This was the first course that I taught at Stanford. It's the differential equations part of the multivariate calculus/linear algebra/differential equations sequence. When taught in the spring quarter, it's a large lecture course; fortunately, I taught it in the fall quarter, and had a relatively manageable 35 students. The class met for 50 minutes every day.
As when teaching 18.03, I wanted to try to use groupwork, though obviously I wasn't going to go as whole-hog as I did there. And I think that it went really well; my experiences here have formed the basis of my notes on groupwork. The format was as follows: I'd present a new piece of material (e.g. a method for solving a new kind of differential equation). Next, I'd give a simple example. Then I'd give the class some problems (usually two, but one wasn't uncommon, though three (or more) was) to work on. Finally, I'd have students present solutions to the groupwork on the board.
For about the first week and a half, we went through two such cycles each class. But then the groupwork started to take longer, and we were in an awkward position where, say, I'd presented a new concept and some exercise but we didn't have time for groupwork. This led to some classes where I tried things that didn't work so well, e.g. starting with group work. If you end a class with an example, it seems to best to just start the next class with the example again (or come up with a new one). And I regretted it when we didn't find time for solutions at the end of class. Sometimes, when that happened, it was because the problems I'd assigned were too hard; in such situations, I sometimes presented the solution myself at the beginning of class the next day. If your problems are more reasonable, you have some flexibility as to when to end the groupwork part: sometimes I let everybody finish, but sometimes I stopped the groupwork when not everybody was finished in order to leave time for somebody to present solutions on the board.
As far as those solutions go: this is one aspect of the course that students had mixed feelings about. Before I started using groupwork, I'd naively thought that students would learn to be confident in their own solutions, and wouldn't need the crutch of outside authority. Many students are that way; many aren't. So it's a good idea to have solutions presented, and I figured that I'd rather have the students present it than do it myself. Some students were bored during solutions, some students really liked it; I think that more students liked it as the class went on. I chose the solution presenters by observing students during groupwork, choosing one who seemed to have solved the problem, and asking her if she'd be willing to present a solution; that way, nobody was put on the spot. I do wish I'd kept a list of who had presented starting from the first day, in order to minimize repeats.
There were weekly problem sets, a midterm, and a final. The grading policy on the problem sets was as follows: on each problem, the students would either get a check, an "ok", or a "rewrite". The check meant that they did everything correctly. The "ok" meant that they made some simple mistake that can easily be indicated on the assignment: e.g. an addition error in the next-to-last line. (If the mistake appeared earlier, they would get a rewrite.) And "rewrite" means that they made a more serious mistake; they should then rewrite their solution and turn it in again.
Both check and ok counted for full credit. Rewrites also counted for full credit once they eventually get a check or ok; this may take several rewrites. The point of this system was to get the emphasis away from grades and on learning: mistakes aren't a big deal, you just learn from them and go on. So I wanted to encourage learning from mistakes; it seemed like the easiest way to do that was to allow rewrites for full credit. But I also didn't think that it made sense for people to rewrite problems with a trivial mistake right at the end; that's where the checks came in.
Also, I graded about five of the problem sets each week myself (with a grader grading the rest). The goal of this was to keep tabs myself on what the students are doing. It worked well with the rewrite system: that way I had an idea of what problems might need rewrites, and also it didn't matter if I graded a problem slightly differently than the grader would, since differences would all come out in the rewrite wash.
All in all, I was very happy with this homework system, and would do it again. It got the focus on learning and away from rating students; students actually looked at the problems that they got wrong, and I never had to spend time arguing with students about whether their solution was worth 10 points or 9 points or 7 points or what. Obviously, reasonable people with different grading philosophies will differ with me on the appropriateness of this method, but I think that it has real merits.
Two other peculiarities of the problem sets: each problem set asked the students to make up and solve two problems, and also asked students to say how long they took. I'm going to keep the latter feature (I think it's really important for professors to be aware of how much time their assignments take; actually, I was often surprised at how little time they took), but I got mixed responses to the former. It seems like an idea with potential, but maybe there are better ways to reach the same goals.
The midterm and final were both take-homes, open book, no time limit, no consulting with other students or with calculators/computers, no rewrites. Thus, they were graded in a much more normal method. This wasn't so much because I believe in more normal grading methods as that I was scared not to have some harder data to back me up when assigning course grades; but I didn't impose time limits, since I feel that tight time limits are fairly artificial constraints that don't have much to do with how people will have to use differential equations in the future. (I could be wrong about that, of course.) The quality of the interaction with my students took a slight but noticeable turn for the worse when discussing grades on the midterm; this made me glad that I hadn't been giving normal grades on the problem sets.
The students did extremely well on the exams, and took a lot longer than expected. I think that some of that time was due to my assigning an overly hard test, some of that was due to their taking extra care to make sure that they got everything right (which is all to the good: we constantly complain that students don't check their answers, but the evidence here seems to be that they do if given time for it), and some of that was probably due to the relative absence of situations in the course where they really had to solve something quickly. I was also pleased to see that most of their mistakes were linear algebra or calculus mistakes rather than differential equations mistakes.
The grade curve for the course was extremely skewed towards good grades; I'm sure people who don't like grade inflation would complain about this, and with some justification. However, I think that the exams show that my students could solve differential equations, and could solve them well enough to earn a good grade; I don't, however, have nearly the same confidence that they could solve differential equations quickly.
At irregular intervals (about every other Friday) I handed out questionnaires to my students to see what they thought about the class. I used Stephen Brookfield's questionnaire; I have mixed feelings about it, but on the whole it got me lots of useful responses. I handed out these questionnaires both because I think it's a good idea in general to get your students' opinions on your classes (which may differ from yours) and because I didn't feel comfortable subjecting them to experiments if they really weren't going well, which I might not realize otherwise.
Actually, though, the questionnaires' main benefit was quite unexpected: the responses were typically extremely positive, espeically about the groupwork, so I was much happier with continuing that than I would have been otherwise. Also, my students typically were happy to have classes taught in non-traditional ways: they saw this as an overt manifestation that I cared about teaching them, and appreciated that. I also got some useful feedback of the sort that I was expecting (e.g. this day went well, that day didn't).
On the last day of class, I handed out a much more detailed questionnaire asking what they felt about various aspects of the class. This was very useful; for example, I'm not sure that I would have had an idea about what students felt about their presenting solutions on the board otherwise. In the future, I'm going to add more detailed questions to questionnaires handed out during the middle of the course, as well. We also spent some time in the last day of the class talking about what worked and what didn't. In the next-to-last day, they filled out the Stanford-provided course evaluation forms; I got good ratings on those. I have copies of the questionnaires that I handed out, and the course evaluations, if anybody wants to look at them.
All in all, I had a blast this class. It can be really exciting when something you're experimenting with turns out well. Also, I liked the students in my class very much, and would like to thank them for helping make my quarter so enjoyable.
Last modified: Wed Jun 28 17:35:39 PDT 2006