Math with Bad Drawings

The State of Being Stuck

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Last year, I got the high school math teacher’s version of a wish on a magic lamp: a chance to ask a question of the world’s most famous mathematician.

Andrew Wiles gained his fame by solving a nearly 400-year-old problem: Fermat’s Last Theorem. The same puzzle had captivated Wiles as a child and inspired him to pursue mathematics. His solution touched off a mathematical craze in a culture where “mathematical craze” is an oxymoron. Wiles found himself the subject of books, radio programs, TV documentaries—the biggest mathematical celebrity of the last half-century.

And so, having lucked into attending a press conference at the Heidelberg Laureate Forum in Germany, where Wiles was an honored guest, I asked him:

The essence of Wiles’ answer can be boiled down to just six words: “Accepting the state of being stuck.”

For Wiles, this is more than just a vague moral, an offhand suggestion. It’s the essence of his work. It’s an experience at once excruciating, joyful, and utterly unavoidable. And it’s something desperately misunderstood by the public.

“Accepting the state of being stuck”: that’s the keystone in the archway of mathematics. Without it, we’re left with nothing but a pile of fallen bricks.

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Wiles began his answer, like any good mathematician, with a premise everyone can accept: “Many people have been put off mathematics,” he said. “They’ve had some adverse experience.”

It’s hard to argue with that.

“But what you find with children,” he continued, “is that they really enjoy it.”

In my experience, it’s true. Kids love games, puzzles, learning to count, playing with shapes, discovering patterns—in short, they love math. So how does Wiles account for our alienation from mathematics, our loss of innocence?

“What you have to handle when you start doing mathematics as an older child or as an adult is accepting the state of being stuck,” Wiles said. “People don’t get used to that. They find it very stressful.”

He used another word, too: “afraid.” “Even people who are very good at mathematics sometimes find this hard to get used to. They feel they’re failing.”

But being stuck, Wiles said, isn’t failure. “It’s part of the process. It’s not something to be frightened of.”

Catch me and my teacher colleagues any afternoon, and—if you can get past the “sine” puns and fraction jokes—you’ll likely find us griping about precisely this phenomenon. Our students lack persistence. Give them a recipe, and they settle into monotonous productivity; give them an open-ended puzzle, and they panic.

Students want the Method, the panacea, the answer key. Accustomed to automaticity, they can’t accept being stuck.

Wiles recognizes this fear, and knows that it’s misplaced. “For people who carry on,” he said, “it’s really an enjoyable experience. It’s exciting.”

Wiles explained the process of research mathematics like this: “You absorb everything about the problem. You think about it a great deal—all the techniques that are used for these things. [But] usually, it needs something else.” Few problems worth your attention will yield under the standard attacks.

“So,” he said, “you get stuck.”

“Then you have to stop,” Wiles said. “Let your mind relax a bit…. Your subconscious is making connections. And you start again—the next afternoon, the next day, the next week.”

Patience, perseverance, acceptance—this is what defines a mathematician.

“What I fight against most,” said Wiles, naming an unlikely enemy, “is the kind of message put out by—for example—the film Good Will Hunting.”

When it comes to math, Wiles said, people tend to believe “that there is something you’re born with, and either you have it or you don’t. But that’s not really the experience of mathematicians. We all find it difficult. It’s not that we’re any different from someone who struggles with maths problems in third grade…. We’re just prepared to handle that struggle on a much larger scale. We’ve built up resistance to those setbacks.”

Of course, Wiles isn’t the first to name perseverance as the key to mathematical progress. Others have analyzed the same challenge—albeit through different conceptual lenses.

One prevailing framework is grit. Under this approach, perseverance is a partly a matter of personality, of exhibiting the right characteristics: tenacity, determination, a sort of healthy native stubbornness. When the going gets tough, grit-less kids bail, whereas gritty kids keep working—and thus prosper.

But recently, the currency of “grit” has fallen among teachers. It’s not that the idea lacks psychological validity. It’s more the weight of its educational connotations. Grit has become an excuse to romanticize poverty as “character-building.” It has devolved into a vague catch-all at best, and at its paradoxical worst, a reason to write kids off as lost causes.

These days, the educational conversation revolves instead around Carol Dweck’s grit-related concept of mindsets.

Some people exhibit a fixed mindset. They believe that one’s intelligence and abilities are unchanging, stable traits. Success, to them, is not about effort; it’s about raw ability. To struggle is to reveal your intellectual shortcomings. They can accept the state of being stuck only insofar as they accept the state of being visibly and irrefutably stupid—which is to say, not very far.

By contrast, those with a growth mindset believe that effort fuels progress. The harder you work, the more you’ll learn. To be stuck is a transient state, which you overcome with patience and persistence.

Wiles is no educational theorist, of course, but I find that he offers a resonant and compelling third path. For him, perseverance is neither about personality (as with grit) nor belief (as with mindset).

Rather, it’s about emotion.

Fears and anxieties come to us all. You can be a nimble mathematician, a model of grit, and a fervent believer in the human potential for growth—but still, getting stuck on a math problem may leave you deflated and disheartened.

Wiles knows that the mathematician’s battle is emotional as much as intellectual. You need to quiet your fear, harness your joy, and cope effectively with the doubt we all feel when stuck on a problem.

Perhaps it’s only a folk psychology of perseverance.  But I’m drawn by its potential to explain how students behave—and to motivate them to strive for more.

For example, take Wiles’ musings on the value of forgetfulness. “I think it’s bad to have too good a memory if you want to be a mathematician,” Wiles said. “You need to forget the way you approached [the problem] the previous time.”

It goes like this. You try one strategy on a problem. It fails. You retreat, dispirited. Later, having forgotten your bitter defeat, you try the same strategy again. Perhaps the process repeats. But eventually—again, thanks to your forgetfulness—you commit a slight error, a tiny deviation from the path you’ve tried several times. And suddenly, you succeed.

Wiles has a nifty analogy for this: it’s like a chance mutation in a strand of DNA that yields surprising evolutionary success.

“If you remember all the false, failed attempts before,” said Wiles, “you wouldn’t try. But because I have a slightly bad memory, I’ll try essentially the same thing again, and then I’ll realize I was just missing this one little thing.”

Wiles’ forgetfulness is a shield against discouragement. It neutralizes the emotions that would push him away from productive work.

Of course, immunity to fear isn’t enough. You need a positive incentive, something to strive for. And here, Wiles understands the delicate emotions of discovery better than anyone. He knows the immense release, the inner fireworks, of solving a problem at last. His problem, after all, took seven years of daily grind. Centuries, if you count the generations of mathematicians who tried and failed before.

“You find this thing,” Wiles said. “Suddenly you see the beauty of this landscape.” Before, “when it’s still some kind of conjecture, it seems really far away.” But now, with a solution in hand, “it’s like your eyes are open.”

For Wiles, doing mathematics is not merely the flexing of an intellectual muscle. It is a long and harrowing journey, so rich and involving that it becomes tactile, sensory, literal.

Listening to Wiles, you feel this. Beneath his gentle poise, you can sense the ten-year-old boy, pouring hours into Fermat’s Last Theorem, undeterred by the centuries of failure that have come before, unafraid of the decades of work ahead.

If you hold one mental image of Wiles, he wants it to be this: not the triumphant scholar with the medal around his neck, but the child learning to glory in the state of being stuck.

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