Okay, here’s a life regret: No one has ever stopped me on the street, grabbed me by the collar, and demanded that I explain to them the essence of mathematics.
I’ve envisioned it many times, though.
What math teacher hasn’t?
Me: So, you want to get math?
Assailant: Obviously! Why else would one human being violently accost another, if not for the acquisition of knowledge?
Me: Easy, then! All you need to do is listen to Sgt. Pepper’s Lonely Hearts Club Band.
Assailant: [arches eyebrow] You can’t be serious. The Beatles album?
Me: [easing out of their grip, brushing my collar] Naturally! The whole album is trippy and spectacular, of course. But I’m talking about the final moments of the final track, a song that Rolling Stone has hailed as the Beatles’ greatest: “A Day in the Life.”
Assailant: [listening on an iPhone] This better be good, or I’m going to pound you into a fine math teacher carpaccio.
Me: Patience, assailant, patience! Wait until three minutes and fifty seconds in. That’s when a cacophonous noise begins. It’s the sound of a 40-piece orchestra playing absolute gibberish.
Assailant: [brow furrowing] This music sounds like I’m losing my mind.
Me: Exactly! Producer George Martin had some very odd and vague instructions to the musicians. Start quiet; end loud. Start low in pitch; end high. “You’ve got to make your own way up there,” he said, “as slidey as possible so that the clarinets slurp, trombones gliss, violins slide without fingering any notes. Most of all, don’t listen to the fellow next to you because I don’t want you to be doing the same thing.”
Assailant: [shaking his head] This is the strangest pop music I’ve ever heard. It gathers like a tornado.
Me: Yes – without melody or harmony.
Assailant: Intensifying as it ascends, like a swirling flock of terrified birds…
Me: Constantly growing higher and louder and higher and louder…
Assailant: Wait! It stopped!
Me: Yes. At 4:19, it abruptly ends, and at last from the madness there emerges a single chord, an E major. It lasts for fifty-three seconds. By the end, you can hear the soft ambient noises of the recording studio: breathing, rustling, the creaking of chairs.
Assailant: It’s triumphant… yet disarming. Revelatory and simple.
Me: It is the very sound of finality.
Assailant: [once again wrapping a strong pair of hands around my neck] Not so fast! I told you to show me the soul of mathematics! If I’d wanted an exegesis of 1960’s pop composition and studio recording techniques, then I’d have attacked a music teacher instead!
Me: [unperturbed] Patience, friend. You see, this one-two punch, the chaos and the chord, is the best musical example I know of a simple and powerful theme: how order lives within disorder. How structure arises from chaos. How clarity is born from the haze. One of the most discordant and alarming sonic passages in the Beatles’ catalogue gives way to one of the purest and most elemental. And with this, I shall finally answer your question.
Assailant: [eager to the point of bursting] Yes? YES?!
Me: The crescendo—wild, dissonant, and defiantly complex—is reality.
Me: The final chord—pure, structured, and definitive—is mathematics.
Assailant: [drops me, begins to cry with joy]
Me: The experience you get listening to “A Day in the Life” is precisely how it should be to learn math. First, you should encounter the teeming mess of reality, in all its sprawling incoherence. Then, you should feel the hot wind as mathematics blasts away extraneous details and distracting features, leaving only a clear, singular vision.
Assailant: [on bent knee, whispering] Mathematics is the chord buried in the cacophony, the music hidden in the mayhem. Why did I never see it before?
Me: Because, sadly, that’s far from the experience that math class actually offers. If real math is a wild animal, then school math is its taxidermied corpse.
Assailant: I see now. School math presents a polished final product, and obscures the process by which it was created.
Me: It’s a punchline without a joke
Assailant: A moral without a story.
Me: Actual mathematics—the endeavor that has captivated humans for thousands of years—is quite different: partial and halting, full of false steps, vain hopes, and dead ends, a grand and endless struggle to formulate patterns that can make sense of reality. If mathematics is ‘A Day in the Life,’ then school math leaves out the entire song to play you—in total isolation—only the final chord.
Assailant: We never get to hear what comes before.
Me: No, we miss the centuries of struggle, the heap of discarded ideas, the wild arguments as the orchestra’s players (accustomed to following sheet music) learn on the fly how to improvise.
Assailant: So mathematics and reality are… opposites?
Me: Mathematics isn’t reality, but mathematics is born from reality. Math class ought to explore the relationship between parent and child: how the one gives rise to the other, and what the second gives back to the first.
Assailant: [strolling away, lost in thoughts] I see it so clearly now. The duty of math class… is to play you the whole song.