Forget the history of calculus. Write me a paper on the calculus of history.

You won’t be the first. In *War and Peace*, Tolstoy compared civilization to a vast integral. Only by summing all “the individual tendencies of men,” Tolstoy wrote, “can we hope to arrive at the laws of history.” His was a true people’s history. Each peasant and prince gets the same weight in Tolstoy’s great Riemann sum. To give the monarchs disproportionate weight (thereby silencing the masses) would be a perversity, a paradox. No delta functions in Tolstoy’s mathematics.

History as an integral. That’s one way to see it.

Or imagine history as an infinite series. Each day adds a new term to the massive sum that precedes it. The question arises: Does history converge? Are we inching, year by year, towards some fixed destination? Will history roll slowly to a stop? Or will it diverge—oscillating between two extremes, or perhaps cascading slowly out of control, millennium after millennium? Will the decades ultimately add up to something unrecognizable?

Or perhaps the sum is finite, and the human story will end abruptly.

Another approach would take history as a solution to a vast set of partial differential equations. First, distill civilization to a set of variables—aesthetic trends, political wills, technological breakthroughs. Second, chart the ways the variables change, their dependences on one another. Third, summarize these interactions with a complex system of relations. The history of the world must be a solution to this system.

But is this solution *unique*? Or could it be merely a particular solution, one of many?

In other words, was our timeline inevitable, or could some other arrangement have satisfied the forces of history? Are we missing out on an entirely different version of human civilization, with alternative institutions, powers, and lifestyles?

Or tell me about limits. Are there discontinuities in the human experience? Does life advance from one moment to the next in smooth and fluid motion, offering no true surprises, every aspect of the future buried somewhere in the derivatives of the present? Or does it occasionally jump, like a historical step function, the next moment completely unlike the last?

The history of calculus? Heck, anyone can tell me about how humans discovered the mathematics of continual change. It’s right there on Wikipedia.

I want you to make something new. Tell me about the calculus of history.

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Two weeks later, I posted an essay called What It Feels Like to Be Bad at Math, about my struggles with topology. It was stubbornly hard to write. I spat out 500 words of excuses and hedges (which I later deleted) before I could bring my fingers to type the truth.

Then the post started getting passed around. Thanks to a friend from college, it got picked up at Slate, where 100,000 people read it—and for the first time in 15 years of wishing, I felt like a writer.

Best of all (and how often do you get to say this about the internet?) were the comments. The “hard pavement” I’d feared turned out to be a warm and gentle crowd of people sharing their stories, wisdom, and vulnerability. Their collective eloquence dwarfed my original essay.

A year later, I’ve pulled out some of my favorite comments, and framed them with questions. Thanks, to all of you, for reading this blog and letting me share my stories and jokes and thoughts with you. Your warmth (and, when necessary, corrections!) have meant a lot to me. So here it is: mathematical anxiety, in your words.

The subject seemed like a smooth rock face, I didn’t even know the questions I should be asking. I still have the textbook, unread for all these years, like a totem: I really should try and learn that. – Paul Ramsey

“I got my first arithmetic assignment handed back to me with nothing but red marks on it, and the sense of humiliation and disappointment has been with me almost my entire life.” -Brandon

“You can see it in their eyes – they really don’t understand. They will pretend that they get what you say to them, but you can tell they don’t. But it doesn’t matter, tomorrow is a different lesson and you will leave today’s lesson behind – until it comes back in an even less comprehensible form in the future.” –Laura Lynn Walsh

“When we teach math, we organize things for clarity. Unfortunately, this creates the illusion that there’s always an obvious, linear progression of concepts. It hides the messy truth of how the knowledge was actually acquired: through many twists and turns in a process of creative exploration. And, as you progress in mathematics, you have to bring more and more creativity to bear. This has many implications, but the most germane is that it’s very difficult to be creative when you’re in a state of panic.” – Brandon

“Grad seminar in logic: Dry mouth, rapid heartbeat, inability to look anyone in the eye. I’m your poster-boy. Took 14 yrs to get my Ph.D. Now emeritus professor of math & computer science.” –Gene Chase “As a grad student, I used to feel stupid. Now as a professor, after having proved a few results of my own, I know I’m not stupid. If I don’t understand something, it’s only because I haven’t thought about it hard enough.” -Victoron

“It’s like navigating a huge landscape, but certain hills are higher than others. You can tell the height because it starts making you feel dumb…. It’s a large part of what makes research so difficult.” –tor_and_ext_are_torture_and_extortion

“Math is a very peculiar subject in that it builds on itself so linearly and so completely. Therefore there’s a sort of unstable equilibrium when it comes to understanding it. You can understand steps 1-100, but if you don’t get step 101, and you never put in the work to do so, then you won’t get steps 102-10,000 and you’ve effectively ‘fallen off the cliff.’ Then you start spiralling out of control.” –True Beauty of Math

“With math, because of its cumulative nature, you have to learn it in sequence. With a subject like art history, while there is clearly a progression of styles and a chronological nature to the historical events behind the art, it is entirely possible to have a meaningful discussion about art of a certain period without possessing a thorough understanding of what came before it.” – Peter T.

“Advice a teacher gave me: ‘You never understand Course N until you’ve taken Course N + 1.” – Gene Chase

“I used to play ping pong every week with a computer science professor. He was a very smart, no-nonsense guy. I told him about my tendency to ask a lot of questions, and how it was sometimes a bit much. What he told me has stuck with me till today: some of the smartest people I know ask some of the dumbest questions I’ve heard. Because they want to be absolutely, 100 percent sure that they get it.” –Nir Friedman

“I was labeled ‘gifted’ in second grade and was lucky enough to leave the grind of classes once a week to learn atomic theory, classical art, logic problem solving and other very cool things. I loved it all. Then I had 4 days in regular class. And I started experiencing the downside of the label.

I was struggling with long division and decimals, and I still (many more years than I care to admit) remember vividly, the teacher exclaiming in front of the whole class, ‘You have two professor parents and you’re in the gifted program, and you can’t get this? I don’t understand. Your sister never had these problems.’

I was done with math from then on – smote by that triple whammy.” – Carolyn

“What a difference it makes when a teacher breathes into our lives and compassionately understands. My son has a lovely math teacher and it’s doubtful he’ll ever be a star student in her class, she has cared and praised him for his efforts.” – Olive Shoot Institute

“We work hard in math, our class motto is ‘Work is synonymous with Opportunity.’ And though I strive, an edutopia I do not have. They rebel, they slump, and they watch the clock as it gets closer to lunch… but they also say great things about math and most importantly, they know not getting math doesn’t mean a person is less intelligent, in fact, if we are not hitting walls when engaged in math, walls that require pounding, diligence, sweat and tirades, then we are not doing math. And oh the celebrations when that wall comes tumbling down.” – Debra Taylor

**I want to finish with a story from my sister, Jenna, a K-8 math coach and (despite her meager Twitter following) a far better teacher than I am:**

Today, I watched a very capable student reduced to tears during my math lesson. She is ordinarily a student that catches onto the material quickly, and even requires an extra challenge.

Today, however, she looked confused, yet insisted she did not need any help. “I hate this game! I hate math!” She continued to protest. “I don’t want to work with my partner! This game is boring! When are we done?” She is six years old.

I pulled her into the hallway to discuss the content of [your blog post]. (How timely!) We talked about the symptoms, and about developing a growth mindset even when all we want to do is pretend we understand. I showed her the math content in a slightly different way — we were working on the inverse relationship between addition and subtraction — and her tense brows started to relax. “Oh, it’s like that?” She agreed to try again another day.

These feelings of failure can start early. I won’t pretend that my one conversation with this girl — a smiling, happy, bright 6-year-old girl — will prevent this from reoccurring. At least, after reading and giving this post some thought, I recognized the symptoms faster the usual. While it’s better to be proactive than reactive, some reactive treatment of math anxiety is better than nothing. Until we figure out a better way…

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For years, I gave nonsense or scattershot answers. “Logic and critical thinking are paramount.” “I care more about conceptual understanding than computational skill.” “A balanced, student-centered approach is always best.” “We buzzword to buzzword, not for the *buzz*word, but for the buzz*word*.” At best, each of my disjointed half-theories captured only a piece of the puzzle.

Worse still, none of my replies explained why I devote so much class time to plain old practice. If I was such an enlightened liberal educator, why did I assign repetitive computations for homework? On the other hand, if I was a traditionalist at heart, why did I fall head-over-heels for high-minded progressive rhetoric? Was I an old-school wolf, a new-school lamb, or some strange chimera?

Well, I’ve finally got my answer, and it only takes eleven words: **Math is big ideas, approached from as many angles as possible.**

What do I mean by “big ideas”? Well, here are a few examples:

And what do I mean by “many angles”? I mean that, in our best moments, my students and I come at these ideas like undergraduates approaching a dessert buffet: relentlessly, purposefully, and from all sides.

First: the historical angle. Even when the names-and-dates history doesn’t fit into my lesson plans, I try to contextualize each idea as part of a long lineage, to show how it answers a question, unlocks a door, fills a hole. I want my students to see each idea as one scene in a grand narrative of mathematical discovery.

Second: the verbal angle. English class isn’t the “opposite” of math class, as too many students think. Rather, good language skills empower us to discuss ideas of all types and stripes, especially mathematical ones. A precise and evocative vocabulary is beyond precious. Language allows us to debate productively, to learn as a collective, to think as a team.

Third: the scientific angle. Math’s most explosive ideas send shockwaves throughout the sciences. Physics, obviously—but also economics, biology, geology, chemistry, even psychology and sociology. Math has a symbiotic relationship with the sciences: it furnishes them with a powerful toolkit, and they provide it with concrete examples, a corporeal form for its abstracted soul.

Fourth: practice. Math without any computational practice is a mushy math, a math with no spine. To understand what makes, say, linear equations tick, you’ve got to solve ‘em, graph ‘em, play with ‘em in a hundred different ways. You can’t grasp patterns until you’ve worked through examples. Without multiplication facts at your fingertips, you’re unlikely ever to apprehend deep truths about the distributive property. If you’ve never spent a day multiplying out products of the form (ax + b)(cx + d), then you’ll never internalize the methods for factoring quadratics.

So there you have it. Big ideas from many angles.

I don’t always succeed from every angle—I might botch the history, or shortchange students on practice, or be just plain ignorant of the relevant applications. But that’s why I’m glad my students have *other* teachers—each with their own philosophy, each in their own distinctive way enriching our students’ understanding of those crucial big ideas.

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*Question #1: Which is the best ‘i’ in Mississippi?*

*Question #2: If humans could replenish their teeth forever, like sharks do, would life be better or worse?*

*Question #3: Which Kardashian sister is most likely to…*

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Mathematical tools are similarly distinctive. They harness industrial-strength power—think of Taylor series, or completing the square. Mathematical tools shine floodlights into dark corners. They unlock doors, solve problems, and make attentive students utter, “Whoa, deep.” They often come with complex instruction manuals, requiring weeks (or months (or years!)) of technical training to master.

Mathematical toys… not so much. They’re simple to grasp, fun to handle, and not much substantive good to anyone. Think of Sudoku puzzles, or differentiating cos(cos(cos(cos(x)))). We might get a kick out of poking and prodding such problems, but solving them won’t teach us anything fundamental about the workings of the universe or the necessities of logic. Toy problems aren’t floodlights; they’re more like flashlights dangling off of a keychain.

But just as the Incas mistook the wheel for a mere toy, sometimes mathematicians get it wrong. Sometimes what seems to be a toy is, in fact, a powerful tool.

Sometimes a toy is just a tool in waiting.

Think of the perfect numbers. Despite the flattering name, they’re not particularly central to number theory. They’re integers whose proper divisors add up to the original number itself. For example, take the factors of 6:

Add them up, and you get 1 + 2 + 3 = 6. Perfect! Or consider 28. Its factors are:

Add them up, and you get 1 + 2 + 4 + 7 + 14 = 28. Again—perfect!

Perfect numbers strike me as a quintessential example of a mathematical toy. They’re easy to explain and recognize, yet surprisingly hard to track down. (The next two are 496 and 8128.) They have no obvious conceptual significance or physical application—only a cute name, a clear definition, and just enough mystery to tickle the mind.

But you can also argue that perfect numbers have the potential to be a powerful tool. They combine two of the most elementary operations in mathematics: factorization (i.e., listing proper divisors) and addition (summing those divisors). That means they occupy prime real estate in the mathematical realm: a special vortex at the intersection of addition and multiplication. So perhaps finding perfect numbers isn’t just a silly puzzle for idle minds.

Perhaps the game really means something.

There are two big unanswered questions about perfect numbers. First, do they go on forever? They clearly grow sparser and sparser (after 8128, the next one is 33,550,336). But a family of numbers can continue forever, despite growing increasingly scarce—think of the primes. So do the perfect numbers constitute an infinite family? Or is there, somewhere out there, a final and largest perfect number, the king of them all?

The second question is: Are there any odd perfect numbers? The answer appears to be no—mathematicians have proven that any odd perfect number would need to satisfy absurdly strict conditions—greater than 10^{1500}, not divisible by 105, and so on. (They’ve stopped just short of proving it would need to float like a butterfly and sting like a bee.) The mathematician James Joseph Sylvester went so far as to declare:

So an odd perfect number would be semi-miraculous. But no one has yet proved it impossible.

The answers to these two questions may shed light on whether the perfect numbers are tool or toy, meaningful or meaningless. Maybe the proofs will reveal deep truths about the integers, a hidden significance to the perfect numbers. Or maybe the perfects themselves will remain a cheap trinket, but the techniques developed to work with them will find applications far and wide—so that the perfect numbers will, in the end, open doors to higher truths.

Or maybe the proof will reveal nothing exciting at all, and the perfect numbers really are just a silly game.

Part of math’s joy—and its frustration—is that it’s hard to tell the tools from the toys in advance. The perfect numbers may look like an Easy-Bake Oven for now. But one day, some clever mathematician may use them to cook a beautiful soufflé.

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I’m talking about tax brackets.

Tax brackets enact a simple idea: that not all income should be taxed equally. The first $8000 that I earn is likely to be precious—rent and grocery money.

For me, income above $30,000 is a little less precious—it’s vacation, savings, and new computer money. And if my teaching/blogging career somehow pulled in over $250,000 (I can dream, right?), those last few thousand bucks would be nice, but not terribly precious.

That’s why the tax code grabs a larger share from higher income levels. The more money you have, the less precious an extra dollar becomes. But this is where tax brackets have been maligned and misinterpreted. They don’t apply to *people*. They apply to *money*.

To see the difference, consider a simple two-bracket world, where all income below $8000 is taxed at 10% and all income above $8000 is taxed at 20%.

If I earn $1, I pay that low 10% tax rate, which comes out to $0.10.

If I earn $2, I’m still paying just 10%, so my total tax is $0.20.

If I earn $3, I’m still paying 10%, so I pay $0.30 total.

You get the idea. This continues until I earn $8000. All that income is taxed at 10%, so my total income tax bill comes out to $800.

But then my boss comes up and says, “Hey, great work teaching/blogging, Ben! Here’s an extra dollar!” Now I’ve made $8001.

I assume this means I pay 20% tax now… but 20% of $8001 comes out to roughly $1600. That’s double the total tax I was paying before the measly bonus. Did an extra $1 in earnings really just cost me $800 in extra taxes?!

No. That’d be crazy.

The 20% tax applies *only to my last dollar earned*. That’s $0.20. So my total tax didn’t skyrocket from $800 to $1600. It just nudged upward slightly, from $800 to $800.20.

As I said, brackets don’t make distinctions among people. They only distinguish between different types of money. The first $8000 I earn will *always* face a 10% tax rate, no matter how much more I go on to earn. Even if I earned a billion dollars, the first $8000 would get taxed at 10%. You, I, and Bill Gates all pay that same 10% on our first $8000.

This system avoids perverse scenarios where earning extra income actually winds up costing me money. In the good ol’ US of A, earning an extra dollar will always increase your taxes, but it’ll always increase your after-tax income, too. When people fret, “Oh no, that’ll bump me up into the next tax bracket!” they’re either joking, crazy, or confused about how taxes work.

When Congress debates, say, a 40% tax rate for people earning more than $250,000, this doesn’t mean a family making $250,001 has to pay 40% of their income in taxes. They’ll only pay 40% on that very last dollar earned.

Does this mean a progressive income tax is better than other systems, like a value-added tax? Not necessarily. But while you’re grumbling about taxes this month, remember to tip your cap to the humble tax bracket, the unsung hero of the IRS.

*P.S. Check out HAL 10000′s comment below for a more nuanced take on the rare situations when earning $1 extra really can increase your taxes by more than $1.*

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“Well, that explains why he’s totally sober. It’s hard to get much liquid in those things.”

“How’s the new iPhone?”

“Eh—it’s diffeomorphic to the last one.”

“Everybody, back up! Donnie’s going to show us a trick, and he needs six dimensions.”

“Here’s your gift! You’ll never guess what it is.”

“A coffee mug.”

“HOW DID YOU KNOW?!”

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