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.