(see posts 1 and 2)

Somehow, I suspect I wouldn’t survive long on the frontier.

Drop me in the American West, circa 1850, and I fear my math-blogging and bad-drawing skills might not carry me far. I need indoor plumbing. I need the rule of law. I need chain coffee shops. I’m not cut out for the frontier.

And yet the frontier is exactly where I found myself the other day, when I came across this formula in the wonderful *Penguin Book of Curious and Interesting Numbers*, by David Wells:

I decided to play around with this product a bit. After all, what are products for, if not playing around?

(Go ahead and play with your Apple products. I’ll play with my infinite ones. We’ll see who has more fun.)

I felt like there should be an easier way to write this expression, exploiting the repetition of factors, so I gave it a shot, and created this:

Then my brain exploded and the universe dissolved around me, because I had just punched logic in the face, and it had punched me back.

The left side of that equation is π/2. It’s roughly 1.57.

The right side of that equation, however, is a product of many numbers—*all of them below 1.*

What happens when you multiply two numbers smaller than 1? You get *another* number smaller than 1.

How the heck could that equal 1.57?

Somehow, law had broken down. I was stranded on the frontier, with no cavalry or sheriff to come save me. This was a dispute I’d have to resolve myself, by hook, by crook, or by showdown at high noon.

At this point, I faced three grim possibilities:

- The cool book where I found this formula was wrong.
- Logic was wrong.
- I was wrong.

Pitting my own abilities against (1) those of David Wells and (2) the tensile strength of the fabric of the cosmos, it became pretty clear where the error lay. So I set out to find where I’d gone wrong.

Going back to the original equation, I came up with an alternative way to rewrite it:

This one was clearly greater than 1. (After all, it was an infinitely long product, with each of its factors greater than 1.) Checking a few terms in Excel verified that it was approaching π/2.

Then I found another way to rewrite the original:

This one starts at 2. Then it gets smaller and smaller, by tinier and tinier adjustments. Once again, it seemed to approach π/2.

So what had I done wrong in my original approach?

Soon enough I realized it. I’d made a mistake that seemed perfectly innocuous at the time but had, in fact, contained the seed of my own annihilation. I’d made a move that doesn’t matter with *finite* products, but which is forbidden with *infinite* products.

I’d forgotten to multiply by 1.

It perhaps goes without saying, but when you’re working with infinities, you need to be careful. Notation that’s usually harmless can explode like dynamite, making rubble of everything.

Infinity is the frontier. Law breaks down.

Back home, in the realm of finite products, multiplying by 1 makes no difference. 3 x 2 is the same as 3 x 2 x 1 (or as 1 x 3 x 2 x 1 x 1, for that matter). Finite products—that is, lists of factors that eventually end—can be rearranged to your heart’s content.

Not so with infinite products. Just look what else I can do, if I allow myself to play around with 1’s:

Throwing in extra 1’s allows me to rearrange the numerators and denominators virtually however I please. It seems I can make this product as large or as small as I want!

I should also confess that steps like the second one above – where I take a long product of fractions, and turn it into a single fraction with a long product on top and another long product on bottom – are a little dubious in this setting. With finite products? Fine. With infinite ones? Not so much.

In short: the frontier is a strange place.

I’m working here with products, but the same dangers apply with sums. Take this one, called the *alternating harmonic series*:

This sequence starts at 1.

Then it jumps backwards to ½.

Then it jumps forwards to 5/6.

Then it jumps backwards to 7/12.

Then it jumps forwards to 47/60.

As you go, the sum keeps jumping forwards and backwards, forwards and backwards… but by smaller and smaller steps. Looking carefully, you can see that it’s hovering around a certain destination—a place it will never in our lifetimes reach, but which it *approaches*.

That point is smaller than 1.

It’s bigger than ½.

It’s smaller than 5/6.

It’s bigger than 7/12.

In fact, it turns out to be roughly 0.693, or more precisely, ln(2).

So far, so good. But what if I rearrange the terms, like this?

We start at 1.

Then we add something positive.

Then we add something *else* positive.

And so on.

Clearly, the answer has to be larger than 1—which 0.693 is decidedly not. So I’ve changed the result sum, simply by rearranging the terms.

That’s not supposed to happen with addition! 3 + 4 is the same as 4 + 3. Making your family members stand in a different order shouldn’t make your family bigger, smaller, smarter, or taller—it’s the same family, isn’t it?

But with infinite things to add, the order in which you arrange them turns out to matter a great deal. In fact, this sequence can be rearranged to form literally any number you like.

Why do we explore this mathematical frontier? What drives us to abandon the safe coastal cities that we’ve always known, and throw ourselves into the unknown interior of the mathematical continent? Aren’t we afraid? Mightn’t we get eaten by bears, swallowed by rapids, or tangled in paradoxes? Why do we risk it?

Simple: because humans are explorers.

It’s as true of mathematicians as it is of Lewis and Clarke. We want to chart new landscapes, to find fresh challenges, to go where our old ways don’t necessarily help or hold true. We enjoy the thrill of surviving on our wits alone.

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(see also part 1)

A few weeks ago, the webcomic Saturday Morning Breakfast Cereal posted a cartoon about the harmonic series.

(Obviously it’s a mistake to post an actual cartoonist’s work alongside my own second-grade-quality scrawl, but hey, maybe I’ll benefit from a math humor cheerleader effect.)

Now, what is the harmonic series? It’s this:

The sum never stops. It goes on forever and ever. Lovely, yes, but does it—in any meaningful sense—“equal” anything?

That is to say: Does it level out at some value?

Or does it just keep rising, forever and ever, eventually exceeding a million, then a billion, then a trillion, and so on, surpassing any ceiling or limit we might imagine?

Perhaps your first thought is this: “You silly man; you’ve already answered your own question. You said the sum goes on forever. So it must be infinite.”

Not so fast.

Take a look at this series, for example:

We start at 1. The next term brings us halfway to 2. The next term brings us halfway *again* to 2. The next term brings us halfway *AGAIN* to 2.

And so on.

This sum is “infinite” in the sense that it goes on forever, with no final term. But it’s finite in the sense that no matter how many numbers you add, you’ll never exceed 2. Sure, you’ll get close—achingly, painfully, infinitesimally close—but you’ll never surpass it. We say that the sum “converges” to 2.

Now, back to our original sum, the harmonic series. What should it equal?

Well… let’s look at some running totals as we go along.

Where is it going to settle down?

It isn’t.

To be clear, this total grows *incredibly* slowly. In fact, the word “slowly” fails to evoke its agonizingly incremental pace. Instead, imagine that you’re waiting in line at the DMV.

And there’s only one employee.

And that employee is one of those **talking trees** from Lord of the Rings.

And he’s stoned.

And the line includes all 7 billion people on earth.

That line you’re in? It moves like ball lightning compared with the growth of this series.

And yet… this series never settles down. This sum eventually exceeds a million, then a billion, on its way to the stars. In mathematical language: the series *diverges*.

This is all weird enough. But now we get to the truly strange part of the whole ordeal, the truth that prompted the inimitable** Zach Weinersmith** of SMBC to build a punchline around it:

If you throw out the numbers with 9’s in them, the series is small enough to converge.

“Nonsense, you gullible old toad!” you are perhaps shouting to your screen. “Why should throwing out the numbers with 9’s make such a difference? We’ve still got *all the other numbers*!”

Again, I say: not so fast. You’re making a classic mistake. When you think of “numbers,” you’re only picturing *little* numbers.

“No I’m not!” you may say. “I’m thinking of big numbers. Huge numbers. Like 9 million, or 47 billion, or 228 trillion.”

Exactly my point: small numbers.

You see, the longer a number gets, the harder it is to avoid a 9. Every time you add a digit, you add a new opportunity for a 9. That might not feel like a big danger—after all, those 9’s will pop up only 10% of the time. But look what happens:

It’s as if, every time you type a digit, there’s a 10% chance that a giant numeral “9” falls from the sky and crushes you. For short numbers, with just 2 or 3 digits, you’re not very worried. The chances are in your favor.

But now you begin to type a 100-digit number. How do you like your chances? Sure, that giant “9” probably won’t fall on *this* digit… nor on *this* digit… but how long do you think your luck will last? Eventually you’ll get unlucky, and that “9” will come plummeting from the sky.

In the long run, *most* numbers have 9’s in them. Virtually all of them, in fact. So if you throw out these numbers from the harmonic series, it’s no surprise that it now converges. You’ve thrown out almost the entire series!

Want to know the weirdest part? The same logic applies to a longer sequence of digits. Say, 999.

At first, most numbers won’t have this sequence. But picture a billion-digit number. Just writing this number out—in size 8 font, double-siding the printing to save trees—takes a stack of paper as tall as a house.

Surely *somewhere* in there the number is bound to have the digits “999” in that order, right?

The same is true of *every* equally big number. And MOST numbers are this big! After all, there are only finitely many *smaller* numbers, and infinitely many *bigger* numbers. So, again, we’re throwing out almost the entire series.

Thus, the harmonic series *also* converges if you throw out all the numbers that include the digits “999.”

Perhaps you can see where this is going. (If so, you may be experiencing vertigo and/or nausea; this is normal.) The logic above works for *any string of digits* you can possibly think of. Even, say, a sequence of a million 9’s in a row.

So, in conclusion, the harmonic series *diverges*. It eventually outgrows any ceiling you’d put on it.

But simply throw out the numbers that happen to include a string of a million 9’s in a row… and suddenly the series *converges*. It plateaus. There’s some value that it will never surpass.

Somehow, by excluding only those numbers with a million 9’s in a row, you’ve changed the nature of the series.

In the words of Weinersmith:

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**A.**The threat of Grexit**B.**The bittersweet knowledge that someday, when all of this has passed, we’ll have fewer opportunities to use the amazing word “Grexit”**C.**People thinking functions are linear when they’re SO NOT LINEAR**D.**Other (e.g., cat bites)

If you answered C, then congratulations! You are probably a teacher of math students ages 13 to 20, and we all share in your pain.

For everyone else (including you poor cat-bitten D folk), what are we talking about? We’re talking about errors like these (warning—mathematical profanity ahead):

What’s wrong with these statements? Well… everything.

I don’t blame the kids, of course. These errors are a natural—perhaps even inevitable—byproduct of the way we teach mathematics.

Every rule in secondary school math is actually a statement about objects—shapes, numbers, whatever. But they don’t *feel *like that to students. They feel like purely symbolic manipulations, rules for how x’s and y’s move around a page, not for how actual things relate to each other. When you see rules that way, it’s easy to fall prey to certain systematic errors.

Why is this particular error so tempting? Perhaps because it closely resembles a *real* rule students have learned:

This is called the Distributive Law, and it’s actually a deep fact about the numbers, an essential link between addition and multiplication.

You can discover it through numerical examples:

Or you can understand it with arrays:

Or you can use my colleague’s wonderful phrasing and think of an expression in parentheses as a “mathematical bag”:

The distributive law is crucial. It underlies most of what we do in algebra, such as factoring and “gathering like terms.” So what’s the problem?

It’s that students don’t learn the distributive law as a fact about *numbers*. They learn it as a fact about *parentheses*.

And unfortunately, mathematicians use parentheses in two very different ways: first, to group numbers; and second, to designate the inputs of a function.

The notation “f(x)” doesn’t mean “f times x.” It means “what I get when I put x into the function.”

But our faithful symbol-pushers don’t always catch the distinction. They see parentheses, and think, “Oh yeah! That rule I learned should apply here.”

What’s the solution? Here’s my current game plan:

**Teach the distributive law more carefully**. Draw pictures. Work examples. Talk about “bags.” Make sure they understand the meaning behind this symbolism.**Teach function notation much more carefully**. Give them the chance to practice it. Think like Dan Meyer and seek activities that create the intellectual need for function notation.**Keep stamping out the “everything is linear” error when it crops up**. Like the common cold, it’ll probably never be entirely eradicated, but good mathematical hygiene should reduce its prevalence.

What’s the only thing giving me pause here, making me doubt my nice, tidy theory of this error? In two words: Jordan Ellenberg.

In *How Not to Be Wrong*, he catalogs a variety of cases where people falsely assume “all curves are lines.” His examples are never just symbolic. Each time, people are messing up on a conceptual level. They’re not just pushing x’s and y’s around a page, but genuinely believing their own wrong statements.

So is my approach off-base? Are people actually acting on false beliefs about functions, or (as I’ve posited) are they failing to think about them as “functions” at all?

I’d love to hear what you think. And if you’ve got solutions for the Greek economy or cat bites, well, I’m all ears.

*UPDATED*: edited to correct the fact that I do not know the difference between numbers and letters. Oh well, like I always say, “C strikes and you’re out.”

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If there’s one thing about math that people love—and to make it through the average day, I have to believe there’s at *least* one—it’s infinity.

Throw the word into a math lesson, and ears perk up. *Infinity? Did he say infinity? *It’s like a distant celebrity, the subject of endless gossip and rumor. “I heard infinity isn’t even a number!” “Only the universe is *really* infinite.” “My last teacher said infinity times two is the same as infinity.” “I can use infinity to prove that 1 = 0!”

Infinity is a sound too high for our ears, a light too bright for our eyes, a taste so sweet that it would tear through our tongues like acid. Basically, it’s mathematical Mountain Dew.

Tellingly, all of our words for infinity define it by what it isn’t. *Infinite*: not finite. *Unlimited*: not limited. *Boundless*: without bounds. It’s hard to articulate what infinity *does*, so we settle for naming what it *doesn’t*: end. Infinity is the Anansi of mathematics, a trickster spider weaving baffling webs of paradox and contradiction.

Take this example: which has more numbers, List A or List B?

Well, they’re both continue forever, so they’re both infinite. But the first list includes every number on the second (2, 4, 6, 8…), and lots more as well (1, 3, 5, 7…). Thus, clearly, List A is the bigger group of numbers. Right?

Wrong. The lists are exactly the same size.

To see why, just go through list A and double every number on it, like so:

Now, doubling each number shouldn’t affect the size of the list. We haven’t added any items, or taken any way. It’s like taking a group of people and giving them fake beards—you don’t create or destroy any humans in the process.

And yet… without changing its size, we’ve somehow turned list A into list B. That can only mean that the two lists were the same size to begin with.

What’s happening here? Throughout our lives, we’ve relied on a simple rule for comparing the sizes of groups: *If one group has all the members of another, plus some others, then it must be bigger.* For example, if your list of Facebook friends includes all of mine, *plus *you’re friends with Kanye West, then you have more friends than I do (and probably cooler weekend plans, too).

And that’s perfectly true, for finite groups. But infinite groups… those are another matter.

Infinity shreds our intuitions. The startling truth is that all infinite lists are the same size, regardless of whether one happens to include the other.

The mathematician David Hilbert captured this paradoxical truth with the parable of Hilbert’s Hotel. Imagine there’s an infinite hotel, in which every room is currently occupied. (There’s a very popular convention in town, apparently.)

You show up, asking for a room. “Sorry, they’re all occupied,” says the innkeeper—who’s probably a little frazzled, what with the infinite calls for room service coming in.

“Yes,” you say, “but just ask everyone to move down one room, and there will be space.”

It’s true. The person in Room 1 goes to Room 2. The person in Room 2 goes to Room 3. The person in Room 136,004 goes to Room 136,005. And so on. Everybody has someplace to go.

Now, whether it’s good manners to show up at a hotel and demand that a literally infinite number of people suffer a minor inconvenience just so you can avoid trying the Best Western down the road—that’s another matter. You’re probably a bad person for making this request. But you’re a fine mathematician.

Can we do this without inconveniencing an infinite number of people? Unfortunately, no. For example, if you take Room 1, and send that guy packing, he’s going to have the same problem as you: every room is now occupied. As long as you’re shuffling a finite number of rooms, normal math applies, and somebody will be left out. Your infinite problem demands an infinite solution.

It gets crazier. What if you’re coming straight from the Infinite Man March, and you’ve got an infinite line of new friends in tow? Can the hotel hold you all now?

Impressively: Yes.

This time, each current guest needs to look at their room number, double it, and move to that doubled room. So the chap in Room 1 moves to Room 2. The gal in Room 2 moves to Room 4. The fellow in Room 3003 moves to room 6006. And so on.

At this point, which rooms are filled? Only the evens. So you and your infinity friends fill the odd rooms, and voila! Problem solved.

(Unless you’re the bellhop in charge of setting out the continental breakfast; then, the problems have just begun.)

At this point, infinity is perhaps beginning to feel like a squishy, flexible, silly-putty sort of idea. Infinite lists? They’re the same size. Infinity plus one? Still the same size. Infinity plus another infinity? Yup, still the same size. It raises the provocative question: Are *all* infinities the same size? Tune in next week for the answer.

Oh, who am I kidding? The answer is no. Some infinities are *much* bigger than others.

Tune in next week for *why*.

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**Logarithms**

**Algebraic Thinking**

**Infintesimals**

**Dress Codes at Mobius High School**

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What’s the difference? Well, in the UK, a “rubber” is an eraser, and in the US…

Oh! You mean for “grades.” Well, **in the US, “grades” are given by teachers**. They aim to assess the quality of your work throughout an entire term. **In the UK, your “grades”—the ones that matter to universities—are your scores on a handful of high-stakes exams**.

In the US, the scores you get from your teachers form the bulk of your permanent academic record; in the UK, those scores don’t even appear.

In my experience, both American and British educators react with utter horror to the opposite system. And, weirdly, many objections are mirror images of one another.

**Objection #1: Your System Drowns Students in Testing**

*The American shock*: “Wait… so years of work come down to a single test? Doesn’t that put tectonic pressure on the kids, and give them a distorted, blinkered view of education as nothing but glorified test prep? How do you convince them there’s more to learning than test scores?”

*The British horror*: “Wait… so you’re giving meaningful tests *every few weeks*? Doesn’t that put kids under constant pressure, the perpetual threat of judgment—25 hours of testing, spread throughout the year, instead of just 4 hours at the very end? How do they ever find time to learn?”

*My take*: Both accusations ring true, because both systems are testing-driven. True, the word “test” carries different connotations—in the US, it’s less definitive and final, more a landmark signifying the end of a topic. (Many teachers allow you to raise your score with corrections or retakes.) But ask the students in either country, and they’ll tell you: the tests are what count.

**Objection #2: Your System Misjudges Student Achievement**

*The American cry*: “There’s so much a test can’t capture! You’re trying to judge years of work by a single day—and not only that, but you’re employing a narrow, time-pressured measure of achievement. Your universities are basing their decisions on a woefully incomplete picture.”

*The British lament*: “So your grades are just in the hands of… whoever happens to teach you? What if my teacher is particularly stingy, or overly generous? How do you ensure fairness across the system? Your universities are basing their decisions on whims and opinions, not real data.”

*My take*: Neither accusation holds water here.

Yes, the British tests are imperfect. But I find they’re a hell of a lot better than most of their American cousins. There’s no multiple-choice to be found; every question requires a written answer. Also, they assess specific curricula and skills—far more akin to AP tests than the nebulous, intimidatingly vague SATs.

As for the American system: yes, Brits, some teachers are arbitrary, inconsistent graders. But each kid’s transcript includes dozens of grades. Some may be unfairly high; some, unfairly low; some (probably most) about right. Average it all together, and you’re likely to get a pretty fair assessment of student achievement.

How can such different systems give equally valid results? Well, teachers, think back to your last round of exams. Did any students bomb after doing well all year? Did anyone ace the exam after tanking all year? Or—as always happened with my classes—did your final exam scores show a shockingly clean alignment with exactly what you’d seen from students all year long?

Neither system is perfect. But both are perfectly functional.

** **

**Objection #3: Your System Warps the Student-Teacher Relationship**

*The American diatribe*: “In your system, teachers are powerless. They’re glorified test prep tutors, with no opportunity to inspire, or to share their love for the subject. All that matters is the standardized test. How can they leave any impression on the students’ lives? How can they forge any meaningful bond?”

*The British harangue*: “In your system, teachers are despots. They write and grade tests themselves, with no external check. Won’t this allow them to bully the poor students, whose academic futures they singly control? And wouldn’t this turn students into ghoulish hyenas or terrible sycophants, gnawing and sucking at their teachers for any extra points they can get?”

*My take*: Partial truths on both sides here.

First of all: Yes, Brits, that’s exactly what happens in the US, and yes, it’s awful. Students feel cheated by teachers, and teachers feel hounded by students—not always, but often enough to cause headaches. I’m not sure this is the fault of our grading system, though. Instead, I blame our litigious instincts. We’re all salesmen and lawyers at heart. If there’s an advantage to be had, we’ll press for it, no matter what the institutional framework. “Deference to authority” is not the American way.

Second: Yes, Americans, I don’t think the Brits quite share our image of the “inspirational teacher.” Just compare the websites of Teach for America (US) and Teach First (UK), organizations with nearly identical missions of sending bright, ambitious young people to teach in the nation’s neediest schools.

The British version begins: “How much you achieve in life should not be determined by how much your parents earn.”

The American version begins: “Change and be changed.”

Whatever your feelings on such soaring rhetoric, it’s clear that Britain has a slightly narrower vision of a teacher’s role. Teachers aren’t expected to enmesh themselves in students’ lives—and perhaps it’s the students’ loss. But again, I don’t think this is a question of grading systems. It’s culture. The British are not a terribly sentimental people: the “stiff upper lip” remains a cultural ideal (albeit a fading one). “Change and be changed,” with its mushy American hug-it-out emotionality, is a simply un-British notion.

So, yes, American and British teachers relate differently to their students. But I don’t think it’s the fault of the grading systems.

** **

**The Actual Difference: Singular vs. Plural**

So, to recap, I’ve covered three mutual objections that Americans and Brits might levy against each other:

- “Your system over-emphasizes testing!” Yes, but so does yours.
- “Your system misjudges student achievement!” Not really.
- “Your system warps the student-teacher relationship!” From your perspective, perhaps; but it’s an issue of culture, not grading system.

Where does that leave us? Can these two systems, which feel so fundamentally different, actually be equivalent—even interchangeable?

Of course not! Don’t be silly.

To grasp the essence of the difference, look no further than the names of the countries. Both are “United,” but one is a singular “Kingdom,” and the other is a plural patchwork of fifty distinct “States.”

Brits expect standardization and nationalization. They expect their country to act as a single unit, evenhanded and fair. Their entire college application system, for example, runs through a centralized hub. You can apply to precisely six schools, including either Oxford or Cambridge, but not both. If you’re an American thinking, “How strange—that’s like if you had to pick between Harvard and Yale!” then you don’t realize the half of it. It’s more like if you could only apply to one school in the entire Ivy League.

In the UK, everyone surrenders minor personal preferences here or there for the sake of cohesion. They don’t think twice about it. It’s what “United Kingdom” means to them.

America is different. We remain individualists, frontiersmen at heart, and we expect local control. When teachers find themselves forced to deliver scripted curricula, following every step of a prescribed course, it almost always leads to revolts. We expect to write our own tests and grade them however we like, serving our own private visions of how our subjects should be taught. We take a diversity of approaches for granted. “Of *course* your algebra class wasn’t precisely the same as mine; we went to different schools!”

I’m not trying to take sides on Common Core here; in my view, it represents a fairly modest degree of nationalization. But the bitter opposition against it speaks to our powerful American belief that education is a local matter.

It’s strange being an American in the British system. I often find myself objecting to the quirks of how British exams are scored. “If a question is posed in degrees, an answer in radians gets no credit? That’s so silly!” “Saying a probability is ‘1/6’ is fine, but ‘1 in 6’ receives no points? That’s so unfair!”

(Both these examples are from last Friday. This happens all the damn time.)

In moments like this, my colleagues don’t know what to make of me. Following such conventions is second nature for them. It takes no effort. It’s as if I said to them, “What’s that stupid symbol you guys use for 7? I’ve always drawn a little devil face instead! It’s mathematically equivalent!” *Well, okay Ben, that’s fine, except you can’t teach that to your students because no one will have any idea what the hell they’re talking about.*

I stumble over my own convictions, my ingrained belief that I shouldn’t have to bow to anyone else’s approach. Teaching is full of a thousand arbitrary decisions, forks in the road where either direction is really fine. I’m accustomed to picking my own path, according to my own taste. It’s strange, now, needing to pay careful attention to the road signs, lest I steer my students against the flow of traffic.

Which system is better? I don’t think there’s a definitive answer, any more than it matters whether a country drives on the left side of the road or the right. What I’m learning, though, is this: Whatever country you’re in, you’d better make sure you’re driving on the same side as everyone else.

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In February, I was throwing together a geometry test for my 12-year-olds. I wanted a standard angle-chasing problem, but – and here’s the trick – I’m lazy. So I grabbed a Google image result, checked that I could do it in my head, and pasted it into the document.

But when I started writing up an answer key, I ran into a wall. Wait… *how* did I solve this last time? I trotted out all the standard techniques. They weren’t enough. A rung of the logical ladder seemed to have vanished overnight, and now I was stuck, grasping at air.

Eventually, some colleagues and I solved it by with industrial-strength tools: the Law of Sines, the Law of Cosines, and some fairly sophisticated algebra. Yet the problem looked so elementary, I felt sure that our approach was overkill—that we were bashing down the door, whereas a defter hand could simply pick the lock.

Apparently, I’d actually chosen a famous gem of recreational mathematics, born in 1922 from the mind of Robert Langley, and known since as “Langley’s Adventitious Angles.”

And, as I suspected, the niftiest solution requires no trigonometry or algebra, just a single ingenious move: construct a line here, at a 20^{o} angle to the base.

It’s a masterstroke. It pops the lid right off of the problem. And, at least to me, it’s utterly un-guessable. If you had a thousand monkeys with a thousand typewriters and a thousand protractors, you’d get full verses of Shakespeare long before any of our furry friends stumbled upon this solution.

Sometimes the general techniques fail, and you need a sneaky trick. That’s life.

To me, a “trick” is a disposable insight, a funny little key that opens one particular closet door. And a “technique” is something more useful: a skeleton key, fitting the lock for a whole hallway of rooms.

But is the difference always so clear?

There’s a certain algebraic move that I first encountered in high school, which you might call “multiplying by the conjugate.”

You use it here, to simplify an expression with square roots:

And here, with complex numbers:

And here, again with square roots, to solve a limit (without recourse to the heavy machinery of L’Hopital’s Rule):

And here, to boil down a trigonometric expression:

At first glance, it feels like a trick: a little too slick, a little too cute. It’s hard to imagine it blossoming into a general technique. But it eventually proves useful in a shocking variety of situations, cutting across the whole secondary mathematics curriculum. This funny little key, which barely looks like it should open a single door, turns out to open hundreds.

Ultimately, I can’t find any categorical difference between a trick and a technique. They’re both problem-solving innovations, lying along a continuum that runs from “almost never useful” to “useful all the darn time.” A technique is simply a trick that went viral, and a trick is simply a technique that fizzled out after a single use.

It makes me wonder how to define the job of the mathematician (and of her little spiritual cousin, the mathematics student). Is it to master existing techniques? To cook up new tricks from scratch? Or to recognize how the tricks of today can be nurtured and expanded into the techniques of tomorrow?

Yes to all of that. But of all the aspects of the mathematician’s job, I think the ability to birth truly new ideas is overrated. Leaf through the pages of history, from Pythagoras to the present, and you’ll find a stunning supply of tricks, techniques, and everything in between. In the five millennia we’ve been working on this, humanity has accrued a remarkable wealth of mathematical ideas. Yes, we want our fresh young minds to make new deposits into the bank of our knowledge; but we also want them to know how to access the existing funds.

I would never, in a million years, come up with the trick to solve Langley’s Adventitious Angles. But I don’t need to. Someone else already has.

By the way, as promised, here’s the rest of the solution to Langley’s Adventitious Angles:

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Math class shouldn’t be a mishmash pile of facts, thrown together haphazardly, like an academic version of *The White Album*. It should be a perfectly interlocking tower of truths, climbing upwards with singular purpose—an academic *Sgt. Pepper* or *Abbey Road*.

A good class isn’t a greatest hits record. It’s a concept album.

In that spirit, I’ve been taking each topic in the secondary math curriculum—algebra, geometry, calculus, etc.—and trying to boil it down to its one-word essence. Here are the rules of the game:

*You must choose a single word to complete the sentence, “[Branch of math] is the mathematics of _____.”*

For example, you might say, “Topology is the mathematics of dinosaurs,” or “Category theory is the mathematics of abstraction,” or “Combinatorics is the mathematics of sadness.” (To be clear, only one of those is remotely accurate; you have my sympathy, combinatorists.)

*You must pick a word that laymen would understand.*

No fair saying “Algebraic geometry is the mathematics of varieties.” What the heck is a “variety”? Like, varieties of breakfast cereal? You shouldn’t need training in a subject just to understand its definition.

*Your word should encompass as much of the subject as possible.*

It might not be a perfect fit. It’s okay if you don’t nail every detail. (Even *Sgt. Pepper* has lots of songs that don’t really fit the theme.) But you want to capture the spirit and scope of the subject.

So you could say, “Algebra is the study of equations,” but that’s weak sauce. Yes, 90% of algebraic work involves creating and manipulating equations. But it totally misses the essence. First, inequalities and expressions can be just as “algebraic” as equations; and second, lots of equations (like 2 + 2 = 4) aren’t remotely algebraic.

That’s the game. If you want to spend some time thinking of your own before being poisoned by my answers, then read no further.

Now, here’s what I’ve got:* *

This one’s a slam dunk. Calculus is already the most beautifully unified course in the high school curriculum, so the one-word treatment fits well here.

The first half of calculus deals with *derivatives*, which tell us how a quantity is changing at a specific moment. “That car is moving 80 miles per hour.” “The city is growing by 80,000 people per year.” “The clown is getting 10% angrier every time you insult his shoes.” It’s all about change. The second half of calculus deals with *integrals*, which allow us to aggregate lots of little, incremental changes to see their cumulative effect.

If you haven’t already cued up David Bowie’s “Changes” on YouTube, then what are you waiting for?

I don’t mean John-and-Yoko relationships, or even John-and-Paul relationships. I’m talking about relationships between quantities.

Pretty much all of elementary algebra boils down to the question, “How do *y* and *x* relate?” You explore different *types* of relationships: linear, quadratic, exponential. And you represent those relationships different *ways*: graphs, tables, equations.** **

“Shape” is a pretty good answer, too. Certainly, you spend most of geometry class studying shapes: triangles, rhombuses, spheres… But I like to think that geometry begins with more elemental ideas. Points. Lines. Planes. Right off the bat, you reckon with abstract concepts like “dimension.” That stuff’s more fundamental than the idea of a shape.

Look around, wave your hands in front of your face, and think about the weirdness of this three-dimensional universe that we occupy. Geometry is the mathematics of *that*. Yes, shapes are important in geometry, but its essence is deeper—and freakier.

My first thoughts were “triangles” and “waves.” But those two are pretty darn different, and that gap points to trouble. How are we going to reconcile such disparate halves? Is this class going to be our version of *Let It Be*—so promising in its conception, but a hot mess in its result?

Fear not! The coherence is there. Fundamentally, trig is about two different ways of understanding position.

First, you can count east/west steps (the x-coordinate) and north/south steps (the y-coordinate).

Or second, you can give a direction (the angle θ) and a distance (the magnitude *r*).

This captures basically all of trig. The SOHCAHTOA stuff, for example, is all about manipulating and integrating these two notions of position. Circle trig, meanwhile, is about translating between them: the sine and cosine functions swallow your direction angle, and then spit up the *y* and *x* coordinates.

To clinch the matter, what is the most powerful historical application of trigonometry? Navigation—which is to say, finding your position on earth.

Sometimes we know exactly what’s going to happen.

For everything else, there’s probability.** **

Maybe I’m overthinking this one. Maybe I should just say “data.” (And maybe the Beatles shouldn’t have released the insufferable song “Wild Honey Pie,” but there’s no going back now.)

But you know what? There’s something vacuous about picking “data.” It’s too obvious. Too technical. It rubs me the wrong way.

So let me make my case for “populations.” Notice how “data” always comes in the plural? Statistics is, fundamentally, about understanding groups. Think of mean, median, mode, range: these are all ways of summarizing a population, of boiling a diverse group down to a few illustrative numbers. Or think about percentile and z-score: these are ways of specifying an individual’s place in relation to the population. And finally, think about hypothesis testing: there, we’re using samples to draw inferences and conclusions about whole populations.

When mathematicians call a statement “true,” they mean something very specific. It’s different from what scientists, artists, plumbers, and politicians. (To be honest, I’m not sure *what* the politicians mean.) They mean that the statement in question follows irresistibly from others that we have already laid out.

So what is logic? It’s the study of how mathematicians use that little word: “true.” As you might expect, it underpins every other branch of mathematics, and it’s a worthwhile field of inquiry in its own right, too.

So, to summarize:

Agreed?

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Yet here I find myself, in Birmingham.

Not Alabama. England.

In some ways, it’s not so different. As my friend John advised before I moved: “The British speak English, care about money, and yell about politics. You’ll barely notice you’ve left.” But it’s not quite like home: the spellings, the roundabouts, the big red buses—and, most relevant for a teacher like me, the sometimes startling differences in the ways our two countries educate kids.

In less than a year of teaching, I’ve encountered some surprising disparities, each of which prompts the obvious question: Which way is better, the British or the American?

I have nothing to gain by these comparisons. If I favor America, my judgments will be dismissed as jingoism (just like 97% of the other things I say). And if I favor Britain, I will have my surrender lorded over me for months to come by English teenagers. Thus, I find myself in a predicament similar to many facing America today: a lose-lose situation. And, in the true American spirit, I shall plunge forward anyway.

In the future, I hope to tackle trivial little issues, such as university majors and the nature of secondary education. But this week, I began with the big stuff: mathematical naming conventions.

These are interchangeable. If you find anyone claiming a strong preference for one or the other, tell them to do take a deep breath, walk away, and go do something beautiful. Watch a sunrise, listen to a Beatles album, punch an investment banker—anything to remind you that life has meaning beyond misplaced pedantry.

*Winner*: Draw.

Yeah, this isn’t a mathematical difference so much as a linguistic one, but z is the third-most common name for a variable, so it comes up daily. Six months in, I’m already loving the “zed.” The alphabet is overloaded with “ee” letters (B, C, D, E, G, P, T, V). “Zed” is fresh. It’s different. It’s like the single onion ring in your order of French fries.

*Winner*: UK.

Welcome to my nightmare.

Multiplication comes up a lot in mathematics. Like, a *lot* a lot. To save time, we’ve developed the convention of omitting the multiplication symbol altogether (hence “2y” for “2 times y”). Sometimes, of course, this isn’t possibly (you can’t write “23” for “2 times 3”), but writing traditional symbol X would be confused for the variable *x*. So we use a simple dot instead.

Here’s the problem: My British textbooks still use that central dot to represent a decimal.

What does this mean? It means that multiplication haunts my days. When I tried to introduce the dot for multiplication, my students freaked out. (Even if they hadn’t, I’d still be wary of putting them at odds with their own countrymen by forcing different conventions on them.) So I settle for X instead. To avoid confusion I must (grudgingly) change the way that I write the letter *x*. I hate it.

After seeing this post, some colleagues (aged 23 and 28) pointed out to me that they never put the decimal in the middle. “Maybe it’s a generational thing,” they suggested. But if that’s the case, then it’s not just *me* who prefers the American way; it’s British people too.

*Winner*: US.

Many problems in America are diffuse. Cultural. Institutional. It can be hard to assign blame to specific people. But there’s a single villain behind the word *trapezoid*: Charles Hutton.

Historically, “trapezium” referred to a four-sided shape with one pair of parallel sides, while “trapezoid” referred to a four-sided shape with no parallel sides at all. As the erudite John Cowan explains: the “trapezium” looks like a trapeze, and the “trapezoid” has a shape “analogous to, but not the same as, a trapeze (as with *humanoid*, *planetoid*, *factoid*).”

So what’s Hutton’s problem? When he wrote his (otherwise wonderful) mathematical dictionary in 1795, he switched the two words. Americans inherited the blunder.

(Or, as the British would say, “blundre.”)

(Yes, I know the British don’t say that. I’m just trying to divert attention towards a debate where we’re on better footing than the trapezium thing.)

*Winner*: UK.

You know who uses this notation? Scientists, that’s who.

And you know what standard numbers look like? They look like numbers, British people. There’s nothing “standard” about this.

*Winner*: US.

This is unforgivable. The word “index” already has a perfectly distinct mathematical meaning, and the word “exponent” has none. It would be like referring to “division” as “subtraction.” We already have a meaning for that word, British people. You do Shakespeare a dishonor.

(In their defense, the British also recognize the word “exponent,” but they seem to default to “indices.”)

Also, most gratingly, the plural “indices” is often gives rise to the bastard singular “indice.” What the heck is an indice, Britain? Will you please stop saying that?

*Winner*: US.

Americans revise their essays and review for tests.

The British review their essays and revise for tests.

It’s a disarming adjustment to make when you hop the Atlantic. But since “vision” and “view” mean basically the same thing, it’s hard to pretend that this makes one iota of difference.

*Winner*: Draw.

This happened with friends: Some Americans and some Europeans were sitting around, shooting the breeze, when the topic of measurement came up. “How many feet in a mile?” a European asked. “It’s about 5000, right?”

“Five thousand, two hundred, and eighty,” the Americans all recited in unison. The Europeans broke into laughter that was spontaneous, derisive, and utterly justified.

There’s no defense. We are fools.

Yes, it’s hard to change systems, but the rest of the world managed it, whereas we in America continue to rear the next generation in our own scientific filth. 12 inches per foot, 3 feet per yard, 1760 yards per mile… 8 ounces per cup, 2 cups per pint, 2 pints per quart, 4 quarts per gallon… Our teachers have to *teach* this stuff. Our kids have to *learn *it. I love America’s spirit of independence and individuality, but this is ridiculous. Mothers like to ask, would you jump off a bridge if everyone else was doing it? The question here is, why are we jumping off the imperial measurement bridge when *no one* else is doing it? I guess we want to be special, distinctive little bridge-jumpers.

Yes, I expect to have my passport revoked in retaliation for such anti-American blasphemy, but there we are.

But, before we bow down before our British overlords, guess what? They don’t always use the metric system either! Yes, they use Celsius instead of Fahrenheit, but that one doesn’t really matter, because in daily life you never do unit conversions involving temperature. When it comes to the big ones, the Brits are wildly inconsistent. They mostly use kilograms, but will occasionally bust out “stones” (which are 14 pounds each). They use “metres” for short distances, but revert to miles for long ones. Yes, they’re “more metric” than Americans, but has no one told them that the whole purpose of standardized units is, you know, standardization and consistency? At least we’re sticking with our system, instead of hopping back and forth.

*Winner*: Draw. Or, well, both nations lose, but in different ways.

Whereas Americans refer to “The Pythagorean Theorem,” the British simply invoke “Pythagoras.”

I don’t really care. The British version is less of a mouthful, I suppose. It’s also wonderfully insane: When you say, “We can solve this using Pythagoras,” you’re effectively suggesting that a Greek man who’s been dead for 2500 years will personally stop by to help solve your math problem.

*Winner*: Draw.

**THE FOLLOWING ADDED 5/21/2015… or is it 21/5/2015? ANYWAY, THANKS TO COMMENTERS/COMMENTRES FOR THE SUGGESTIONS:**

What I love is that these both capture the insanity of irrational numbers. Americans think they’re radical; Brits think they’re absurd; and according to legend, the Pythagoreans find them so disturbing that they’ll kill anyone who dares discover them.

These words also perfectly capture the flavor of their respective countries, which is to say, the US is a nation of skateboarders, and the UK is a nation of Latinists.

*Winner*: Draw.

To the British, ( ) are just a special case of the many types of brackets, also including { } and [ ]. To the Americans, they’re different.

Yes, the singular “parenthesis” is often mislaid by Americans. But I like the adjective “parenthetical,” as in “a parenthetical remark” or “when it comes to discussing important mathematical issues, the choice of symbols is, at best, parenthetical.”

*Winner*: Draw. I apologize for my indifference; it’s far too British of me, innit?

I’ve been proctoring tests for years, and only this year did I get the distinct pleasure of invigilating one.

It’s delightful, invigilating. You hold vigil. You, in fact, *bring* the vigil. You *inject* vigil into an otherwise vigil-less exam room. It’s like being an Invigorator.

*Winner*: UK.

You hear both words in both places, but the UK leans towards -ising, whereas the US is happier to use “factor” as a verb.

If you care about this, then congratulations! You are probably the sort of person who can turn on any sport – up to and including the Little League World Series and youth cricket – and immediately find yourself a passionate supporter of one team or the other. Which is to say, you care too much about stuff, dude.

*Winner*: Draw.

Hey Britain, what’s the point in having an easy-to-remember mnemonic if everyone uses a different one? BIDMAS, BODMAS, BEDMAS, BIMDAS… yes, you could argue that the variety of acronyms reflects the arbitrariness of the convention, but that’s a cheap argument. You’re better than that.

*Winner*: US.

**FINAL SCORE: USA wins! **At least, until someone points out to me another difference that “favours” the UK, at which point it will be back to a comfortable stalemate.

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