I just taught my last class of linear algebra and multivariable calculus yesterday, so it seems like a good time to summarize my thoughts about teaching. I taught the same class last year, but I think I did so quite differently.
One of the first things I asked myself about teaching was whether it is acceptable to teach while barefoot. I didn’t do so last year, but not because of any objection to doing so; rather, last year I was teaching in the winter, and I rarely wanted to be barefoot anyway. But in the fall, things are different. Until the summer, I didn’t see any problem with it at all. Then, I read an article that suggests that sometimes students get distracted by peculiarities in the teacher’s behavior, and that may make learning a bit more difficult. I wasn’t sure about that, so I decided that it would be safer to wear shoes to classes, and I did so.
On the other hand, it’s a good idea for the students to believe that the teacher is approachable. One of the best ways to seem unapproachable is to display a lack of personality. Going to class barefoot would be a reasonable way of displaying personality, but that one was out, so I had to come up with others.
I think that appearing interesting was one thing I did poorly last year. My classes were usually very serious, almost business-like. That was unfortunate. This year, I felt that my classes were more of a conversation between the students and myself. I think they asked a lot more questions this year than last year, although it’s hard to remember that well what happened last year. Therefore, the class was more geared toward things that students thought would be helpful, rather than what I would teach by default. Given that the class is supposed to benefit them primarily, and not me, that’s a good thing.
After a while, I think the conversational approach that my classes took on helped me to let my personality come through. It is important in all classes for the teacher to demonstrate that the material is interesting, and that the students should care about it. But this is especially important in the first math class that students take in college. Most of them probably have never had a math teacher who actually likes math before, so I felt responsible for convincing them that interesting people can do math and really enjoy it. Naturally, I have to convince them that I am an interesting person if that’s going to work.
I’m not entirely sure how I did that, but I think I did. Perhaps it helped that I occasionally ranted about the unfortunate situation of having to do linear algebra over a non-algebraically closed field. (Okay, so maybe it wasn’t so occasional. But I had to stop myself mid-sentence when trying to say that the determinant is the product of the eigenvalues way more times than I want to think about!) But however it happened, I think they believe that I’m an interesting person even though many of them don’t know that I’m vegan, militantly atheist, opposed to monogamy, don’t like to wear shoes, don’t use telephones, don’t drink, don’t drive, wake up at 5:30 every day, and so forth. But I guess to most people, being passionate about mathematics is enough to be far away from the vast majority of people they know.
It was also interesting to me to see how a single student can radically alter the dynamic of a class. I had intended for my two sections to run rather similarly, but this was very far from what actually happened. The presence of one very outgoing student in my afternoon section made me much more inclined to joke around and be informal in that section as compared to my morning section.
So, how does that affect class performance? I’m not sure how much the quality of my teaching influences students’ scores. I tend to believe that students will study a certain amount anyway, regardless of how well the teacher teaches. At the high and low ends, this is almost certainly true, but perhaps if I teach well I can help the students in the middle do better than they would have otherwise. My morning section was extremely strong; on the first exam they had an average 9 or 10 points above the entire Math 51 average. That was very encouraging, but it’s a small enough sample size that I’m not sure if I can give myself credit for that. (Both my sections did better than the class average on both exams though (and in three out of four cases, far above the class average), and if that continues on the final, maybe that’s enough to be statistically significant.)
My focus was quite different from last year. Last year, I taught the section something like the way I would have wanted to see it had I been a student, with most things proven. But proving things takes up a lot of time, so that cut out time from examples, which seems to be what the students would rather see in sections. So, this year, I gritted my teeth and presented more magic formulae than I would have liked and explained them only if anyone asked. Mathematicians tend to get a bit brainwashed by the presentation in graduate classes and books, so we think that everyone wants to know how to prove things, but this is false. So, I got to do more examples. The way I usually handled examples was to do a problem on some topic myself, and then give another problem for the students to work on, and then have someone present to the class when it appeared that most people were done. This is good for them because explaining how to do a problem is helpful for clarifying their thoughts, and of course “mathematics is not a spectator sport.” But I didn’t want to put anyone on the spot, so I always asked for a volunteer. Some people volunteered a lot, and some people never did, so I never got to see whether the shy ones knew what they were doing.
Another thing I did that surprised some of the students was to tell them (only occasionally!) that they should refuse to do certain homework problems. I always prefaced such remarks by explaining that I was acting in my role as a civilized human being rather than as their teacher in such cases. But once they reach a stage in which they have to do a lot of hideous symbol manipulation and no more actual math, they should stop working on problems. If they’re learning math for the “real world” (to which I assured them from time to time mathematics has no applications), they can find a CAS to finish off the problem. If they’re doing it just because they are interested, then there’s no more interest left in symbol manipulation. So the only reason I can think of for why they’d finish such a problem is because someone else told them to do so. But one should never do anything just because one is told to do so!
Also, I really like teaching! I don’t have that much confidence in my ability to solve hard problems (like my thesis problem) or to understand the literature, but I know that I understand basic linear algebra very well. And, since I get really excited by learning certain things, I like to pass on that excitement to others as well. It’s always nice to give back something to mathematics and the mathematics community, which have provided me with so much enjoyment.