This was the second course that I've taught at Stanford. The basic question that I had when teaching this course was: what should the relation be between the textbook for a course and the lectures? To what extent should the lectures duplicate material covered in the course; to the extent that they do, what's the point, especially in an upper-level course? So I wanted to try to get some more evidence on this question, starting as a hypothesis that there isn't much point.
The class started out with a rather anarchist format. At the end of a class, I would give the students a reading assignment; they would then read that assignment before class, think about it, and work some exercises relevant to that reading if they felt that they understood the reading. Then, in class, I wouldn't lecture on the reading; instead, we'd discuss the reading, and perhaps I'd give a mini-lecture on something relevant that the book didn't cover. The students would then do groupwork, and present solutions. There was no formal homework (other than the reading); however, the students were assigned a final paper.
This didn't work too well. Discussions essentially never materialized; maybe I'd get some specific questions, but not even that. There were some early instances when I gave a mini-lecture on something else: for example, I talked a bit about category theory once when it seemed to shed light on two definitions of the product of a set. Those went over well when they happened, but I could rarely think of something suitable to talk about. As the course went on and the book got a bit more complicated, I found more to say that was based on the book, such as examples or presenting parts of the material in a somewhat different way. Those often went well, especially the examples; the book was rather lacking in that area. Still, it never turned into the sort of active discussion that I'd envisioned; more thoughts on that later.
I was surprised how well the groupwork went, even from the beginning. Honestly, I hadn't expected it to go particularly well in an upper-level math course, but I think it went okay. A couple of times, when we ran into a new kind of structure for the first time, I gave as a groupwork problem "think up all of the examples of this kind of structure that you can". I liked the results of that, though some students commented (and I agree) that that works best with a cap on time, say about five minutes. Many of the other groupwork problems were exercises from the book; I wasn't entirely happy with that, but I couldn't always come up with more suitable exercises. I wish I'd been able to come up with more exercises of the form "work out this concrete special case of what we're talking on".
After a couple of weeks, though, I realized that I had no idea whether the students were actually learning anything. They seemed not to have any questions and doing reasonably well with the groupwork, but neither of those left me confident that everybody understood what was going on, since one person could easily be fairly silent in both of those situations. So I started asking people to write up one specified problem; sometimes it was a groupwork problem, and sometimes it wasn't. That way, I could tell if people really could work out the details properly or if they were only understanding things on a more superficial level. That seemed to work pretty well; it set my mind at ease, if nothing else.
About halfway through the course, I realized that they had a paper due in a few weeks, that they probably had no idea what a paper in a math course should be like, and that I wasn't sure I knew, either! So I spent some time talking to people with experience about this, and I handed out out a list of possible topics to my students. I then asked students to have a general idea of which topic they were doing a week later, and helped students if they needed references. A week after that, I set up meetings with students to get a more detailed idea of exactly what they'd write about. A week before the final versions of papers were due, I had them turn in fairly complete rough drafts; I handed them back two days later, and I tried to make fairly detailed comments on them.
I ended up quite enjoying the papers. It helped that, not only was I teaching a course not in my area of expertise, but also many of my students' main interests weren't in math, so only one of the students ended up writing on something that I actually knew much about. This made reading the papers a lot more interesting than it would have been otherwise. If I had to do it over again, I would hand out a sheet of writing guidelines, since there are various aspects of mathematical style that the students, understandably, weren't aware of. Also, some students didn't like the uneven pacing of the course: as the course went on, the reading got harder and the students had the additional work of the paper to do. (In fact, a few students dropped out as the paper started approaching.) So I'll have to think more about pacing the next time I teach in this way. Some of my students said that they ended up enjoying writing the paper; I don't know if that was the general feeling, however.
As far as the discussions (or lack thereof) went, I had a very interesting discussion with Jack Prostko at the Center for Teaching and Learning about this. He made the quite sensible point that the students didn't know what to expect out of this class format any better than I did, so it was quite unreasonable for me to expect a discussion to materialize out of thin air. One suggestion that he had was to give students a concrete topic to think about when reading a section; that would give us a starting point for the next class.
I tried that out once, with fairly good results. The section in question started off with three proofs that were all quite similar, so I asked students to think about what they had in common and to write some notes on a "proof template" that they all fit. We started the next class with them telling me their proof templates and me writing them up on the board. (So not exactly a discussion, but a lot closer to it than we'd had before.) The groupwork then consisted of their proving another theorem that fit into this proof template; they did a very good job of it (honestly, I was kind of amazed to see a group of students come up with a proof so smoothly together); obviously, they'd understood the important parts of that section. If I teach like this again, I'll try that out a lot more.
The class was quite a bit of fun, and had lots of positive aspects to it. But, probably because I was trying out a couple of things for the first time, there were some things that didn't go too well, especially towards the beginning of the course. (The second week, in particular, was quite bad.) Enough of those aspects improved as the course went on that I think I'll try some of these experiments again the next time I teach an upper-level course.
Last modified: Wed Jun 28 17:35:39 PDT 2006