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.

***

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.

I am a stuck mathematician myself right now – I really needed to read this. Thank you 🙂

This was a timely reminder. I am currently stuck with one of those “It can be shown that” sentences you find in advanced math texts, and the cynic in me is quick to say “It’s one thing to be stuck with a famous problem that no one knows the solution to, quite another to be stuck on something the author thinks is trivial enough to skip completely” but I’ll try to keep the last line from this post in mind and keep at it. Thank you for this! 🙂

Careful here: “it can be shown that” doesn’t necessarily mean it’s easy! It just means the author doesn’t think it’s worth it providing a proof. As a mathematician myself, I would use these words if I thought the effort of explaining the proof is greater than the benefits of the reader’s seeing it. It is not the same as “one sees easily” or “by a routine argument” — which, by the way, sometimes is just lazy writing (as in “I think it’s true but I couldn’t be bother writing down the details, so you do it”).

Fantastic article. Comes to me at a great time. Thank you once more for your wisdom and humor, Ben!

What a beautiful commentary on math and life. Andrew Wiles sounds like an extraordinary human being, a philosopher as well as mathematician. Thanks for sharing.

That is so true. Most people can’t deal with being stuck – I never knew how to motivate my classmates when they get stuck in class. Most could not keep working for more than 5 minutes; they would then immediately give up.

Many have told me (e.g. concerning the final exam) “If my first approach works, then I could possibly succeed at that problem. However, if it does not work, then I cannot possibly solve it.”

I believe though that the main problem is that mostly algorithmic procedure are taught in high school math.

If you want to find extreme points, then it’s always computing derivatives. If you want to find the roots of a quadratic equation, then it’s always the same formula. If you want to solve a system of equations, then it’s always the same procedure.

You just rarely get to be creative in high school math classes. How should the average student learn to deal with being stuck?

i enjoyed the article, thanks for making it

I have to take (slight) exception with Wiles insofar as what he describes applies at the K-12 level (more or less). When giving students non-routine problems over the last quarter century, I’ve noticed that weaker students (not in any ultimate sense) rarely keep track of what they’ve tried, and hence will, in fact, try the same approaches multiple times until they just give up (some give up after one try if that, but I’m looking at perhaps the next level up from those). The notion of treating mathematical problems as scientific experiments (or experiments of invention – I’m thinking of Edison and the light-bulb filament) is utterly foreign to them, so my recommendation that they do things systematically and keep notes of what they do in a lab book strikes them as bizarre or at least as too much work.

So what Wiles describes as useful forgetting is pretty much the antithesis of what I want these students to do. He’s likely thinking from the perspective of someone with good intuitions who might make what Shimura describes in talking about his late colleague Taniyama as a talent for making good mistakes. These sorts of students don’t have well-developed intuition in mathematics; quite the contrary for the most part. If you start with weak intuitions, randomly-chosen, path-of-least-resistance strategies, and keep no record of what you’ve tried (let alone why you thought it would work, why it appeared not to work, where things went south, etc.), it’s not likely to be mutating DNA with “good mistakes” leading to good outcomes. That’s a professional mathematician of the caliber of a Wiles or Taniyama, not some K-12 kid whose first impulse is, “I don’t know how to do this.”

Michael, my strategy is tell the students to work through each thing they know how to do, trying that.

In Geometry, for example, have they looked for alternate, corresponding etc. Have they tried angles subtended from the same arc, angles to centre etc. In Trig, have you tried Pythagoras yet, have you seen if there is some geometry element first etc.

That at least gets them to work through options rather than just staring blankly hoping something will happen.

Often I tell them to just do something, anything (“blunder around” as I put it). Too often then won’t start because they don’t know where it leads, whereas just starting and doing the obvious can work wonders.

The likes of Wiles forget that they have internalised the rules, so that they actually think to work all possible alternatives. We need to explicitly teach that as a stragegy for a lot of our students.

Let me add to the above that for such students, it helps for them to have successes at tasks they can recognize as meaningful challenges for them (and in the beginning, that might be pretty undaunting from a teacher perspective. But that’s perfectly okay with me).

Great post.

A teacher should be qualified to know how to get a student ‘unstuck’ but that shouldn’t be the definition of teacher’s job; rather it’s to manage and increase a student’s tolerance for being stuck, learning when and how to change the level.

Absolutely! So many times teachers have gotten me stuck and had a very hard time getting me unstuck. It never ends well.

Teaching is like math: blank look, blank look, blank look, eureka! And the best line in the world, ‘I got this, Mrs.Maxcy!’ Perhaps it is that we want the solved in 30 minutes plot line, and math is more like the never-ending plot line. (Downton Abbey, Walking Dead, Game of Thrones?) I love the idea of getting used to being stuck!

I was stuck at my job. So I quit. And today, I restarted a new engagement with the same company – exactly at the point I got stuck. I wasn’t sure if it was the right thing to do – and I know this doesn’t have anything to do with mathematics or magicians – but this post gives me hope that my decision to restart at exactly the point that made me give up may not necessarily be disastrous. Thank you!

As a high school and university math teacher, I constructed tests and exams to reflect the level and type of questions my students and I worked through together in class and those that were assigned for homework. I avoided the “challenge questions” that went beyond those variants. Now “challenge” is subjective. Why? At one extreme, some would claim changing sin to cos or a cubic to a quartic is spoon feeding. At the other extreme, some would lament, “I had never seen anything like that before and was completely stuck!” Over the years, a few questions have prompted both responses!

Even so, here is my take based on my, ahem, pretty good test and exam performance in math elementary through graduate courses and thirty five years of, ahem, successful teaching. I do not (well, retired now so did not) put questions on tests and exams that require, “INGENUITY ON DEMAND.” I have no idea what mental processes are percolating so that the problem that stumps me one moment becomes tractable in an astonishing insightful flash while doing something else hours or days later.

I know what my learning expectations are for my students. I try to communicate them, we work together to master them, and I test them on those. The very nature of the questions (including theorems/proofs and constructing examples and counter-examples) settles somewhat into easy, medium, and difficult categories. Because some of those problems are inherently difficult, they become the discriminators that inform the students as to the level of their mastery. In my opinion, the place for those ingenuity on demand questions is on an assignment, ideally unique for each student or student group. And that, in high school and upper year classes in university, was, as a student, where I found them, and as a teacher, where I put them. In first year university classes, with classes of several hundred, well…

“Stuckness shouldn’t be avoided. It’s the psychic predecessor of all real understanding. An egoless acceptance of stuckness is a key to an understanding of all Quality.”

— Robert M. Pirsig in Zen and the Art of Motorcycle Maintenance

Pirsig rocks and that’s a very useful quotation. Just remember that kids in math class are rarely egoless. Schooling makes that state hard to come by.

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Being stuck seems to come in different forms. Some are clearly positive. Are they all positive? At least in my experience, the more you know (and the more confident you are) the easier it is to be happy about being stuck.

The feeling of being stuck is an uncomfortable feeling. But, confronting the stuckiness, banging your head against it, having the “ah-ha” and coming out the other side is quite satisfying.

My sense is that students who are good at math are those who can tolerate being stuck for more than a few minutes. Most give up far too soon.

Wiles showed an abnormal ability to deal with being stuck, even by the standards of the best mathematicians. He was able to continue to work on the same problem for years. Sometimes he took a pause to working on other projects, later returning. But, he spent large portions of his life confronting the same problem, getting nowhere.

I find myself in the same boat of being stuck alot, getting tired of the math is a class by class thing. Only are you pushed to go on with solving the next and the next and the one after that. Math is for many to be a unicorn among studies, always searching for things that were never truly real to them. The gap starts after kids get to more complicated equations such as using pi to measure something. It becomes this drag that not many are willing to undergo: asking questions like ” When will I use this in real life?” or “Why do I have to learn this even if I don’t want to be a *insert job title here*?”

Being a software developer is also about continually trying different approaches until you get it right.

“Success Is Going from Failure to Failure Without Losing Your Enthusiasm”. Churchill

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Hi

This is the best essay I have read in a long time— and a public service. I am also stuck (again) and I thank you.

I had tears in my eyes when I finished reading this. For a long time in school I thought I was stupid and bad at math because I was slow and didn’t learn concepts right away like my classmates. I asked a lot of questions. It took a very long time and the right mix of teachers (surprisingly, not math teachers, but math education professors in college) to help me develop “grit,” and to understand that that is what makes me a good mathematician.

I am a high school math teacher now and I strive to show my students that mistakes are valuable and that it’s okay to be stuck. I am definitely showing this to my students. Thank you!

“all good mathematics involves sitting alone in a room, being confused almost all the time.”

quoted from

http://mark.reid.name/blog/fisher-information-and-log-likelihood.html

hello, I am a visual artist who occasionally looks at math. I was both scared and scarred by my primary education in this subject until a wonderful (female) teacher in college helped me to see that math has beauty too. reading this article reminds me that creating/solving requires both love (what less emotional people call interest) and courage (the ability to go ahead even though you don’t know what the next step will produce). I’m so glad to read about a mathematician who values the same approach and is not afraid to admit to ‘noodling’ when they’re at at sticking point! thanks for this very interesting report.

I like math because it’s a language used to describe things. I didn’t like math in school because it was rushed! Hurry and learn, hurry and finish the test, hurry! hurry! hurry! test! test! test! No wonder many dislike math. It’s like introducing someone to Filet Mignon as the best steak they’ll eat but they have to cut it up into large pieces and swallow them whole as quickly as possible.

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When students feel pressure to achieve a certain grade point average to advance to the university program of their choice, they feel incredible pressure to get “unstuck” before they are tested on math concepts. Testing emphasizes getting the right answer and does not reward the persistent student who hasn’t figured it out yet; those students rarely get the opportunity to become research mathematicians. Is it any wonder students end up fearing being stuck and having no patience for it? What is the solution for that?

This is the truthiest truth I’ve ever read. How many times have I been stuck only to set it aside for awhile while I go run or read or cook or work on another problem? And this is a great argument against the timed assessment which precludes any student from getting stuck.

Hi Ben,

This is the first article of yours that I have read. I will say that you have a new fan, and I look forward to going back in your archives to learn more from you.

I liked this article very much because I understand what it means to be stuck, at times. In fact, I recently tried to explain the feelings I experienced in a recent competition. I was completely stuck in neutral before making tiny, incremental progress that built up to a surprising ending. Here is the article: http://wp.me/pBJNE-6Br.

Thanks,

Ken

I like this article because now I know the real meaning of being stuck as a mathematician. I was stuck doing a particular assignment for days. l could not think right because the assignment will be due the next class. luckily for me I went ahead and asked questions from a professional colleague who knows more about mathematics.So l think being stuck in mathematics is a normal thing all you need to do is to practice more and do not stop asking questions about what you do not know

I think of being stuck as just not knowing. I’ve never thought of being stuck as being this impossible thing. Not to say I have never freaked out over a math problem that was due the following day. No, I see it as the question I haven’t asked yet. And I make sure to ask many of them. I only feel stuck because I don’t know, and when I don’t know, I have fear or worry; only when you are certain does a problem become solvable.

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What the math-guy, Wiles, is describing is precisely Carol Dweck’s “growth mindset”. So there is that.

And the thing about Dweck is that she has studied 1000 guys like Wiles to arrive at her conclusions. A non-specialist (a mathematician?) proposing a whole new theory of learning based on an obviously biased interpretation of one guy’s very brief testimony is not a good method for arriving at a good explanation.

Still it was a good try and I can see that you were trying to communicate what is of value to you in the process of doing maths. And clearly it resonated with your readers, which is also important. However, you might want to look more closely at how what Wiles says is very closely aligned to what Dweck says the growth mindset is like.

Definitely! The post mentions Dweck. Her research is certainly the place to turn if you want an actual theory, grounded in actual empirical work. I think Wiles’ thoughts make for a nice personal exploration of the issues, though

(Also, my sense of Dweck’s work is that it places more of an emphasis on belief, putting the emotional aspects that WIles focuses on in a more secondary role.)

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Reblogged this on Singapore Maths Tuition and commented:

Nice story (with drawings) on Andrew Wiles and the Fermat’s Last Theorem.

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This is what I tell my undergraduate and graduate students in mathematics. I remind them of the movie “This is Spinal Tap”, where the band claimed that it was the loudest rock group, because their amps go up to “11” rather than “10”. I then tell them that in mathematics, the degree of difficulty goes up to aleph-null. If you are not stuck on your main problems, then you are not asking yourself hard enough problems.

Reblogged this on mathematics matters… and commented:

Fascinating interview – so important to recognise the nature of doing mathematics.