"Teaching"

I noticed the other night, just before leaving campus, that in Abstract Algebra I'm about two weeks "behind" (about three handouts, each roughly equivalent to two days of class) the place where I was at this time in Fall 2008, the last time I taught the course.

My immediate reaction, which, fortunately, dissipated almost immediately: "Holy crap...how can I catch up?!?"

I looked over the handouts separating now from then. They were filled, for the most part, with technical lemmas and other minutiae about groups, facts like "the condition that g^-1 be a two-sided inverse is redundant" and "to check that H is a subgroup it need only be shown that gh^-1 for all g and h in H." I thought about my students, and their aims and ambitions, and I realized almost immediately that they could do without "covering" these lemmas. At best, they'd memorize them for several days, work a contrived homework problem or two meant to test them on the results, and then forget them.

Meh. We'll skip those handouts. Instead, we'll move on to the good stuff: subgroups, homomorphisms, more meaningful structural results that are powerful, intriguing, and beautiful.

Meanwhile, the homework is clearly kicking my students' butts. Even the more experienced students are struggling with it. It was only after thinking about it for a bit that I realized why this is: instead of asking them to pantomime the proof of some result that differs only slightly from something we've talked about in class or put the polish on a theorem I read out loud to them, I'm having them build from scratch most of the canonical examples; I'm having them introduce and analyze the most important definitions.

It would take me fifteen minutes to "teach" them all there is to know about the subgroups of the integers, but in doing so I'd guarantee that nine out of ten of them would smile (or scowl) and nod their heads, take careful notes, commit what I'd said to memory, and not understand a lick of it. I'd rather they take a few hours outside of class to do it for themselves, struggling with every step, pounding their heads on the Math Lab's countertops in frustration, cursing me under their breath, gaining intuition all the way. The best human computers of all time, from Napier through Gauss to Ramanujan, built their skills by living with numbers, loving them, spending their days with them cheek to jowl. It's hard work, but you'll gain so much for it, mathematically speaking. You'll gain intuition, and, ultimately, understanding.

I spent an hour or two with several of these students in the Math Lab this afternoon, and the progress they made was incredible. I'm so impressed by their intelligence and determination.

Keep it up, folks! I'm proud of you all.