Bags of Tricks (idle thoughts about their place in instruction)

I read a very provocative piece today, written by the late Gian-Carlo Rota, about the failures of undergraduate differential equations courses. I took DiffEQ as a summer course immediately after my freshman year of college, and I was struck by how effective it was. I had heard complaints about DiffEQ being a completely impenetrable, unmotivated course, but that was not my experience--and now, I know why. The instructor fully admitted to the methods being hacks, and I appreciated it.

In a perfect world, yes, we want our students to learn math (substitute physics, or astronomy, or basketweaving, or whatever else), but we also want them to internalize how to think. I agree that learning about integrating factors makes the light go out in students' eyes, but it's possible to both motivate those "tricks" and use them.

I ran into this when I was a TA for intro astronomy, as well: it was challenging to formulate well-motivated examples, in a way that illustrated that solving a word problem was not just about trying different hammers and pounding away until an answer emerged. Rather, we want students to (a) develop a correct physical intuition; and (b) convince them that mathematics holds the answers. The issues come from getting mired in counter-productive bookkeeping, of the kind we so often see in our "mathematical methods/intro" classes.