August 31, 2021

Today I was helping a university student with a proof of the following statement.

Let \(G\) be a group and suppose that \((ab)^2 = a^2b^2\) for all \(a\) and \(b\) in \(G\). Prove that \(G\) is an abelian group.

You can prove this with just a few equations, like so:

\[ \begin{aligned} (ab)^2 &= a^2b^2 \\ abab &= aabb \\ a^{-1}ababb^{-1} &= a^{-1}aabbb^{-1} \\ ba &= ab \end{aligned} \]

This is a good proof; it’s concise and easy to understand. But on the other hand, it’s just a series of equations. And the kind of student who always writes like this is bound to write lots of terrible proofs. So when I saw that my student had written his proof just like the one above, I advised him to add some English. I also suggested he read Don Knuth’s lecture notes on mathematical writing. And this got me wondering how a typical student learns to write a good proof. As far as I can tell, it’s rare for this sort of thing to be taught at all (Knuth’s course being a notable exception). Students are, by and large, just expected to figure it out on their own. And this is a slow process that doesn’t always end in success, as anyone who has graded proofs will testify.

What is to be done? The obvious thing is to have everyone who majors in a STEM subject take a class on mathematical writing. And how will students find the time to take yet another required class? By getting rid of “gen ed” requirements! Instead of wasting everyone’s time with a class they don’t care about and won’t benefit from, try this: Everyone meets in a room for one hour to read part of Knuth’s lecture notes, discuss, and then read and critique some proofs. Homework: write a proof. Repeat once per week until the end of the term.