John Urschel Learns The ‘Secret Code’ Of Calculus

The following is an excerpt of Mind and Matter: A Life in Math and Football by John Urschel and Louisa Thomas.

When my dad and I didn’t go to the gym, we’d go to the library at the University of Buffalo before I headed home to my mom’s.

One day when I was in eighth grade, while I was doing some homework and my dad was working on a problem set for his economics master’s program, I asked him what he was working on. This is a matrix, he said, showing me a rectangular arrangement of numbers in rows and columns held together by long brackets on the left and right. Then he pointed to the numbers inside the brackets. The numbers are the elements of the matrix. He explained how to add, subtract, and multiply different matrices together. They’re pretty useful, he said. See how this matrix is shaped like a square? We can learn some of its properties by calculating something called the determinant. That’s what I’m working on now.

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How do you calculate the determinant? I asked. He showed me how to use the well‑defined determinant of a 2 × 2 matrix—a matrix with two rows and two columns—to determine the determinant of a matrix of any size.

A few minutes later, I was playing with matrices—adding them together, turning them upside down, and trying my dad’s homework problems myself. He stared at me, surprised.

That summer before eighth grade, my mother enrolled me in a summer camp for kids who were interested in engineering, as she had for several summers in a row. This year, though, I was bored. I liked the other kids in the class, but I couldn’t understand their enthusiasm for the projects we were doing. We were making rockets propelled by vinegar and baking soda, and building models with balsa wood and glue. Everyone seemed to be having fun except me. I was miserable. It seemed to me that we weren’t doing real math or science. We weren’t learning anything exciting or difficult or new. We certainly weren’t being challenged in the way that I expected and wanted to be challenged. Baking‑soda rockets? We might as well have been following the instructions from a kit available at a toy store. I had no problem with games—I loved games—but the projects we were given made me feel as if we were being treated like children.

I have an idea, my dad said after I complained to him. He used his University of Buffalo student ID to enroll me as an auditor in a calculus course for business students during the summer semester. His name was John Urschel; my name was John Urschel. And I could pass for someone much older than my age, since I was nearly six feet tall.

There were about thirty people in the class, mostly business majors. The course was taught by a graduate student who seemed almost as nervous as I was. He would try, awkwardly, to connect with the students by talking about muscle cars before the start of class. The other students mostly ignored him—and, for a while, ignored me. My nerves at being found out quickly disappeared. I thrilled to the challenge of fitting in—to doing the work well enough that no one was the wiser. I liked the anonymity of passing for someone older, someone else—another John Urschel, a John Urschel who could have been any regular college student.

Except that your average college student probably does not enjoy doing calculus homework.

Learning calculus was like learning a secret code. If you were ignorant of it, much of the physical world was indecipherable. But if you knew calculus, you could describe the orbit of planets or the spiral of a football. It didn’t occur to me that calculus was supposed to be too hard for me—and it wasn’t. I’m convinced that most other thirteen‑year‑olds wouldn’t find it too hard either, if they weren’t conditioned to think that calculus was way too advanced for them. In fact, a ten‑ year‑old could understand the underlying themes, if not perform the calculations. The question that calculus begins with is basic: what happens if we think of a smooth curve as a straight line?

Just that question was enough to make some of the mathphobic people in my class turn off their brains. You could see their eyes glaze over, even before they had to solve a single problem. It sounds like a contradiction: a curve as a straight line? But it’s very easy to visualize. I imagined a cannonball shot straight across the face of the earth. I would be able to see only a short bit of its path through the air as it flew by, and so to my eye it would look like it was following a flat trajectory. If it were possible to see the cannonball from a long distance away, though, I would be able to see its whole path. As long as it didn’t rip into something, eventually I’d be able to see the projectile curve toward the ground as gravity pulled it toward the earth (which, of course, itself looks flat but is actually round). Then I thought of a football moving through the air. I know that a thrown football follows an arc. What if I could see only a small segment of its path? If it was thrown hard and directly, it would look like it was moving in a straight line. If you drew the arc of a football on a piece of paper and then zoomed in close to a segment of the arc, the same thing would happen: the curve would flatten, and then flatten more, until it looked like a straight line. The closer you zoomed, the straighter the line would be. If you zoomed in so close that the segment was infinitesimally small—smaller than any size you could imagine, but bigger than zero—then it turns out the curve wouldn’t just look kind of like a straight line. It would be infinitesimally close to one.

That, more or less, is what Isaac Newton realized, back in the seventeenth century. The slope of that line (a calculation of its steepness) is called the derivative, and it forms the basis of one of the two main branches of calculus, differential calculus. (The process of finding the derivative is called differentiation.)

I learned that the derivative can be a very useful way of measuring change. It lets us relate the position, velocity, and acceleration of a football—or an airplane, a missile, a planet, a speck of dust. Or— since I was in an economics class—it lets us represent and explain economic behavior, like cost minimization and profit maximization. The other branch of calculus, integral calculus, is related to differential calculus (that, in fact, is the fundamental theorem of calculus), but it deals with questions of area, volume, and displacement, by letting us calculate the area under a curve. It’s fairly easy to calculate the area of anything bounded by straight lines—just divide it into a bunch of triangles—but harder when you’re dealing with the area under, say, a roller coaster. It turns out that you can divide the area under curves into a bunch of infinitesimally thin rectangles and then add them all together. Calculus, I quickly grasped, let me move from a world that was static and frozen to a world that could move and flow. It was as if I had stepped out of a photograph and into a movie. Suddenly, time could pass, speed could quicken, gravity could pull, economies could grow.

It was easier to understand calculus as another language because some of it was in another language. There were new symbols to learn, some of them from Greek. The countless hours of doing puzzles and brain teasers had helped me learn to move between numeric and symbolic thinking. Besides, there was something exciting to a kid about seeing math as a kind of code. Learning it would give me the ability to describe a half‑hidden world—our world, which we rarely stop to see. It wasn’t that it was extremely easy for me to compute differential equations and integrals. I struggled with the more advanced techniques as did everyone else. But I hadn’t internalized the common idea that the word “calculus” is basically a synonym for something impossible. Instead, I saw it as a way of describing the physical world as we all experience it. Calculus was a way of turning the physical world into puzzles, and I had never stopped loving puzzles.

I couldn’t solve the puzzles fully yet, of course. I didn’t understand everything in that class. Totally grasping the idea of something infinitesimally small or thin, for instance, or adding infinitely many things together, required an understanding of the concept of infinity that I would not achieve until college—if then. I took for granted that I knew what “infinity” was, until I was forced to manipulate it in more advanced ways. But I didn’t need that level of comprehension to do well in calculus. I thought the subject was accessible, in part because I was too young to have it hammered into me that it shouldn’t be. I didn’t dread calculus. It excited me. It felt like real math. It was revealing something fundamental about the world. And perhaps it revealed something fundamental about me too. I wanted that kind of challenge. I liked countering expectations. I wanted to do things that people assumed someone like me—in this case, a kid who hadn’t even started high school—couldn’t do. And I wanted to learn things that were rigorous and interesting and surprising. I was willing to put in the work.

I would sit quietly in the back of the classroom, hoping not to attract attention. I didn’t try to make friends. But after a while, a few of the other students started to notice that I was doing well. When the teacher called on me, I was ready with an answer. When he passed back problem sets and exams, my grades were consistently high. One day, as I was waiting for a lecture to start, one of my classmates came over and spoke to me. Hey man, he said, did you finish the problem set yet? I could use some help. He slid into the chair next to me, and I walked him through the work. As the semester wore on, more students came over to me and asked me for help with homework or to clarify something. No one seemed to notice anything unusual about me, except that I had a good grasp of the coursework—which is how I liked it.

Adapted from Mind and Matter: A Life in Math and Football, published by Penguin Press, an imprint of Penguin Publishing Group, a division of Penguin Random House LLC. Copyright © 2019 by John Urschel and Louisa Thomas.