Teaching

Communicating anything gracefully, mathematics and software included, is hard! I'm certainly not done figuring it out. These are a few ideas I've found productive in my own practice. Reading this short page won't make you a better teacher on its own, but it may plant a few seeds that your experience with students grows into stronger teaching habits. Try out some of the ideas below and let your students' reactions guide you. There's more than one way to teach well, and watching your students carefully gives you the evidence you need to become your own best teacher.

People feel first and think second

This morning (September 2026) I got a phone call from an HVAC company trying to collect on an invoice for a job they hadn't finished. It wasn't a pleasant call. When I got to work afterwards I was distracted, frustrated, and foggy. I was cutting with a dull knife.

People do their best thinking when they feel good. Ideally, students are well fed, rested, and haven't recently been threatened with collection agencies by an HVAC company. You can't control the emotional energy students bring with them, but you can shape and nurture what shows up. Jokes, a smile, and a genuine greeting go a long way. In group settings, encouraging students to engage with each other does similar work.

Asking a question, and drawing attention to your own misunderstanding, takes trust and comfort. Students get that from their relationship with the instructor, not from the instructor's expertise in the subject.

We learn the most when we work the hardest (and still succeed)

Like Goldilocks' porridge, there's a sweet spot in how hard to challenge a student. A question that's too easy or too hard leaves their skills stagnant: either they succeed too quickly, or they can't make meaningful progress and give up. Too hard is especially costly because it dents confidence, which is lost more quickly than it's gained. Ideally, a student works on a problem at the furthest reach of their abilities, and succeeds (see the zone of proximal development). You can tell a student is working on a "just right" problem when they put in sustained, effective effort.

To that end, give a student as little support as possible while still letting them succeed. Once a problem is finished, a student can look back on the path their mind took to get there, step outside that particular problem, and generalize what they learned so it's useful on future problems. In the middle of a problem, they haven't "got it" yet. Offer support (a hint, some advice, or a pointer to a relevant warm-up problem) so that they get there.

On the other hand, there's no free lunch in learning. Offer too much support and the problem becomes so easy that the skills never develop. So how do we find Goldilocks' sweet spot, with just enough support for success and no more? Stop talking, and listen to your students. Give them room to try a problem or idea first, and see how they respond. In whole-class instruction I often open with a warm-up question to get students talking and activate whatever ideas they already have before I take the lead and develop an idea for them. In a small study group working on homework, a TA can ask students questions that help them pin down exactly where they're stuck. This sets the expectation that the student leads the discussion toward some insight, and it lets you listen for which ideas need developing so you can offer a little support where it's needed. As long as a student is making progress, it's often best to just get out of the way (this is especially true in study groups).

The attention bargain: what do students get for their attention?

At its core, students trade their attention and effort for knowledge and skills. What exchange rate are they getting? If a student listens to you for 5 minutes, what do they get out of it? If our instruction isn't "just right" for them (see above), or we ramble about something unrelated, or administrative details eat up too much time, students notice they're not getting a good deal. Eventually they stop spending their attention and effort on what we're offering. As best you can, maximize the learning your students get for their attention.

In whole-class instruction, this usually means choosing ideas that are "just right" for as many students as possible (no small feat when abilities vary and we can't check in with each student beforehand!). In small groups, it helps to learn what motivates each student. Do they have an appetite for a nuanced explanation of some subtlety, or are they trying to get an A with as little effort as possible? Trim the extras for the second student. For the first, make sure any new idea you offer is worth the time and effort it takes to explain.

Study group tips

Encourage students to help each other

• Students often find it easier to ask a peer for help than someone with more authority (a TA or instructor).
• Students who help others reinforce their own understanding (or find the holes in it) by having to explain it out loud.
• You can't help everybody at once. Expecting to can lead to busy days where students are left waiting for you.

Don't re-teach the lecture

Keep whole-group instruction to a minimum; it isn't productive for students who already picked the material up in class. If you suspect everybody has the same issue, you can offer a quick (under 5 minutes) intervention, but see how much of it you can get the students to lead themselves.

Stick to the course materials

There are different conventions, notations, and entirely different approaches out there, and their quality varies. There's usually a reason we approach a topic the way we do, so read our class notes to get in sync with how we tackle it. Avoid bringing in approaches or materials from outside the course. If you have something you think is an improvement, run it by me first so I can benefit too: I learned a way to represent Dijkstra's algorithm from a TA, and it's now how I teach the topic in class.