I’ve been having a little argument with five-years-ago me. The question is this:
Five-years-ago me? Throw his drink in your face. He’d tell you that rote thinking is the bane of his working days, that deep understanding is the whole point of learning mathematics. He’d tell you: No black boxes, ever.
Today-me is less convinced. Don’t working mathematicians, from the ground floor all the way up to Andrew Wiles, sometimes use black boxes? Isn’t it common sense that sometimes you need to use tools that you can’t build for yourself?
I’m still wary of equipping students with black boxes, but these days I’m willing to do it, so long as three conditions are met. I hesitate to share this crude checklist, knowing my colleagues out there in the profession will have wiser ways to frame the tradeoffs. (After all, aren’t checklists too binary, too black-and-white, for an idea as elusive and shaded as “understanding”?)
Nevertheless, my checklist goes something like this:
When I taught my 17-year-olds the product rule for derivatives last year, I didn’t give them a proof. We talked through a few examples, and that was it.
“How do you think we’d prove this?” I asked later.
“Limit definition?” they said.
I nodded, and we left it at that. The proof I know is a clever algebraic trick; satisfying, but not terribly illuminating. I don’t really care whether students know the product rule’s origin story, so long as they know that it has an origin story.
By contrast, take my 11- year-old students as they begin secondary school. Many know lots of impressive “maths” (as they adorably say): they can divide fractions by fractions, subtract negatives, and state the nth term of an arithmetic sequence.
And if you ask them to explain why a technique works, they just describe the technique again.
In their view, mathematical methods aren’t rooted in reason, emerging by natural processes of logic. They’re plastic flowers popping out of the pavement like magic. It’s not just that they don’t know why these methods work; they’re fundamentally unaware that “why” and “how” are different things.
To use a black box safely, a student needs to know there’s something they don’t know. If that isn’t happening, then I shun black boxes like I shun black bears.
Some techniques are not that enlightening—but you need them anyway.
I’m thinking of a three-act lesson where students estimate the number of pennies used to build a massive pyramid. Working from first principles, they can mentally dissect the pyramid, breaking it down into layers of various sizes. But once they’ve done that, they still won’t know how to total the number of pennies.
They need a formula: the one for the square pyramidal numbers.
Deriving it would be an impossible chore in the confines of a short lesson, and wouldn’t play to the learning goals. We’re left with two choices: (1) Deny students the formula, thereby forcing them through a long, tedious, repetitive computation, or (2) Supply students with the formula, a handy shortcut they don’t totally understand.
I’m comfortable choosing Door #2. After all, part of being a mathematician is tapping into the wisdom of those who came before.
This year, in an ambitious move, I tried to teach my 12-year-old students about square roots. In particular, I hoped they could learn to flexibly employ the rule √ab = √a√b, to simplify expressions like √300, or √72/√2 or √20 + √45 + √180.
In the immortal words of Rick Perry: Oops.
I pushed them too quickly into technique, and then watched them rehearse a rule they didn’t understand. All struggled; many rage-quit. They came to see square roots like an Old Testament plague. Luckily, there’s a simple solution:
Don’t make them simplify square roots.
They have no practical or intellectual need for this technique right now. They need to build numerical and geometric intuition about square roots first. No reason to thrust them into the deep end of this quasi-algebraic pool.
This is a surprisingly common tale in mathematics education. We rush headlong into technique, trying to outrun an imaginary time-monster. So I’m always reminding myself: Be patient. Build context. Go concrete before you go abstract.
To recap, I’m comfortable with students using a technique they can’t justify only if all three of these conditions are met:
Now the real question: when are these conditions met?
If you ask me: Almost never. Basically, it occurs when you’re teaching sophisticated students a piece of mathematics not for its own sake, but for its applications. Engineers, psychologists, and environmental scientists don’t necessarily need to trace the derivatives of sin(x) and cos(x) back to the squeeze theorem.
But I know this isn’t how most black boxes get deployed.
More often, it happens when your back is against the wall: Students arrive at your door unprepared for an immovable high-stakes exam. The shortcut to decent scores leads away from understanding. You face two repugnant paths: forsake the students’ learning to preserve their economic opportunities, or vice versa.
Many of us seek a middle way. We try to carry both treasures up the steep mountainside. But all too often, we arrive at the top to find that the learning is gone, vanished from our hands. We look back and see it scattered along the path. Step by step, we let it slip from our fingers, not even realizing.
To the teacher on that lonely mountaintop, I offer neither applause nor condemnation. Just sympathy.
I’ve been there.