I find that lots of students are really good at how.

Like, how do you factorize a quadratic? How to you differentiate a cubic? How do you solve a system of simultaneous linear equations? How do you poach an egg?

(Apparently you need a gentle whirlpool to get the egg moving. Whirlpools: the unsung hero of the breakfast table.)

Why are they so skilled at how? It’s because students like procedures. They like certainty, clarity, the feeling that you know exactly what to do at every moment.

But they struggle with why. And – even more basically – they struggle with what.

For example…

I find that questions like this elicit one of two responses from students. Either this:

Or this:

These aren’t questions students are accustomed to answering in math class. In history, perhaps, where they have to write IDs of historical figures and events; or even in science, where they have to understand each component’s role in a theory.

But not in math. We math teachers tend to ask lots of how questions, and not so many what questions.

If you ask me, that’s sort of sad. They’re experts in how, and they can’t even tell you what the how is for.

And in this case, it turns out, there’s a pretty satisfying answer. Continue reading →