*In this section, Dr. Snowden explains the mathematical goals of this undergraduate math seminar course and the implications of teaching mathematical content under this course structure.*

## Mathematical Goal of the Course

I assumed the students 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.

— Dr. Snowden

Because of the undergraduate math seminar structure of the course, the experience of learning mathematical content in this course was different from that of many other math courses.

The mathematical goal of Seminar in Topology was to introduce students to algebraic topology. I wanted to show the students the landscape of a subject they'd likely see in a first-semester class in math graduate school; I thought it would be good to expose them to this material.

We followed Allen Hatcher's textbook on algebraic topology fairly closely. We covered the fundamental group and covering spaces, then homology, and then cohomology and Poincaré duality. For the final 2 weeks of the course, students lectured on the topics they had explored in their final papers.

I assumed the students 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.

## Student Learning Through Lecturing

Students took turns presenting lectures in this course. Each student presented roughly once every two weeks and was responsible for teaching about one-tenth of the course content to the rest of the class.

In terms of learning content, this structure was great for the student presenters. Student presenters who put solid effort into their lectures would typically be extremely engaged in the material they were lecturing about because they had to know it very well.

## Challenges in Learning from Classmates' Presentations

Unfortunately, the inexperience of the student presenters impeded the class's experience in learning the mathematical content of the class.

The students were learning the material for the first time. Most of the students, no matter how talented at math, didn't have enough mathematical background and experience to have the broader vantage point when it came to what to present, how to present it, and what side remarks to make.

Furthermore, a lot of the students had never lectured before. There were many times when I thought experienced lecturers would have chosen to present things differently, using slightly different material, and the students in the audience didn't have the advantage of seeing a better presentation. On the student evaluations of the course, the lowest scoring area concerned how much the students learned from other students' lectures. That's the tradeoff.

I think the quick pace in my particular seminar may have exacerbated this issue. In the future, this problem might be partially solved by moving more slowly through the content.

## Challenges in Content Presentation Stemming from Course Structure

The rotating lectures were a significant challenge in this course. Each student presented only roughly half an hour out of every five or six hours of lectures.

Students sometimes had trouble preparing for their lectures because they hadn't learned the content requisite for understanding the material they had to present. A few students who struggled with this would come to my office a few days before their lectures. I would try to teach them whatever math they needed to know in preparation and help them plan.

In addition, the class's understanding of a topic sometimes depended directly on a previous lecture, and if the person who gave the previous lecture didn't do such a good job, then it would be difficult for students to understand the subsequent material. Other times, a student presenter would assume someone had covered something when they hadn't, making part of their lecture incomprehensible to the audience. There were lots of little things like that.

When that kind of thing happened, I would try to step in and say something, but you can't give a whole back story through a little comment in a lecture. So, that was a problem.

## Using Problem Sets to Help Students

Although the focus of the class was on the student lectures and the final paper, I thought it was important for students to do homework. A lot of the students didn't seem to treat the class like a normal class when they weren't presenting. For example, I don't think they took notes as faithfully as they might have in a more typical class.

The problem sets provided another way for the students to actually learn the material. I gave three problem sets, each consisting of about 5 difficult problems. I gave the students two weeks to do each set. I wanted them to just actually do some of this math.

## Exploring a Single Topic in Depth via the Final Paper Assignment

In every undergraduate math seminar at MIT, students are required to write a final paper. In this class, students each picked a topic from a list of suggestions. This gave students the opportunity to explore a single topic in algebraic topology in depth. The writing process is described in Helping Students Learn to Write Mathematics Papers.