At UIUC, my ICES ratings have been good and improving, with all ratings above 4.0 after my first semester here fulltime. In Spring 1999 my ratings for Math 323 (Differential Geometry) made me one of only four math professors included in the Incomplete List for excellence in an undergraduate course.
When lecturing on almost any mathematical subject, I like to bring out the geometric intuition behind the results under discussion. I find that a slightly informal style, with plenty of diagrams, conveys best those aspects of the material that are hardest to get from a textbook. Different students have different styles of learning, and thus it is best to give varied perspectives (say, geometric, analytical, and numerical) for each topic.
My favorite courses have been ones where I do not lecture every class period, but instead get the students actively involved in the work. Several years ago I designed a course Geometry and the Imagination, which conveys to students the basic notions of surface curvatures, symmetries, and orbifolds. All of this is done without calculus, largely through hands-on group work with polyhedral models. The course was inspired by one of the same name first taught at Princeton by Conway, Doyle and Thurston, and I used some of their ideas while developing the syllabus for my course.
In a typical class, after a brief review of earlier material, the students break into groups to explore that day's topic, guided by a written handout. Many of the discussions involve models: we have used Polydron triangles as well as tennis balls, masking tape, mirrors, and various vegetables. At the end of the term, each student does an independent project. Past projects have included models of four-dimensional polytopes like the 120-cell, sets of mirrors forming kaleidoscopes for the Platonic point groups, and computer programs for drawing symmetric designs in any of the 17 wallpaper groups.
By now I have taught this course several times, for instance to liberal arts honors students, to high-school students in a summer program, and to advanced undergraduates mostly majoring in math. The course has been enthusiastically received, with comments from students like "This was a brilliant course!" and "He challenged everyone while making sure everyone understood! Wonderful!" At UIUC, I offered this course in Spring 1999 as an honors seminar, Math 296. It was very popular, and I plan to teach it again this coming spring.
Computers are another good way to let students participate actively in learning. Several years ago, when teaching an undergraduate differential geometry course, I developed software for interactive exploration of curves and surfaces, with graphics and symbolic computation of curvatures. In such a course, computers are indispensable for showing a wealth of examples that would otherwise be inaccessible, though it would be a mistake to let this distract students from learning the theory. Many times, some students in a course are interested in helping develop software, and of course these students will get even more out of the computer experience.
My final year at Minnesota, I taught a new course to future math teachers, Technology in the Geometry Classroom. This course covered material about mechanical linkages, dynamical systems, and planar symmetries, while showing the students how to use computers in the classroom. We used Geometer's Sketchpad, various java applets, and special-purpose software like Kali. The students learned how to use these programs, how to teach with them, and how to create webpages for their final projects.
This past January (thanks to support from the campus Teaching Advancement Board) I participated in an MAA workshop on Geometry in the Classroom in the Next Millennium. There I demonstrated (to an audience of almost one hundred) my own java applets for stereographic projection and parallel transport on a sphere. However, I also demonstrated how actual physical models are much more instructive and convincing than computer simulations. At that meeting, I made contacts with many professors elsewhere who are interested in renewing the teaching of geometry.
Those contacts have led to my current work (with Prof. Susan Tolman) on revamping UIUC's course on non-Euclidean geometry, Math 302. We have switched from a dry axiomatic approach (as found for instance in Martin's book) to an interactive approach, following Henderson's text Experiencing Geometry. The new course uses small group work extensively in class, and makes the students come to grips with the basic notions of curved spaces in a hands-on fashion. Although the semester is only just beginning, we have already had a number of favorable comments from students about the new course. We expect that they will finish the course with a much better intuitive feeling for hyperbolic space than they would get from the more traditional course, thanks in part to physical models and interactive computer programs.
In conjunction with this work on redesigning Math 302, I have been in contact with Prof.~Bridget Arvold of Curriculum and Instruction. With her, I am participating in a new grant, the Illinois Professional Learners Partnership. This partnership (with four other universities in Illinois) has just this week been funded by the Department of Education, at over two million dollars per year. I will work with Arvold to improve the training of future teachers and to ensure that the technological skills they gain in courses like Math 302 will transfer to their high schools.
Too often, math majors, at the undergraduate level, see only established results, and get no real taste of the mathematical discovery process. The several excellent REU programs have started to change this, by involving undergraduates in mathematical research. I co-led one REU program at Five Colleges in 1993, where the students learned knot theory and helped compile a table of energy-minimizing knots. Another summer, I guided an undergraduate who wrote a program to study symmetries of electron configurations on a sphere. I would like to work in the future to increase the level of undergraduate research here at Illinois.
I feel that my effectiveness as a teacher is due in large part to my excitement about the material, and my ability to convey this enthusiasm to my students. I have also brought my excitement about mathematics to a wider audience in several ways. I have given many colloquium lectures at various undergraduate institutions, at industrial research labs, at interdisciplinary conferences, and in engineering departments here at UIUC and at other universities. Also, my research results have often involved computer simulations, and I have been able to make attractive computer graphics pictures to illustrate them; these images have been widely published and serve to increase the visibility of mathematics. Finally, I have written several expository papers explaining my research and related mathematics to a general audience.