Mathematical Communication is a developing collection of resources for engaging students in writing and speaking about mathematics, whether for the purpose of learning mathematics or of learning to communicate as mathematicians.

Topology–Andrew Snowden

In spring 2011, I ran the 18.904 seminar (Seminar in Topology).  Below are some comments about how I ran the seminar and recommendations for people who will run it in the future.  See also the course webpage.


Organization. The class met for one hour three times a week.  Each time, two students spoke for about 25 minutes each.  This worked very well.  I think it’s preferable to having one student speak for the entire time for a few reasons:  for instance, it gives the students less to prepare for their lecture and if someone is not doing so well at least they’re not talking for very long!  I planned out the topics for the lectures ahead of time and gave the students a fairly detailed overview of what they should cover in their lectures. See the syllabus and semester schedule.


Final paper. The students were required to write a final paper of about 10 pages in length, on a special topic.  I came up with a list of possible topics and put it on the class webpage to give them ideas.  I didn’t require them to use one of these topics, but I think everyone did.  I made them choose a topic by March 7, hand in a draft on April 11 and then had the final paper due April 25.


Homework. I gave three problem sets, each consisting of about 5 difficult problems.  I gave the students two weeks to do each set.


Material. We followed Allen Hatcher’s textbook on algebraic topology fairly closely.  We spent February and the beginning of March on the fundamental group and covering spaces, March through the beginning of April on homology and the rest of April on cohomology and Poincare duality.  The two weeks in May I had the students lecture on their final papers.  I think I was a bit too ambitious, and tried to cover too much.  I think doing the fundamental group, covering spaces and homology would have worked well, but doing cohomology and duality was just a bit too much.

Instructor Observations

In student evaluations, students rated this class favorably. When asked about the factors that contributed to the success of the class, the instructor responded,

“First, I put quite a bit of effort into the class:  meeting with students, carefully grading their work, attempting to organize things precisely without micromanaging them.  Second, I treated the students basically as if they were graduate students (almost as peers):  I assumed they were smart and knew the math they said they had learned, I didn’t correct every little mistake they made, I gave them hard and interesting problems to work on and I moved the class at a fast pace.  Also, I kept them informed about the decisions I was making regarding the class and asked for their opinions.  I think the students were very cognizant of these points, and very much appreciated the respect and attention.

“The class was not perfect though.  I think I did move too fast.  It seems like a delicate balance — I had a few students tell me both that they really liked the class because it was so hard but also that it was a bit too much.  So I think if I were to do it again, I’d cut the material back 75% or so.

“I think the lowest score in the responses of the students concerned how much they learned from other students’ lectures…It certainly seems like the main drawback of this style is that there will be a lot of subpar lectures, since the students are inexperienced.  I think the fast pace of my particular seminar may have somewhat exacerbated this issue — another reason to cut back a little!”

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