Astute as those three words are, they miss the basic strangeness of Wiles’ life story. For all his calm, elegance, and precision, the guy is also a unicorn, a sasquatch, a one-of-a-kind creature from the pages of myth. He is, if you will, a walking oxymoron.

He is a celebrity mathematician.

When he proved Fermat’s Last Theorem in 1994, it captured the public imagination in a way that few mathematical breakthroughs ever have. Usually, great mathematicians earn fame in their professional circles, but can walk down streets unnoticed. Wiles was different. He became the subject of bestselling books and primetime documentaries. He became “Sir.”

The difference, of course, was the problem he solved. Fermat’s Last Theorem can be explained to a high school student, but actually proving it took three centuries. It’s like the old board game commercial says: “a minute to learn, a lifetime to master.”

Or in this case, five lifetimes. Maybe more, if Wiles hadn’t come along.

Last autumn, Wiles spoke at the Heidelberg Laureate Forum in Germany—a conference putting young researchers in math and computer science into contact with living legends such as Wiles.

After describing the history of Fermat’s Last Theorem—and emphasizing its mathematical importance over the romantic tale of his work on it—Wiles answered their questions.

In the process, and without naming them as such, he articulated three clear and compelling rules for how—and whether—to tackle a famous problem.

Famously, Wiles hid his work on Fermat’s Last Theorem. He labored alone in an attic office, and covered his silence by discreetly releasing a trickle of research he’d saved up.

But this wasn’t what Wiles originally planned.

“I didn’t actually embark on it in secret,” Wiles says. “I did tell at least one or two people at the beginning.”

Wiles soon regretted that confidence. “It’s like issuing a weather forecast on the BBC. They wanted an update every hour and a half.”

That’s when Wiles went into intellectual hiding. “The secrecy was very much to give myself peace to work on it.”

For top researchers tackling famous problems, it’s the only way. “If you’re working on the Riemann hypothesis,” Wiles says of another famous problem, “there’s no point in telling people. They’ll just hound you.”

At his talk, an audience member asked Wiles about how to get started on a similar problem—such as proving that there are infinitely many perfect numbers.

“To be perfectly honest,” Wiles said, “I would say, ‘Don’t do that one.’”

Wiles’ reasoning relied on a key word: *responsibility*.

“I worked on Fermat[‘s Last Theorem] as a child, because I just loved the problem,” Wiles said. “But when I became a professional mathematician, I thought it was irresponsible.”

Glorious old problems like these are seductive. Prove a famous conjecture, and you might score not only a publication in the *Annals of Mathematics* but a profile in the *New York Times* or a BBC documentary: Wiles-level fame.

But in that seduction lies the danger. You can waste years chasing the pot of gold, and wind up empty-handed.

So why did Wiles return to Fermat’s Last Theorem as an adult? Because other researchers proved it was closely linked to a deep problem in algebraic number theory: the Taniyama-Shimura Conjecture.

“Then,” Wiles said, “I knew it was responsible to work on it. This was a problem that *had* to be solved, in the middle of mainstream mathematics, with lots of structure to it.”

That’s the other key word: *structure*. Deep connections to other ideas of value.

“Pick a problem that really appeals to you,” Wiles advises, “but pick one that has some structure—so even if you don’t succeed, you will prove other things.” Even if you never reach the pot of gold, you want to gather some coins along the way.

Perfect numbers don’t fit the bill. “Don’t be irresponsible and pick something where after two thousand years, there’s still no more structure to it [than when it was first stated].”

Finally, Wiles echoed the wisdom of a popular mathematical canard: *Theorems are proved by believers*.

“It’s a very odd thing in mathematics,” said Wiles, “that if you know something is true, it’s much easier to prove it.” There’s a mystery of human psychology here. “Having the choice between it being ‘true’ and ‘not true,’ you’d think, well, you spend half the time on each. But it doesn’t work that way.”

Researchers might try to prove a statement true for years—and then start looking for counterexamples, and find one quickly. Or they might spend ages searching for counterexamples—then, briefly supposing that it’s true, they’ll happen swiftly upon a proof.

Somehow it’s hard to entertain both thoughts at once.

Wiles, who has learned this lesson better than anyone, puts it succinctly: “You have to really believe.”

**© Heidelberg Laureate Forum Foundation / Flemming – 2016**

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ME: What do you think of these drawings?

MY WIFE: Hey, a cat with a mustache. What’s not to like?

ME: That’s not a mustache. It’s whiskers.

MY WIFE: Okay. I’m not going to tell you what to call your cat’s mustache.

Cats have a symmetry group of order two, because there are two ways to transform a cat while preserving its basic structure: reflect it in a vertical mirror, or leave it alone.

Most cats prefer the latter.

A cat’s activity can be modeled by a delta function. That’s a function whose value is zero everywhere, except at a single point, and yet whose integral is 1. Similarly, the cat is motionless except when it is destroying furniture in the space of a single Planck time.

Note: a delta function is not really a function, just a distribution with good branding.

Are cats more like pure mathematicians, or applied?

Well, like pure researchers, they are aloof from reality. But like applied ones, they benefit off the hard work of others. Best of both worlds, really.

I’m not sure the words “logic” and “cat” belong in the same sentence, except perhaps for sentences about the incongruity of unifying those two words in a single sentence.

And if that sounds too self-referential for you, well, what is logic if not the art of careful self-reference?

There is no theoretical limit on the number of people that a cat can scratch in the span of a minute. Millions, billions… anything is possible. It’s just (thank goodness) very unlikely.

Aw, how cute! The double torus is playing with its toy.

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**On a Small Planet**

**The Two Capes**

**The Layover**

**Invisibility**

**The Enchanted Book**

**In the Waiting Room**

**Flowers for Louis**

**The Self on the Shelf**

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Ten minutes isn’t very long, but it is roughly 297 times their average attention span, and I didn’t want to bore them. So I asked my class of 12-year-olds: What should I talk about?

Here’s how the conversation went:

Then they sort of yelled indiscriminately for a while, which I suppose was my own fault for riling them up. You might as well feed them sugar right before bedtime.

Seeking greater pliability and innocence, I asked my class of 11-year-olds. This is how that went:

Then they spent the afternoon giggling, for which, once again, I can only blame myself.

After this, I realized what I wanted to talk to them about: them. After all, it’s not just me who finds that this age-group ping-pongs between “charmingly ungovernable” and “utterly feral.” Everybody seems to share that feeling.

The world isn’t quite sure what to do with 11-to-13-year-olds.

Here in England, they’re secondary school students, thrown in with the big guys. In June, you’re sharing hallways with 6-year-olds; come September, you’re staring up at passing 17-year-olds. It’s a hell of a transition, and though my school has a wonderful support network for the lil’ guys, it’s not always easy on them.

These 11-to-13-year-olds aren’t prepping for high-stakes tests. They’re not eyeing university admissions. They’re passengers in a school built for a very different clientele.

At the other end of the spectrum, you’ve got schools like the one where my sister teachers: what Americans call “K through 8’s.” Here, my lil’ guys are the big dogs, constituting a primary school’s most senior wing.

Some schools manage this beautifully. But this too is an awkward fit. A 12-year-old needs greater challenges, new horizons, subject specialists. They’re nothing like 6-year-olds, and it’s a strain for a school to serve both needs at once.

A third option is what I (like most Americans) experienced: a “middle school,” exclusively for ages 11 to 13.

In short: quarantine.

When I talk to adults, many name middle school as childhood’s unhappiest stage. I’m not sure the “lock ’em in a room and wait until the diseased age has passed” model is to blame, but it sure doesn’t help. When I taught high school in California, I looked upon my middle school colleagues with admiration and pity. Their days were grueling. The kids’ days were just as tough.

I don’t know quite why this age is such a bumpy ride. But one theory I like comes from psychologist Erik Erikson.

Erikson imagines life, from birth to death, as composed of eight stages. Each is defined by a signature struggle, a characteristic crisis. (As you might guess, this theory is more about “novelistic resonance” than “scientific falsifiability,” but hey, so is life.)

In the first stage, as infants, we ask: *Can I trust the world? *It’s a battle between trust and mistrust.

In the second stage, as toddlers, we ask: *Do I feel at home in this body, in this world?* It’s a battle between a sense of autonomy and a plague of shame and doubt.

In the third stage, as preschoolers, we ask: *Do I feel comfortable taking action*? The conflict plays out between a sense of initiative and a shadow of guilt for taking initiative.

Next comes primary school, and a conflict between “industry” and “inferiority.” At this age, we ask: *Can I do it?* *Can I succeed? Or am I doomed to flounder and fail?*

I see this struggle in vivid technicolor every time one of my students receives a low test grade. They’re not thinking, “Did I study hard enough?” or “Do I understand these topics?” or “Did Mr. Orlin screw me over by forgetting to teach a whole topic?” They’re thinking, “Am I any good at math?”

It’s the signature struggle, playing out in an instant. The stakes are stratospheric.

During secondary school, a new stage of life opens up, and the focus shifts. Now we ask: *Who am I?* *What defines me? What is my role in this world*?

Questions of ability give way to questions of identity.

I see this in my 11-to-13-year-old students, too. They’re starting to ask tough questions, to contrast themselves with others, to seek role models, to rebel. They’re beginning the process of identity creation that makes being a teenager so exhilarating and bizarre.

In short, these guys are tackling two fundamental life questions at once: *Can I do it? *and *Who is this “I” who’s doing it, anyway**?* They’re caught in a moment of transition. It’s painful and confusing and full of possibility. And this helps me to understand the nagging problem of what to do with 11-to-13-year-olds.

It’s a hard age to *teach* because it’s a hard age to *be*.

I said all this, more or less verbatim, to the school’s cohort of 11-to-13-year-olds. They listened patiently, because they’re stars and I love ’em all. And I closed with a story.

In 2013, I was a stubble-chinned California man, living the surfer life, when my wife accepted a three-year job in Birmingham, England. I figured I’d need to do more than sit around feasting on crumpets (especially given that I wasn’t sure what crumpets were), so I started looking for teaching slots.

I was a perfect specimen of American ignorance. I didn’t know an A-Level from an O.W.L., and had no clue what “GCSE” stood for or what dark magic it signified. (Still don’t, frankly.) And when I landed a job, I was surprised to find that I wasn’t just teaching the 16-to-18-year-olds I expected.

I was teaching the little guys, too.

You can imagine my surprise. It is dwarfed only by my surprise, three years later, to look back and find that these little guys have offered the greatest challenges and rewards of my English teaching experience. They’ve made me laugh, scream, apologize for screaming, and smile so wide I forgot that I’d ever screamed to begin with. Thanks to them, I’ve had three years of relentless growth and new experiences. I’ve had to ask *Can I do it?* and *What kind of teacher am I? *They’ve helped me frame the questions and find the answers.

I thanked them onstage, and I’ll say it again here: Thanks, guys. I only hope that your three years can be as wonderful and rewarding as mine have been.

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**Tennis**

**Soccer**

**Golf**

**Baseball**

**Hockey**

**Marathon**

**Tee Ball**

**Cricket**

**Skiing**

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Five-years-ago me? Throw his drink in your face. He’d tell you that rote thinking is the bane of his working days, that deep understanding is the whole point of learning mathematics. He’d tell you: *No black boxes, ever*.

Today-me is less convinced. Don’t working mathematicians, from the ground floor all the way up to Andrew Wiles, sometimes use black boxes? Isn’t it common sense that sometimes you need to use tools that you can’t build for yourself?

I’m still wary of equipping students with black boxes, but these days I’m willing to do it, so long as three conditions are met. I hesitate to share this crude checklist, knowing my colleagues out there in the profession will have wiser ways to frame the tradeoffs. (After all, aren’t checklists too binary, too black-and-white, for an idea as elusive and shaded as “understanding”?)

Nevertheless, my checklist goes something like this:

When I taught my 17-year-olds the product rule for derivatives last year, I didn’t give them a proof. We talked through a few examples, and that was it.

“How do you think we’d prove this?” I asked later.

“Limit definition?” they said.

I nodded, and we left it at that. The proof I know is a clever algebraic trick; satisfying, but not terribly illuminating. I don’t really care whether students know the product rule’s origin story, so long as they know that it *has* an origin story.

By contrast, take my 11- year-old students as they begin secondary school. Many know lots of impressive “maths” (as they adorably say): they can divide fractions by fractions, subtract negatives, and state the n^{th} term of an arithmetic sequence.

And if you ask them to explain why a technique works, they just describe the technique again.

In their view, mathematical methods aren’t rooted in reason, emerging by natural processes of logic. They’re plastic flowers popping out of the pavement like magic. It’s not just that they don’t know why these methods work; they’re fundamentally unaware that “why” and “how” are different things.

To use a black box safely, a student needs to know there’s something they don’t know. If that isn’t happening, then I shun black boxes like I shun black bears.

Some techniques are not that enlightening—but you need them anyway.

I’m thinking of a three-act lesson where students estimate the number of pennies used to build a massive pyramid. Working from first principles, they can mentally dissect the pyramid, breaking it down into layers of various sizes. But once they’ve done that, they still won’t know how to total the number of pennies.

They need a formula: the one for the square pyramidal numbers.

Deriving it would be an impossible chore in the confines of a short lesson, and wouldn’t play to the learning goals. We’re left with two choices: (1) Deny students the formula, thereby forcing them through a long, tedious, repetitive computation, or (2) Supply students with the formula, a handy shortcut they don’t totally understand.

I’m comfortable choosing Door #2. After all, part of being a mathematician is tapping into the wisdom of those who came before.

This year, in an ambitious move, I tried to teach my 12-year-old students about square roots. In particular, I hoped they could learn to flexibly employ the rule √ab = √a√b, to simplify expressions like √300, or √72/√2 or √20 + √45 + √180.

In the immortal words of Rick Perry: Oops.

I pushed them too quickly into technique, and then watched them rehearse a rule they didn’t understand. All struggled; many rage-quit. They came to see square roots like an Old Testament plague. Luckily, there’s a simple solution:

*Don’t make them simplify square roots*.

They have no practical or intellectual need for this technique right now. They need to build numerical and geometric intuition about square roots first. No reason to thrust them into the deep end of this quasi-algebraic pool.

This is a surprisingly common tale in mathematics education. We rush headlong into technique, trying to outrun an imaginary time-monster. So I’m always reminding myself: *Be patient. Build context. Go concrete before you go abstract.*

To recap, I’m comfortable with students using a technique they can’t justify only if all three of these conditions are met:

Now the real question: when are these conditions met?

If you ask me: Almost never. Basically, it occurs when you’re teaching sophisticated students a piece of mathematics not for its own sake, but for its applications. Engineers, psychologists, and environmental scientists don’t necessarily need to trace the derivatives of sin(x) and cos(x) back to the squeeze theorem.

But I know this isn’t how most black boxes get deployed.

More often, it happens when your back is against the wall: Students arrive at your door unprepared for an immovable high-stakes exam. The shortcut to decent scores leads away from understanding. You face two repugnant paths: forsake the students’ learning to preserve their economic opportunities, or vice versa.

Many of us seek a middle way. We try to carry both treasures up the steep mountainside. But all too often, we arrive at the top to find that the learning is gone, vanished from our hands. We look back and see it scattered along the path. Step by step, we let it slip from our fingers, not even realizing.

To the teacher on that lonely mountaintop, I offer neither applause nor condemnation. Just sympathy.

I’ve been there.

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NOTE: These are 100% subjective and 110% definitive.

Historians will look back at this period and ask, “What mass lunacy gripped these people, that so many of them sought pleasure in running long distances?” Their books will have titles like “The 21st-Century Illness: How Marathons Brought Civilization on the Brink” and “26-Mile Masochism: Had They Not Heard of Cars and Bicycles?” and “Running in Giant Meaningless Circles: You Were Right All Along, Ben.” Then they will go play dodgeball, because the future is a better place.

Were I a better man, I would be emptying my upper-right quadrant as fast as I can, while never thinking to touch the bottom-left. But I am not a better man. I am a man whose phone has a Twitter app but no “critically acclaimed fiction” app. I am a sorry man, a broken man, a man who epitomizes his times. But judge me not, lest ye be tweeted about.

My colleague Richard will think less of me for ranking the Central Limit Theorem as sub-average for beauty, but this graph was going to look pretty biased if I stacked everything in the top two quadrants. My motto: The credible blogger must occasionally dis.

Yes, I’m indifferent between giving forgiveness and giving noogies, which creates occasional madcap mix-ups where I respond to an apology by grinding my knuckles into the apologizer’s hair. Also, love is great, but it is strictly dominated by high-fives, a fact that is evident if you replace the word “love” in any Beatles song title with the word “high-five.” (Try it. “She High-Fives You.” “And I High-Five Her.” “High-Five Me Do.” “All My High-Fivin’.” You kinda high-five it, right?)

I’m not saying we have a *bad* national anthem. I’m just saying Itsy Bitsy Spider would be easier to sing, while no worse to hear. Imagine if, before every sporting event, a singer grabbed the microphone and hummed the Star Wars Theme. Or, better yet, what if the whole crowd joined in singing the theme song to Arthur? Picture the swell of patriotism as we say, “Hey! What a wonderful time of day.”

I’m willing to take my lumps here. Geckos are wildly underrated. Guinea pigs are just fat rats that provoke less anxiety. Fish are like wetter and marginally cuter rocks. Monkeys are strictly better than dolphins. And turtles are almost as good as cats. That’s right; come at me, internet.

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