You Don’t Hate Math, William Heard Kilpatrick Does

I hate math. I find it tedious. As a kid I failed algebra in high school twice. Numbers, fractions, systems of equation — all of them, disgusting. And I am not unique: at least 40% of high schoolers rank math as their least favorite subject, a plurality of students. And yet, in my graduate work I interact with math daily, often with concepts that are much more complex and unintuitive than mere algebra. Not only that, but I find the struggle here engaging, and I keep hunting for more complexity, more nuance, more depth. So which is it — do I hate math, or do I love it? And what about the rest of us?

The reason Americans struggle with mathematics is not because the subject is difficult (it’s not any harder than English grammar), nor because they’re stupid (although their teachers might be). No, the primary reason American’s hate math is because they are taught math backwards, starting from the most tedious and arbitrary applications and only after years, if they are very lucky and persistent, maybe being given a morsel or two of theory. Americans don’t understand math because they literally are not taught fundamentals of mathematics, sometimes not even in college. The American fetish for arithmetic and algebra has crippled their logicking abilities and taught them that math is some fearful, Lovecraftian tome of cryptic incantations that are impossible to grasp without a PhD and a freakishly powerful brain. Nothing could be further from the truth. Math is simple, intuitive and enjoyable, and should be taught as such.

It’s one thing to identify a problem, but quite another thing to understand it. To begin our diagnosis of broken maths, let me introduce the villain of the piece: William Heard Kilpatrick.

Kilpatrick was a student and sometime colleague of John Dewey, and inherited Dewey’s project of transforming the American education system into a industrial drone production facility. Previously, education was (1) not universal and (2) focused on training mind and attitude, cognition and class. The traditional education model was/is focused on the student’s academic ability and cultural heritage. Kilpatrick was determined to exchange this model for progressive education, which sees schools as a social service and practical preparation for real life. Real life, in this case, meant menial labor. Geometry isn’t necessary to run a cash register, arithmetic is.

But arithmetic is tedious and, well, menial. Geometry, on the other hand, is visually interesting, conceptually straightforward and full of creativity. But to Kilpatrick, geometry and even algebra were detrimental to students (read: future wageslaves) and should be stricken from the books:

Reflecting mainstream views of progressive education, Kilpatrick rejected the notion that the study of mathematics contributed to mental discipline. His view was that subjects should be taught to students based on their direct practical value, or if students independently wanted to learn those subjects. This point of view toward education comported well with the pedagogical methods endorsed by progressive education. Limiting education primarily to utilitarian skills sharply limited academic content, and this helped to justify the slow pace of student centered, discovery learning, the centerpiece of progressivism. Kilpatrick proposed that the study of algebra and geometry in high school be discontinued “except as an intellectual luxury.” According to Kilpatrick, mathematics is “harmful rather than helpful to the kind of thinking necessary for ordinary living.” In an address before the student body at the University of Florida, Kilpatrick lectured, “We have in the past taught algebra and geometry to too many, not too few.”

This idea was, of course, fiercely opposed by actual mathematicians. But Kilpatrick had the greater influence among policymakers, and progressive education became the standard model. Prog. ed. advocates also drew on the work of behavioral psychologist Edward Thorndike to build their model, using stimulus-response patterns to train students into proper arithmetic practices, as opposed to a model of critical inquiry and exploration which is natural to higher math. This cemented math as a field of rote memorization and rule-based problem solving, guaranteeing that students would hate it for decades to come.

There was a brief glimmer of hope in the 1950s and -60s called “New Math”, in which mathematicians tried once again to make the case for the academic side of their field. The following quote is from Jerome Bruner, a Harvard psychologist, and is taken from the same link as the blockquote above:

I am struck by the fact that certain ideas in teaching mathematics that take a student away from the banal manipulation of natural numbers have the effect of freshening his eye to the possibility of discovery. I interpret such trends as the use of set theory in the early grades partly in this light — so too the Cuisenaire rods, the use of modular arithmetic, and other comparable devices.

But the idea was quashed in the 70s, it’s only legacy being the introduction of calculus to high school curriculum. But a formulaic calculus, or a formulaic geometry, stripped of conceptual discussions and thought-provoking proofs, is merely an extension of the tedious bore that is elementary arithmetic.

Kilpatrick’s ideas were for an industrial society, but we live in a post-industrial one. His vision of math education has been rendered totally obsolete by the calculator, since all he ever wanted to do was turn human beings into calculators themselves. How long do you think it takes to learn how to operate a cash register? Not twelve years, that’s for sure. It is possible to equip children with practical “real life” arithmetic and such without losing sight of the true vision of mathematics — that it is a playful, inquisitive study, focused around exploration and clever thinking, not memorization and applying algorithms.

If American’s valued their children, they would make some curriculum reforms immediately: promote geometry and algebra to elementary school — I promise you the concept of variables is not too difficult for a nine year old to grasp. In every class, focus first on concepts and theory, and only get around to applications during the second half of the year. Aside from powers of 2 and perhaps the multiplication table, there’s little need for memorization. Allow calculators as frequently as possible. Have students read and write proofs of all sorts throughout their education. Teach formal logic and digital logic, both of which will be much more practical skills than either algebra or calculus. Attack math concepts visually first, rather than writing out a formula to memorize and only getting around to building intuition if there’s a lull in the semester. It’s absolutely criminal that virtually every high school calculus class begins by assuring the students that they won’t spend any time addressing the Fundamental Theorem of Calculus. Too often, the conceptual underpinnings of math are abridged or left out of the curriculum entirely, when they should be the core focus of the class. These concepts are not too hard for students; what is too hard for students is trying to do math without them. Imagine trying to use a keyboard without labeling the keys. That’s what it is to solve math problems without understanding the tools you’re using. These are only a smattering of suggestions, and they practically rolled off the top of my head. Stick a few mathematicians together in a room, ask them to reinvent math curricula from the ground up, and you’ll get something wildly different — and vastly superior — to what we have now.

So now you know. The reason you hate math isn’t because math is hard or you’re just not cut out for it, it’s because a misguided psychologist who thought the purpose of education was to condition children like rats and a malignant reformer who wanted to turn the U.S.’s general population into a robotic slave race teamed up to wreck the education system forever. Unfortunately, it’s too late to punch Kilpatrick right in his ogreish nose, but it’s not too late to undo the destruction he wrought. We can rebuild math education in a way that does a better job of teaching children actual mathematics while also doing a better job of giving them practical skills they will need in a digital world. All we need to do is upend the bureaucracy that has agglomerated around the education system like an ingrown nail. Easy, right?