my contribution to the Convention of Mathematical Flavors – check it out!
Every teacher has tics. I don’t mean facial twitches (though a few rowdy, beloved classes have driven me there). I mean teaching tics: instincts, fallbacks, go-to-moves. The habitual ways that we frame questions, or receive answers, or structure lessons. I’m now going to recount one of my tics—one that crops up, for better or worse, in every grade, on every topic, in rain or in sun or in humidity so damnable that my sweat is sweating.
The tic of “give me.”
Give me six different triangles whose area is 10 cm2. Give me a pair of functions so that f(g(x)) is always bigger than g(f(x)). Give me five data points so the mean, median, and mode are all different. Give me, give me, give me…
Am I selfish, asking for so much? Am I selfless, giving my students the gift of giving (which, after all, is better than receiving)? No and no. I’ve converged on this question style because it’s quick, it’s carefree, and it encodes several of my mathematical and pedagogical priorities.
What am I hoping to convey with “give me”? Here’s a partial list:
- In math, we create just by thinking.
It’s worth asking: How do you “give me” a rectangle? Do you forge it from iron? Sculpt it from clay? Buy it off Amazon Prime?
No: you just describe it. Specify a length and a width. Maybe sketch a wobbly picture. No construction permits needed.
In mathematics, description is creation. A description is the blueprint of a thought, transferred from your brain to mine. That’s the principal activity of mathematicians: the conjuring of ideas in one another’s skulls. “Give me” is an invitation to join that process.
- Answers live in multiplicity.
I ply my students with plenty of closed-form questions. “Compute this area,” “simplify those fractions,” “go wrassle them integrals.” Such questions are easy to assign, easy to grade, and easy to tailor for specific skills. Some even make for thick, juicy puzzles.
But mathematics is not a kingdom of single answers. It’s a realm where truth comes in families, where answers live in glorious multiplicity. Take this standby: “Give me a pair of rectangles, the first with a bigger perimeter and the second with a bigger area.” A dozen students could supply a dozen replies. Your answer needn’t look anything like your neighbor’s.
- A good technique is a gift that keeps on giving.
Here’s another favorite: “Give me a fraction between 10/11 and 11/12.”
Finding your first example may take some elbow grease. But once you’ve got one, it’s easy to generate others, because good mathematical techniques aren’t one-trick ponies. They’re fertile, birthing answer after answer.
For example, I saw one student note that each original fraction has “one piece missing.” The first is missing a bigger piece (1/11) than the second (1/12). To find something in between, pick a middle-sized piece (like 1/11.5, or 1/11.4, or 1/11.78). My student didn’t just find an answer; he found a path that led to infinite answers.
- The math is for you, not for me.
The way school is structured, students seem to perform for the teacher’s amusement. I say “jump”; they jump. I say “differentiate these functions”; they plug them into Wolfram Alpha. I say “studentwithextrahomeworksayswhat”; they say, “What?”
So it’s liberating—for me and for them—to do work that’s a little freer. Work that’s not graded by resemblance to some master key. Work with room to swing your elbows and let your mind romp, work that even lets you—dare I say it—express yourself.
I’ve rarely seen my 7th-graders so energized as when writing their own challenges and questions, to be shared with the rest of the class. Instead of working to please a single authority figure, they now play for a crowd of 25 peers. “Give me” questions—selfish as they may sound—help flip the usual script.
- In math, we sift the possible from the impossible.
Sometimes I’ll make a trollish demand. “Give me three numbers,” I’ll say, “so that the sum of any two of them is odd.” Before long, students are yelping in outrage. And that’s where the learning happens.
First—and most important in life—they learn never to trust me.
Second, they learn that math is obsessed with possibility and impossibility. More than any other subject, it’s the dancefloor of the possible, where we see how far logic can bend without breaking. (Caution: do not try this on an actual dancefloor.)
In the sciences, “impossible” is a contingent idea, subject to revision. Try asking Newton about gravitational lensing, or Einstein about whether God plays dice. But in math, “impossible” is a tangible presence. There are no three numbers whose pairwise sums are all odd. Period. The attempt to find one brings you into contact with impossibility—and thus with certainty, with proof, and with the other qualities that make math so delightfully and epistemically math-y.
I could go on. But I’ve given you enough, haven’t I? Now that I’ve got you here, I want you to give me your “give me” questions. I’m always looking for fresh ones—and I suspect that you’ve got ‘em. Transfer them from your mind to mine, please. That’s what math is all about.