I view teaching at Berkeley as a wonderful challenge. Because there are so many courses in the Department of Mathematics, I can teach different subjects every year. I never teach the same course two years in a row, and I prepare every lecture from scratch, "reinventing" material, and constructing new examples to illustrate theory.

To teach mathematics well, you have to understand that students in lower-division, upper-division, and graduate courses are different. Lower-division students must learn how to calculate, and they may have great difficulty following even the simplest proofs. It's like learning to walk: you need to do it before you can make it effortless. Hence, I try to present the material so clearly that students can apply it without much difficulty.

I don't let my students take notes. My lectures are intended to be listened to, and notes will never be as good as required books. It makes some students very uncomfortable; they have had years of practice taking notes to avoid thinking. Instead, I am asking them to watch, think, question, and ask.

I am also old-fashioned in not using overhead projectors. There is a tendency to put too much information on transparencies, and typical students cannot get anything out of the lecture as it flies by. Instead, I use the blackboard, slowly creating ideas before their eyes. I keep them involved by insisting that they help me perform routine calculations. And students delight in catching my mistakes.

When students begin upper-division classes, the character of mathematics changes. The importance of calculations diminishes and the attention becomes directed toward theory. It is exhilarating but frightening: grinding examples take a back seat to the abstraction and generality of mathematics. Since it is important to know definitions by heart, I ask the students to repeat them in unison. Sometimes it sounds like religious service: "What is continuity?" (they answer), "What is a compact set?" (they answer).

In graduate classes, students should learn how to think. I often teach the graduate numerical analysis and scientific computing class. In this course, students learn how to solve complicated problems on the computer. Just as in real life, my problems may have several answers. This irritates everyone; students want precise, tidy problems. But my job is to teach them how to take messy, vague questions and transform them into a precise model that can then be attacked.

Sometimes, students get stuck. To unleash their creativity, I ask "What is the dumbest way we can solve this problem?" After I have asked the question two or three times, it becomes a game in which everybody is willing to offer suggestions because of the playful nature of the question. We can then sift through the suggestions, weed out the bad ones, and study the good ones.