You know what’s often missing from math class? Yes, candy bars, but even more important than that: coherence.
Math class shouldn’t be a mishmash pile of facts, thrown together haphazardly, like an academic version of The White Album. It should be a perfectly interlocking tower of truths, climbing upwards with singular purpose—an academic Sgt. Pepper or Abbey Road.
A good class isn’t a greatest hits record. It’s a concept album.
In that spirit, I’ve been taking each topic in the secondary math curriculum—algebra, geometry, calculus, etc.—and trying to boil it down to its one-word essence. Here are the rules of the game:
- You must choose a single word to complete the sentence, “[Branch of math] is the mathematics of _____.”
For example, you might say, “Topology is the mathematics of dinosaurs,” or “Category theory is the mathematics of abstraction,” or “Combinatorics is the mathematics of sadness.” (To be clear, only one of those is remotely accurate; you have my sympathy, combinatorists.)
- You must pick a word that laymen would understand.
No fair saying “Algebraic geometry is the mathematics of varieties.” What the heck is a “variety”? Like, varieties of breakfast cereal? You shouldn’t need training in a subject just to understand its definition.
- Your word should encompass as much of the subject as possible.
It might not be a perfect fit. It’s okay if you don’t nail every detail. (Even Sgt. Pepper has lots of songs that don’t really fit the theme.) But you want to capture the spirit and scope of the subject.
So you could say, “Algebra is the study of equations,” but that’s weak sauce. Yes, 90% of algebraic work involves creating and manipulating equations. But it totally misses the essence. First, inequalities and expressions can be just as “algebraic” as equations; and second, lots of equations (like 2 + 2 = 4) aren’t remotely algebraic.
That’s the game. If you want to spend some time thinking of your own before being poisoned by my answers, then read no further.
Now, here’s what I’ve got:
This one’s a slam dunk. Calculus is already the most beautifully unified course in the high school curriculum, so the one-word treatment fits well here.
The first half of calculus deals with derivatives, which tell us how a quantity is changing at a specific moment. “That car is moving 80 miles per hour.” “The city is growing by 80,000 people per year.” “The clown is getting 10% angrier every time you insult his shoes.” It’s all about change. The second half of calculus deals with integrals, which allow us to aggregate lots of little, incremental changes to see their cumulative effect.
If you haven’t already cued up David Bowie’s “Changes” on YouTube, then what are you waiting for?
I don’t mean John-and-Yoko relationships, or even John-and-Paul relationships. I’m talking about relationships between quantities.
Pretty much all of elementary algebra boils down to the question, “How do y and x relate?” You explore different types of relationships: linear, quadratic, exponential. And you represent those relationships different ways: graphs, tables, equations.
“Shape” is a pretty good answer, too. Certainly, you spend most of geometry class studying shapes: triangles, rhombuses, spheres… But I like to think that geometry begins with more elemental ideas. Points. Lines. Planes. Right off the bat, you reckon with abstract concepts like “dimension.” That stuff’s more fundamental than the idea of a shape.
Look around, wave your hands in front of your face, and think about the weirdness of this three-dimensional universe that we occupy. Geometry is the mathematics of that. Yes, shapes are important in geometry, but its essence is deeper—and freakier.
My first thoughts were “triangles” and “waves.” But those two are pretty darn different, and that gap points to trouble. How are we going to reconcile such disparate halves? Is this class going to be our version of Let It Be—so promising in its conception, but a hot mess in its result?
Fear not! The coherence is there. Fundamentally, trig is about two different ways of understanding position.
First, you can count east/west steps (the x-coordinate) and north/south steps (the y-coordinate).
Or second, you can give a direction (the angle θ) and a distance (the magnitude r).
This captures basically all of trig. The SOHCAHTOA stuff, for example, is all about manipulating and integrating these two notions of position. Circle trig, meanwhile, is about translating between them: the sine and cosine functions swallow your direction angle, and then spit up the y and x coordinates.
To clinch the matter, what is the most powerful historical application of trigonometry? Navigation—which is to say, finding your position on earth.
Sometimes we know exactly what’s going to happen.
For everything else, there’s probability.
Maybe I’m overthinking this one. Maybe I should just say “data.” (And maybe the Beatles shouldn’t have released the insufferable song “Wild Honey Pie,” but there’s no going back now.)
But you know what? There’s something vacuous about picking “data.” It’s too obvious. Too technical. It rubs me the wrong way.
So let me make my case for “populations.” Notice how “data” always comes in the plural? Statistics is, fundamentally, about understanding groups. Think of mean, median, mode, range: these are all ways of summarizing a population, of boiling a diverse group down to a few illustrative numbers. Or think about percentile and z-score: these are ways of specifying an individual’s place in relation to the population. And finally, think about hypothesis testing: there, we’re using samples to draw inferences and conclusions about whole populations.
When mathematicians call a statement “true,” they mean something very specific. It’s different from what scientists, artists, plumbers, and politicians. (To be honest, I’m not sure what the politicians mean.) They mean that the statement in question follows irresistibly from others that we have already laid out.
So what is logic? It’s the study of how mathematicians use that little word: “true.” As you might expect, it underpins every other branch of mathematics, and it’s a worthwhile field of inquiry in its own right, too.
So, to summarize: