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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Depending on the school, I suspect you’ll find a superficial consensus (*cheating is terrible! and, thankfully, our students do it very rarely!*) masking deep rifts. Is the problem with cheating that it undercuts your own learning? That it steals glory from classmates in the zero-sum competition for grades? That it betrays the teacher’s trust? Are all acts of cheating equally terrible, and if not, what does that mean for “zero tolerance” policies?

We all know cheating is bad. But we seem unable to talk honestly about *why*.

So, I offer up these dialogue-starting cartoons, a few badly drawn meditations on the most basic question: *Why do students cheat?*

Is cheating a crime of character, or of opportunity?

Talking to teachers, I find they talk a lot about virtues like honesty and integrity. Good students have ’em, and cheaters don’t.

Talking to students, you hear a lot more about circumstance. “I wasn’t sure how to do it.” “I was just looking over to check my answer.” Cheaters are people who work hard at their lies; so if my copying is effortless and victimless, then how can I be a cheater?

Moral intuition is a funny thing.

If I hurt someone else to help myself, you’ll probably judge me morally deficient. Fair enough. But I’m willing to wager that your judgment depends heavily on one detail:

Am I harming that person in order to fulfill a wish, or merely to relieve a pain?

Harming someone else to bring myself joy (e.g., stealing money so I can buy a fancy car, or beating someone up so I can gain status among my friends) makes me an antisocial jerk. But harming someone else to spare myself harm (e.g., stealing money so I can afford painkilling medication, or beating someone up to avoid a beating from my “friends”) is more sympathetic. The former is moral black; the latter, moral gray.

In many classrooms, teachers would have you believe that cheating is a rare and terrible crime, like a small-town homicide. When it happens, it is so horrible and conspicuous that the criminal is brought to swift and certain justice.

There’s no two ways about it: So far as I can tell, that’s wrong. Cheating happens in every school. It’s the nature of an educational system where the assessments are both high-stakes and game-able.

Meanwhile, though, students (especially those caught in the act) would have you believe that the whole class was doing it, that cheating (particularly on homework) is as mundane as going 3mph over the speed limit.

This, also, strikes me as wrong. Cheating may be every*where*, but it ain’t every*one*.

Sometimes students are lazy. I know this because students are people, and I am a people, and sometimes I am too lazy even to decide whether I am plural or singular.

But sometimes students are bewildered and afraid to ask.

I know this because… well, students are people.

One of the most fascinating justifications I’ve heard from a cheating student (via anonymous online op-ed, not in person) is that he felt perfectly righteous in cheating whenever he felt his teachers were failing to educate him.

The logic is pretty simple: “You cheated me first. I owe you nothing.”

I can think of counter-arguments. But to me, it’s a reminder that those arguments are necessary. Even to bright students, the “wrongness” of cheating isn’t self-evident.

It’s lovely to think that students are driven by curiosity alone.

It’s also lovely to think that Flintstones cars are driven by motors.

But look down at those little scurrying feet, and you’ll see what’s really providing the forward momentum. Grades are so economically important that, for most kids, they are inevitably the primary carrot and the #1 stick.

“They only care about grades” isn’t a fair knock on students, any more than “They only want to work here because we’ll pay them” is a fair knock on job applicants.

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That’s a quarter-thousand! It’s double five-cubed! It’s 1/4000 of the way to a million! If posts were days in a pregnancy, this blog is… kinda almost birth-ready! If posts were meters, I’d have walked… to the corner store! If posts were nickels, I’d have… uh, $12.50!

Okay, now that you’re duly impressed, I have a confession: I’ve been hiding comics from you.

Since late 2016, I’ve been posting one-off cartoons to the Math with Bad Drawings Facebook page. I encourage you to “Like” the page so you can see them all. But in case you’re some kind of conscientious objector to social media (if true: thanks for exempting WordPress from your boycott!) here’s a (partial) retrospective collection.

I find it funny when people claim that “percentages over 100% are impossible.” They’re not just possible – they’re easy! For example, the United States National Debt is currently 105% of the United States GDP.

Now, how hard is it to rack up a little debt? A trillion here, a trillion there, and it adds up surprisingly fast. Easy.

Not depicted: the graffiti itself, which is *also* a feeble statistics pun.

(That’s what the purple parent is really mad about, obviously. Puns aren’t even jokes, really; they’re just a lazy mind eating its own language. That’s the real deviancy here.)

*And the Sigma’s Greek glare!*

*The terms added in air!*

*Gave proof through the night*

*that our sum was still there…*

*Oh, say does that star-indexed banner yet wave…*

[*2 hours later*]

“Don’t tell me. *Another* unanticipated 5-minute delay?”

“No, this delay is only 45 seconds! But I’ll come clean: this one we actually anticipated.”

Honest mistake.

d(Instruction Method)/d(Student) = 0 is a differential equation that applies to an awful lot of lessons, I’d imagine.

The real problem with “capital numbers”: are the ones we have now the *upper*-case or the *lower*-case version?

8 feels super upper-case to me, but 3 could really go either way.

Side point: I find two-tailed hypothesis tests pretty inexplicable in many contexts where they’re used.

“We think our drug will make you a lot better. Or worse. We haven’t decided yet so we’ll hedge our bets on the statistical analysis.”

“We expect people who have suffered a recent trauma to show more physiological signs of stress. Or – who knows – maybe fewer signs. We’ll see how it plays out.”

Anyway, I think this student reporting the p-value from the two-tailed test is pretty sensible, although it’s admittedly burying the lead.

Ah, a time capsule: I drew this one in October, back when it felt like “breaking news” was a thing news organizations said to make you care about something unimportant but recent.

Maybe less relevant now. It’s been a newsy few months.

My wife hates this one. Sorry, Taryn! But hey, it’s not *my* fault that my drawings achieve such a haunting photorealism.

My own ill-formed impression of flipped classrooms? They make a lot of sense for college, but not really for high school.

In college, you’ve often got 50+ students to a class, and lots of content is delivered by lecture. Might as well videotape those lectures, let kids watch ’em at home, and spend class time interacting and asking questions and doing cool papier-mache activities.

In high school, though, with classes of 25 and attention spans measured in milliseconds, lecture isn’t a useful tool anyway. Even direct instruction needs some element of interaction and responsiveness to be more than 2% effective. So watching videos at home isn’t a natural substitute; it’s a shoddy one.

Little-known fact: Zeno’s original phrasing of the paradox was just a long rant about why “The Hobbit” should have been a single film.

Three years in, this is still how I feel about England.

I looked up the etymology, and it’s wildly unsatisfying. At some point “terrific” just started meaning “good.”

I say we try this with a new word. For example, I’d like “idiot” and “idiocy” to keep their current meanings, but for “idiotic” to mean “sizzling with genius.”

For example: thanks to all my idiotic readers for your idiotic comments. Honestly, I know I’ve become a real idiot about replying, but I read and appreciate everything you guys write, and I’m hoping to overcome my own idiocy and engage in the conversations more.

Y’all are terrific, and I feel so lucky and grateful that my little jokes and sketches and musings have a place in your Internet life. Your replies are always wonderful and your enthusiasm means worlds to me.

Thanks for checking out the first 250 posts. Here’s to 999,750 more.

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