Halfway through, the idea of “exponent” takes a sudden turn.
And—fair warning—so does this post.
A drumroll and an awed hush, please! Here’s my teaching load for this year:
Though it makes my Yankee eyeballs melt and dribble out of my head, this is a fairly typical schedule here in England. The one aberration—a scheduling concession my Head of Department graciously made—is that instead of a group each of Year 8 and Year 9, I’ve got two of the latter.
That means I get to focus (so to speak) on that critical year when “elementary” math (the stuff every citizen needs) yields to “advanced” math (the gateway to specialized professions and fields of expertise). And what proud little gatekeeper stands at this fork in the road, welcoming those students who understand its nature, and vindictively punishing those who don’t?
Why, the exponent, of course!
Exponents start pretty simple. Exponentiation is just repeated multiplication. The big number tells you what you’re multiplying, and the little parrot-number on its shoulder tells you how many times to multiply it:
Sometimes we multiply them together, like this:
From this pattern you can glean a simple rule, the kind of tidy and easy-to-apply fact that we lovingly expect from mathematics class:
But this is when exponents take a sudden turn. Without much warning, we rebel against our original definition—“exponentiation is repeated multiplication”—and start complaining about its flaws.
Specifically, that definition makes perfect sense for values like 54 or 227, or even (-3.5)14. But what about when the exponent is negative? Or zero? Or a fraction? What would it mean to compute, say, 91/2—i.e., to multiply 9 by itself “half of a time”?
To say “exponentiation is repeated multiplication” is perfectly pleasant. But it takes us only so far. It opens up the world of whole-number exponents, but leaves other realms locked behind soundproof doors.
And so we renounce this definition, and begin to worship a new one: Exponentiation is “the thing that follows the rule abac = ab+c.”
It’s a weird change of game plan. We’re abandoning a clear-cut explanation of exponentiation in favor of a more nebulous one. Instead of defining the operation by how you actually do it (“multiply repeatedly”), we’re defining it by an abstract rule that it happens to follow.
And any number to the –n must equal the reciprocal of that number to the n—because our rule says so.
These new statements represent a funny sort of mathematical fact. They’re not just arbitrary and capricious, as students might grudgingly maintain. But nor are they 100% natural and inevitable, as teachers might optimistically insist. Rather, these truths depend on a leap of faith, a change of heart, an extension of the exponent into terrain where it could not originally tread.
We tear out the first page of our exponentiation bible, and replace it with a rule that, when we first encountered it, felt merely peripheral or secondary.
I celebrate this as a magnificent sleight of hand, an M. Night Shyamalan twist that reconfigures your sense of everything that came before.
When you meet exponentiation at a cocktail party, and ask it what it does for a living, it replies, “Oh, I’m just repeated multiplication.” But it’s only being modest. It has a secret identity, as the all-important operation that translates fluently between addition and multiplication.
Why am I so smitten with this? Well, because weirdly enough, it strikes close to home.
I’m half a decade into my teaching career, and to be honest, I scarcely remember why I originally got into the profession. To impart truths? To change lives? To “give back”? I doubt my reasons—whatever they were—carried enough oomph to sustain me for long.
But over time, my reasons have transformed. These days, I love this job because it’s equal parts social and intellectual. What other job puts you in such close contact with people and ideas—not just one or the other, but both of them, constantly?
My core reason for doing what I do—just like my notion of exponentiation—switched somewhere along the way.
I’m hoping some of my students can experience that same evolution. Many arrive in 6th grade as “math kids,” accustomed to top marks, easy A’s, and plentiful praise. They often cite math as their favorite subject, and I can guess why—because it makes them feel smart. All told, that’s a good thing. It’s perfectly natural to enjoy something that makes you feel like a star.
But this momentum has its limits. When you find yourself surrounded by equally talented peers, you lose heart. You don’t feel so smart anymore. It’s a straitjacket sort of success that depends on the failure of others.
Math’s saving grace, though, is that it can make us feel smart for another reason: because we’ve mastered an ancient, powerful craft. Because we’ve laid down rails of logic, and guided a train of thought smoothly to its destination. Because we’re masters—not over our peers, but over the deep patterns of the universe itself.
Above all, I hope my students learn this lesson: that, regardless of how slowly or quickly you achieve it, and regardless of how you compare to the kids surrounding you, mathematical mastery is a badge of intellect. It makes you smart. It is your glorious gain, at no one’s expense.
And if they don’t learn that, I hope they at least learn that abac = ab+c.