Karim: [offering a lesson plan attempting to defang the wolverine, using iPad pricing as a hook for linear equations]

Zeno: What reason is there to think there would be a linear relationship between the storage capacity and the price of an iPad?

Karim: Great question! If you wanted, that could actually be the hook, no? [offering other remarks on expanding mathematical access]

Zeno: [offering nothing; no response]

Zeno isn’t wrong to ask for proof. If you get students in the habit of extrapolating any two points into a linear model, you’re setting them up for a whole lot of pain later. On the other hand, if you insist that middle schoolers justify every linear extrapolation or (for another example) define every polygon as “a simple closed plane curve composed of finitely many straight line segments,” you’re positing mathematics as a 400-pound wolverine with fur like razor wire and teeth like broken glass, which makes you kind of a monster.

Most educators, I think, understand instinctively the tension between access and correctness, the difficulty of extending one while insisting on the other.

There is a demographic, though, that feels little tension along that line. Call them “wolverine wranglers.” These people handle dangerous animals like you and I can’t believe. They’re gifted and there aren’t a lot of them. Their most striking feature, though, is their conviction that wolverines are dangerous and you are not taking that seriously enough. Work up the nerve to approach a wolverine and the nearest wrangler will remind you of all the ways that could go wrong.

I find their motivations mystifying, though, certainly, if I were much good at wrangling wolverines, I would find it tempting to remind people by means both subtle and obvious that they needed my wrangling skills.

[Virtual Conference on Soft Skills]

Many of my students have been failing math for as long they’ve been assigned grades. It’s been necessary, then, to disabuse them as fast as possible of the negative, self-defeating misconceptions about math they bring to my classroom on the first day of school. Here’s one:

My students give math a wide berth. They treat math like it’s some kind of injured wolverine. Like, if you treat it just-so, if you come to class every day, keep your eyes low, and write your name neatly at the top of your homework, then it might let you sneak past it for a C-. If you attempt to engage the wolverine, though, it might kill you. Down that road lies pain, brother.

That’s the misapprehension. The truth is that math is a happy, gregarious, cuddly wolverine. It only looks scary, and we should heap scorn on parents and teachers who convince students otherwise. (“I tell TF that I always hated math when I was a kid,” said the parent to TF’s slack-jawed math teacher.) In point of fact, math loves to play.

Throw it a stick and it brings two back. Throw it two sticks and it brings four back. Don’t be scared to speculate what’ll happen when you throw those four sticks. Whatever happens will happen and that’ll inform your next hypothesis. In the meantime, you won’t ever find math impatient or angry. It’s always eager to play.

In practice, if we’re graphing 3x + 2y = 12, I’ll ask a student for a solution. The student will reply “I don’t know,” because, well, I mean, look at the fangs on that thing.

So I ask for two numbers. Any two numbers. The student replies “one and three.” And we’ll evaluate (1, 3). And it doesn’t work, but that incorrectness is properly perceived as a gift. We put an x on the board at (1, 3) and we express our gratitude to the student for helping us narrow in on a solution by identifying something that isn’t.

The student relaxes a bit, realizing that the wolverine isn’t going to retaliate. It’s slobbering but not because it’s mad. It just wants you to throw the stick again.

“Okay, so (1, 3) didn’t work. Find me five pairs that do work.”

That’s where I stop and they begin, students of every ability, playing with an animal they previously assumed hostile. And there am I, surprised and grateful that my paid job was to make that introduction.