Teaching is full of compromises. This is a story about one small compromise that I refused to make, a stubborn act that paid off, though I didn’t expect it to. The setting is a Calculus classroom, but I hope the story will resonate with anyone who spies something dubious in the rigid and widespread assumption that learning can be endlessly itemized, carefully quantized, and instantaneously measured. This story has a moral, which I’ll tell you up front: Some lessons don’t sink in right away.

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By my third year of teaching, I expected my classes to go all right. Not great, mind you: I might stumble over a definition, or botch the phrasing of a question, or optimistically allocate 5 minutes for an example that takes 15. Many days, I still made a minor idiot of myself. But I had put the fiascos of my first year behind me: no more droning 20-minute lectures, no more kids nodding off in the front row, no more pleading for their attention or castigating them for losing focus, as if my sloppy lessons were their fault.

Best of all – perhaps my only real strength as a teacher – I knew the terrain of their minds, how much mathematical territory we could cover in a day together.

So I was perfectly confident when I allotted one day for the Intermediate Value Theorem. The IVT captures a perfectly obvious idea: If at one time you’re 4 feet tall, and later on you’re 6 feet tall, then at some point in between you must be 5 feet tall. In other words: if you reach two different values (e.g., 4 and 6), you must also reach any “intermediate value” between them (e.g., 5, or 4.2, or 5.97).

Of course, the theorem frames this in the rather technical language of the mathematician: Continue reading →