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home / Yet another perspective on math education

published June 19, 2023 in math, education

status: written for school. it’s… ok, but not great.

Before high school, I did not understand why math was useful. When my dad took me to the AoPS learning center across the street to be interviewed for taking their pre-algebra course, I questioned how this world of abstract nonsense could ever be purposed for the bigger picture — how does fudging symbols ease world poverty? What more did you need to get rich or cook food than the basic arithmetic behind accounting and recipes? The instructor pulled out his phone, wondering which of the two deductions I was stuck at: whether technology is built on math, or if it is useful at all. But because I already loved programming, the trick didn’t work on me, since hardly anyone will ever compute astrophysical simulations or need to implement a cryptography scheme, and I could already access the useful parts of current gadgetry with high-level Javascript anyways. I desisted from continuing the debate, and obviously didn’t get selected for the class.

By the time I was applying the quadratic formula, I was failing algebra and recruited a math tutor to literally do the work for me. We spent a month trying to catch up, but in the end I retook the class.

Everyone agrees that math education in the U.S. sucks. Some people complain that it’s repetitive or uninteresting, and on the other side, there are those who believe that learning calculus and up is otiose and overcomplicated. From my naive perspective, the folks who believe Math is Beautiful try to make New Math and get the traditionalists on board to learn about generatingfunctionology and whatnot, but this just provides more fuel for the fire, because clearly nobody needs to know Galois theory to live logically. In either case, both sides are turned away from math, as taught in school, at an early age, though both are right: it’s futile to teach math for practical use and the current curriculum is too focused on its utility. As Paul Lockhart wrote in his essay, “A Mathematician’s Lament,” the whole curriculum needs to be “scrapped.”

So what is math good for? The process of learning math aids abstract understanding. Math is easily generalized across subjects, and the physical sciences are especially mathematized. Math education attainment is well-correlated with the infamous g factor, and it almost stands without proof that if you can comprehend calculus, you’ve an adept, rational mind. Even the founder of AoPS claims that doing contest math helped him succeed in college biochemistry. Proof techniques like induction build the skill of arguing correctly. Much of the benefits math provides come in the ability to solve more general problems and reason logically.

George Pòlya, an eminent mathematician, continues this philosophical thread in his series on Mathematics and Plausible Reasoning, exposing how key insights come from reasoning “plausibly.” Using tools like analogy to seek patterns, good math consists of thinking laterally of what could be true, and passing beyond ordinary logical deduction. Pòlya asserts that jumping to conjectures is what makes math special — the abundance of prime gap conjectures theorizing different upper bounds on the gaps between primes, for instance, shows that the human mind has considerable strength in devising novel, often unbroken, and sometimes proven claims. You couldn’t prove anything that a computer couldn’t in less time without the power of human inference to reason plausibly. This is what sets mathematical arguments above plain syllogisms or computations, giving awesome math the qualities of an excellent riddle.

Of course, math also has applications: calculus helps you find the area under a curve, and therefore machine car exteriors. But if you aren’t doing math professionally, you are doing it recreationally. Solving puzzles is fun, and as a bonus, math is historically rich. You can read the criticisms Newton faced with fluxions, or how the ancients derived Heron’s formula without trigonometry. The beauty of human ingenuity has deposited as much in the past in mathematics as it has in other genres of nonfiction.

Take, for instance, Japan’s tradition of Sangaku. Distributed across their country in temples, these geometric puzzles were completely detached from physical applications. Instead, they challenged worshippers to solve their tricky enigmas. While today, we would happily use the quadratic formula to find the position of a circle tangent to three others — Descartes’ theorem — the Japanese cultivated their own techniques to solve this problem that became far more beautiful than bland algebra. Mostly separated from symbolic manipulation, this artform is distant from the “useful” math we see today. Instead, the Japanese taught geometry for its creativity and elegance.

Math contests and math circles are truer to the purpose of mathematics. They encourage insightful solutions and foster curiosity through challenging problems. Taking a semester-long class from AoPS last year, however, cost around 500 dollars, if I remember correctly. I think we can do better. Engaging math education should be inside every school.