Last April 13th, I emailed a few friends to let them know I was starting a blog. “I’m a little afraid it will land with a dull thud against the hard pavement of the internet,” I wrote.
Two weeks later, I posted an essay called What It Feels Like to Be Bad at Math, about my struggles with topology. It was stubbornly hard to write. I spat out 500 words of excuses and hedges (which I later deleted) before I could bring my fingers to type the truth.
Then the post started getting passed around. Continue reading
I’ve always dreaded being asked for my “teaching philosophy.”
For years, I gave nonsense or scattershot answers. “Logic and critical thinking are paramount.” “I care more about conceptual understanding than computational skill.” “A balanced, student-centered approach is always best.” “We buzzword to buzzword, not for the buzzword, but for the buzzword.” At best, each of my disjointed half-theories captured only a piece of the puzzle.
Worse still, none of my replies explained why I devote so much class time to plain old practice. Continue reading
I don’t usually struggle to distinguish toys from tools. Gas stove? That’s a tool. Easy-Bake oven? That’s a toy. Bricks? Tools. LEGO bricks? Toys.
Mathematical tools are similarly distinctive. They harness industrial-strength power—think of Taylor series, or completing the square. Mathematical tools shine floodlights into dark corners. They unlock doors, solve problems, and make attentive students utter, “Whoa, deep.” They often come with complex instruction manuals, requiring weeks (or months (or years!)) of technical training to master.
Mathematical toys… not so much. They’re simple to grasp, fun to handle, and not much substantive good to anyone. Think of Sudoku puzzles, or differentiating cos(cos(cos(cos(x)))). We might get a kick out of poking and prodding such problems, but solving them won’t teach us anything fundamental about the workings of the universe or the necessities of logic. Toy problems aren’t floodlights; they’re more like flashlights dangling off of a keychain.
But just as the Incas mistook the wheel for a mere toy, sometimes mathematicians get it wrong. Sometimes what seems to be a toy is, in fact, a powerful tool.
Sometimes a toy is just a tool in waiting. Continue reading
A pessimist’s take