Practice Math Like a Baby

I’ve always felt conflicted about repetitive practice.

On the one hand, I see how vital practice is. Musicians repeat the same piece again and again. Soccer players run drills. Chefs hone their chopping motion. Shouldn’t math students do the same: rehearse the skills that matter?

But sometimes, I backtrack. “This is just going to bore them,” I fret, scanning a textbook exercise. “I’m emphasizing the rote aspects of math at the expense of the creative ones. They’re going to forget this skill anyway, and be left only with the insidious impression that math is a jackhammer subject of tooth-grinding repetition.”


(Then I assign the exercise anyway, because class starts in five minutes and— despite my repeated petitions—the administration has denied me access to a time turner.)

These two trains of thought suffer daily collisions in my mind: repetition is dull, but repetition is necessary. This inner conflict takes for granted the idea that repetitive practice is a separate endeavor, a distinct stage of the learning process. First, you learn the concept. Second, you practice it. In this view, practice is like cleaning up after a picnic: absolutely essential, but not much fun.

But this summer, a very wise teacher showed me a path forward, a way to reconciliation.

I’m referring, of course, to a two-year-old named Leo.


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Why I’ve Stopped Doing Interviews for Yale

Last year, I conducted alumni interviews for Yale applicants. It’s an easy gig. You take a smart, ambitious 17-year-old out for hot chocolate, ask them about their life, and then report back to the university, “Yup, this is another great kid.”

I recently got an email asking me to re-enlist. Was I ready for another admissions season?

I checked “No,” mostly because “Aw, hell no” wasn’t an option.

Why my reluctance? No grudge, no beef, no axe to grind. It’s just that the whole admissions process is so spectacularly crazy that participating in it— even in the peripheral role of “alumni interviewer”—feels like having spiders crawling out of my eyeballs.

In the last 15 to 20 years, Yale’s applicant pool has gone from “hypercompetitive” to “a Darwinian dystopia so cutthroat you’d feel guilty even simulating it on a computer, just in case the simulations had emotions.”

I don’t fault the admissions office. For every bed in the freshman dorms, twenty kids are lining up, at least five of whom are flawless high-school rock stars. From that murderer’s row, they face the impossible task of picking just one to admit. There’s no right answer.

But two things freak me out about this process.

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13 Trig Functions You Need to Memorize Right Now

In 2013, The Onion thrilled the nation’s math teachers by, at long last, mentioning them:

It’s a headline that bites right to the center of the Tootsie Pop that is our mathematical curriculum. If math is a deafening barrage of arbitrary things to memorize, then trig cranks the volume of gibberish up to 11. I mean, cot2(x) + 1 = csc2(x)? Why do I care again?

Of course, all of trig really just boils down to two functions:

This pair of functions has many offspring, several of which we ask students to learn. Evelyn Lamb lays out some more obscure cousins in a wonderful post:

As Lamb explains, every function in this motley crew once served a useful purpose, back in the days before automated computing. Now, they’re mostly obsolete. So that means we can narrow our attention to the essential few, like sine, cosine, and tangent, right?

Wrong, I say! In the spirit of life imitating The Onion, I propose that all humans memorize the following utterly essential trig functions:


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The Exponential Bait and Switch

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.

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The Smartest Dumb Error in the Great State of Colorado

Entering the town of Gold Hill, Colorado, you encounter one of the most extraordinary posted signs in the entire USA:

The founding year: 1859.

The elevation: 8463 feet.

The population: 118 people.

And at the bottom—oh, the glorious bottom—these three numbers have been added together, yielding a total of 10,440. You can check it yourself: 10,440 is exactly correct. The arithmetic is flawless.

It’s perfectly right… and profoundly wrong. It’s a memorable token of a common mathematical mistake: carrying out an operation without investigating its meaning.

I could easily spin out 1000 words bagging on this poor sign-maker. But I’m not going to. (For one thing, there’s a chance the error was a deliberate joke, and even if it wasn’t, there’s enough ridicule out there for bad math.) Instead, I want to argue the opposite.

This error isn’t brainless, stupid, or contemptible. Rather, in several ways, the Gold Hill error is a uniquely sophisticated and modern one.

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