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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In grad school, my wife took a class that assigned no homework. The topic was an advanced, hyper-specific area of research—the only plausible problems to give for homework had literally never been solved. Any answer to such a question would have constituted novel research, advancing the field and meriting a publication in a professional journal. The professor assigned no homework for the simple reason that there was no practical homework to assign.

This tickled me. I’d never thought of good questions like a fossil fuel. A nonrenewable resource. Built up over eons and consumed in minutes.

But the thought kept popping back up: *Good questions are a resource*. And in this new light, something started to make sense, an uncomfortable little fact that had nagged at me since my first year teaching.

I lurched sideways into the profession, like a bowling ball that hops into the next lane and knocks over the wrong pins. I knew nothing about pedagogy, other than whatever naïve intuitions I’d gathered as a student. So that year, whenever we encountered new types of problems, I simply told my geometry students exactly how to answer them.

It felt natural; it felt like my job. Need to prove these triangles are congruent? Do this. Need to prove that they’re similar? Do that. Need to prove X? Do Y and Z. I laid it all out for them, as clean and foolproof as a recipe book. With practice, they slowly learned to answer every sort of standard question that the textbook had to offer.

Months passed this way. But something wasn’t clicking. I kept seeing flashes and glimpses of severe misunderstandings—in their nonsensical phrasings, in their explanations (or lack thereof), in their bizarre one-time mistakes. Despite my best intentions, something was definitely wrong. But I didn’t know what.

And, more worryingly, I didn’t know how to find out.

I’d already coached them how to answer every question in the book. How, then, could I diagnose what was missing? How could I check for understanding? For every challenge I might give them, every task that might demand actual thinking, I’d equipped them with a shortcut, a mnemonic, a workaround. The questions were like bombs defused: instead of blasting my students’ thoughts open, they now fizzled harmlessly.

Good questions are a resource, and I’d squandered mine.

I couldn’t articulate it then, but I began seeing questions differently. They were not just obstacles to overcome, or boxes to check. Instead, they were my fuel. My kindling. My only real chance to ignite reflection, curiosity, and the impulse to seek out deeper truths.

Questions were not just things to answer; they were things to think about. Things to learn from. Giving the answer too quickly cut short the thinking and undermined the learning.

Good questions, in short, are a resource.

Solving a math problem means unfolding a mystery, enjoying the pleasure of discovery. But in every geometry lesson that year, I blundered along and blurted out the secret. With a few sentences, I’d manage to ruin the puzzle, ending the feast before it began, as definitively as if I’d spat in my students’ soup.

Math is a story, and I was giving my kids spoilers.

Now, I don’t believe in withholding the truth from kids indefinitely, or forcing them to discover everything on their own. But I’m aware now of a danger: if I tell them too early how the film will end, they may turn it off halfway through, and I’ll have a hard time convincing them of everything that they’re missing.

Questions aren’t merely targets that need to be hit. They’re also our arrows for hitting bigger, more elusive targets: concepts, connections, ways of thinking.

Questions aren’t the enemy. They’re the ammo.

As I move through my career, I find myself increasingly overcome with the opposite difficulty: too many good questions. Too much cool stuff. A sprawling wealth of worthwhile problems, such that my students and I couldn’t possibly get around to half of them.

But sometimes, I still slip up. I think my students are learning deep concepts and strategies, but they’re only parroting phrases or mimicking procedures. Too late, I realize what’s happening… but by then the suitable questions are used up. They can answer without understanding. And it’s back to the drawing board for me.

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My parents’ generation, on the other hand, didn’t encounter the 21^{st} century until they were full-grown adults. They’d settled into their habits when this digital tide began rising around them: Facebook, Twitter, viral videos, actual computer viruses, Android, Snapchat, gifs, Reddit…

And so was born that tragicomedy of 21^{st}-century life: young people trying to explain technology to their parents. It’s frustrating both for the kids (“Why are you so incompetent?!”) and for the parents (“Why do I need this stupid device anyway?!”).

“This is so easy. Why can’t you do it?” vs. “This is so hard. What’s the point?” Now, why does that sound so familiar…?

Oh, that’s right! Because I’m a math teacher.

I have plenty of experience in conveying technical details to people who are skeptical of their value. I know that practice is important, that learning is scary, and—most of all—that it helps to understand what exactly it is you’re doing.

So I hereby offer, on behalf of generations already here and generations yet to come, three basic steps for explaining technology to your elders.

After several lessons, my grandmother still couldn’t use her new flatscreen TV. “I’ll never learn,” she laughed, half a taunt and half a lament. “I’m too old and stubborn. Your father is the computer man.”

I pressed on. What was she missing? Yes, she knew that the cable box showed lots of channels. Yes, she knew that the TV had two remotes. Yes, she even knew the appropriate sequence of buttons to push. And yet she kept complaining about a darkened screen that wouldn’t turn on.

Eventually I realized the problem. She didn’t realize that the information comes *in* through the cable box, and is then projected *out* through the television screen. She knew there were two machines, cable box and TV. But she didn’t realize that you need them both to be on.

We put down the remotes and talked through this basic idea. “What happens if the TV’s on, but the cable’s off?” “What if the cable’s on, but the TV’s off?” It took time and some animated hand gestures. (Though sharp and witty as ever, she was inescapably 89 years old.) But eventually she got it.

In the classroom, I find concepts wonderfully fun to teach—they deliver the light-bulb moments, the “Eureka!” smiles when something clicks. But that’s not enough. Ideas alone won’t update your antivirus software, change your email password, or reprogram your DVR.

Mastering the concept of “bread” doesn’t mean you know how to bake a loaf. You need to learn some procedures, too.

With my grandmother, this meant recording the exact steps she needed to follow. Once I’d written instructions for her, she smartly insisted on jotting down her own, additional comments in the margins. This worked well, for the same reasons I ask students to take notes in class: (1) Note-taking is active, forcing you to consider each step in turn; and (2) It gives you a chance to annotate and elaborate upon the tricky bits.** **

Next came the hardest part: Letting her try it herself.

Watching her fumble with the remotes, I felt the urge to step in. A three-second intervention—my hitting a single button—could have saved her two minutes or more. No grandson wants to watch his beloved grandmother spend her precious time on Earth locked in a losing struggle with a cable box.

But making mistakes isn’t just a *part* of learning. Making mistakes *is* learning. In math class, knowing the principles for solving an equation doesn’t mean you’ll be able to actually solve any. You’ve got to get out there and do it, erring and tripping along the way. Success becomes automatic only after a bit of drill and feedback.

And so it is with cable boxes. Ten minutes of guided practice can save you an hour of confusion and heartache down the road.

The whole lesson, from start to finish, took perhaps forty minutes. By the end of it, my grandmother’s new TV had transformed from a taunting demon into something a little more approachable—something that maybe, with a little luck, she could work by herself.

To achieve mastery, all three steps are crucial. (1) Learn the concept; (2) Break down the technique; and (3) Practice the skill.

In classrooms and living rooms alike, I find that the “concept” step is most often overlooked. Too often, we ask our students (be they teenagers or octogenarians) to start practicing skills before they have any sense of their context and purpose. Unsurprisingly, they forget or mix up the steps, growing frustrated at the maddening arbitrariness of it all.

We teachers must be wary of “expert blindness.” When our knowledge is effortless and automatic, it can be hard to unpack it for novices. We take for granted that the pupil understands the purpose of the machine, or its fundamental features—when that basic fact may be precisely what’s missing.

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and some of your favorite non-statistical issues, too!

**Tracking the Polls**

**User Response Rate**

**Amazon’s Algorithms**

**How Do You See Your Fellow Travelers?**

**Race Against Time**

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This system is okay. But also, it’s kind of crazy.

Why 60 minutes per hour? Why 60 seconds per minute? It goes back to Babylon, with their base 60 number system—the same heritage that gives us 360 degrees in a circle. Now, that’s all well and good for Babylon 5 fans, but our society isn’t base-60. It’s base-10. Shouldn’t our system of measuring time reflect that?

So ring the bells, beat the drums, and summon the presidential candidates to “weigh in,” because I hereby give you… *metric time*.

Now, this represents a bit of a change. The new seconds are a bit shorter. The new minutes are a bit longer. And the new hours are quite different—nearly two and a half times as long.

So why do this? Because it’d be so much easier to talk about time!

Here’s one improvement: **analog clocks are easier to read**. At first glance, the improvement may not be so obvious—we’ve simply reshuffled the numbers a bit.

But notice, the minute hand makes more sense now. When it’s at the 2, we’re 20 minutes past the hour. When it’s at the 7, we’re 70 minutes after the hour. And so on.

Second, **times are no longer duplicated**. For example, instead of needing to distinguish between 6am and 6pm, we can simply say “2:50” and “7:50.” (This is, of course, how “military time” currently works.)

Third—this is a big one—**the time tells you how far through the day you are**. The time 2:00 is exactly 20% of the way through the day. At 8:76, we’re exactly 87.6% of the way through the day.

Fourth, consider the moment when we’re 99.9% of the way through the day. In the new metric system, we get to watch the clock **roll from 9:99 back around to 0:00**. Isn’t that nicer and more conclusive than 11:59pm rolling around to 12:00am?

Fifth, it’s so **much easier to talk about longer times**. Two and a half days? That’s 25 hours. Three days and 6 hours? That’s simply 3.6 days. Since an hour is now a nice decimal fraction of a day, these conversions become easy.

Will there be adjustments to make? Certainly! But the adjustments are half of the fun.

Let’s start, as all good things do, with **television**. Whether you enjoy half-hour sitcoms or hour-long dramas, the length of your favorite shows is probably going to change. Why? Because, under our new system, what we now call “half an hour” will be 20.83 minutes. What we now call “an hour” will be 41.67.

There’s nothing magical about these “half-hour” and “hour” lengths, obviously. They were chosen simply because they were nice round numbers. But under the new system, they aren’t! Since it’d be silly to divide the TV schedule into 21-minute intervals, presumably television networks would tweak the lengths to go more evenly into an hour.

If so, they’d have two choices: 5 blocks per hour (i.e., two dramas, plus a sitcom), or 4 blocks per hour (i.e., two dramas).

If you choose the former, **shows will be 4% shorter** than today, leading to accelerated storytelling. (It’s the same change that’s unfolded over the last 20 years, as increased ad time has squeezed the shows themselves to be shorter.)

And if you choose the latter, **shows will be** **20% longer**. They’ll perhaps unfold at a slower, more cinematic speed. Either way, expect the pacing and rhythm of TV shows to change.

Sports run into the same issue. **Football** will probably opt for **four quarters of 10 minutes each**, which shortens the game by 4%. Expect slightly diminished scoring as a result. (And, if we’re lucky, diminished concussions.)

**Hockey**, meanwhile, might go for **three periods of 15 minutes each**, which actually makes the game 8% longer. It might give someone a chance to tackle Wayne Gretzky’s scoring records (but then again, probably not—he’s way out of reach right now).

I’d expect **soccer** to select **two halves of 30 minutes each**, which (as with American football) shortens games by 4%. If you thought soccer was too high-scoring already, you’re in luck (and also in a very small minority, I suspect).

When it comes to sports, the lengths of games won’t be the only thing changing. We also need to reconsider record running times.

Usain Bolt’s world-record for the **100-meter dash** (currently 9.58 seconds) would be, under the new system, **11.09 metric seconds**. Doing the 100m in 11 metric seconds might be achievable in the future, but 10 seconds? Perhaps never. (That’s the equivalent of 8.64 of our seconds!)

What about **the mile**? Well, it’s a little funny to imagine a world with metric time still worrying about that strange unit of distance (5280 feet? Really?), but the famed 4-minute mile would correspond to a **2.78-minute mile**.

This is weird because, for top runners in the 1940s and 1950s, the barrier to running a 4-minute mile may have been less physiological than psychological. Would the **2.8-minute mile** have felt as intimidating? Would the **3-minute mile**? Perhaps it’d be the 2.5-minute mile, seeing as the current world record (3:43 in our old system) is **2.58 metric minutes**?

And we might as well mention the **marathon**, where the world record time (currently 2:02:57) is now under an hour: **85 minutes, 38 seconds**. I suspect that the 1-hour marathon would be a real badge of honor, something that every distance runner aspires to.

Leaving sports aside, what about food?

Restaurants would open for **breakfast** at perhaps **3:00** or **3:50**. (Of course, coffee shops like Starbucks might open as early as 2:50.)

You’d get **lunch around 5:00**—that is to say, noon. Under our current system, I feel silly eating before 11:30, which is **4:80** under the new system. But I wonder—would I feel comfortable grabbing **lunch at 4:75**? Perhaps even **4:60** (even though that’s earlier than 11am under our current scheme)?

Eating is psychological, and how we number our hours might steer our behavior.

As for **dinner**, I suspect **7:50 to 8:00** would be the preferred time (although the famously late-eating Spaniards might hold off until **8:75 or 9:00**).

Other numbers change, too. Take **speed limits**: the typical 65mph limit on many highways translates to **156 mph** under the new system; I suspect we’d see that bumped up to **160 mph** or down to **150 mph** for the sake of roundness (which translates to 66.7mph or 62.5mph under our current system).

The **speed of sound**? Not 340 meters per second any longer; it’s now just **294 meters per second**. Meters haven’t changed, of course, but seconds have gotten shorter!

And the **speed of light**? Unfortunately, we lose the lovely number 300 million meters per second; instead, it becomes roughly **260 million meters per second**.

Speaking of light, on the **equinox**, you get **5 hours of light **and** 5 hours of dark**.

The **winter solstice** is pretty grim: in London, you’d see just **3 hours, 25 minutes, and 25 seconds of daylight.**

The **summer solstice** is nice, though: London would get **6 hours, 93 minutes** of sun.

Okay, time to come clean: I propose this without a single iota of seriousness. It’d be insane to ditch our current system. We’re used to it. We’ve agreed upon it. We’ve built our lives around it. The hassle of a change far outweighs the gains.

But I still love the thought experiment. It asks you, in some small way, to reimagine your life. How do you spend your time? How do you measure the success of a day? When you plan your hours, are you conceding to the arbitrary dictates of a quirky clock, or are you truly giving your tasks the time that they deserve? If I scrambled your sense of time, relabeling all your moments, would it change the way you feel about them? Do the numbers we assign to times *matter*? Or are we just scratching lines on the shifting dunes of eternity?

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(He later confessed that he was just curious if he could play puppet-master with this blog. The answer is a resounding yes: I dance like the puppet I am.)

So, do we have ceilings?

The traditional orthodoxy says, “Absolutely yes.” There’s high IQ and low IQ. There are “math people” and “not math people.” Some kids just “get it”; others don’t.

Try asking adults about their math education: They refer to it like some sort of NCAA tournament. Everybody gets eliminated, and it’s only a question of how long you can stay in the game. “I couldn’t handle algebra” signifies a first-round knockout. “I stopped at multivariable calculus” means “Hey, I didn’t win, but I’m proud of making it to the final four.”

But there’s a new orthodoxy among teachers, an accepted wisdom which says, “Absolutely not.”

You’ve got to love the optimism, the populism. (Look under your chairs—*everybody *gets a category theory textbook!) But I think you’ve got to share my pal Karen’s skepticism, too.

Do we have a ceiling, Karen?

Karen works hard. Karen asks questions. Karen believes in herself. And Karen still feels that certain mathematics lies beyond her abilities, above her ceiling.

The chasm between students (“everybody’s got a limit”) and teachers (“anyone can do anything!”) seems unbridgeable. A teacher might say “You can do it!” as encouragement, but a frustrated student might hear those words as an indictment of their effort (or as a delusional falsehood). Is there any way to reconcile these contradictions?

I believe there is: the **Law of the Broken Futon**.

In college, my roommates and I bought a used futon (just a few months old) off of some friends. They lived on the first floor; we were on the fourth. Kindly, they carried it up the stairs for us.

As they crested the third-floor landing, they heard a crack. A little metallic bar had snapped off of the futon. We all checked it out, but couldn’t even figure out where the piece had come from. Since the futon seemed fine, we simply shrugged it off.

After a week in our room, the futon had begun to sag. “Did it always look like this?” we asked each other.

A month later, it was embarrassingly droopy. Sit at the end, and the curvature of the couch would dump you (and everyone else) into one central pig-pile.

And by the end of the semester, it had collapsed in a heap on the dusty dorm-room floor, the broken skeleton of a once-thriving futon.

Now, Ikea furniture is the fruit-fly of the living room: notoriously short-lived. There was undoubtedly a ceiling on our futon’s lifespan, perhaps three or four years. But this one survived barely eight months.

In hindsight, it’s obvious that the broken piece was absolutely crucial. The futon *seemed* fine without it. But day by day, with every new butt, weight pressed down on parts of the structure never meant to bear the load alone. The framework grew warped. Pressure mounted unsustainably. The futon’s internal clock was silently ticking down to the moment when the lack of support proved overwhelming, and the whole thing came crashing down.

And, sadly, so it is in math class.

Say you’re acing eighth grade. You can graph linear equations with perfect fluidity and precision. You can compute their slopes, identify points, and generate parallel and perpendicular lines.

But if you’re missing one simple understanding—that these graphs are simply the x-y pairs satisfying the equation—then you’re a broken futon. You’re missing a piece upon which future learning will crucially depend. Quadratics will haunt you; the sine curve will never make sense; and you’ll probably bail after calculus, consoling yourself, “Well, at least my ceiling was higher than some.”

You may ask, “Since I’m fine now, can’t I add that missing piece later, when it’s actually needed?” Sometimes, yes. But it’s much harder. You’ve now spent years without that crucial piece. You’ve developed shortcuts and piecemeal approaches to get by. These worked for a while, but they warped the frame, and now you’re coming up short. In order to move forward, you’ve got to *unlearn* your workarounds – effectively bending the futon back into its original shape – before you can proceed. But it’s well nigh impossible to abandon the very strategies that have gotten you this far.

Adding the missing piece later means waiting until the damage is already underway, and hellishly difficult to undo.

This, I believe, is the ceiling so many students experience. It’s not some inherent limitation of their neurology. It’s something we create. We create it by saying, in word or in deed, “It’s okay that you don’t understand. Just follow these steps and check your answer in the back.” We create it by saying, “Only the clever ones will *get* it; for the rest, I just want to make sure they can *do* it.” We create it by saying, “Well, they don’t understand it now, but they’ll figure it out on their own eventually.”

In doing this, we may succeed in getting the futon up the stairs. But something is lost in the process. Sending our students forward without key understandings is like marching them into battle without replacement ammo. Sure, they’ll fire off a few rounds, but by the time they realize something is missing, it’ll be too late to recover.

A student who can answer questions without understanding them is a student with an expiration date.

EDIT, 4/15/2015: What a response! The comments section below is infinity and beyond. It’s like eavesdropping in the coffee shop of my dreams. I wish I had time to reply individually; please know that I read and enjoyed your thoughtful replies and discussion.

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*Or, Humility in the Face of Weirdness *

You’re on an alien spaceship orbiting the planet Newcomb. Don’t worry—the air is perfectly breathable. They’ve even got magazines in the waiting room.

As part of their research into human behavior, the aliens have placed two boxes in front of you: a **transparent** box containing $1000, and an **opaque** box, whose contents remain a mystery.

You’re allowed two options: **take both** boxes, or take **just the opaque** box.

It seems obvious. You’d be crazy to leave the guaranteed $1000 behind, right?

Not so fast. The aliens have observed humans for decades and have run this experiment thousands of times—so they’re very, very good at predicting which people will settle for the opaque box, and which people will greedily snatch both. And that’s where things get tricky. The aliens hand you a note:

*We made a prediction about your behavior. We won’t tell you what we predicted, but we will tell you that in this experiment, we fill the opaque box based on our prediction for each human.*

*If we predict the human will take only the opaque box, we put $1 million inside (to reward the human’s restraint).*

*But if we predict that the human will take both boxes, then we leave the opaque box empty (to punish the human’s greed).*

*We made your prediction earlier today, placed the correct amount in the opaque box, and sealed it. The choice is now yours. Will you take both boxes, or will you take only the opaque?*

The classic answer to this question—from mathematicians, economists, and other rational-minded folks—is to take both boxes. After all, your actions now cannot affect what the aliens have already done. No matter what’s in the opaque box, you’ll be $1000 better off if you take the transparent box, too. So you might as well take it.

I’m not so sure.

This is a hypothetical world that contains semi-psychic aliens and million-dollar boxes. It seems to me that the humble thing to do is to respect our lack of understanding about extraterrestrial matters and leave the transparent box behind, rather than trying to squeeze an extra $1000 out of an already quite generous alien. (Besides—if the aliens are observing you right now, and you commit to taking both boxes, they’ll know, and won’t give you the million.)

Such alien game-show scenarios are admittedly hypothetical. (Though if you’re observing me, aliens, I’m sorry I called you hypothetical.) Even so, we often encounter systems and situations in life that are nearly as foreign to our experience and intuition. Sometimes, we respond to these bizarre or unfamiliar scenarios with the greedy profit-maximizing approach of an economist or a Ferengi from Star Trek. But these are precisely the situations where a little caution and humility go a long way.

Given a morsel of information, we often try to scheme and optimize. We rarely consider the 4000 facts that we don’t have. To quote the satiric film *In the Loop*: “In the land of truth, the man with one fact is king.” We ought to stop—or at least take a brief pause—to consider the multitude of things we don’t know, and the distinct possibility that our entire framework is wrongheaded.

I say leave the transparent box. Will you really regret leaving $1000 on the table, when you’ve got $1 million to play around with?

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No one can fathom her wrath.

She wants us to do the impossible:

She wants us to

How can you study for something

where talent is so black-and-white?

You get it, or don’t.

You’ll pass, or you won’t.

It’s pointless to put up a fight.

Her mind must have leaked out, like water,

and slipped down the drain of the bath.

I might as well “read up on breathing”

as study for something like *math*.

Math’s an implacable tyrant,

a game that I never can win.

And even if I stood a prayer of success,

how would I even begin?

My teacher, the madwoman, told me:

“**First, list the things that you know.**”

Her mind’s gone to rot.

Still, I’ll give it a shot,

though I’m sure that there’s nothing to—

oh!

Well, now that I glance through my papers—

my homework,

my notes,

and my quizzes—

I see that I’ve learned a few things in my turn,

though I still fall far short of the whizzes.

**“Second, list things that you don’t know.”**

That’s what my teacher said next.

I’ll follow her words,

just to show she’s absurd—

the last thing that lady expects.

Hmm… well, it’s funny to notice,

but as I revisit my work,

I find a few bits

where nothing quite fits,

where the math goes all strange and beserk.

I’m starting to feel a bit dizzy.

She’s lured me somehow, down this path.

Her craziness scrambled my thinking—

she’s making me study for math!

The third thing my fool teacher told us,

was, “**Fill in conceptual gaps.**”

And this is her looniest notion of all,

a sign of her mental collapse.

How can I teach myself something

that she failed to teach me herself?

Does she assume that I keep in my room

a magical math-teaching elf?

All that I’ve got is the textbook.

Oh, and the internet too.

Plus a few friends I could text for some help—

well, I *suppose* that’ll do.

Still, once I’ve got the big picture,

how will that help on the test?

A test’s full of puzzles and problems to solve,

with answers not easily guessed.

She thinks it’ll help if I *study*?

That illness, I fear, has no cure.

The last piece of counsel she gave us was this:

“**Practice until you feel sure**.”

Clearly, the lady is batty.

Her brain’s made of ketchup and flies.

Still, with a sigh, and a roll of my eyes,

I gather some problems to try.

The next day I sit through the test,

hoping like heck that I’ll pass.

The day after that,

the test’s handed back,

and hey—I scored best in the class!

My teacher is crazy as crayons.

Her mind’s rolled away like a ball.

Doesn’t she see that I’m great at this stuff?

I don’t need to study at all!

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I never did the reading for math. You know, my major.

I’m not proud of it, but I know I’m not alone. As students from primary school to PhD have discovered, mathematical writing is a different beast. It’s not just a matter of jargon, equations, or obscure Greek letters. It’s something more basic about the way mathematical texts are structured and paced.

The trick is this: In mathematics, you say things precisely once.

(And no, I’m not going to repeat that.)

A talented colleague of mine once asked in frustration why her students refused to read the textbook. Her background was in biology, where the book—dense and difficult though it may be—is an irreplaceable source of learning. Now she was teaching Algebra II, and was losing patience with her students’ incapacity to glean anything, anything at all, from the text. “Why do they need it all spoon-fed to them?” she asked.

“I see what you’re saying,” I said. “But I’m not very good at learning math from a book myself. It’s a skill for 21-year-olds more than 15-year-olds.”

You see, ordinary writing has a certain redundancy to it. It *needs* redundancy, because English (lovely language though it is) can never capture a complex idea with perfect precision. In any phrasing, some shade of meaning is lost or obscured. A subtle, complicated thought must be illuminated from many angles before the reader is able to sift reflections from reality, or tell the shadows from the thing casting them. Thus, typical prose is full of pleasing repetition—paraphrase, caveats. You can skim, and even if you miss a few details, you’ll walk away with the gist.

Math is different. Unlike English, mathematical language is built to capture ideas perfectly. Thus, key information will be stated once and only once. Later sentences will presuppose a perfect comprehension of earlier ones, so reading math demands your full attention. If your understanding is holistic, rough, or partial, then it may not feel like any understanding at all.

Single words are saturated with meaning; immense focus is required; the diction is exactingly precise… more than anything, reading mathematics is like reading poetry.

This is why good mathematicians always read with a pencil in hand. Passive perusal of mathematics is pretty much worthless. You need to investigate, question, and probe. You need to fill in missing steps. You need to chew for a long time on every sentence, fully digesting it before you move on to the next course of the meal.

My indifferent, shrugging approach towards reading math in college may explain my struggles with a certain topology class. That class—a good simulation of first-year graduate school—demanded that I learn from the book, rather than from a professor’s lectures. I was unpracticed and unready for such a challenge.

I hope to equip my own students a little better.

*Wisdom from the comments*:

Phil H.** **proposes “a **math-reading class**, where you take a paper and walk through it, elucidating on the steps and answering questions.” He points out that reading slowly takes – among other virtues – **humility**. Reading fast is the token of a clever mind; but reading slow builds the wealth of a wise mind. (*hackneyed aphorism mine*)

John Golden points towards a relevant cartoon: http://abstrusegoose.com/353

Ariel finds math harder to read than physics or biology, and makes a wonderful observation: “**In biology… you do the science in your lab, and just describe it [in the paper]. In math, you do the science in the paper [itself]**.”

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