I decided to become a mathematician because I find math beautiful and exciting. I try to show my passion for the material of the course, but I also convey this enthusiasm through quick ``fun math facts'' that might be completely unrelated. During the name game on the first day, I ask students' birthdays as well as their names -- setting up the surprising fact that just 23 people in a room are enough for a better than even chance of a birthday match. Other classes begin with fun math facts on topics like the Euler characteristic or fractals -- providing quick glimpses into a world of math beyond the particulars of any course.

A focus on different learning styles. When I was preparing to give my first talk in a summer research program, my advisor taught me to ``write everything you say, and say everything you write.'' This was good advice for a research talk, and it also proved highly useful in the classroom. Writing everything on the board works to slow down my speech -- which can get quite fast when I'm excited -- and also creates a written record for students to take down. It also has the effect on catering to both visual and auditory learners. The first time I taught a class on my own, I had both a blind student and a deaf student in the room -- providing great motivation to show everything both ways.

In fact, most concepts have many more than two approaches. Functions in Calculus can be described both algebraically and graphically, but also numerically or with a verbal narrative. Different students prefer different descriptions, and most students can use practice translating among them. Thus when I teach a concept, I try to remember the perspective of someone who doesn't understand it yet, and get at it from several points of view. If a student (or, more typically, multiple students) is confused after the first explanation, I try to find another approach.

Active learning. Research indicates that even the most dedicated student listening to the most engaging lecturer will tune out after about twelve minutes. To avoid this effect, I work to involve students in the learning process. When I teach, I ask questions at multiple levels: both ``what's the derivative of this?'' but also ``how do we approach this problem?'' and ``why do you think this is true?'' Challenging questions do not get instant answers, so I have learned to wait -- longer than felt comfortable at first -- for students to think. If I don't hear an answer after 10-15 seconds, I might rephrase the question or give hints, or ask the students to brainstorm with their neighbors. One way or another, my students quickly learn to expect to grapple with hard questions.

Another useful tool has been to have students attempt problems in class, either individually or in small groups. Even in a 50-minute lecture or section, I have found that devoting some time to problems often does more to convey the underlying concepts than straight lecturing. This year, with a two-hour accelerated Calculus section, I've had the luxury of spending the entire second hour on problem-solving. In a recent unit on the applications of integrals, I had the students attack an unfamiliar problem about measuring a heart patient's cardiac output from the concentrations of a dye dissolved in the blood. While I circulated around the room to provide touch-up help, the small groups figured out a way to set up the problem as an integral. Within each group, even the sharpest and quickest students receive the benefit of explaining their approach to the problem.

Active learning is highly effective in other settings. For three summers in a row, I have led a seminar to prepare graduate students for qualifying exams in Complex Analysis. We met twice each week: once to cover conceptual background, and a second time to go over problems from past exams. At first, I lectured on the background days -- but we quickly discovered that everyone learned more when the new students took turns presenting background topics. Each week, the student presenting had to learn the material especially well, and everyone else stayed alert with questions. On the problem-solving days, the students took turns showing solutions on the board -- although the explanations often turned into active discussions in the middle of a problem.

Communicating quantitative ideas. Until they reach college, the only practice that many students receive with writing mathematics involves ``showing their work'' on a problem, so that someone already familiar with the concept can check if they did it right. On the other hand, the scientific disciplines that many of our students pursue -- from psychology to economics to civil engineering -- often expect them to both speak and write cogently about quantitative ideas. I believe that writing about math not only helps to bridge this gap, but also solidifies students' understanding of the concepts.

When I had the opportunity to teach my own Calculus course in the winter of 2002, I designed an applied project in place of one of the midterm exams. The students had to work in groups to recommend a chemotherapy dosing schedule for an advanced cancer patient -- a schedule optimized to aggressively treat the cancer without destroying the patient's immune system. These tough competing constraints made for a challenging problem. Once they settled on a schedule, the students in each group had to jointly write a paper explaining and justifying their solution to hypothetical doctors. Several commented afterwards that the process of explaining the model forced them to understand it much better.

Consulting experience. As I have grown more confident in my own teaching, I became first a liaison and then a consultant with Stanford's Center for Teaching and Learning (CTL). In these roles, I have been able to promote and spread some of the wisdom that the CTL's staff have accumulated. I have organized several teaching lunches, in which faculty and students gather to discuss how to teach the notion of limits, how to write effective exams, or other teaching-related topics. I have planned workshops for training new teaching assistants -- first at the department level, then University-wide. In the past year, I have also started holding videotape consultations to help instructors improve their teaching. I have facilitated small-group student evaluations that provide highly specific feedback in the middle of a course.

These planned activities, as well as simple informal conversations with the other consultants and CTL staff, have taught me an increased awareness of learning styles and a number of new tricks to use in my own teaching. For example, I have learned to replace the standard question of ``do you have any questions?'' -- which most students treat as rhetorical -- with ``what questions do you have?'', an easy switch to a much more effective invitation to participate. I have learned to use ``minute papers'' at the end of a class to quickly determine the topics that my students find most confusing. My growth as a teacher is by no means complete, and I look forward to learning more in a new position.

Note: A teaching video and teaching portfolio are available upon request.