I’ve taught at two schools in my career.

The first, in California, had a dozen teachers in total. I adored my colleagues, but we each had our own domain. I handled Trig, Precalc, Calculus, and Stats. Soon I fell into unquestioned habits, built on assumptions I didn’t know I was making.

My second school, in England, had a dozen *math* teachers. It was as if, after years of playing bass in my empty garage, I had suddenly been recruited into an actual band.

My first winter, I took myself on a tour of the department, observing a lesson from each of my new colleagues. I came away convinced that there’s no one way to teach mathematics, that our methods are necessarily as diverse as our goals.

Take two very different teachers: Simon and Tom.

Whereas I’m always fretting about students who can execute procedures without understanding them, Simon wastes no such worrying; he simply weaves the two together far better than I do. His notes at the board model clear and disciplined thinking, and he gives written comments every bit as careful and analytical as the work he expects. (He is also one of the two most competitive sportsmen I have ever met.)

Tom couldn’t be more different. At the front of the room, he’s less a lecturer than a provocateur: the mischief-maker in chief, whose highest goal is to create space for mathematical exploration. That means open-ended problems, and multiple days spent wrestling with a single provocative question. He incites debates, sets traps, and shines a spotlight on the students’ own thinking. (He is also one of the two most competitive sportsmen I have ever met.)

But the diversity in math teaching runs deeper than the Simons vs. the Toms, traditionalists vs. progressives.

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