May 11, 2024

For the Love of Math

A zoomed-in portion of the Mandelbrot set
A zoomed-in portion of the Mandelbrot set

By Elias Leventhal

I was first exposed to math in sixth grade. I wasn’t raised in a cave, or by wolves; my childhood was perfectly normal. But sixth grade was when I discovered Mathcounts and saw the true beauty of the subject for the first time.

Mathcounts is the nation’s largest middle-school math competition: a tweenage nerd Olympics. Every year, more than 100,000 students compete for the right to represent their school or county or state in progressively higher levels of competition. But more importantly, Mathcounts is one of the leading standard-bearers of a radical new approach to math pedagogy. 

The problems Mathcounts uses are impervious to prepackaged formulas. They draw from a huge range of mathematical topics, including fields that are not typically part of a K-12 curriculum like number theory and combinatorics. And they are difficult — not because they rely on any advanced background knowledge, but because they demand so much creativity and insight.

After years of memorizing my way through my school’s curriculum, I was completely unprepared for problems that made me think so deeply. But, with some parental urging, I pushed through my initial frustration. I became familiar with the mechanics of these problems and less intimidated by their unfamiliar notation. Before long, I was a convert.

I never performed very well in Mathcounts, but it hardly mattered. The competition was simply a gateway drug. Through self-study, online courses, and friendships with other math nerds, a strange and wonderful universe started to come into focus. I spent hours playing with visualizations of the mind-bending and beautiful Mandelbrot set, a fractal derived from the properties of complex numbers. I worked my way through pure math textbooks for middle-school students. I learned the basics of mathematical proof, and I gained a new appreciation for the profound difficulty of rigorously expressing simple ideas.

But the more captivated I was with this new way of interacting with math, the more frustrated I became with the version I was taught in school. In class, our minds were crammed full of formulas and “rules of thumb,” and we were instructed to apply them to pages of repetitive exercises. Proofs, which I knew by now to be the foundation of math’s incredible power, may as well have not existed.

As a result, I saw more and more of my classmates grow to hate math. I continued to love it only because I knew that our “math class” was misnamed. Like my classmates, however, I lost all desire to engage with the curriculum.

Then I got lucky. In high school, I was blessed with a flexible guidance counselor who encouraged me to follow my passion. From 9th grade onwards, I was left to my own devices to study the subject I loved. I went on to spend my next four years exploring topics like fractal geometry, computational biology, graph theory, linear algebra, and game theory. There is nothing inherently difficult about these subjects — they require no prerequisites beyond basic algebra — but all of them are excluded from a standard high school curriculum. In their place is a dreary march through a fixed list of topics most students have no interest in and will probably never use.

I know that my experience is personal, and that what worked for me won’t work for everyone. Still, in conversations with other math-loving students and adults, I’ve found that I’m not alone. The most passionate devotees of this subject are often the most frustrated by the way it’s taught in schools. With American test scores at abysmal lows, teachers and leaders are scrambling for new ways to communicate math. I suspect that teaching students to view the subject as an art form, and not merely a practical tool, could go a long way.

I’d like to close with two examples. Consider the following problem from the 8th-grade level of the Common Core curriculum:

Which of the following is a prism?

  1. Cylinder
  2. Cube
  3. Sphere
  4. Pyramid

If you are an 8th grader confronted with this problem, you face one of two possibilities. You either remember the definition of a prism and solve the problem in a fraction of a second (it’s B), or you’re doomed to take a wild guess. You can’t recreate a definition out of thin air. And as you struggle through questions like this one, you could quite possibly internalize the idea that you are “bad at math,” making you increasingly unmotivated to apply yourself to the subject. And the vicious cycle spins round and round.

Now, consider a related problem from a regional-level Mathcounts exam:

A rectangular prism of size 3x4x5 is made from 60 unit cubes. Including the full 3x4x5 prism and the 60 unit prisms, how many total rectangular prisms can be found in the large rectangular prism by taking a subset of the 60 cubes?

If you know the definition of a prism, you can solve this problem — at least in theory. You start by counting the sixty 1 x 1 x 1 unit cubes mentioned in the problem statement. You then move on to 1 x 1 x 2 prisms, carefully cataloging each of the dozens of ways that such a solid can fit within the confines of the larger prism. Then you move on to 1 x 2 x 2 prisms, and so on…

But by now, it is quickly becoming obvious that the number of possibilities is far too large to manually count. If you try to brute-force this problem, you will go insane. Which is the point.

So instead, you visualize. You hold the image of the 3 x 4 x 5 prism in your head, and you try to understand the process of extracting smaller subprisms from within it. With a bit of imagination, you see that by making a pair of vertical slices, a pair of horizontal slices, and a pair of lateral slices to the 3 x 4 x 5 prism, you can always extract a unique subprism. Suddenly, the problem is almost solved. You can work out that there are 15 ways to make two slices along the length-5 axis, 10 ways to make two slices along the length-4 axis, and 6 ways to make two slices along the length-3 axis. The answer emerges naturally: 15 x 10 x 6 = 900.

That’s math. No one will deny that it’s difficult, but it’s also fun and dizzying and profound and beautiful, and maybe one of the most important things that humans have ever created. I think that students deserve a taste of the real thing.

Elias is a graduating high-school senior from Shelburne, Vermont, planning to major in math and physics at Yale University next fall. Outside of math, he enjoys meditating, hiking, gardening, piano, basketball, and talking to his cat.

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