Advanced Math Education

When taking “basic math” courses, these are usually pretty optimized by centuries of teachers having thought about and experimented with teaching methods. We’ve generally found that a bunch of computational exercises are a great way to incrementally build up a student’s skills.

Get to proof-base math and BAM no more incrementalism. No more exercises, much less pedagogy in general. Definition - theorem - proof, as the slogan goes. It can’t be too surprising that this is also about the point at which people drop out of math classes or a math major—if only because so many such students explicitly name proof-based courses as what convinced them that they’re not cut out for math.

But I also sympathize for the textbook writers. For one thing, they don’t usually have much training in pedagogy. For another thing, it would be hard for them to get good at pedagogy when they are doing advanced math and spending all their time on research. For yet another thing … well … just how would you do it better even if you had an author interested in trying? Computation is mostly out the window, so all that remains for exercises is proving theorems.

One idea: Fill in the blank problems.

Yep. Fill in the blank problems. Remember in middle school, you’d use context clues or something like that to fill in a blank in a sentence? Or like on the SAT?

Sounds pretty childish right? Incongruous with the dignified arena of university mathematics. Imagine giving a proof, with strategically blanked-out sentences or sentence clauses, or chains of equalities. That’s not an easy thing to fill in, when you’re first learning a new and theoretical subject. And it offers incrementalism. Usually we do our best with incrementalism by giving proofs that range from easy to hard. But usually only the very easiest proof exercises are actually reasonable for a first-timer in a theoretical subject.

Moreover, the medium and hard proof exercises often involve some trick or fact that the student could not reasonably be expected to think of. I think most of the time, we expect the student to see the professor after class or in office hours to get over this hump. But fundamentally that’s a sloppy stop-gap that we accept as standard operating procedure. It still fundamentally means that the student was assigned an unreasonable problem, and that can be thoroughly demoralizing. There’s value in struggle, but not in struggle that breaks your back.

A further idea: Two-column proofs. Remember those from high school geometry? Statements on the left, reasons on the right? It gets formalized further in course on Symbolic Logic and then we never see it again. Why don’t we keep doing that in advanced math? We kinda do if we’re interested in automated theorem provers like Coq or the various other softwares on the market, but this doesn’t do anything to the higher math novice. Not yet anyway.

So why not also give two-column proofs with either statements or reasons (or both or neither) blacked out?

One last thought on the topic: One of the really valuable things about computational exercises is that you can change small parts here and there to make them gently different or harder. Seeing these variations-on-a-theme really helps a student to zoom out and see the general pattern. We should try to have banks of exercises that similarly give exercises that are all very similar to each other, but for small modifications either in the theorem or its proof.