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It’s happened to every teacher. It’s Thursday, but your students don’t seem to remember Wednesday or Tuesday, and you’ve got three times as much material to cover if there’s any chance of Friday’s lesson working. Finally, you gather them together. “Not all of you have figured this out yet,” you say, “But I’m going to show you how this works so you’ll be ready for tomorrow.”
I’ve done this more than I’d like to admit. And every time, I regret it. I see the students slump in a combination of relaxation and helplessness. I’ve just taught them that the teacher’s there with the right answer, and it doesn’t matter if they understand or not. Weeks of work are being undone. And worst of all, it doesn’t work! Friday comes, and the students still don’t know what they needed to know. Now I’ve got to go back to Monday, and start the whole project again, but without the advantage of novelty. “I’ll come back to this topic later,” I think. What a failure.
The best I can say is that it’s my failure, and not my students. I mis-estimated how much time they’d need, and then gave them less rather than more. I succumbed to the old false idol of teaching: if you just tell the students the answer, then they’ll know. It would be so sweet if it worked. But for almost all students, it doesn’t, and then we’ve got to resist the impulse to blame the students. I rushed them. Math takes time, and there are no shortcuts, “no royal road to mathematics.”
No matter how I try to resist, I make this mistake occasionally. But I do think there is a perspective that keeps me honest. A virtue I can cultivate.
A lesson plan is predicated on a lie: that students will “understand” the idea you’re teaching by the end of the lesson. But what we don’t say is that all understanding is partial. No matter how elementary the topic, I guarantee that it’s just a whisper away from unsolved questions in mathematics. It’s not just that I can’t fully understand, or that my students can’t: it’s that most (and it’s possible to make that word “most” quite mathematically precise) mathematical questions are outside of anyone’s reach. The greater the understanding, the more aware of all you don’t know.
And this humility, this modesty, is for me, a cure for rushing. Because where are you going, really? If there’s no end, why are you in a hurry?
This is true even for really simple topics. Consider addition.
- The question of how many ways there are to break a number up into an addition problem—the so-called partition problem—was unsolved for centuries. It was only just solved.
- When Russell and Whitehead tried to ground arithmetic in set theory, it took them 362 pages to prove that 1+1 = 2.
Nobody can know everything about anything. There’s no such thing as completely “covering” a topic. And knowing that keeps me honest. I won’t rush my students. I’ll give them time, and with it, the chance to get somewhere real, to attain actual understanding, instead of forcing them to participate in the lie that agrees with my lesson plan. They can’t understand everything, but they’ll have some real understanding, and just maybe, start to know what they know, and what they don’t, and how one becomes the other.