Here is a selection of courses I've taught over the years, and some of my thoughts about them.
I have yet to teach this, actually. But I'm excited for it, especially the statistics part, because I know very little statistics. There are two things I want to try to incorporate. The first is material from Nate Silver's FiveThirtyEight, a website about politics from a statistical slant. There are lots of great articles on current topics, but there is also a lot of talk about understanding polls and statistics, and I want my students to appreciate how subtle and difficult this can be. Pure mathematics (and the human mind) is enamored of causation; it will be nice to try to play the correlation game a little. The second thing I'd like to talk about, at least a little, is developments in topology and statistics, specifically, persistent homology. While I think this isn't the right class for understanding homology, I do think the ideas of topology are useful and natural in statistics, and I'd like to expose my students to this mode of thinking, because it looks like a promising new field.
I've always felt the material in this class to be something of a hodgepodge. On the one hand you cover integration, techniques of integration, and applications, and on the other hand, you talk about sequences, series, and Taylor series (if you're lucky; it seems Taylor series gets the short end of the stick at lots of places). The theme that unifies this class for me is approximation. I like to cover the error formulas both for things like the trapezoid and midpoint rules, as well as those for series and especially Taylor series. I think the students find this material difficult, but I think it's important to understand how powerful it is. We often give in to the temptation to live in a perfectly computable world in class, because many of us live in this world in our own research.
One thing I really like about teaching this course at Wellesly is the extra time. We meet for 70 minutes three times a week, which gives me an extra hour each week with my students. Some of this translates into covering more material, but not much. I like to spend time each class period giving my students something to work on while I wander around and help people, and fortunately the small class size (around 25) makes this possible. I used to go to these teaching lunches comprised of some science faculty while I was at Stanford, and one thing that will always stick with me was a study that was presented showing a marked difference in retention between those students who immediately practiced what they were taught and those who did not. The extra time gives me this opportunity, and I and my students are grateful for that.
This was a difficult class for me to teach. I am not an expert in this area, and although I was teaching mostly first-year graduate students, they were more than able to keep up with me. I am something of an expert on spaces of embeddings, but I felt like quite the outsider teaching this course. I used a wide range of material: Lickorish's GTM and Rolfsen's book, some handwritten notes of my own on the linking number (about which I have written a paper), another set of notes from Ismar Volic on finite type invariants, and some papers of Bar-Natan about Khovanov homology. I wanted to give the students some more recent work in addition to classical stuff, not only because I was curious to learn it, but because it is important to expose them as early as possible to questions people are actively thinking about. I was lucky to have a very motivated and engaging group of students.
This is probably my favorite undergraduate class I have taught. This is the course where I learned how to incorporate in-class work with students in a successful way, because it was an integral part of the design of the course. It also taught me the value of interaction with one's students. When they see that you're invested in them, they get invested too. Math Xa and Xb (I think now renamed) are basically a year-long course which ends up covering the material from Calculus I. To describe it in terms of its content is to do it a huge disservice, however. It was taught using a "first principles" philosophy. Every function and concept was introduced from its proper definition, and we carefully deduced each of its properties. This gave students a really solid foundation in logical reasoning, and it also taught them how to ask questions. Every day we had a worksheet for these students consisting of challenging problems that required my and each others' help, and it was great to have two really tremendous undergraduate TAs to help out.
I also had a graduate student sit in a help for a week and then teach for a couple of classes as part of his training for being in the classroom, and it felt a little funny to be the one giving advice on teaching. I really enjoyed watching someone start to find his voice and approach as a teacher. I was lucky to have very motivated students whom I got to know very well because of our constant interaction. I think this class perfectly suited my style, which is to let the students work and struggle and only give a push when they need it.
As part of my NSF postdoctoral fellowship I had proposed to give a course on calculus of functors, and Stanford was nice enough to let me do so. Since Stanford is such a mecca of topology, I had a pretty large enrollment consisting of some people I am very grateful to now call colleagues. Although I am an expert on this material, I really had to be a stickler for the details when presenting to this group of students. I have yet to learn how to be dismissive of unanticipated questions that might be better left for reading, because my reaction always was "Well, that's a good question, let's think about it." I wasn't always able to pull off an on-the-spot answer, and as a result I certainly didn't cover as much material as I would have liked. However, I do think that exposing what are going to be some of the best topologists out there to the ideas of calculus will ultimately prove fruitful for me and them.
I taught this course with Ben Brubaker (now at MIT) when we were both postdocs at Stanford. Ben wanted to learn some of this stuff for his own research, and he roped me into helping. Basically, I did taught the Lie groups material, and Ben taught the Lie algebras material. We had a very large and lively class, and it was very challenging. This is not material I have a very good perspective on, so it was difficult at times to find the right tone and emphasis. But it was worth it, and I really still think this is something every graduate student should learn, because it ties together so many things they learn early in their studies. I'd love to teach this material again.
This was a small class of advanced undergraduate students, and really equivalent to a beginning graduate course. This is some of my favorite material to think about, and I was very excited to be able to talk about the linking number, which was something I had discovered had connections to calculus of functors, and give my students a fresh perspective on it. Someday I'd like to write a book for this course. It's tough to compete with Milnor's amazing book, however, as well as Guillemin and Pollack's exercises.
This was the first course I taught after graduate school. I had a very diverse group of students, ranging from graduate students in engineering and political science to undergraduate non-math majors. I taught out of Ken Ross' fantastic UTM. I learned from this book as an undergraduate, and is still one of my favorite math books. It is incredibly carefully written, and a wonderful way to introduce students to proofs. This was really a course about understanding the power of a careful definition of limits and the interplay between algebra and geometry (functions and their graphs). I also incorporated some literature to reflect some of the themes, taking readings from some of my favorite authors (like Haruki Murakami and the poet Wallace Stevens) and having a couple of literature discussions. I wanted to do something a little more than the usual dry math class and give students the sense that math is a living and breathing expression of the human mind. It was a lot of fun.
Last change: 9.09