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.

Toddlers have always fascinated me. They’re clumsy little people who laugh spontaneously, nap at random, and feast on Cheerios—basically, my heroes. And I love imagining a complex inner life for them, full of discovery and improvisation. I think they experience a far more voluminous river of thought than can trickle through the narrow spout of their language skills.

I spent an afternoon with Leo and his parents. When we got to the playground, Leo wanted to go down the slide.

Then, Leo wanted to go down the slide.

Then, Leo wanted to go down the slide.

Then, Leo wanted to go down the slide.

Then, Leo wanted to go down the slide.

Watching Leo’s joyful, oblivious repetition of the same activity, over and over and over, I could see why babies get a bad rap as theoreticians and intellectuals. I couldn’t help thinking—as Leo ascended the stairs and descended the slide for the seventeenth consecutive time—that maybe there just isn’t much going on inside a two-year-old’s thoughts.

But then, something changed: Leo wanted to go down the slide.

Okay, fine, no change there. What surprised me wasn’t the down, but the *up*. Instead of the usual steps, Leo sought out a ladder on the opposite side of the structure. From there, he could reach the slide only by crawling through a sort of cage-tunnel, full of wide gaps, through which his whole body could easily fall.

“Oh, lovely,” his dad groaned. “Now you want to do the danger path.”

By Leo’s fourth journey through the cage-tunnel, his father was getting a little worn down from the heightened vigilance of supporting him. So I offered to step in as Leo’s spotter, and began watching more closely.

Every moment brought a tentative new movement. Probing forward with his feet. Shifting his balance slightly. Releasing and re-gripping the bars above. From a distance, he must have looked merely squirmy. But up close, I could see him challenging his kinetic abilities.

It dawned on me that he’d chosen this path specifically because it taxed and excited him. Because he desired a new frontier.

Soon, we came to a particularly tricky step. Leo had exhausted his repertoire of wiggles. Up to this point, he’d paid me no notice, but now—without looking up—he grabbed by hand and pressed it against his chest. He needed help. This was his way of asking for it.

It struck me in that moment: Leo’s repetition on the structure isn’t mindless. It’s a deliberate path to mastery.

Leo is practicing.

No one needs to command or force Leo onto the climbing structure. It’s instinctive. He wakes up every day with four funny limbs attached to a silly little body, and—naturally—he wants to learn how to move the whole apparatus around in a coordinated, effective way. He wants mastery.

Sometimes, that means repetition.

Sometimes, that means trying a “danger path,” something hard and novel.

Sometimes, that means pausing and reflecting.

Sometimes, that means asking for help.

Repetitive practice doesn’t belong in quarantine. It’s not some separate chapter in the book of learning, to be consulted only at one precise spot in the sequence. Instead, practice functions best as part of an integrated and organic whole, enmeshed and woven in with other aspects of learning.

Practice is the best way to hone and solidify a skill. When students want the skill, practice comes naturally.

I ought to be striving for that same ideal in teaching mathematics.

Now, that’s not necessarily easy. Whereas Leo can easily gaze upon the entire play-structure, math is less tangible. Often, students possess only a hazy, uninspected idea of what they’re building towards. They may see their own purposes as obscure, opaque—unlike Leo, who knew exactly what he wanted.

All this helps define my job. I need to help supply a vision of what mathematics is, a sense of the powers they can acquire. I need to paint them a compelling picture of the play-structure.

A goal-motivated human is a powerful, capable thing.

Even if that human naps at random and feasts on Cheerios.

No, not “even if”—*especially* if.

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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.

You may have heard this chestnut: “The hardest thing about getting a Yale degree is getting accepted in the first place.” For me, it rings true. Thousands upon thousands of the rejects from Yale would have thrived there, if they’d just gotten the thick “yes” envelope instead of the thin “no” one. (That includes the five totally amazing kids I interviewed last year, none of whom got accepted.)

Dozens of people have asked me, “Wow, how did you get into Yale?”

Not a single one has ever asked, “Wow, how did you manage Yale coursework?”

With so many uber-qualified students lining up, top colleges don’t—as you might expect—look for the “very best.” They don’t even operate on a single, well-defined notion of what “best” means. Instead, they pick and choose. They go for balance. They’re just trying to fill their campus with a dynamic, diverse cohort of freshmen. Consistency and “fairness”—whatever that would mean—have nothing to do with it.

It’s like making a trail mix. I don’t care whether this particular peanut is more “deserving” than that particular chocolate chip. I’m just choosing high-quality ingredients to strike a nice balance of flavors. Nothing more.

It might not be “random” from the university’s perspective. But it is from the students’. One year favors trumpeters, the next favors bassoonists, and kids have no way of knowing whether their particular skills will be in demand this time around.

All this wouldn’t be particularly troubling, except when coupled with this fact:

Just look at the demands of the Common App. “Write me a confessional essay. Document your leisure activities in meticulous detail. Muse on a philosophical question. Tell me what you love about my school. Give me testimonials from your teachers.”

The application becomes an autobiography, an audit of your whole self: ambitions, achievements, convictions. The process feels customized, personalized, complete. Before they make a decision, Yale insists on peering into your very soul. (Either that, or they’re gathering the data to build your robot doppelgänger.)

I get why they want all that information. But all this data puts a mask of intimacy on what is fundamentally a factory process. No matter how sincere their intentions, the Yale admissions team is beholden to grim statistical reality: 94% of students are getting rejection letters, period.

Being rejected by a university ought to feel like getting swiped left on Tinder. There’s nothing terribly personal about it. They don’t really know you. The university is just looking out for its own interests, and you don’t happen to fit into the picture.

But between everything—campus tours, information sessions, supplemental essays, test scores, transcripts, letters of recommendation, and alumni interviews—the application process becomes a lengthy and weirdly romantic courtship.

Rejection feels less like turning down a first date than getting left at the altar.

Long story short, that’s why I’m not doing Yale alumni interviews anymore. As much as I loved my college education, it drives me crazy to be the face of a process that’s unpredictable, opaque, and (at least 94% of the time) disappointing.

I find myself compelled by the so-crazy-it’s-gotta-be-right proposal of the psychologist Barry Schwartz: run admissions by lottery. Says Schwartz: “Every selective school should establish criteria [for admission]…. Then, the names of all applicants who meet these criteria would be put into a hat and the winners would be drawn at random.”

Before you write Schwartz’s proposal off, remember this. Currently, we’ve got a random process, disguised as a deliberative one.

Why not take off the mask?

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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, cot^{2}(x) + 1 = csc^{2}(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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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 5^{4} or 22^{7}, 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, 9^{1/2}—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 a^{b}a^{c} = a^{b+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.

Why bother? Because suddenly we can make sense of statements like 9^{0}, 9^{1/2}, and 9^{-2}.Any number to the zero must equal one—because our rule says so.

Any number to the ½ must be the number’s square root—because our rule says so.

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 6^{th} 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 a^{b}a^{c} = a^{b+c}.

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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.

Hop in a time machine. Tour the range of human societies. You’ll find that most consider the idea of “mindless” computation paradoxical, nonsensical—a contradiction in terms.

That’s because, for most of human history, computation was really hard.

As recently as 1000 years ago, simple multiplication—the stuff we teach to 9-year-olds now—required expert equipment. Determining a square root was the special provenance of academics and well-trained merchants.

Back then, “simple” math didn’t feel so simple.

Now, of course, we’ve got calculators, iPhones, and—most importantly—the Hindu-Arabic numeral system. This way of writing numbers, familiar and obvious to us today, arrived as a revelation. It’s far easier to compute with our friendly numerals than it was with, say, Roman numerals.

For most of our history as mathematical thinkers, the Gold Hill error would’ve been tremendously unlikely. Computations took too much effort for you to waste time performing a silly one!

Math and science teachers chide students for “forgetting their units.” Omitting units is like leaving a number naked and unclothed. Do you mean 7 feet? 7 meters? 7 miles? Or 7 jars of jelly?

That’s the basic problem with the Gold Hill sign. The three numbers have very different units (years, people, and feet). These can’t be added.

But for many mathematical cultures, “forgetting” the units would have been unimaginable, because numbers (and operations) were inextricably tied to physical representations.

That is: numbers *needed* units.

Take square numbers. Why is multiplying a number by itself known as “squaring”? Simple: it’s equivalent to finding the area of a square (with the original number as a side length).

What about “cubing”? Likewise, that’s just finding the volume of a cube (with the given side length).

For Greek mathematicians, a single number represented a length. The product of two numbers stood for an area. And the product of three numbers represented a volume.

What about the product of four numbers? It’d be nonsensical, impossible. In Euclid’s famous geometry book, he carefully avoids ever multiplying four numbers together. For him, it would have violated common sense, gone against his geometric conceptualization of number.

What’s the point of all this? It’s that our very ability to conceive of numbers without units is a pretty remarkable abstraction. The Greeks would find our notion of numbers, even as practiced by little kids, a bit eerie and ethereal. Their vision of math was beautiful but immensely concrete, closely tied to geometric and visual meanings.

And so to a Greek, the Gold Hill sign would be… well, like Greek.

Most of the math that we teach to our students comes from long, long ago. Integers and fractions have been studied since prehistory. All of our geometry was known to the Greeks. Simple algebra dates back to medieval Arabia.

But one particular branch of mathematics is incredibly new—almost shockingly modern in comparison to the lumbering dinosaurs with which it cohabits our curriculum. That young, upstart branch?

Statistics.

Although it’s often used as a synonym for “number,” the word “statistic” in fact has a specific and little-discussed meaning. It refers to a single number that summarizes a whole sample. For example, if I grab five people off the street, ask them how annoyed they are (on a 1-to-10 scale) at being grabbed by me, and then average those numbers, that average is a statistic.

The thing about statistics is that a lot of numbers go into them. Just as honeybees need acre upon acre of flowers to produce a single jar of honey, statisticians need lots of raw data to produce a simple summary.

But where past societies lacked for data, we’ve got a surplus. Heck, it’s 2015; we’re practically swimming in it. If you clicked on this post through Facebook, Google, or Twitter, then your click is another data point in some massive collection in Silicon Valley. We’re so immersed in numbers—drowning in them, really—that we *need* statistics. We need summaries.

We need totals.

Now, I don’t mean to suggest that finding a total is a new idea. The number at the bottom of the Gold Hill sign is a simple sum. It’s the result of a basic addition operation—a computation as old as time.

But the attitude that produced the Gold Hill error—the practice of summarizing data immediately after presenting it—is a new development in our mathematical culture, a strange necessity created by our glut of information.

So yes, it’s easy to see the Gold Hill sign as an emblem of everything that’s wrong with our mathematical culture. We push students into mindless computations and unnecessary abstraction, and the result, too often, is gibberish.

But if you look deeper, you can see the Gold Hill sign as a bizarre but encouraging artifact of the advances we’ve made in mathematics. We’ve developed (and propagated) technologies for computing with ease. We’ve built (and popularized) an impressively abstract concept of number. And we’ve grown so prolific at gathering and processing data that we practically do it in our sleep.

The Gold Hill sign is still wrong. But it’s wrong for fascinating reasons.

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I submitted the requisite form, giving all of them A’s. My chairman was indignant. “How can they all be A’s?” he asked. “Is this some kind of joke?”

I said, no, it wasn’t a joke, but that the more I got to know each student, the more he seemed to me distinctive. My A was not some attempt to affirm a spurious equality but rather an acknowledgment of the uniqueness of each student. I felt that a student could not be reduced to a number or a test, any more than a patient could. How could I judge students without seeing them in a variety of situations, how they stood on the ungradable qualities of empathy, concern, responsibility, judgment?

Eventually, I was no longer asked to grade my students.

Dr. Sacks is a neurologist. His expertise ranges so far and wide (he has written on autism, Tourette’s, migraines, colorblindness, sign language, musical hallucinations) that the word “specialization” no longer fits.

Now, I’m a teacher, not a doctor. But reading Sacks’ autobiography, I’m struck by how teachers and doctors both feel a crucial tension, confronting the same fundamental choice in how to define our professional selves. Am I a narrow specialist, applying my expertise to address a specific need of the pupil or patient?

Or am I generalist, embracing the full complexity and interconnectedness of the human before me?

Sacks embeds his answer in practically every paragraph. His purpose, always, is the health and flourishing of the human being. This doesn’t mean jettisoning professionalism. Insofar as this is a debate, Sacks refuses to take a “side.” Instead, through his work and writing, he marries the clinical and the human, the scientific and the spiritual, the pragmatic and the poetic. His prose bubbles and eddies with insight and compassion, rich currents of thought cascading in long sentences that layer adjectives like river sediments, each deposit deepening and shifting the fluid organic whole.

Sacks does not choose between being *scientist* and *humanist*. Instead, he weds the roles together, merging them into one.

How does this process begin? For Sacks, it starts with listening to his patients. He lends not only an analytic eye but a sympathetic ear. Sacks grew up with three brothers; two became doctors, but the third, Michael, was troubled by schizophrenia. Over time, Michael found he could confide only in his younger brother Oliver:

[Michael] had begun to think of… the entire medical profession as determined to devalue or “medicalize” everything he thought and did, especially if it had any hint of mysticism, for they would see it as an intimation of psychosis. But I was still his little brother, just twelve years old, not yet a medicalizer, and able to listen sensitively and sympathetically to anything he said, even if I could not fully understand it.

Much of neuroscience aims to reduce our minds to mechanisms. We seek the brain region responsible for X, the neurotransmitter that underlies Y, the biological mechanism for Z. (Never mind if X, Y, and Z happen to be our most precious expressions of self.) Shining in this way, the light of neurology is bright and deadly. It explains and sterilizes us, turns us into labeled anatomical diagrams.

But Sacks resists this “medicalization.” People are people: gloriously unique, irreducibly complex. They are not bundles of symptoms. He gazes into the deepest, most mystical parts of the human psyche, and—far from extinguishing the living mystery of experience, of selfhood—Sacks’ science embraces and nourishes it.

So it must be with teachers. Recent decades have given us a wealth (you might say a clutter) of tools and checklists. Our shelves overflow with tables of state standards, banks of test items, standardized assessments. Like doctors, we’ve got a plethora of diagnostics at our fingertips.

But we must not “medicalize” our students. These tools begin a conversation; they do not end it. To learn any subject—math, history, Spanish, even neurology—is to undertake a fabulous and singular journey, to exercise your humanity at its highest level.

That’s not on the checklists; rather, it’s what the checklists are there for.

To be sure, specialization has its benefits. As a math teacher, my job is quite specifically to help my students learn mathematics, not to cook their breakfast or to counsel them on romance. (Neither my cooking nor my counsel would do them any favors.) For Sacks, this goes double: a neurologist is nothing if not an exquisitely trained specialist.

But when Sacks began working at a headache clinic, he found that his patients’ needs forced him outside of this mindset:

[Seeing patients] gave me a feeling of what seemed wrong with American medicine, that it consisted more and more of specialists. There were fewer and fewer primary care physicians, the base of the pyramid…. I found myself feeling not like a super-specialist in migraine but like the general practitioner these patients should have seen to begin with. I felt it my business, my responsibility, to enquire about every aspect of their lives.

Later, of his patients at Beth Abraham hospital, he writes:

I lived next door to the hospital and sometimes spent twelve or fifteen hours a day with them. They were welcome to visit me; some of the more active ones would come over to my place for a cup of cocoa on Sunday mornings.

This might seem to threaten the idea of clinical distance. In patient/doctor and pupil/teacher relationships, we are rightly afraid of impropriety. So we grow to fear attachment, enmeshment—any sort of excess sympathy—as a sign that lines are being crossed, balances disturbed. Instead, we stay cool. Removed.

“Professional.”

But to Sacks, propriety is not at odds with intimacy. In fact, the whole purpose of professionalism is to create space for a clear and purposeful connectedness. He writes of his own psychoanalyst:

I still see Dr. Shengold twice a week, as I have been doing for almost fifty years. We maintain the proprieties—he is always “Dr. Shengold” and I am always “Dr. Sacks”—but it is because the proprieties are there that there can be such freedom of communication. And this is something I also feel with my own patients. They can tell me things, and I can ask things, which would be impermissible in ordinary social intercourse.

Reading Sacks, I’m always moved by the extraordinary empathy he lends to every patient. In those crippled by illness or disorder, he finds hidden strengths, adaptive resilience. In those dismissed by others—as psychologically diminished, intellectually null, or beyond treatment—he finds untold complexity and richness. He writes:

I find every patient I see, everywhere, vividly alive, interesting and rewarding; I have never seen a patient who didn’t teach me something new, or stir in me new feelings and new trains of thought.

I wish, desperately, that I could say the same of every student I’ve taught. But the day is only so long; owing sympathy and attention to all of my students, I inevitably shortchange most of them. They each face unique puzzles and struggles, carry with them distinctive strengths and personal motives—but hell if I manage to discover half of these stories before they graduate and move on.

I’ll never understand my students as deeply as Sacks understands his patients. Still, it’s an ideal I can strive towards.

As it is, my “knowledge” of my students is often statistical. I know how they’ve performed on homework, quizzes, and tests. I can predict how they’ll fare on the high-stakes exams that will circumscribe their opportunities for the future. Our educational system is increasingly a statistical one.

In medicine, statistics are also ascendant, and in a footnote (it’s always a footnote, with Sacks—the notes for *Awakenings* originally ran to twice the length of the text itself) Sacks relays one memorable anecdote on the topic:

“How many patients do you have on L-dopa?” he asked me.

“Three, sir,” I replied eagerly.

“Gee, Oliver,” Labe said, “I have three hundred patients on L-dopa.”

“Yes, but I learn a hundred times as much about each patient as you do,” I replied, stung by his sarcasm.

Sacks doesn’t deny the value of statistics. (“All sorts of generalizations are made possible by dealing with populations,” he remarks.) But he insists that they are not enough. Alongside the cold and towering arsenal of data, he supplies an utterly necessary complement: a renewed humanism, a depth of caring and compassion, a creative receptiveness to the variety of human strengths and experiences. He doesn’t rebel against the statistical regime; he simply denies that it can substitute for older and more personal forms of wisdom.

That’s the balance I seek in my teaching. The word *essential* springs to mind, in its two distinct senses: the newer one of “very important,” and the older one of “capturing the essence.” Test scores, statistics, student data—these things are very important, but they are not the essence.

Tests are only stethoscopes. They are not the heartbeat itself.

Sacks is dying of cancer. In a February essay, he wrote, “It is up to me now to choose how to live out the months that remain to me.” When he passes, the world will lose an extraordinary being, a form of life every bit as distinctive and wonderful as the patients to whom he lent his sympathy, his Sunday afternoons, and—perhaps most lasting of all—his words.

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Dear Benedict Carey,

I very much enjoyed your book *How We Learn*. It blends the vast and varied harvest of research on learning into something light, flavorful, and nutritious. A psych-berry smoothie, if you will. It’s a lovely summer read for a math teacher like me.

But I’m also a blogger—which is to say, a cave-dwelling troll, forever grumping and griping. And so I’d like to dive into your chapter on practice (“Being Mixed Up: Interleaving as an Aid to Comprehension”). In it, you purport to remain impartial in “the math wars,” but it’s my view that you come down distinctly on one side.

It’s towards the end of the chapter that you hit the culture war in mathematics. On the one hand (you explain), there are top-down **progressives**, who urge children “to think independently rather than practice procedures by rote.” On the other, you’ve got bottom-up **conservatives**, who put their faith in “the old ways, in using drills as building blocks.” All fair enough.

Then, however, you land on a rockier claim:

This clash over math was (and is) about philosophies, and in math of all subjects it is results that matter, not theories.

Maybe so. But it’s far from obvious *which* results matter. That depends on what philosophy you subscribe to.

For example, one teacher might care a lot about standard textbook questions like these:

Another might scowl and spit upon those, preferring more open-ended questions:

Others might embrace all these questions. Or despise them all.

The disagreement isn’t just about the most effective path to excellence. It’s about what “excellence” even means.

At one extreme, you can view math as a fixed and specific body of knowledge, a pre-existing library of tools, techniques, and numerical skills. Or, at the other extreme, you can view mathematics as a frame of mind, more about habits of thought and broad problem-solving approaches. Like most teachers, I fall somewhere in the complicated middle. I have my own views about which tasks are meaningful, and which are rubbish—which are math, and which are mush.

In the end, it’s silly to say that “results” matter and “theories” don’t. Our choice of the former is inextricable from the latter.

Next, you dive into a study (promising that it consists of “real math” and implying that it ought to transcend the debate between mathematical progressives and conservatives). In the experiment, kids learn rules for computing the numbers of faces, edges, corners, and angles that a prism has, given the number of sides in its base:

Unfortunately, this is a task that many progressives would reject outright.

Here, students must recognize a code word (*face*, *edge*, *corner*, or *angle*) and match it with an arithmetic operation (*+ 2*, *x 3*, *x 2*, or *x 6*). To me, this represents a narrow and overemphasized aspect of mathematics: signal-triggered computation. You characterize it as high-level (“We are not only discriminating between the locks to be cracked; we are connecting each lock with the right key”) but I suspect I’m not alone in considering it rather drab and rote, not to mention irrelevant to the primary purpose of mathematics education.

What’s the alternative?

Well, first off, mathematics is about framing questions. So what is this “prism” thing we’re discussing—not to mention “faces” and “edges”? What definitions are we using? What properties are we focusing on? How are these 3D objects similar to (and different from) 2D objects? Are there borderline cases that are hard to categorize? For example, which of the following shapes would count as “prisms”? (And are there any that *aren’t* prisms, but still follow the rules?)

Now, as for those rules: sure, it’s nice to use them, but can we explain *why* they’re true? Could we have uncovered them for ourselves? Can we test the boundaries of their applicability, rooting out exceptions, or proving to our satisfaction that no exceptions exist?

These are the mathematical activities I care most about.

Framing questions.

Reasoning.

Contrasting.

Extending.

Explaining.

Understanding.

I don’t mean to hate on the study. Yes, I find the specific task pretty silly, but I buy the essential conclusion that interleaved practice (where you mix all four types of questions together) is more effective than blocked practice (where you practice each type in isolation). Insofar as my teaching is conservative and skill-focused (which it sometimes is—certain skills, I believe, demand automaticity), that’s useful guidance.

But I worry about your presenting this as a consensus vision of “real” math. Many of us believe that math is about far more than memorizing and applying procedures, just as we believe that a song consists of more than its bass line. It’s a shame to see someone trying to transcend the debate over the aims of the math curriculum, but then focusing entirely on how to accomplish the aims championed by just one “side.” It’s if someone said, “I won’t take a position on whether aliens exist. Now, here’s a method for contacting extraterrestrials that’s 39% more effective.”

I should be clear: This matter occupies only a few pages of what is, to be clear, a fun and practical book, from the story of Winston Churchill failing his classics exam to your delicious images of the creative process (“For me, new thoughts seem to float to the surface only when fully cooked, one or two at a time, like dumplings in a simmering pot”). So I thank you for the enjoyable book you’ve written.

In any case, I hope I’ve been able to offer a more detailed (if not entirely novel) perspective from the muddy trenches of these ongoing “math wars.”

Best,

Ben Orlin

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(see posts 1 and 2)

Somehow, I suspect I wouldn’t survive long on the frontier.

Drop me in the American West, circa 1850, and I fear my math-blogging and bad-drawing skills might not carry me far. I need indoor plumbing. I need the rule of law. I need chain coffee shops. I’m not cut out for the frontier.

And yet the frontier is exactly where I found myself the other day, when I came across this formula in the wonderful *Penguin Book of Curious and Interesting Numbers*, by David Wells:

I decided to play around with this product a bit. After all, what are products for, if not playing around?

(Go ahead and play with your Apple products. I’ll play with my infinite ones. We’ll see who has more fun.)

I felt like there should be an easier way to write this expression, exploiting the repetition of factors, so I gave it a shot, and created this:

Then my brain exploded and the universe dissolved around me, because I had just punched logic in the face, and it had punched me back.

The left side of that equation is π/2. It’s roughly 1.57.

The right side of that equation, however, is a product of many numbers—*all of them below 1.*

What happens when you multiply two numbers smaller than 1? You get *another* number smaller than 1.

How the heck could that equal 1.57?

Somehow, law had broken down. I was stranded on the frontier, with no cavalry or sheriff to come save me. This was a dispute I’d have to resolve myself, by hook, by crook, or by showdown at high noon.

At this point, I faced three grim possibilities:

- The cool book where I found this formula was wrong.
- Logic was wrong.
- I was wrong.

Pitting my own abilities against (1) those of David Wells and (2) the tensile strength of the fabric of the cosmos, it became pretty clear where the error lay. So I set out to find where I’d gone wrong.

Going back to the original equation, I came up with an alternative way to rewrite it:

This one was clearly greater than 1. (After all, it was an infinitely long product, with each of its factors greater than 1.) Checking a few terms in Excel verified that it was approaching π/2.

Then I found another way to rewrite the original:

This one starts at 2. Then it gets smaller and smaller, by tinier and tinier adjustments. Once again, it seemed to approach π/2.

So what had I done wrong in my original approach?

Soon enough I realized it. I’d made a mistake that seemed perfectly innocuous at the time but had, in fact, contained the seed of my own annihilation. I’d made a move that doesn’t matter with *finite* products, but which is forbidden with *infinite* products.

I’d forgotten to multiply by 1.

It perhaps goes without saying, but when you’re working with infinities, you need to be careful. Notation that’s usually harmless can explode like dynamite, making rubble of everything.

Infinity is the frontier. Law breaks down.

Back home, in the realm of finite products, multiplying by 1 makes no difference. 3 x 2 is the same as 3 x 2 x 1 (or as 1 x 3 x 2 x 1 x 1, for that matter). Finite products—that is, lists of factors that eventually end—can be rearranged to your heart’s content.

Not so with infinite products. Just look what else I can do, if I allow myself to play around with 1’s:

Throwing in extra 1’s allows me to rearrange the numerators and denominators virtually however I please. It seems I can make this product as large or as small as I want!

I should also confess that steps like the second one above – where I take a long product of fractions, and turn it into a single fraction with a long product on top and another long product on bottom – are a little dubious in this setting. With finite products? Fine. With infinite ones? Not so much.

In short: the frontier is a strange place.

I’m working here with products, but the same dangers apply with sums. Take this one, called the *alternating harmonic series*:

This sequence starts at 1.

Then it jumps backwards to ½.

Then it jumps forwards to 5/6.

Then it jumps backwards to 7/12.

Then it jumps forwards to 47/60.

As you go, the sum keeps jumping forwards and backwards, forwards and backwards… but by smaller and smaller steps. Looking carefully, you can see that it’s hovering around a certain destination—a place it will never in our lifetimes reach, but which it *approaches*.

That point is smaller than 1.

It’s bigger than ½.

It’s smaller than 5/6.

It’s bigger than 7/12.

In fact, it turns out to be roughly 0.693, or more precisely, ln(2).

So far, so good. But what if I rearrange the terms, like this?

We start at 1.

Then we add something positive.

Then we add something *else* positive.

And so on.

Clearly, the answer has to be larger than 1—which 0.693 is decidedly not. So I’ve changed the result sum, simply by rearranging the terms.

That’s not supposed to happen with addition! 3 + 4 is the same as 4 + 3. Making your family members stand in a different order shouldn’t make your family bigger, smaller, smarter, or taller—it’s the same family, isn’t it?

But with infinite things to add, the order in which you arrange them turns out to matter a great deal. In fact, this sequence can be rearranged to form literally any number you like.

Why do we explore this mathematical frontier? What drives us to abandon the safe coastal cities that we’ve always known, and throw ourselves into the unknown interior of the mathematical continent? Aren’t we afraid? Mightn’t we get eaten by bears, swallowed by rapids, or tangled in paradoxes? Why do we risk it?

Simple: because humans are explorers.

It’s as true of mathematicians as it is of Lewis and Clarke. We want to chart new landscapes, to find fresh challenges, to go where our old ways don’t necessarily help or hold true. We enjoy the thrill of surviving on our wits alone.

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