- to develope your own geometric questions, conjectures and projects:
- to prepare and present some material in class, and in a
written project;
While the topic of this project must be discussed with me,
I encourage you take your own questions and ideas seriously. Propose a project which is significant to your own learning!
In addition, I encourage you to use the resources of the internet to track
information and discussions about geometry. I can suggest several electronic news groups as well as the following web sites linked on my page
of
interesting geometry sites.
I may require you to sign onto one of these lists,
for a few weeks, and comment on the potential
of such lists as a resource.
Texts: We have one text for the course, plus many supplimentary materials.
We will begin working from the text at the second class,
but will not cover all of this text.
- David Henderson: Experiencing Geometry on Plane and Sphere
(Prentice Hall), 1996; ( on reserve in Steacie, as PC1802
You are also encouraged to develope your own 'model spheres' for
explorations at home. Some suggestions and examples will be
offered during the first few classes.
Evaluation:
Graded work will be something like:
- regular assignments, including proofs, conjectures, (approx. 40%)
- progressive development of responses for open ended problems (10%);
- participation in class, including in class 'lab (10%);
- oral presentations, written projects (minimum 30%);
- reflections on learning and geometry (selections from
a geometry journal) (5%);
- possibly a 'visual / geometry journal' (5%)
Every assignment should end with a page (or so) of your current
questions, or responses to an ongoing dialog with the instructor, provoked
by previous questions. (See the handout on evaluation standards .)
At this level (Masters program) I hope your own curiousity and questions about the mathematical material are bubbling to the surface. I strongly encourage you to ask 'What if ... ?' about any or all of the material. I assume that any good assignment will leave you with more questions and a sense of other possibilities when you 'finished' than when you started! In the marking scheme, there will be a deduction of one grade point, if you fail to include this last page of questions/dialog/responses.
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