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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They blinked hopefully, as children do.

“Who was it?” I demanded. “Who taught you – or should I say, *failed* to teach you?”

“Um…” they hesitated. “You?”

“Well, whoever it was,” I said, “he denied you the chance to experience the beauty and centrality of the distributive property. For this, he shall have my undying scorn.”

The philosopher Alfred North Whitehead once described European philosophy as “a series of footnotes to Plato.” Twist his arm enough, and perhaps he’d have consented to describe algebra as “a series of footnotes to the distributive property.”

We use it frequently in arithmetic, but rarely name it as such. For example, try calculating 17 x 6 mentally.

One common strategy is this:

Another approach is this:

Both are just applications of the distributive property. You’re exploiting the fact that “a+b” groups is the same as “a groups” + “b groups.”

In fact, to *avoid* using the distributive property, you’d have to do something a bit unusual, perhaps like this:

Now, my 6th graders are pretty slick at the numerical version of the distributive property. They gobble up problems like this:

But they falter at the same step that haunts students worldwide: converting numerical instincts into algebraic generalities.

The distributive property is like most algebraic facts: just the crystallization of a familiar thought pattern from arithmetic. But that’s not how most students learn it. The easier short-term path is to see it as a new, disconnected rule for manipulating symbols: **a(b+c) = ab + ac.**

I fear this purely symbolic understanding is a broken futon, liable to collapse under enough strain. In particular, it seems to breed the “everything is linear” mistake. (One of my recurring nightmares.)

It’s crucial to get this right, because (to mangle the words of William Carlos Williams) “so much depends upon the distributive property.” Just look:

Now, most of my colleagues aren’t as worried about this as I am. They see a natural order for learning math: First, become adept at symbolic manipulations. Then, later, come to understand the deeper meaning. Trying to do it all at once – manipulation *and* meaning from day one – leaves students befuddled, befogged, and belligerent. Better to reach for the big ideas only once you’re comfortable with the mechanics.

They’re not wrong. Lots of effective mathematicians learned their craft like this.

But I can’t figure out how to teach that way. “No black boxes” has long been my motto. Intuition before formalism. Never charge forward with symbolic manipulations until you understand what those symbols actually symbolize.

This approach generally works for me. It means puzzling out notions of area before introducing compact formulas like “A=bh/2.” It means playing around with prime factorization before developing an algorithm for finding a highest common factor. It means baking understanding directly into a student’s thinking, rather than waiting until the cake is settled and then trying to sprinkle a little understanding over the top.

But when it comes to the distributive property, I still don’t have this figured out. Area models don’t seem to do much for my students. My wordy explanations (“it’s seven bags, each containing x + 7!”) are little better. I’m still seeking tasks that can sharpen and hone their thinking about distribution.

But I guess that’s the perpetual state of the teacher: still seeking.

*On a different note:*

A few weeks ago, the folks at Brilliant.org asked me to help announce their “100 Day Summer Challenge,” a free collection of math puzzles targeted at high schoolers to keep them sharp and math-enthused over the summer. I was about to say “no thanks” (as I do to all such publicity requests) when I realized that the problems are actually quite slick and fun. And what is this blog for, if not peddling addictive stuff to high schoolers?

So here’s the cartoon I drew for them. I encourage you to check it out!

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If your jaw is not on the floor, it’s because (A) you’ve spent shockingly little time browsing the list of Fields Medal winners, and (B) you’re not Vietnamese.

A helpful Vietnamese journalist I met explained to be that Châu is “the biggest celebrity in Vietnam.” Châu won his Fields Medal in 2010 for proving—hands inside the vehicle, please, because this is a wild ride—a key relationship between “orbital integrals on a reductive group over a local field” and “stable orbital integrals on its endoscopic groups.”

In Vietnam, that relationship is apparently the one sizzling on tabloid covers.

Châu is not your prototypical superstar. Even in Vietnam, apparently, he is a cryptic figure; not a chatty TV celebrity, but a silent legend. At the press conference where I met him, at the Heidelberg Laureate Forum, he gave some journalists terse one-sentence answers. Not because he was being standoffish, but because a mathematician like Châu never proves in ten lines what he can prove in just one.

I didn’t know what to ask him. I’m not a research algebraist and have never been mistaken for one. So I asked about his education, his youth in Vietnam, his mathematical coming of age.

How does Ngô Bảu Châu get to be Ngô Bảu Châu?

In his soft, low voice, he told a beautiful story. It cut against the romantic myth of the mathematician (lone genius, predestined, operating in a sphere beyond mortal reckoning). In fact, it resonated with the themes I encounter as a teacher: self-doubt, missteps, the need for helping hands.

Here, in three acts, is Châu’s tale.

“When I was in the sixth grade,” Châu confided, “I was not particularly interested in mathematics.” He lacked discipline. “I was in a very new-style school. We were free to do whatever we want.”

And what did he want? Well, not mathematics. Not yet.

When an opportunity came to join a gifted class, Châu leapt at it—but failed the entrance test. “I thought I was very good,” he remembers. “I was not.”

Rejection stung, and the sting drove him. He refocused his efforts. “There were so many problems I couldn’t do,” he says, but he drew inspiration from those who had walked this path before—and from the awe and reverence in which classmates held them.

“I heard stories about the older students who were in the Olympiad,” Châu says, referring to the international math competition. “Mathematicians had mythic status.”

As a teacher, I downplay the competitive side of mathematics. I preach patience and collaboration and encourage students to end proofs with “Kumbaya” as an alternative to “QED.” When they compare test scores, I cringe; I don’t want math to feel like a zero-sum tournament. But for many children (including, if I’m honest, my younger self) competition is part of the draw. Kids love what they feel good at. Measuring yourself against peers is how you know where you stand.

And soon, Châu stood atop the mountain. He won back-to-back golds in the Olympiad. No Vietnamese student had ever done that. It was only a few years from test-day failure to local legend.

“I do well with competition,” he says, smiling.

In a word, that’s the first phase of Châu’s mathematical journey: competition. Math was a ticket to an elite club—and now, Châu was in.

Châu earned a scholarship to study combinatorics in Hungary. Then, suddenly, the Berlin Wall fell. Borders were redrawn. Instead of combinatorics in Budapest, he found himself studying algebraic geometry in France.

The transition wasn’t easy. “I suffered quite a bit. This abstract algebra was very unfamiliar.”

I challenge you to find a group theory student who doesn’t identify with that.

Combinatorics is a concrete, problem-solving branch of math, concerned with the counting and rearrangement of objects. Abstract algebra is… abstract. Notoriously so. It studies nebulous collections of slippery objects defined not by their substance but by their properties and interactions.

“It was really hard for me,” Châu recalls. “I couldn’t get it. It was very painful.”

Says the Fields Medalist.

Châu hid his struggle below the surface. “My professors thought I was a fantastic student,” he remembers. “I could do all the exercises. I did well on exams.” Then he shakes his head, half-smiling. “But I didn’t understand anything.”

So it went throughout his undergraduate degree: a pantomime of understanding, and an anguish below. It’s an all-too-common tale for the mathematics student: you copy patterns blindly, not comprehending, until one day the patterns evade you, and your mathematical days are through.

Châu feared falling into that same trap.

It was his PhD that saved him. “I had one of the best advisors in the world,” Châu glows. That’s Gérard Laumon, with whom he later collaborated on his breakthrough work.

“I would come to his office every week. He would read with me one or two pages every time.” They went line by line, equation by equation, settling for nothing less than full comprehension.

Châu embraced this slow, deliberate pace as a gift. “It was very revealing for me.” For me, this strikes a resounding chord. Early in my teaching career, I saw my students as mired in meaningless symbols; I wanted them to step away from the notation and think about the ideas. More recently, I’ve come to believe the ability to read notation is vital: but you’ve got to *read* it, not just push it around the page. The work to invest mathematical symbols with meaning is slow and painstaking but utterly central to the project of becoming a mathematical thinker.

And that meticulous project of unpacking mathematical writing is how Châu spent his PhD. Week by week, he built his understanding, growing under Laumon’s mentorship from a competitor into a scholar.

Châu soon began work on the famous Langlands Program, which I think of as the transcontinental railroad of modern mathematics.

Almost fifty years ago, Robert Langlands laid out a sweeping vision for how to connect several distant branches of higher math. (Which branches? I’ll let you read up on it.) The project has drawn generations of ambitious mathematicians like Châu into its orbit.

Châu found himself attracted to a particularly vexing piece of the Langlands Program: proving the “Fundamental Lemma.”

(The name is something of an oxymoron; a “lemma” is typically an intermediate fact, proved on the path towards a grander, more exciting truth called a “theorem.” But Langlands named the “fundamental lemma” before its immense subtlety and difficulty became clear.)

By the time Châu arrived on the scene, the Fundamental Lemma stood as a crucial choke-point in the Langlands Program. It was the bottleneck stifling further progress.

So were rival mathematicians jockeying for primacy, racing to be the first to prove it? Was it the Olympiad all over again?

No, says Châu.

“I was helped a lot by people in my field,” he says. “Many people encouraged me to do so, in a very sincere way. I asked them for advice, and they would tell me what to learn. It was very open. I did not feel competition.”

And, within a few years, Châu managed to prove the Fundamental Lemma.

This feat earned him individual glory (fame, fortune, Fields) but did not come by individual labor alone. As NBA squads have learned, handing the ball to a superstar is no match for a fluid and cohesive offense. Individuals don’t win championships; teams do.

So not only did Châu build on past work, but he turned to his colleagues for guidance throughout the process, drawing on their expertise to patch holes, to strategize, and to overcome snags. In this, his work resembles pretty much every great human accomplishment I’ve ever witnessed, from my students’ best math to my colleagues’ best teaching to the kitchens at my favorite restaurants. People thrive thanks to other people.

Châu has thrived through competition, grown through mentorship, and advanced the field through collaboration. Now, he perhaps begins the fourth act of his mathematical tale: Leadership.

Though Chicago is his home now, Châu spends his summers back in Vietnam. He helps to run the Vietnam Institute for Advanced Study in Mathematics, and reaches out to young people to encourage them into mathematics. In the years after his upbringing (when Olympiad winners were revered as true Olympians) the cachet of mathematics had fallen in Vietnam. But after hitting a low point in the 1990s and 2000s, it is now on the rebound.

Thanks to Châu, mathematics is cool again.

Meanwhile, the Langlands program has entered a new phase. Of the earlier work, Châu says: “It was very structured. There were very clear task divisions. It all fit together.” It was as if Langlands had composed the table of contents—a crisp outline of ideas—and had left the other researchers to produce the text.

But with that phase complete, work today is different. “Now, there is no such division. [Langlands’] description is more like poetry. It is not programmatic.”

Châu pauses. “Now we are on our own.”

Only 44 years old, he has decades ahead of him in which to push mathematics forward. Younger researchers will be following his lead. And Ngô Bảu Châu, for one, knows what they can expect on the journey ahead.

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

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It comes from a man as brilliant as he is grumpy. Last week in the staff room, one of my all-time favorite colleagues busted out a dozen-line proof of a world-famous theorem. Though I doubt it’s original to him, I shall dub it **Deeley’s Ditty **in his honor**:**

This argument is sneaky like a thief. I admire but don’t wholly trust it, because it solves a differential equation by separation of variables, a technique I still regard as black magic.

What I can’t deny is that it’s about 17 million times shorter than the proof I usually show to students, which works via Taylor series:

You might call this one **Taylor’s Opus**. It builds like a symphony, with distinct movements, powerful motifs, and a grand finale. It takes familiar objects (trig and exponential functions) and leaves them transformed. It’s rich, challenging, and complete.

It’s also as slow as an aircraft carrier making a three-point turn.

Here we have two proofs. The first is a nifty shortcut, persuasive but not explanatory. Watching it unfold, I’m unsure whether I’m witnessing a scientific demo or a magic trick.

The second is an elaborate work of architecture. It explains but perhaps overwhelms. If an argument is a connector between one idea and another, then for some students, this will feel like building the George Washington Bridge to span a creek.

The question it prompts, to me, is: *What do we want from a proof?*

Here’s another example. Week #1 of teaching here in England, I posed a classic challenge to my 14-year-olds: *Prove that **√**2 is irrational*. To my delight, it took only 90 seconds before one of them produced this clever argument, which I’ll call **Dan’s Ditty:**

I loved it, finding it slicker and more satisfying than the standard proof I’d seen a dozen times:

Of course, I could also see the ditty’s downfall. It relies on the Fundamental Theorem of Arithmetic: the idea that each number has a unique prime factorization. That’s a nontrivial result, one I didn’t encounter until group theory in college. No such machinery is needed for the standard proof, which Hardy and Erdös (among others) hailed as one of the loveliest and most perfect in all of mathematics.

Holding the two ditties side by side, some themes emerge:

What do we want from a proof? I say it depends on the spirit in the room.

In the somber mood of scholarship, clad in academic gowns and posing for our portraits, we prize rigor and depth. The “standard proofs” are standard for good reason. They convey unambiguous truths through careful logic. They’ve stood the test of time better than just about any other work of the human mind.

But in playful moods, holding coffee in one hand and chalk in the other, there’s a lot to be said for the ditties. They’re fun. They provoke. They refresh. They’re like trying a new path on the commute home; coming at the street from the other side, you see a slightly different world.

Here’s a last and favorite example, which I heard from my boss Neil, who heard it from whoever he heard it from: A proof that all higher-order roots of 2 are irrational.

I find it hard not to smile at that one.

It’s as if Andrew Wiles has arrived at the top of Everest, only to notice that I’ve been riding piggyback the whole way. “Hey, what are you doing here?!” cries Sir Wiles, and by way of response, I grab a chunk of fresh Himalayan snow and drop it into my drink. “Needed ice,” I explain.

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Let me be very clear, as clear as the vials of tears that I keep on my desk: This story is a long and sad one. It converges to no happy ending, and perhaps does not converge at all, although as you read, you will find your own joy and sanity both converging swiftly to zero.

If you were to abandon this text and go read about something pleasant, like butterscotch pudding or statistical sampling, I would applaud your good judgment, and humbly beseech you to statistically sample a pudding on my behalf.

As for me, I am compelled to tell this tale to its sour end, because I am an analyst—a word which here means “someone who fusses over agonizing details, bringing grief to many and enjoyment to none.”

But if you insist on reading further, then you ought to meet the three poor students at the heart of this tale:

Though all enrolled in the same course covering mathematical series, each came for a personal reason. Violet was drawn by the practical applications of series; Klaus, by their central role in the birth of modern mathematical thought; and Sunny, because this seemed the next logical step for her education, where “logical” means “expected by her parents and the HR departments at large corporations.”

Each came with good mathematical preparation and fresh enthusiasm. But their hearts all fell when they met Professor Olaf.

“We shall begin with arithmetic series,” said Olaf, not bothering to say so much as “hello” or “welcome to class” or “here are some reasons why learning about series is a good use of time that might otherwise be spent playing guitar or preparing lasagna or watching films about elephants.”

I am sorry to say that this is how teachers often are.

“Now,” said Olaf, “it is obvious that the sum of a finite arithmetic series is half the sum of the first and final terms, multiplied by the number of terms.”

“But why?” said Violet. “Can you offer some intuition?”

“Silence, undergraduate!” cried Olaf. “It is true for reasons any fool can see.”

“We can’t,” protested Klaus, “and we’re not fools.”

“Well,” said Olaf with malice, “I’m afraid you’ll need to find a fool to explain it to you. I cannot be bothered to hold the hands of dimwitted undergraduates orphans.”

Sunny rolled her eyes.

“Now,” said Olaf, “on to infinite geometric series.”

The word “geometric” here means “having nothing to do with geometry.” As you might guess, this topic offered no great clarity.

On it went. By the time the class ended, Olaf had filled the board with inscrutable notes about series of all kinds: arithmetic and geometric, convergent and divergent, harmonic and amelodic, alternating and direct, Taylor and Swift, McLaurin and McGuffin.

Then, for homework, he assigned a delta-epsilon proof, a phrase which here means “symbolic manipulations that persuade no one of anything.”

Now, if you are anything like me, you would have met this onslaught by filling new vials of fresh tears and filing a hasty application to change majors, perhaps to English or Art or simply Cowering Under the Bed Studies. But the three students, far braver than I, instead formed a study group.

Violet, pragmatic and efficient in the manner of all good engineers, found a trick for adding up certain long strings of numbers.

“Just imagine that there are two copies of the string,” she explained to the others. “And write the second one in reverse order, just below the first.”

“Now, we have a string of pairs, and each pair adds to the same amount.”

“This makes it easy to find the total simply by multiplying.”

Klaus and Sunny were very impressed, although Violet pointed out that this only works when the original string of numbers share a common difference. “Otherwise,” she explained, “each pair will add to a different total, and the whole method will unravel.”

(This trick, of course, had been known to many others, including the great Carl Gauss, and the far less great Professor Olaf. Indeed, beautiful ideas sometimes fall into the clutches of ugly minds, and do not always manage to brighten their surroundings.)

Meanwhile, Klaus—attracted by big ideas and conceptual shifts, as all great historians of mathematics are—found an infinitely long string of numbers whose total was, miraculously, *not* infinite.

“Look at this,” Klaus said, “and imagine the numbers go on forever, growing smaller and smaller.”

“Now, it might seem that infinite numbers should add up to infinity. But they don’t! Every new number brings us half of the remaining distance to 1. And so, no matter how far you go, the total can never exceed 1.”

“Instead, going further brings you closer and closer to 1, until you are less than a hair’s breadth. In some sense, the ‘final’ sum must be 1 itself, although this will happen only at the end of eternity, and that is well after our curfew.”

(In his exposition, Klaus joined many centuries of mathematicians who had puzzled over precisely this paradox—a word which here means “a mathematical oddity that prompts you to think two contradictory thoughts at once.”)

Finally, Sunny grabbed a piece of paper and wrote down the following strange observation:

“Really?” said Klaus.

“Why is that true?” said Violet.

Sunny shrugged and said, “Taylor,” which her companions understood to mean, “Proving this striking claim would require mathematical machinery beyond our command, but it certainly whets my appetite for the further study of series.”

“Wow,” said Violet as the study session ended, “we’ve learned a lot.”

“Yes,” said Klaus, “but we still haven’t completed the homework.”

Violet sighed. “I don’t understand deltas and epsilons at all.”

“Greek,” said Sunny. She did not mean “delta and epsilon are letters in the Greek alphabet,” although they are. Rather, she meant, “Don’t despair! The highly technical 19^{th}-century framework of deltas and epsilons would have been alien to the inventors of calculus, and entirely baffling to the great Greek mathematicians of antiquity.”

“You’re right, Sunny,” Klaus said. “Even Cauchy, the scholar credited with our modern understanding of convergence, didn’t develop the language of deltas and epsilons. It’s no knock against us to stumble over these subtle, taxing ideas.”

“But what about Olaf?” Violet said. “And our homework?”

The three students put their heads together—a phrase which here means “exchanged ideas, without actually bringing their foreheads into contact”—and came up with a plan.

When they arrived at the next lecture, Olaf collected their homework. “What is this?” he said, gazing at the pages full of strange symbols.

“It’s an omega proof,” Violet said.

“And mine is a lambda proof,” Klaus added.

“Alpha,” Sunny said, describing her proof.

“You malodorous undergraduates!” roared Olaf. “I don’t know any of those proof styles! I only know about delta-epsilon proofs, because that’s what’s in the textbooks.”

“Greek letters are only symbols,” Violet said. “It doesn’t matter which ones you use.”

“Is your thinking so brittle that you can’t handle a mere change of notation?” Klaus asked.

“Alpha,” Sunny said, commenting on Olaf’s personality type.

“Omega proofs? Lambda proofs? Alpha proofs?!” Olaf cried. “I’ll tell you what I think of your precious Greek letters, you verminous undergraduates.”

And on a fourth piece of paper, he wrote a giant letter F.

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This September in Germany, between talks at the Heidelberg Laureate Forum, I managed to catch a few minutes with Cornell professor John Hopcroft.

He’s a guy with bigger things on his mind.

“I’m at a stage in my life,” he says, “where I’d like to do something which makes the world better for a large number of people.”

Skimming Hopcroft’s C.V., you start to wonder: Um… hasn’t he done that already?

Born to a janitor and a bookkeeper, he grew up to become a foundational figure in computer science. Exhibit A: His textbooks on automata, algorithms, and discrete math have been adopted across the world. (His most recent one—on data science—is free online.) Exhibit B: He has a distinguished research record, highlighted in 1986 with a Turing Award— the closest thing to a Nobel for computer science. And finally, Exhibit C: During a decorated teaching career, he was twice named Cornell’s “most inspiring” professor.

With all this, you’ve got to figure he’s done at least a *little* good for a *few* people, right?

Well, Hopcroft has a larger number in mind: 1.3 billion.

Hopcroft has become an advisor to Li Keqiang, the Premier of China. He describes this as “the opportunity of a lifetime”: to transform Chinese education for the better.

“They have one quarter of the world’s talent,” Hopcroft says, “but their university educational system is really very poor.”

What makes Hopcroft—working-class Seattle-ite turned Ivy League professor—think he can leave his mark on a country as vast, distant, and internally diverse as China? Isn’t this like a swimmer trying to steer an aircraft carrier?

“A couple of things are going in my favor,” he says. First, he is apolitical. “I don’t have any special agenda to push in China,” Hopcroft explains. “I’m pushing education.”

The second is subtler, and carries echoes of Hopcroft’s engineering background.

“I understand the scale of the problem,” Hopcroft says.

It was Hopcroft’s wife who helped throw the magnitude of Chinese education into stark relief. Hopcroft had begun teaching at an elite program in China, instructing 30 students a year. “But my wife tells me, ‘That’s not going to have any impact on China.’

“China has one million faculty in universities, and 30 million students,” Hopcroft explains. “It’s a totally different scale.”

You can’t treat an aircraft carrier like a rowboat.

Recently, a fellow member on a committee proposed that they focus on funding small, elite programs. Hopcroft pushed back.

“I said, ‘Look, if they had a thousand of these, each with 100 students, that would be a hundred thousand students,” Hopcroft says. “What about the other thirty million?”

Rather than narrow his focus, Hopcroft hopes to nudge the entire system forward.

“The real leverage point,” Hopcroft explains, “is if the Premier writes 1500 university presidents and tells them it’s their job to improve undergraduate education. If 10% of them do, that will impact the lives of millions of people.”

In other words: to steer the boat, you need the ear of the captain.

And what do you tell that captain to do, exactly?

In three words: Change the incentives. “The metrics are wrong,” Hopcroft says.

Currently, universities measure their success in two primary ways: by research funding, and by number of papers published.

“If you’re a university president,” Hopcroft says, “you can tell people, ‘I raised the research funding from #15 up to #5,’ and they view it as impressive. But it has absolutely nothing to do with quality of education.”

Instead, Hopcroft proposes a new basis for evaluation. First, look at the content being taught. “Is the material in the course up to date?” he prompts. Second—and more ambiguously—look at the instructor.

“Does the faculty member show up?” Hopcroft asks. “Or do they send graduate students to show the PowerPoint slides? Does the person really know the material? Are they excited about it? Are they engaging these students?”

I butt in with a query: Doesn’t this come down to an aesthetic judgment? At least in part?

“The whole thing is,” Hopcroft agrees. He describes it as “kind of like scoring an ice-skater.”

That is to say: aesthetic, but not arbitrary.

“Could you evaluate a course if it was in Mandarin, and you didn’t speak Mandarin?” Hopcroft asks. Yes, it turns out: when Hopcroft sat in on the classes of two faculty members, his judgment matched with that of Mandarin-speaking evaluators. “Not only that,” Hopcroft says, “but I was able afterwards to give them advice as to how to improve their teaching.”

The measures may be subjective. But they’re detecting real qualities.

All this is outlined in the proposal Hopcroft has on the Premier’s desk.

“He understands what all the problems are,” Hopcroft says. “But they’re very hard to change because they’re driven by culture. I think that’s why [the Premier] likes me: I don’t tell him what’s wrong, I tell him how to try to fix it.”

Hopcroft has already helped advocate for the transition from a contract system to a tenure system for Chinese professors. Now, from among seven proposals, he’s focusing on teacher evaluation.

“We can’t evaluate a million faculty at once,” Hopcroft admits. “So I said, let’s take the top ten institutions, and in each one, pick five top disciplines, and in each of those, pick the three key courses.” That’s 150 teachers to evaluate in all.

“The university president is not going to know who we’re going to be evaluating, so he’s got to improve the whole system.”

Hopcroft knows there is danger in over-stepping. “[My proposal is] not saying *how* to improve education,” he says. “It’s telling 1500 university presidents, ‘It’s your job to improve it. You go figure it out.’”

In some sense, Hopcroft is fighting against the very quality that gives him such enormous access: centralized control.

The Premier has enormous influence on China, far more than any American politician has on policy. But now, Hopcroft wants him to deploy that power to encourage university presidents to stand on their own feet.

He wants the system to act less like a single aircraft carrier, and more like a fleet.

While American politicians fret about the economic and political threats that China may pose, Hopcroft sees the relationship differently.

“To me,” he says, “this notion that we should constrain China doesn’t make sense. We’re never going to be able to do that. We should be working with China and building the kind of relationship we have with Europe.”

It’s an awfully ambitious vision for a computer scientist living in upstate New York. But at this stage in his life, that’s exactly what Hopcroft wants.

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