How to give a good math talk…

(…or a good talk of any kind.)

I recently read an excellent article called Giving Good Talks, by Satyan L. Devadoss, in the Notices of the AMS. The timing of the article was good, because just last week I gave an invited research talk at the University of South Carolina. I was able to use the advice given to improve my talk beforehand. The Devadoss makes many good points, and I recommend reading the article. Here, I’ll just list my take-aways.

Tell a story. Do not deviate from that story, even if the deviation seems interesting to you. Stay focused! Your audience needs to be able to follow along without too much jumping around.
Do the work for your audience. Distill major points down to their essence, so that they are clear and memorable. It’s okay to ignore subtleties and exceptional cases. Use intuition and images over precise definitions and explanations.
- A note here: it is very tempting when creating a slideshow based on a research paper to cut-and-paste from the paper into the slides. Do not do this. Language that is appropriate for a paper is often too verbose for a slideshow. Your audience can’t go back and re-read definitions or mull over the statement of a theorem, so it needs to be presented simply and memorably the first time.
Use your lecture slides wisely. Just because you can cram more info onto a slide, or run through many slides quickly, doesn’t mean you should. Less is more here (and pretty much everywhere else).
The most important one: Speak slowly and end on time. If you’re running behind, do not just talk faster. Rather, leave out some stuff (in fact, decide beforehand what can be left out). Do no go over your time. DO NOT GO OVER YOUR TIME.

Most of these lessons apply to giving any type of talk, not just a math talk. Also, they apply when lecturing to a class. When I teach, I try to connect the lesson to what we’ve learned already and what we ultimately want to learn, so the lessons build a kind of story. I first present intuitive explanations and pictures (often through an example) before giving technical definitions and theorems. And of course, I try to speak and write slowly, remembering that my students are hearing this information for the first time and need time to process it.

Communicating math

What’s the point of solving a math problem if you can’t explain the answer to someone? Nearly every other discipline encourages its students to practice written and oral communication, but math courses often forgo this practice so they can focus on covering more material. This is a mistake.

I am thinking about the importance of math communication since I have just attended the QTM Math Circle poster presentation. At the end of each summer, students enrolled in QTM Math Circle complete a capstone research project and present their work at a poster presentation. For many of these students (who are all high schoolers), this is their first time having to explain math to someone. Of course, many of them can explain steps of a calculation–“then you divide by 2 and take the square root to solve for x,” e.g.–but this was their first time explaining what problem they tried to solve, what their approach was, and why it worked.

The best presentations were those that included helpful visuals. For example, one team studied screensaver patterns. You know the style of screensaver that floats around the screen, bouncing off the walls, until it finally satisfyingly hits a corner? The students on this team studied how many times the box will bounce off a wall between corner-touches, and how far the box will travel, and how many times it will cross its own path. A few clear example pictures on their poster were enough for me to understand the problems they wanted to solve and predict some of the patterns. If an audience member doesn’t clearly understand your problem, you’re wasting their time with everything else you tell them.

Also, many good presentations focused on the idea behind the proof over the step-by-step calculations of the proof. Another team, for example, studied strategies for finding your way out of a forest if you are lost (assuming you know the dimensions of the forest). They wanted to minimize the worst-case escape time. Their solution involved evaluating a few integrals, and these were shown on the poster so anyone who was really interested could see the details, but when they presented their work, they just explained the high-level strategy, then noted, “our calculations show that this is how long you would need to find your way out.”

I have notice many professional mathematicians still struggle with communicating their results. I cringe a little every time I watch a math talk where the presenter shows a slide full of equations, as if the audience members will have time (or energy) to digest each line of calculations. These talks would be so much better with a few helpful images and an outline of the method used, without getting too deep into the weeds. As a graph theorist, I find pictures to be worth a thousand equations, and by using a picture to explain an idea from a proof, the real reason for the result becomes more clear to me.

When working with my Calculus students at Emory, I have started focusing more on communicating math. I have assigned a few take-home assignments (an example is here) that involved interpreting their results. Also, many quiz and exam questions ask them to explain their final answer in terms of the problem/scenario given. When I return quizzes to students, I always have those who have solved problems correctly write their solutions on the board and explain them to the class, but this is mostly about explaining the calculations. I would like to introduce into my classes opportunities for the students to explain math ideas to each other and practice this skill. I will try this in the coming semester of Calc I I’m teaching.

A quick note on board-work when teaching

A simple hint to improve your teaching: when you’re writing on a board and you run out of space, you should take time to slowly and completely erase part of the board. It is natural to try to erase the board quickly so you can get back to writing notes, but this is a mistake in my opinion, for two reasons.

First, students need a moment to process your lesson as you teach. When you take twenty seconds to erase the board (rather than ten seconds), you give them a better opportunity to catch up with what you’ve been saying. They can finish copying down the last thing you’ve written, or can think about whether the lesson makes sense so far. Students appreciate this short break just to catch their breath during a lesson.

Second, quick erasing is often sloppy erasing. Do yourself a favor and clear the next section of the board completely of all stray marks. Now you’ll have a clean slate (literally) on which to write. Your students will have an easier time reading what you’ve written if there are no straggling serifs sprinkled throughout nor ghosts of what was previously written haunting the background.

Also, taking a bit more time to clearly erase the board gives you a second to breathe as well. You can gather your thoughts and prepare for the next section of notes you want to write, so you can give the best lesson possible.

Emphasizing intuition over algebra

I just saw this great YouTube video giving an intuitive explanation of why e^(i*pi)=-1. The creator doesn’t use any trigonometry or calculus, or really any algebra. Just basic arithmetic and lots of great visuals. Give it a watch here:

I often find myself relying on rigorous notation and calculation when trying to explain an idea–both in the classroom and when writing papers. Of course, being correct is a priority. But being convincing is nearly as important. If a student or reader walks away thinking, “I guess I buy it. I mean, all the algebra checked out,” then I’ve given them very little basis for growth.

I want to start including more intuitive explanations in my teaching. I can still give the formal proof of a concept, but let me first try to justify the outcome naturally.

Why does sin(x)/x approach 1 as x approaches 0? Not because the squeeze theorem says it does, using two cleverly chosen other functions. But because sin(x) has values very close to x near the origin.

Why does the formula for Simpson’s Rule work? Again, not because we can do a bunch of arithmetic and change the indices on a summation. It’s because it’s a weighted average of two other approximation rules, one that over-estimates and one that under-estimates.

I can also focus on giving intuition when writing my research. Before I define some new variable that seems to come from nowhere, and before I start some algorithm on a graph that decomposes it for some reason, let me invest a couple sentences saying “Here is the goal. Here’s how we’ll reach it.” Now the reader has some idea where we’re headed and may suddenly recognize the route we’re taking to get there, instead of spending the entire proof scratching their head. It’s not enough that they agree that all the variables add up the way I claim; the reader should walk away feeling like they’ve learned a method they can reuse.