MATH3050.06 Course Outline:
Introduction to Geometries 2006-07.

Course Director: Walter Whiteley
Office: South 518 Ross
Telephone: 736-5250 Extension 22598
E-mail: whiteley@mathstat.yorku.ca
WWW: http://www.math.yorku.ca/~whiteley
Office hours: TBA (currently by appointment)

Classes: Tuesday 6:00-9:00 Ross S525
Note the room change

Tutor Bernd Schulze
Office Ross N611
E-mail: michiganbernd@hotmail.com
Tutorial To be scheduled.

Geometry has an important classical side: Euclidean Geometry as practices by 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 related to 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 applications, 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 approaches.
Geometry also has important modern applications, to such areas as Computer Aided Geometric Design, computer graphics, computational geometry, robotics, modern physics, biology and engineering. Even how geometers such as myself practice geometry (and teach geometry) is changed by new computer programs, such as The Geometer's Sketchpad. This program has now been purchased for all schools and all teachers in Ontario, and its use is required in the new grade 9 curriculum.

In this course we will explore these plane geometries multiple settings. However, underneath this content, I have some more basic and far reaching objectives: this course is designed to further reflection on the practice, the learning and the teaching of geometry in particular and mathematics in general. As a geometer, and an educator, my deepest hopes for this course include that it will:

• change what you see (in geometry and elsewhere);
• change the questions you ask (and how often you voice them);
• change how you learn mathematics;
• change how you think in mathematics;
• improve your communication of mathematics in many modes and forms;
• improve your geometric reasoning and spatial visualization.

To be more modest, my expectations are that by the end of the course you will:

• Experience "seeing geometrically" and know that you can change what you see);
• Be reflectively aware of how you learn mathematics and how others learn mathematics);
• Be willing (and able) to ask your own mathematical questions, in general, and ask geometric questions in particular, with your own 'voice';
• Be better able to investigate open ended problems which have a geometric basis);
• Be able to communicate your questions, your methods, and your answers orally, visually, and in writing to other students);

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; I encourage students to use study groups for this course (and other courses). There is some supporting material at Study Group Guide for Students
• work with and build physical models in class (such as plastic spheres for spherical geometry, plastic "polydron" for polyhedral models, kaleidescopes, origami, etc.);
• work with a dynamic plane geometry program: The Geometer's Sketchpad (which is installed in the many labs, including the Gauss Lab which we will use for some classes, and which is also accessible in our classroom Ross S525). Under the provincial license all students are entitled to a copy of this program at home. For pre-service teachers, they should also be available through your host school, as well as from the Education Resource Center. For other students I will circulate my copy, or you can contact the library at your high school.
• Spherical Easel, a java based program which replicates the constructions of Geometers' Sketchpad for the sphere. It can be used on the web, or downloaded as a program for your own use, from the web site: merganser.math.gvsu.edu/easel/.
• develop your own geometric questions, conjectures and projects.
• 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 that is significant to your own learning! Asking geometric questions is a core activity of any geometer.

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. For any people preparing to become mathematics teachers, these resources will be important support for your practicum courses and for your teaching.

Text: We have one text for the course, plus other supplementary materials. We will begin working from the text at the second class, but will not cover all of this text.

• David Henderson and Daina Taimina: Experiencing Geometry: Euclidean, Non-Euclidean with Strands of History (Prentice Hall), 2004;
• You can also use the 2nd edition of David Henderson: Experiencing Geometry(Prentice Hall), 2000, which may be available used.

These are for sale in the bookstore. We will cover, at least, the first 11 chapters of the first edition, during the fall semester.

Further materials are in the course cabinet in the classroom and can be borrowed from the instructor.

Evaluation: Graded work will be something like:

1. regular assignments, including proofs, conjectures, (approx. 50%)
2. reflections on learning and geometry (selections from a geometry journal) have now been included in each assignment;
3. the major project, marked in stages, including preliminary proposals and drafts, presentation in class, and written (drawn?) final form, with reflections (approx. 40%);
4. participation in in-class work (sheets from in class activities turned in, with signatures) (10%)

Every assignment should end with a page (or so) of your current questions, or responses to an ongoing dialog with the instructor and the tutor, provoked by previous questions. As was noted above, developing your voice to ask geometric questions is an essential objective of the course. (See the handout on evaluation standards .)

Some additional background material on visual reasoning in mathematics, and my own take on the recent history of geometry is available on my main web page, and I repeat some of the links here.

• I participated in a working group supported by SIGGRAPH creating a White Paper on Visual Learning in Science and Engineering. You can find
our report at the SIGRAPH site.
• Further discussion occured at the recent International Congress on Mathematics Education 10, in Copenhagen, where I co-chaired Topic Study Group 16 on Visualisation in the teaching and learning of mathematics.
• My own paper, distributed at ICME10, was about Claims and Questions towards a Research Program in Visualization. Download a PDF file of this paper.
• Dynamic Geometry and the Practice of Geometry, for distribution at ICME9 Tokyo, July 2000.
  Download a PDF file of this paper.
• Teaching to see like a mathematician, to appear in the proceedings of the conference on Visual Representation and Interpretation conference in Liverpool England in September 2002. download
• The Decline and Rise of Geometry, which appeared in the Proceedings of the 1999 CMESG Conference
  download a PDF file of this paper.