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.
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:
Statements like this usually baffle students. (Don’t worry if they baffle you, too. I don’t expect readers of this blog to speak that kind of mathematical language, any more than I expect them to speak Finnish.) Still, these juniors seemed comfortable with abstract logic, and they possessed a firm grasp on mathematical truths far subtler than the IVT. I anticipated no trouble.
“Turn to your neighbor,” I said after presenting the theorem, “and explain why f needs to be continuous.”
Silence. A bad sign. In pairs, they’re typically chatty and playful; silence means they lacked the confidence to venture a guess aloud, even to a single partner.
“C’mon, give it a shot,” I said. “Tell your neighbor: What would happen if f wasn’t continuous?”
They reluctantly turned to one another, arching their eyebrows, smiling in minor panic. A few students fumbled their way to an explanation, without confidence or consensus.
Next, I guided them through a proof, and then asked them to prove a similar fact. As I walked around the room to troubleshoot, they filed complaints.
“Is this right? It doesn’t feel right.”
“I have no idea what you’re talking about.”
“Why are you doing this to us?”
The next day brought little improvement. I couldn’t understand: What made this so hard? Was their understanding of functions shallower than I thought? Were they feigning confusion to slow down our pace and lighten their workload? Had their geometry teacher utterly failed to impart any sense of what it means to prove something? (After all, they’d learned geometry two years earlier from a clumsy and underprepared rookie teacher – me.)
That’s when I got stubborn. We put in several days of sweat and frustration, mine and theirs. I ran through examples at the board; they practiced at home; I fielded questions in the afternoon. It was, by any reasonable measure, a waste of time, hours devoted to a minor topic that merits half a class at most – and worse yet, it still wasn’t clicking.
Eventually, I did what one does in such situations: threw up my hands, gave them a too-easy test, and called it a win, though it felt like a loss.
We returned to the IVT six months later, in Calculus. I dragged my feet. I couldn’t imagine that their ascent to 12th grade had brought any new gift for abstract reasoning; more likely, I knew, I’d find them in the exact state I’d left them in.
The quizzes were phenomenal.
I kept interrupting my fiancée (a math PhD student), pulling the headphones out of her ears to tell her about another sterling answer. “Look at his counterexample!” I’d exclaim. “Check out her phrasing in her proof – those are her own words. They actually get it.”
She sighed jealously. “I wish my students could do that.” She’d taught the same class to Berkeley undergrads – arguably the best college freshmen in the state of California. Their ability with proof didn’t stack up against these Oakland 16-year-olds.
I glowed for the next few days. And months later, when we ran up against meatier, more challenging theorems, the students mastered them with ease, accomplishing in two days what my previous Calculus classes had struggled to achieve in a week. It left me to wonder: How had those original lessons on the Intermediate Value Theorem gone so right, when they felt so wrong?
This is where we hit the moral of the story. There’s a lot to like about this data-driven era of teaching. We zero in on concrete, highly specific objectives; we follow standardized curricula and assess frequently; we do our best to become careful pedagogical bookkeepers, starting every day with the question, “What exactly should my students learn?” It makes sense. Every lesson ought to bring progress, and progress ought to be measurable.
The problem is that progress is sometimes invisible – at least at first.
Teaching often feels like throwing fistfuls of sand into the ocean. You watch the particles spread and sink, a pointless cloud, and you feel like a fool for ever imagining it would go better. You try again the next day, to the same result: nothing. You keep at it, bucket after bucket of sand, and it never seems to make any difference, until one day, you toss in one more cup of sand, and… there it sits on the surface, a tiny newborn island. You realize it then. Those fistfuls of sand – they were settling in heaps on the ocean floor, rising every week, hidden out of sight. Grain by grain, you’ve built yourself a little continent, only now protruding above the waves.
As a teacher, you’re a cartographer of your students’ minds. You know by instinct where the islands are supposed to be. And if you navigate your way to the proper set of coordinates, to be greeted only by an empty sea, don’t despair. Don’t give up and retreat back to the harbor. Drop anchor and start throwing that sand.