Math has turned my world on its head many times. I’ve left math classes feeling confused, overwhelmed, enlightened, and even betrayed. A few times stand out in particular from my first 12 years of math classes: being told that the fraction line and division were one and the same, that there are different ‘sizes’ of infinity, or that 3.999999999… = 4.

But one day in particular during my senior year was special. My AP Calc teacher was introducing the concept of integration with Riemann sums. She drew a curve on the coordinate plane, and asked my five classmates and me how we could find the area under the curve.

None of us said a word. I feel like I can safely say that, at least at the time, we were all overachievers and perfectionists, and we didn’t want to guess. We wanted to know. And if we didn’t know we couldn’t answer.

My mind was racing, thinking about the only kind of area formula I knew for a shape that curved. But I couldn’t begin to imagine how to use the area formula for a circle as a launching point.

We all sat there, waiting for our teacher to unveil the mystery formula, to flourish a hand and amaze us all with a brilliant method that would immediately make sense once we heard it, but that we couldn’t come up with ourselves.

But she didn’t. She just… waited. For us to think it through. For us to actually hazard a guess. After a few moments, she prompted us again. She told us to think outside the box. She gave us permission to, and encouraged us to speculate, not caring if we were right or wrong. And of course, this was a math class, so there had to be an answer or two that were right, and a whole bunch of answers that were very much not.

And slowly, timidly, quietly, we started talking about it. Discussing the problem and how to approach it, like a sports team might plan their next play, weighing the pros and cons. Our teacher patiently prodded us along, as we finally let go of our need to leap immediately to a neat solution with all the details worked out. We embraced the notion of approximating. Of getting as close as we could to the actual area, and then closer, and closer, until we were so infinitely close that we had a concrete value. We let ourselves break the problem down into simple, smaller pieces, and worked with those one at a time.

Approximating the area under a polynomial curve using rectangles

For years and years, I’d been learning not just the vocabulary, grammar, and rhetorical structures of mathematics, but also the problem solving skills that allowed me to use my fluency in practice. But as much as I mastered the mechanics of elementary mathematical language, I’d never been asked to create with it. I’d never been let loose with a mission, a toolkit, and absolutely no idea what to do with it. And the experience was freeing.

I think that was the day I started to understand what mathematics was, at least to me. And on that day, in that lesson, it was like therapy.

I was given permission to start down a path even when I didn’t know how to get where I needed to go; permission to start a task even when I wasn’t sure what obstacles I would run into, or if I would even be able to finish. Permission to try without knowing in advance that I could succeed. Permission to be messy, to be close but not quite there, to experiment and approximate. Permission to not be perfect.

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