The Calculus of History

the paper I’d assign to a calculus class if everyone shared my slightly skewed sense of intellectual fun and my excessive fondness for mathematical metaphors


Forget the history of calculus. Write me a paper on the calculus of history.

You won’t be the first. In War and Peace, Tolstoy compared civilization to a vast integral. Only by summing all “the individual tendencies of men,” Tolstoy wrote, “can we hope to arrive at the laws of history.” His was a true people’s history. Each peasant and prince gets the same weight in Tolstoy’s great Riemann sum. To give the monarchs disproportionate weight (thereby silencing the masses) would be a perversity, a paradox. No delta functions in Tolstoy’s mathematics.


History as an integral. That’s one way to see it. Continue reading

Anxiety, Mathematics, and Words of Kindness

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

A Teaching Philosophy I’m Not Ashamed Of

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

Teaching as Self-Sacrifice

Like everyone else, I’m heir to the prejudices of my culture. So even though I know firsthand that teaching is a profession, on some level I see it as an act of self-sacrifice, as a hard path undertaken for the greater good.

This might not sound like a prejudice. Not a damaging one, anyway. It paints a generous portrait of teaching as noble and virtuous—a useful antidote to the too-common caricature of educators as lazy, union-fed bureaucrats. This ethic of “the greater good,” in fact, helped draw many of us into the profession. What we lack in salary and prestige, we make up for in civic-mindedness, in moral conviction, in feeling we’ve taken a high road through our professional life.

But all this high-mindedness has a dark underbelly. In painting the choice to teach as an act of self-sacrifice, I believe that we may harm our schools more than we help them. Continue reading

How to Talk to a Mathematician

In an email, Bonny Becker asks: How do I go about gaining a better understanding of what my math PhD-seeking son is talking about?

This question haunts those of us with a beloved mathematician in our lives. While most professions have a few dark rooms mysterious to outsiders, the field of mathematics feels like a palace in which every room is dark.

Continue reading

Who Cheating Hurts

My first stab at a “cheating is bad” speech came when I realized the magnitude of over-the-shoulder copying on my daily trigonometry quizzes. It was epidemic. So, turning reluctantly from teacher to orator, I sat down the juniors and gave them two basic arguments.

First, cheating is bad for the cheater. It circumvents the very purpose of school, which is learning. When you cheat on homework, you deprive yourself of the chance to learn. And when you cheat on tests, you deprive me of the chance to measure your learning—and thereby to help you learn better.

And second, cheating is bad for others. How you perform in school determines your access to resources and opportunities in life. Thus, when you cheat, you’re lying to steal resources from honest folk.

A sharpshooting student can leave both these arguments riddled with bullet-holes. Continue reading

Confessions of a Math Major

I’m not proud to say it, but I majored in math because I wanted people to think I was smart. I chose the math major as a status symbol, a résumé-topper. I consider myself an innocent dreamer, a lover of curiosity and learning, so the fact that I navigated college with such calculating self-interest isn’t exactly a badge of honor. But that’s not my confession.

My confession is that I don’t think math is better than any other subject. Not in the ways that matter. Continue reading

Undiscovered Math

One day in fifth grade, I was playing with numbers, scribbling down products and quotients—you know, typical cool-kid stuff—when I noticed a pattern. Take any pair of numbers that are two apart (like 6 and 8, or 9 and 11). Multiply them together. Then add one.

You’ll get the square of the number in between them!

This blew my mind. The numbers were hiding secret alliances, passing coded messages amongst themselves, and I’d somehow broken inside. I was a number spy. Continue reading

Blaming the Last Guy

Like most first-year teachers, I couldn’t believe some of the gaps in my kids’ knowledge. My geometry students scowled at word problems, mangled the distributive property, and handled fractions with the same fondness you’d show a steaming diaper. Privately, I grumbled. Why didn’t their old teachers cover this stuff properly? Why is it my job to pick up the pieces?

Two years later, I got the chance to teach the same kids trigonometry. I still couldn’t believe some of the holes in their background. What’s their problem with special right triangles? Where’s their intuition about proof? Why didn’t they learn this stuff? As soon as I found myself blaming the last guy, I realized.

The last guy was me.

Continue reading

Whiteboards: A Love Story

One piece of classroom tech has always vexed and fascinated me. It’s my magic wand, my Batmobile, the love of my teaching life.

I’m a whiteboard man.

My romance with the whiteboard began with embarrassment and miscommunication. (It’s just like any rom-com movie plot, in that way.) You might think it’s hard to screw up a whiteboard; you’d be wrong. I spent months scraping half-dry markers against the shiny surface, too stubborn to admit I was writing in invisible ink. Students in the third row squinted at my writing like it was a high-stakes eye exam.

Continue reading