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.