At Colby College, which I haven’t written about yet, I got some more answers to my question about what characterizes mathematics teaching at Liberal Arts colleges. These answers were consistent with what I had learnt before, but they also deepened my understanding.
Fernando Guovêa was very specific. The difference (as compared to other schools) he said, is the Tellos, the goal. Education at Liberal Arts colleges is not vocational and mathematics teaching is not primarily to prepare for graduate school (PhD studies). In particular in upper level courses there is more freedom to choose topics. And even in basic Calculus classes, even if several parallel section is taught by different professors, they can all have different books and don’t have the same exams.
We also talked about the art of reading mathematics. While hanging out in the corridor I saw an article with the title “How to read mathematics” by Guovêa and Simonson, so I opened our conversation by asking about it. I haven’t had time to read it yet, unfortunately, so I will not comment on it until I’ve read it. I remember that I’ve seen a Swedish PhD dissertation in pedagogics which studied how students read mathematics textbooks. The short review I read essentially said that they jump from formula to formula, skipping the intervening text. This is all very interesting, and I will have to return to it. Remind me if I forget!
I also learned that there has been explicit interactions between mathematics and the humanities for first year students in the so-called Integrated Studies Program consisting of clusters of three courses.
Then I had a very nice conversation with Andreas Malmendier, whose research interests are very close to theoretical physics; gauge theory and string theory. So we talked about physics a lot, in particular about string theory and what happened back in 1984 with the anomaly cancellation stuff. I described a bit about my work on higher spin gauge field theory. I did some name-dropping, mentioning that Lars Brink was my PhD adviser and that I was at Queen Mary College in the mid-80’s when Michael Green was there. So we had a nice chat.
But then Andreas said something that I haven’t thought about: He said that when teaching physics, there is often (or perhaps always) a narrative going on. When teaching Quantum Mechanics, then Schrödinger, Heisenberg and Dirac are all very present. They and the historical context is there surrounding the development of the theory. Same thing with relativity. Albert is there all the time. It’s so obvious that I haven’t noticed it. Theoretical physics is immersed in its history. Just think of the recently discovered Higg’s particle.
Well, you can say that it’s the same with mathematics. Just think of the Riemann hypothesis or Fermat’s last theorem. True, but I still think there is a difference in undergraduate teaching. This needs more investigation.
Another difference between mathematics and theoretical physics that Andreas mentioned was that theoretical physicists in a sense are “braver” than mathematicians. They want to break through barriers, whereas most mathematicians want to work within well-defined borders. Interesting thought – I cannot judge.
Then I talked to Leo Lifshits. He was also very specific. He said that he always uses classroom time for teaching ideas, not techniques. For instance, in Calculus, he uses a standard textbook (Stewart) which has exercises on-line that are automatically corrected. So the students do these as homework. This frees up time for talking about the ideas underlying calculus. I asked if this really works with weak students, and he admitted that there is a self-selection. Students that cannot work like this, or do not want to work like this, take other calculus classes. Still it is an interesting way to work. This way of teaching forces students to think as opposed to doing routine manipulations. That forces a “crisis”. Those who cannot deal with it leaves.
But perhaps it can be done in a milder way. In a humanistic way. I’m sympathetic with the idea of challenging students’ prejudices about what constitutes a mathematics class and what mathematics is about. That’s really the fundamental reason why I’m off on this quest for the perfect mathematics class.