Dan’s comment on this exchange got me thinking about procedures (the mathematical kind, like FOIL, not the general classroom kind, like how to turn in homework). The student in that exchange was a senior who had given up on ever learning to multiply polynomials. I said “Oh, no problem” and began to explain, and she literally covered her ears. She said the words don’t help. So I said “Ok. I’ll write on the board. You ask questions.” I did a silent, large-format, multi-color demo of FOIL, one step at a time. At each step she looked intently at the board for a minute and then I saw her make up a hand gesture for each step. I told her to go practice. The next day she came in for assessment and I put one on the board and left the room. When I returned she had done that one and three others that she’d made up on the spot, including one with huge coefficients, just for fun.

There’s a magic moment between hard and boring. FOIL is a rote procedure — anathema — that for this student had always been hard. When she constructed her own way of organizing it, it became fun. The success was fun. Knowing what to do was fun.

This year I taught few if any memorized formulas or procedures, and I downplayed the ones students presented. I am firmly in the “if you understand it, you don’t have to memorize it” camp. I think Dan’s comment describes precisely why that student finds math “boring or hard.” But, I experienced this student’s mastery of FOIL as a successful educational moment. Put these together and I am confused.

Maybe I need to examine my aversion to teaching rote procedures. Are they like sugar in the diet? Insubstantial and unhealthy as a main course, but in small amounts, an enjoyable and useful way to increase confidence and willingness for more serious inquiry? The hard-line anti-proceduralist in me says “That was a pyrrhic victory. If she hasn’t built up a solid mental model of what it means to multiply sums, it’s just another arbitrary algorithm.” But another part of me saw in her exuberance–finally learning to FOIL–a move from I Can’t to I Can, a step in the direction of smart and free.

So I am left with questions about rote procedures. Ever? When? Which ones? Why? How often? For whom?