November 2023

Carolyn Abbott (Brandeis University)

I’d like to start a conversation about how we teach graduate classes. In the last several decades, there has been a lot of discussion about active learning, in all its many forms, in mathematics classrooms from early elementary through college. There is generally a consensus among those who care about math education that engaging students actively in their own learning leads to better outcomes. Students take ownership over the material, think more deeply about the subject, and make more complex and nuanced connections between topics. Of course, not all educators have embraced active learning at those levels, and I don’t mean to imply that it’s always implemented perfectly. The conversation is ongoing and should continue to be so.

Graduate education, however, is largely left out of the conversation. A standard graduate class is still a lecture. It’s as if we, the larger mathematical community, think those that are strong enough to get into a PhD program should not “need” active learning in their classes. The research shows that an actively engaged classroom encourages *everyone* to engage more deeply with the material, strong and weak alike. Certainly, graduate students *can* learn the material in courses taught in the traditional lecture format; they’ve been doing it for generations. But wouldn’t they learn the material *better* if they engaged more deeply with it, in the classroom as well as when working on problem sets?

As a research mathematician, I think about how I engage with material I am learning, say a paper I’m reading, one whose techniques or results I’m trying to generalize. I read it, not necessarily linearly, but again and again, until I understand what the authors are doing. But I don’t stop there. I engage more deeply with the text, questioning each hypothesis and each step: is the hypothesis truly necessary, or just necessary for this particular method of proof? Is there an alternative way to approach the problem or step? What’s the intuitive, or “moral,” idea behind the proof? Are there connections to other, seemingly unrelated, results I’ve read?

Increasing one’s level of engagement with the material is one step towards becoming a research mathematician. Anyone who has advised PhD students knows that, for many, perhaps most, students, this change does not happen overnight. It is a process, often a long and difficult one. If this is where we want our graduate students to end up, why aren’t we teaching them these skills from the very beginning, in the classroom?

I have some ideas about why this is, which I’ll get into later, but first let me tell you what led me to start thinking about active learning in graduate classes. I taught middle school and high school mathematics for five years in New York City, and during that time I embraced active learning in my classrooms. When I started graduate school and was a TA, I continued using it in the recitations I ran. I even trained entering graduate student TAs at University of Wisconsin in active learning methods. I don’t mean to imply that I’m an expert in implementing active learning in my classrooms – I still have a lot to learn. But it is something that I believe in, and something that I am used to incorporating into my classes.

So I was surprised when I looked back on my first year of teaching graduate courses after becoming an assistant professor and realized that I basically lectured for the whole year. I lectured; the students listened and took notes. There were questions from time to time, but the room was often quiet. Too quiet. I missed the activity, the discussions, the liveliness of a classroom where students were actively engaged with the material. What happened? Why did my whole teaching philosophy seem to go out the window when I started teaching graduate classes?

Last summer, I spent time thinking about how to revise Algebraic Topology I, which is a required course for first-year graduate students. Incorporating active learning techniques presented real challenges, many of which felt more complex than when teaching an undergraduate class. Some of these are real, some (I believe) simply require a change in mindset, and none are insurmountable.

- Student buy-in. Learning via lecture is comfortable for students, perhaps because it requires less engagement. Other first year graduate courses, as far as I know, are being taught in a standard lecture format. I worried that my class would be the odd one out, and not in a good way.
- Complex material. Problems at this level are usually long and difficult! Worksheets, an easy and effective fallback for undergrad classes, are not a good option for a grad class. Even most examples (with some notable exceptions, like first applications of van Kampen’s Theorem) are too long and complex to expect students to work through them in 10 minutes.
- Lots of material. This is a challenge at every level, but the higher the level of the course, the harder the challenge seems. There is just so much to get through every semester. I did not see how to find time for interactive learning.
- My role as a teacher. While students can read the book, the concepts and proofs are difficult and harder to follow than in undergraduate texts. This is somewhat counterbalanced by students being more sophisticated readers of mathematics, but reading the book is still quite challenging. Even more than when teaching an undergrad class, I see my role as the go-between between the text and the students. I’m there to explain, to draw the key pictures, to give the intuition behind the steps, to give the big picture, to explain the history. I worry that spending time on active learning will take away from this role.

In the face of these challenges, I decided to start small. While I always approach summer thinking it contains endless free time, the reality is quite different. There are grant proposals to write, conferences to attend, research to catch up on. Finding time to completely rework the course was unrealistic. My strategy, then, is to take small steps: each time I teach this course, I will make a few small changes to encourage active learning. Maybe after I’ve been doing it for 20 years, I’ll have it all figured out.

The two small steps I took this year are both techniques I use when teaching undergraduate classes: encourage discussion and questions from day one, and start each class with a problem. I know it doesn’t sound like much, but it’s what I’m able to do right now.

Before starting to teach high school, I was told to start the school year the way I wanted it to continue. The advice was given in the context of discipline: while one can easily become more lenient as the year goes on, it is almost impossible to become stricter. But the advice applies in almost every aspect of teaching, as well as life, and I think of it often. If I want to have a lively classroom filled with discussion and debate, I need to model and encourage that from day one. During the first week of class, I went out of my way to say that I am happy to be interrupted with questions, and I paused more often than I usually would to ask if there are questions. I reguarly asked if the class understood. This typically resulted in stares and silence, and so I would follow up by asking if someone could explain it in their own words, and then waiting. Waiting until someone spoke up with an explanation or question. I wanted my students to know that when I asked a question, it was genuinely a question; I was not going to answer it myself.

Starting each class with a question or problem has made a bigger difference than I expected. It has not always been easy to come up with a question, and, to be honest, some days I don’t manage it. When I do, though, the question(s) I choose either foreshadow what we will do that day or reminds students what we did the previous class. For example, one class we defined what it meant for a loop to be nullhomotopic, and at the beginning of the next class I asked the students for a straightforward example of a non-contractible space in which every loop is nullhomotopic. This resulted in a lively debate, including about what counted as a ‘straightforward’ example. A different class ended in the middle of a proof, so I started the next class by sketching what we had done in the proof so far, but with some steps missing. The question was to fill in the missing steps. Once I asked for the definition of a homotopy, which we covered in the previous class. In today’s class, we used van Kampen’s Theorem to reprove that the fundamental group of a torus is Z^2. To start the next class, I’ll ask them to generalize this to a closed surface of genus g, and then to come up with a different way to apply van Kampen’s Theorem to those surfaces.

As can be seen from the examples, the level of difficulty of these questions varies widely. Some simply ask the students to look back in their notes. These serve the purpose of reminding students what we were doing in the last class. I like to use this kind of question on Mondays, since it’s been several days since we last met. Some are much more complicated, like the application of van Kampen’s Theorem. I’ll probably give them about 10 minutes to work it out, and I hope they will be eager, or at least willing, to discuss it with each other during that time.

So how active is my class this semester? Well, not as active as I’d like, but so far, it’s more active than last semester. Incremental changes lead to incremental results. Students seem to be more willing to ask and answer questions, particularly ask questions. And I enjoy starting class by hearing a couple of student voices. At the end of the semester, I’ll look back on the changes I made and think about how I can engage the class even more.

There is nothing groundbreaking about the techniques I chose to implement. They are standard, even expected. They’re certainly not the only way to make a class more active, and I don’t know that they’re the best way. They’re simply what I was able to do this semester. I’d like to hear what other professors are doing and how graduate students are reacting. To get new ideas and learn what has worked and what hasn’t. To push my boundaries as a teacher of graduate students. I want to start a conversation.