Undergraduate math seminars abide by a basic, defined structure, with some room for instructors to customize the courses they are leading. In this section, Dr. Snowden describes some of the ways in which he customized the course structure for the Spring 2011 offering of Seminar in Topology.
Since the Seminar in Topology is an undergraduate math seminar, parts of the course structure were pre-determined. For example, it was required that students give the vast majority of class lectures and that students write at least one substantial math paper with a revision cycle.
As the course instructor, I built upon these requirements and what prior Seminar in Topology instructors had done. I was given a basic set of guidelines. I looked at past course websites for Seminar in Topology that were still on the web. I also discussed ideas with Susan Ruff, the MIT math department's writing instructor, who has been involved in many of the undergraduate math seminars in recent years.
For information about the course's structure, see the course's Syllabus. Some of the ways in which I chose to structure the class are described below.
Picking the Textbook and Topics to Cover
This course was designed to serve as an introduction to algebraic topology. I was allowed to choose the textbook, and I picked Allen Hatcher's textbook on algebraic topology. This was different from what had been used in the past, but I chose it because it was available for free online. That way, students didn't have to buy anything. We followed the textbook pretty closely.
From what I could see of old Seminar in Topology course web sites, it seemed the course typically includes covering spaces and the fundamental group, and maybe some homology. I aimed to be more ambitious and cover more material, so in addition to those topics, we covered cohomology and Poincare duality. I think I was a bit too ambitious and tried to cover too much. Doing covering spaces, the fundamental group, and homology would have worked well, but doing cohomology and Poincare duality was just a bit too much. If I were to teach this course again, I think I would cut the material back to 75% or so.
Keeping Student Lectures Short
Because students gave shorter lectures, they spoke more often than they would have otherwise, giving them more opportunities to get feedback and improve.
— Dr. Snowden
In some undergraduate math seminars, students lecture for a full 50-minute class. In this class, each class session consisted of two students each lecturing for 25 minutes.
This worked very well. I think it's preferable to having one student speak for the entire time for a few reasons. It gave the students less to prepare for each lecture, and if someone wasn't doing so well, at least they weren't talking for very long! Because students gave shorter lectures, they spoke more often than they would have otherwise, giving them more opportunities to get feedback and improve.
Scheduling Lectures
There were 12 or so students in the class. Thus, each round of lectures included 12 lectures, or 6 class periods, typically equal to 2 weeks of class time.
At the beginning of each 2-week period, I would sit down and decide on the material we needed to cover in those two weeks and then split it up into individual lectures and assign it to the students. I didn't do that too far ahead of time; I kind of adapted it as we went along. Initially, I followed past courses a little more closely, but as time went on, both because I needed to make changes and because our course was going more quickly than past courses, I customized it completely.
If I'm teaching a class myself and I'm planning out the next two weeks of material, I can kind of say, "I'm going to do this, this, and this," and that's all the scheduling I have to do. It's quick. Then, preparing the lecture takes a little while longer, but I can adjust as I go. But with something like this course where a different person is lecturing each time, the lectures have to be planned very carefully and put together very well, because the next person has a fixed topic and can't really adjust. So it did take quite a while to carefully plan these things out.
Being Accessible
I think it's good to make yourself accessible to the students so they feel like they can talk to you. To this end, I think it was really helpful that I was only a postdoc and not a real professor. The students seemed to feel really comfortable and liked talking to me and bringing things up, and that's probably good for this kind of class.
I made every effort to be accessible to the students. I made myself available to meet with them and encouraged them to e-mail me questions or comments related to the class.
Before the first round of lectures, I required that each student meet with me and the writing instructor for a practice session. Afterwards, I was always willing to meet with students who wanted extra support in preparing for their lectures. After each student lecture, I e-mailed the student with feedback about the lecture.
For the final paper, the writing instructor and I gave students detailed feedback. I also willingly met with any students who wanted to discuss their final papers while they were working on them.