So i times i is negative one. Good. Now, what’s the square root of -4?
I expected someone to shout out 2i and we’d move on. But no one did. A pause. Then guesses: “2?” “-2?” We had just—I mean 30 seconds earlier—discussed why neither 1 nor negative 1 could be the square root of negative one.
Decision time: I could show them, giving them this tool, and getting on to the problems I’d planned to introduce. Or, I could leave this as an open problem for them to work on, on the theory that moving on won’t have much value until they put this step together for themselves.
I decided to leave it open. That was Friday. Today, Monday, it’s still open. Most students have raised their eyebrows at it and turned away to work on other things. A few are still poking at it: “Mister-nothing works.” “Well, we know it can’t be positive, or negative, so it must have something to do with i.” “Oh, I have to use i?” “Well, try it.”
My internal dialogue on this is punishing:
I’d be an idiot to tell them—a few are perplexed, and they’ll enjoy getting it when they figure it out. The ones that aren’t working on it are more interested in other things, and what’s wrong with that?
I’m an idiot NOT to tell them. The reason most aren’t interested is because they have no fluency with this thing yet. They’re bogged down and need me to give them a push out of the mud so they can get going. And anyway, my job is to make sure they all know how to find the square root of negative numbers, not to let a few of them figure it out while the rest do whatever they want.
So, am I an idiot, or am I an idiot? And why do I love this job so much?