One of the main reasons I’m at the AMS/MAA Joint Meetings this week is to take an MAA short course on discrete and computational geometry. That course is wrapping up this afternoon, and it’s been a good experience. I came into the course with zero knowledge of computational geometry, a within-\(\epsilon\)-of-zero knowledge of algorithms, and an extremely rusty skill set in topology. But I’m coming out with an appreciation for this subject and, hopefully, a basis for pushing farther into the field and eventually contributing something new.

Teachers ought to take courses more often. Apart from being intellectually satisfying, it’s useful to be on the receiving end of academic teaching in one’s own discipline every now and then because it helps you remember what it’s like to be in the shoes of your own students. Here are some things I’ve re-learned about being a student in a math classroom, of sorts:

**1. There’s a discernible physical component to learning**. I started out the minicourse with a lot of energy and was highly engaged. Later in the day I was tired and didn’t click as much with what was going on. Last night I slept poorly and I’ve been struggling to engage with the material all day. After anything having an active component to it in the course, I was more attentive. And so on. In other words, the body plays as much a role in learning as the brain does, which is obvious because the brain is part of the body. I have to remember this as a teacher whose main constituency is 18-22 year old young adults who are not necessarily taking the best care of themselves or their time. Active learning helps (see below), and I need to remember that there is always a backstory — that student who looks totally disengaged may be depressed, or sick, or just needing help in some non-mathematical way.

**2. Learning works best when it’s active.** This morning, for instance, we were learning about perimeter halving, which is a method for folding an arbitrary convex polygon into a polyhedron. The session had been all lecture up to a point, and I felt like I understood it. But then we were given sheets of paper and rolls of tape and asked to implement the perimeter halving methods ourselves, and fold the paper into polyhedra. Turns out I didn’t understand it as well as I felt like I did. It was maddening, in fact. But my brain was engaged; I met my classroom neighbors for the first time; and I’m still thinking about this concept even as I write this. Activity does this. And let’s face it: Lecture does not do this. I learned that I can’t leave it up to my students to self-engage during a lecture. They need, like I needed, something to do that will wake them up and get them thinking.

**3. The fear of looking like an idiot is real, and it prevents learning.** I’m in this minicourse with about 90 other people, at least half of whom are from major research universities, and a few of whom are Big Names in their fields. Why they are taking this minicourse is anybody’s guess. But I can attest to the pressure not to ask or answer questions for fear of looking stupid, or of making my institution look bad. And I have a Ph.D. in algebraic topology from a top-25 research university. So, that fear can be present at all levels and at al times. I have to assume that exists among my students too — some of them all the time, all of them some of the time. There are ways to relieve this pressure if you’re the instructor, and in fact doing so is part of your job. I was definitely reminded that I have to be sensitive to this and to think of creative ways to get students involved in non-threatening ways in the intellectual life of the class, and not just be a fly on the wall.

I’m sure I could think of more, but the last session in the course starts in 10 minutes. So I’ll just say I’m thankful for what I’ve learned in the minicourse and equally thankful for the reminders about learning as I head back into the classroom next week.

How about you? What would you add to the list here of things you’ve learned about teaching by being a student?