I decided to become a mathematician because I find math beautiful and exciting. I try to show my passion for the subject matter 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 to be a useful guide in the classroom. Writing complete sentences 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 study from. This has the effect of 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 demonstrate 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 is confused after the first explanation, I try to find another approach.

Active problem-solving. 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 ``why do you think this is true?'' and ``what tools do we need to prove this result?'' 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 15-20 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 non-trivial questions.

Another useful tool has been to have students attempt problems in class, either individually or in small groups. Even in a 50-minute setting, a problem-solving interlude often does more to solidify the underlying concepts than straight lecturing. In a calculus class, these problems might take the form of short exercises that firm up the concept of the day. In proof-based courses for mathematics majors, I have found that discussing a sequence of true/false questions on theoretical topics really helps students get at the nuances of a definition, and smoothes out the creases of misconceptions before they set in too deep.

I have also sought opportunities to guide students through harder problems in a longer setting. In a real analysis course that I recently taught at Michigan State, most of the students were grappling with their first real exposure to rigorous proofs. To help guide them through the difficult material, I led an optional problem session. Every Monday night, the students worked in groups to brainstorm and formulate the arguments for that week's proof problems. The process of explaining a solution to their peers turned out to be instrumental to solidifying understanding.

Feet-on learning My favorite experiences with active learning have been the ones that give students a tangible grasp of the subject at hand. In a point-set topology class last fall, we once spent over half an hour constructing surfaces out of polygonal pieces of paper. This led the students to intuit the concept of a homeomorphism (what does it mean that two of these are the same surface?), and provided the setup for the notion of Euler characteristic. Another time, while teaching vector calculus, I took my class on an excursion into the hills behind Stanford's campus. With contour topographic maps in hand, the students could compute directional derivatives in the direction of the trail, and then check their answers by looking at the steepness of the actual hill. By tracing out gradient flow lines on the map, we managed to locate an unmarked creek.

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. Their opportunities to speak orally about scientific topics are even more limited. On the other hand, the scientific disciplines that many of our students pursue -- from psychology to economics to civil engineering -- expect them to both speak and write cogently about quantitative ideas.

To help bridge this gap, I have included writing and speaking components into many of my courses. In an undergraduate topology course, I had each of the eight students give a half-hour presentation on topics that ranged from the math behind fractals to the shape of the physical universe. According to student feedback, one challenging and rewarding part of the assignment was to internalize the material well enough to explain it. One great benefit was that we all got to see the connections between topology and other subjects.

Assignments of this sort are helpful even in a lower-level class. During a first-semester calculus course, 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 formulating a written explanation of the model forced them to understand it much better.

Since arriving at Michigan State, I have had the opportunity to mentor several promising undergraduates. In the first week of a core class in real analysis, three students approached me about doing an honors version of the course. For the rest of the semester, we met every other week to gain a deeper insight into the key ideas of the class, while also discussing career opportunities. More recently, two other students initiated an independent study project in point-set topology. Of those five students, three have just entered graduate programs and a fourth is currently applying.

Consulting experience. As I grew more confident in my own teaching, I became a consultant with Stanford's Center for Teaching and Learning. In this role, I have planned workshops for training new teaching assistants -- first at the department level, then University-wide. I have also held a number of mid-semester consultations to help instructors improve their teaching, working off videotaped lectures and small-group evaluations solicited from students. After moving to Michigan State, I have continued to provide this type of teaching advice more informally.

This consulting experience has helpted me to contineu my own growth as a teacher. I am still learning how to pitch problems at the proper level, and how to give hints that are helpful without giving away the whole question. I am still exploring how best to gauge student understanding during a class, in order to react rapidly when a confusing topic needs extra attention. I look forward to learning more in a new position.

Note: A teaching video is available upon request, as are student evaluations, writing assignments, etc.