A nearly-verbatim dialogue, in honor of one of the hardest-working students I’ve ever taught.
Student: Can you explain this problem to me?
Me: Sure, the idea is… Wait. You got it right.
Student: But I don’t feel like I did. I feel like I guessed.
Me: Well, explain how you got your answer.
(He proceeds to give a perfect explanation of the problem, justifying every detail.)
Me: See? You understand it.
Student: No, I just guessed.
Me: You “guessed” a flawless, conceptually motivated solution to the problem?
Student: Yes. You have to explain it to me.
Me: I mean… okay…
(I proceed to parrot back his explanation to him, virtually word for word.)
Student: Ah, thanks. That helps.
Priceless!
Glad you liked it. I wish all teaching was that straightforward!
So is it really helpful to just hear it again, or is it something like a vocabulary problem? I’ve had this same issue in science of *knowing* the procedure (and the instructions recited verbatim) for calculations, but not *understanding* why it works or where it comes from.
I think this kid just needed a little reassurance. He could say, for example, “We set the derivative equal to zero because we want to find an extreme, where the tangent line will be flat,” which either means he understands, or is really good at memorizing things that sound like understanding. But he’s not a typical student – usually when students feel like they don’t understand, it’s because they really don’t.
Yes…sometimes having someone repeat back to you exactly what you just said can be a gigantic confidence boost.
For me, I have the inverse problem when students have trouble with -3^2 vs. (-3)^2, especially when they write -3^2 when they should have written (-3)^2 and then in the next step write 9.
Regardless of the many ways I have of explaining the distinction between the two, invariably, I will have a student who still wonders why s/he had points taken off. After my explanations, the student will day, “I don’t get it.” My response is then, “Exactly.”
Yeah, that one’s a notational nightmare… I usually chalk it up to the convention that exponentiation gets evaluated before multiplication, so in some sense you’re looking at 3, first squared, then multiplied by negative one… do you happen to have a slicker way of justifying it?
It’s not just convention, it’s that exponentiation has higher precedence than multiplication. Don’t you teach order of operations (BODMAS/BIDMAS/)?
Maybe we’re just quibbling over words, but I’d call that precedence a convention. It’s an appealing one (since exponentiation is in some sense repeated multiplication, it makes sense somehow to evaluate exponentiation first). But it’s still just a convention. An alien planet could use the reverse order (BASDMO, for example) and still have a perfectly valid system.
I would hazard a guess: typing on a calculator, -, 3, power, 2; = 9.
Calculators use a different notational convention; but maths teachers haven’t yet taken that on board enough to, at least, explain carefully to students how the calculator notation differs from the “written mathematics” notation (mostly carried over into most computer programming languages) that gets summarised by diverse sequences of initials (that I can never decipher, much less remember; I just know how to parse notations).
I have a student who says to me a lot, “I get it, but I don’t understand it.”
Perhaps this student is a zen master, trying to teach you something via this koan.