**
**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 12^{th} 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 IVT lesson landed on a Friday. The next day, I sat in a coffee shop with a poppyseed bagel and a mocha, grading their quizzes from the end of class.

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.

What an interesting article! Thanks Ben!

Thanks for reading, Nathan!

Ben

Thanks for the beautiful essay. I shared it with my whole department this morning. A spring time pep talk like this is priceless

Thanks for reading, Jim! I’m glad it resonated. It always amazes me how much students can learn if you give them time.

Thanks fro this pep talk, Ben. I teach introductory physics, algebra based, at the community college level. I have been at sand casting many times over 45 years. this semester we’ve been burdened by snow cancellations and the Boston tragedy. Many students reach this level with poor skills and attitudes. The challenge has ever been to try to move them forward a bit. It is not possible often to address these past deficiencies and teach a course based on those expected prerequisites. I only hope that the constant revisiting of ideas and discussions of cool natural phenomena will eventually produce some aha moments. In no event do I want the students to develop a hateful attitude toward science. It is ever a balancing act.

Thanks for reading, Art. It sounds like you’ve got a wise approach, and one that I’m sure benefits your students. Teaching high school (and moving to California, which has a wonderful community college system) has made me realize how crucial community colleges are to social justice in education. I can only imagine how varied (and often gap-filled) the students’ backgrounds are. They’re lucky to have a teacher who embraces that balancing act of filling in background vs. marching forward, and who sees the value of those “aha!” moments.

My best wishes to anyone you know who was affected by the attacks in Boston. Is your community college in MA?

Thank you. Makes me remember the Bruce Cameron quotation, “Not everything that can be counted counts, and not everything that counts can be counted.” Standardized-test-writers focus on content and ignore process. But I would rather have my students learn less content if it meant they would better learn how to: think persistently about a novel problem, read or write a proof carefully, or make connections. Would we ask kids after an English course to answer only quotation-identification questions about the books they read? No, we want to ask them to write an essay, too. It should be the same for math. I agree with your point that content knowledge can invisibly accumulate — I love the sand metaphor — but even if the students did not get the IVT in the end, having them work on reading the dense mathematical language of its statement would not have been pointless! Anyway, thanks again.

Thanks for reading, Russell – well said. I find some standardized tests are better than others, but most that I’ve seen have that bias towards content you mention. Process is, of course, much harder to measure than content – but we can’t give up on the highest goals of education just because they’re hard to measure.

This is definitely something I’ve seen from both sides of the desk. In high school, I encountered high-level math courses like Linear Algebra and Differential Equations, and they made almost no sense whatsoever. Over the next two years I didn’t do much more than flip through the books a couple of times, without really paying too much attention to them. Coming up to this summer, I was worried about having trouble while taking the classes for college, but every single concept clicked and made perfect sense. The whole image of fistfuls of sand really applies from the viewpoints of both students and teachers.

I’ve also gotten to see it from helping classmates with math and physics. Sometimes it seemed like I’d explain something over and over again, without it sinking in. But all of a sudden they would catch on, and I’d finally be able to lean back with a sigh of relief and achievement.

Keep up the good work!

It’s nice when that payoff comes, isn’t it? Even more so because you spend hours increasingly worried that there’s never going to be any payoff.

“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 don’t know whether to feel proud for being from Oakland, or to feel ashamed as a Berkeley student!

I wish I had a high school teacher that wasn’t so stubborn. The lesson was too fast for some, and too slow for others, but he would teach everyone at the same pace. Some students would be discouraged, while others were bored and fell asleep in class. I wished my teacher offered to teach both AB and BC Calculus to those who could handle it.

In any high school version of a course, students spend more time interacting with the teacher on the subject,more hours per week, and a year vs a semester, so naturally high school students would perform better if both teachers were competent.

Oh, I’ve missed you, Yu Hang. 😉

I’ve only seen this from a students point of view. Sometimes I feel like I don’t fully understand the concept, yet when doing the homework and even the test, I realize I do.

Love this article. I took calculus a year before many of my friends in high school, and around halfway through their precal year, I started telling them, “Guys, in a few months, someone is going to tell you that graphs that aren’t straight lines can have slopes. The slopes can be different at different parts of the graphs. But just let that roll around in your head for a while, because it takes a while to make any sense, and you want it to make sense before you learn it in precal.” I had a hunch that relatively passive awareness of something abstract could blossom over time.

Marin: I agree. I had a chemistry teacher (with a PhD in theoretical chemistry) who took care to talk to us about the structure of molecules in terms of fuzzy clouds of electron that could reconfigure themselves into diverse shapes, that could stick together with matching clouds from other atoms, with only a hand-wavy explanation of how quantum mechanics made that happen, which sufficed as long as we got the general idea that bonds could come out of atoms in various configurations of directions. (I’d also read some of George Gamow’s lovely stories of Mr. Thompson.) This “passive awareness” of quantum weirdness proved thoroughly useful, later, when my physics teachers came to the subject of quantum mechanics. The weirdness was intuitively tractable, because I was used to how it worked to make bonds between atoms. I think most students coming to it find quantum weirdness is a major obstacle to even beginning to make sense of the subject. Likewise, I can imagine that exposure to the idea of instantaneous gradient on curves, before the formal treatment of differentiation, can make the ideas more tractable.