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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*or, The World Through Rectangular Glasses*

Now that I’m teaching middle school, I find myself wrestling with the sheer number of area formulas that my students need to know (or at least be passingly familiar with). Rectangles, triangles, parallelograms, trapezia…

The logic is this: A handful of geometric figures keep recurring throughout our world. Once you know how to spot them, they’re everywhere, like the Wilhelm Scream. It’s useful to determine the sizes of these shapes effortlessly, via formulas.

That’s all true, so far as it goes. But reducing geometry to formulas alone can lead to tragic misunderstandings, like when a student asked a friend of mine: “Is there a simple way to remember the difference between volume and surface area?” That’s like asking for a simple way to remember the difference between oceans and deserts: You can only confuse them if you have deep misconceptions about each.

So when I teach these formulas, I try to remind myself of an elegant truth: when it comes to area, everything is rectangles.

And yes, I mean *everything*.

So let’s begin. With **rectangles**, finding area is a simple matter of multiplication. In each rectangle, you’ve got a little array of squares:

In this case, we have an array of 5 square centimeters by 3 square centimeters. That’s 15 square centimeters in all. Or, more generally: A = bh.

Now, what about **triangles**?

Well, just draw them inside rectangles.

Notice that we can divide the rectangle into two sides. The triangle fills half of the left sides, and half of the right. So it must be half of the whole area.

Up next: the **parallelogram**.

Notice that you can chop off one side of the parallelogram, and move it to the other side, giving you…

…a rectangle! So its area is the same as the rectangle’s.

What about the **trapezoid** (or, as my new British neighbo(u)rs quaintly insist, the **trapezium**)? This one’s a little trickier.

One method: double the trapezoid, and look at what it makes: a parallelogram!

The area of this whole thing, per our parallelogram formula, is (b_{1} + b_{2}) x h, but this is two times too big. So we divide by 2 to get our answer.

Alternatively, you can notice that the trapezoid has a little rectangle that fits inside…

…and that the trapezoid, in turn, fits inside a bigger rectangle:

The smaller rectangle has area b_{1}h. The larger has area b_{2}h. And the trapezoid is exactly halfway between them. Hence: the same area formula!

Up next: we go fly a **kite**!

Deploying our by-now-familiar trick, we can fit the kite inside a larger rectangle, and notice that it fills exactly half:

Hence, our formula: the area is d_{1}d_{2}/2.

What about a **rhombus**? The same trick!

If you recall that a **square** is both a special rectangle (because its angles are all equal) and a special rhombus (because its sides are equal), you’ll see that we can find its area in two ways:

Combining these two formulas, we even get a cool result:

(And you don’t need to phone Pythagoras for help with the proof.)

So far we’ve limited ourselves to simple shapes with straight sides. It’s perhaps not surprising that their areas can be found by the clever application of rectangles. But what about something more exotic, something less obviously rectangular?

What about, say… the **circle**?

First, slice your circle up like a pizza. Then, rearrange the slices in an alternating pattern, to get a shape like this:

It looks kind of like a parallelogram, right? It has a “height” of roughly *r* (the radius of the circle) and a “base” of roughly *πr*.

*(Note: If you recall, a circle has circumference 2πr. Notice that this base is half the circumference. That’s πr.)*

Now, slice it up even finer, and repeat the process. What do you notice?

It looks even *more* like a parallelogram. And in fact, it’s starting to look like a rectangle.

This is where you draw upon your imagination. Picture slicing the circle up into finer and finer pieces, like a single pizza being shared among the population of the entire world. Then slice it up even *finer*, so that there’s a slice for every mouse and moose and bacterium on earth. Then slice it up even finer than *that*!

What will we get?

The finer we slice, the closer we have to a rectangle. And if you could imagine cutting it into “infinite” slices—impossible, but bear with me—you’d get a perfect rectangle.

What would its area be? Well, base times height… which is πr times r… which is πr^{2}. Area formula proved! QED! (That’s Latin for “Game, set, match.”)

Okay. So we’ve boiled the following shapes down to rectangles:

But I promised you that *everything* was rectangles. And we haven’t covered everything. We haven’t, for example, found the areas of shapes like these:

So, what can we do? Well, we can try to estimate the area of such a shape using one rectangle, but we won’t get very close:

The one on top is too small, and the one on the bottom is too big. And we’ve got no idea how MUCH too big – it isn’t an obvious fraction.

So what do you do when a rectangle fails? Add more rectangles!

Better, but still not great. What about 8 rectangles?

That’s looking closer. And it points towards a pattern: the more skinny rectangles we allow ourselves to use, the closer we get to the true area.

Summarize the whole thing with some equations, and you get an object familiar to calculus students: the integral!

That’s right: the integral. We’re talking about math’s all-powerful engine for finding the area of any shape you can poke an equation at. And the whole thing is nothing but weaponized rectangles.

So, trying to find an area? Put on those rectangular glasses.

]]>

and to all poets, everywhere

**I.**

All my life

I had known only lines

so when my teacher

drew a parabola

I said,

“Huh?”

**II.**

I took all the numbers,

and squared them.

The big ones grew.

The little ones shrank.

The negative ones

became positive.

Opposites agreed.

It was kinda cool.

** **

**III.**

I watched an object falling,

tracing its arc,

the ink of time leaving curves

on the paper of space—

a perfect parabola.

(Except for air resistance.)

(NO ONE LIKES YOU, AIR RESISTANCE.)

**IV.**

My teacher told us something

about beauty,

and curvature,

and the essence of number.

I took what she said

and plugged it into the quadratic formula:

No real solutions.

Worthless.

**V.**

I was of three minds,

Like a parabola

Which is defined by three points.

**VI.**

I found a cone, and sliced:

a little this way,

I’d have made an ellipse;

a little that way, a hyperbola;

and a little the other way,

I’d have hacked off a finger.

But I cut true, and so,

a parabola.

**VII.**

An equation and a graph

are one.

An equation and a graph and a student

are confused.

**VIII.**

If you walk a narrow path,

never too close to the house,

never too close to the road,

just the same distance from each,

then you will weird people out,

because why are you walking like that?

Nobody walks in parabolas.

**IX.**

The polynomials all babbled

in languages I did not speak,

like beasts, or birds,

or soccer commentators.

I could grasp no one’s words,

except the parabola,

and so I let it speak for them all.

**X.**

I held a mirror to my parabola

and it simply admired itself.

**XI.**

I do not know which to prefer,

The beauty of constancy

Or the beauty of change,

The turn in the parabola

Or just after.

**XII.**

I know logarithms

and trigonometric equations;

But I know, too,

That the parabola is involved

In what I know.

**XIII.**

I asked a cubic for its derivative.

It spoke in parabolas.

I asked a linear for its integral.

It spoke in parabolas.

I asked Jesus for a math lesson.

He spoke in parable-as.

]]>

One of the joys of being married to a pure mathematician—other than finding coffee-stained notebooks full of integrals lying around the flat—is hearing her try to explain her job to other people.

“Are there…uh… a lot of computers involved?”

“Do you write equations? I mean, you know, long ones?”

“Do you work with *really* big numbers?”

No, sometimes, and no. She rarely uses a computer, traffics more with inequalities than equations, and—like most researchers in her subfield—considers any number larger than 5 to be monstrously big.

Still, she doesn’t begrudge the questions. Pure math research is a weird job, and hard to explain. (The irreplaceable Jordy Greenblatt wrote a great piece poking fun at the many misconceptions.)

So, here’s this teacher’s feeble attempt to explain the profession, on behalf of all the pure mathematicians out there.

**Q: So, what is pure math?**

A: Picture mathematics as a big yin-yang symbol. But instead of light vs. dark, or fire vs. water, it’s “pure” vs. “applied.”

Applied mathematicians focus on the real-world uses of mathematics. Engineering, economics, physics, finance, biology, astronomy—all these fields need quantitative techniques to answer questions and solve problems.

Pure mathematics, by contrast, is mathematics for its own sake.

**Q: So if “applied” means “useful,” doesn’t it follow that “pure” must mean…**

A**: **Useless?

**Q: You said it, not me.**

A**: ** Well, I prefer the phrase “for its own sake,” but “useless” isn’t far off.

Pure mathematics is not about applications. It’s not about the “real world.” It’s not about creating faster web browsers, or stronger bridges, or investment banks that are less likely to shatter the world economy.

Pure math is about patterns, puzzles, and abstraction.

It’s about ideas.

It’s about the *othe*r ideas that come before, behind, next to, or on top of those initial ones.

It’s about asking, “Well, if *that’s* true, then what *else* is true?”

It’s about digging deeper.

**Q: You’re telling me there are people out there, right this instant, doing mathematics that may never, ever be useful to anyone?**

A**: ****glances over at wife working, verifies that she’s not currently watching Grey’s Anatomy**

Yup.

**Q: Um… why?**

A**: **Because it’s beautiful! They’re charting the frontiers of human knowledge. They’re no different than philosophers, artists, and researchers in other pure sciences.

**Q: Sure, that’s why they’re doing pure math. But why are we paying them?**

A**: **Ah! That’s a trickier question. Let me distract you from it with a rambling story.

In the 19^{th} century, mathematicians became obsessed with proof. For centuries, they’d worked with ideas (like the underpinnings of calculus) that they knew were true, but they couldn’t fully explain *why*.

So at the dawn of the 20^{th} century a few academics, living on the borderlands between math and philosophy, began an ambitious project: to prove everything. They wanted to put all mathematical knowledge on a firm foundation, to create a system that could—with perfect accuracy, and utter permanence—separate truth from falsehood.

This was an old idea (Euclid put all of planar geometry on a similar footing 2000 years earlier), but the scope of the project was new and monumental. Some of the world’s intellectual titans spent decades trying to explore the rigorous, hidden meanings behind statements like “1 + 1 = 2.”

Can you imagine anything more abstract? Anything more “pure”? Curiosity was their compass. Applications could not have been further from their minds.

**Q: So? What happened?**

A: The project failed.

Eventually, the philosopher Kurt Gödel proved that no matter what axioms you choose to start with, any system will eventually run into statements that can’t be proven either way. You can’t prove them true. You can’t prove them false. They just… are.

We call these statements “undecidable.” The fact is, many things can be proven, but some things never can.

**Q: Ugh! So it was just a massive waste of time! Pure maths is the worst.**

A: Oh, I suppose you’re right.

Of course, the researchers tried to salvage something from the wreckage. Building on all this work, one British mathematician envisioned a machine that could help us decide which mathematical statements are true, false, or undecidable. It would be an automatic truth-determiner.

**Q: Did they ever build it?**

A: Yeah. The guy’s name was Alan Turing. Today we call those machines “computers.”

**Q: *stares blankly, jaw slowly unhinging***

A: Exactly.

This enormous project to prove everything—one of the purest mathematical enterprises ever undertaken—didn’t just end with a feeble flicker and a puff of smoke. Far from it.

Sure, it didn’t accomplish its stated goals. But by clarifying (and, at times, revolutionizing) ideas like “proof,” “truth,” and “information,” it did something even better.

It gave us the computer, which in turn gave us… well… the world we know.

**Q: So the pure mathematics being done today might, someday, give us a new application as transformative as the computer?**

Maybe.

But you shouldn’t hold any specific piece of work to that standard. It won’t meet it. Paper by paper, much of the pure math written this century will never see daylight. It’ll never get “applied” in any meaningful sense. It’ll be read by a few experts in the relevant subfield, then fade into the background.

That’s life.

But take any random paper written by an early 20^{th}-century logician, and you could call it similarly pointless. If you eliminated that paper from the timeline, the Jenga tower of our intellectual history would remain perfectly upright. That doesn’t make those papers worthless, because research isn’t a collection of separable monologues.

It’s a dialogue.

Every piece of research builds on what came before, and nudges its readers to imagine what might come next. Those nudges could prove hugely valuable. Or a little valuable. Or not valuable at all. It’s impossible to say in advance.

In this decades-long conversation, no particular phrase or sentence is necessarily urgent. Much will be forgotten, or drift into obscurity. And that’s all right. What’s vital is that the conversation keeps on flowing. People need to continue sharing ideas that excite them, even—or perhaps *especially*—if they can’t quite explain why.

**Q: So, pure maths… come for the pretty patterns, stay for the revolutionary insights?**

A: That about covers it.

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Have you been **doing** math, but not sure you’re really **learning** it?

Fret and fume no longer! Below, you will find a definitive (*read: not definitive*) checklist. Simply think back to your latest mathematical experience, and **check a box** for each question to which you can answer yes. (*Boxes not provided.*)

- Did you recognize a
**strange pattern**, or the beginnings of a pattern, or the*lack*of a pattern, and say to yourself, “Wait…*what?!*”

- Did you stand
**dumbfounded at the foot of the problem**, unsure how to begin, like a rock-climber staring up a cliff?

- Did you try something, without any confidence that it would prove correct or even useful, but simply
**because it seemed worth a shot**?

- Did you
**take a break**to let your brain chew on the problem in the background (perhaps while you chewed on gummy candies in the foreground)?

- Did you
**recruit a calculator or computer**to do the repetitive work for you, so that you could focus on the big picture?

- Did you get stuck, start explaining your dilemma to someone, and then cut off your explanation halfway, because you’d
**suddenly solved your own problem**?

- Did you receive a
**one- or two-word hint**(like “triangles” or “topological invariant”) that made your**eyes go wide with understanding**, as if you’d just glimpsed the inside of an atom?

- Did you produce a solution that
**you knew was flawless**… and then realize with a sinking feeling that**it was obviously flawed**?

- Did you work
**so closely alongside a partner**that you can’t remember whose ideas were whose anymore, as if you were**sharing a brain**?

- Did you admit that you needed
**expert help**, and turn to a book/teacher/colleague/message board/Magic 8 Ball?

- Did you embrace an expert’s advice… then begin to doubt it… and then regain trust after
**verifying the key steps for yourself**?

- Did you finally solve a
**seemingly tough problem**and think, “I’m a blind fool! It was obvious from the start!”

- Did you finally solve a
**seemingly easy problem**and think, “I’m a fiery genius! It was far harder than it appeared at the start!

- Did you, in writing up or summarizing your work, find
**a simplifying shortcut**that saves buckets of time and effort?

- Did you, after solving a problem, feel
**the giddy impulse to go around explaining**your work to friends, family, baristas, and pet dogs you pass on the street?

If you checked ** anything**, then congratulations! You’re learning math.

If you checked ** nothing**, then don’t worry! Just go out there. Find something mathematical that you want to do, but aren’t sure how. Then start making mistakes!

If you checked ** everything**, then stop checking things! You didn’t actually

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“So, like, I don’t really understand anything you’re saying,” he informed me, “But I can still get the right answer.”

He smiled, waiting.

“Which part is giving you trouble?” I asked.

“Oh, you were talking about this extra stuff,” he said, “like the ideas behind it and everything. I don’t… you know… *do* that.”

I blinked. He blinked. We stood in silence.

“So is that okay?” he concluded. “I mean, as long as I can get the right answer?”

Here it was, out in the open: the subtext of practically every class I’ve ever taught. I’ve grown accustomed to yanking my side of the rope in an unspoken tug-of-war. The teacher emphasizes conceptual understanding. The students conspire to find shortcuts around it. So it always goes.

But I’d never heard a student break the fourth wall quite like this. It was as if Peter Jackson popped up on camera saying, “I know you want a good story, but what about a bloated trilogy full of mind-numbing battle scenes instead? You’ll still buy a ticket, right?”

“Is that okay?” my student repeated. “I mean, I can *get* the right answer!”

He had a point. What else *is* there?

There’s a powerful ideology at work here, one my student has perhaps internalized without realizing: the unshakeable belief that math is all about right answers, and nothing more.

The Church of the Right Answer.

Millions of students worship there, and it’s not because they hate ideas. They do it because schools exist, in part, to sort and label them. “Honors.” “Remedial.” “Vocational.” “College-ready.” And the primary mechanism of this sorting—society’s basic equipment for doling out futures to young people—is grades.

Make no mistake. Grades are a bona fide currency. When I write a score on a test, I might as well be signing a paycheck.

And as for our tests—no matter how well-intentioned, no matter how clever and fair, there will always be a back-road to the right answer. There will be something to memorize—a procedure, a set of buzzwords, whatever—that will function as a fake ID, a convincing charade of understanding.

Some tests are better than others, but all of them can be gamed.

It’s tempting to blame the kids: these grade-grubbers, these incurious mercenaries, shuffling through our schools, forever demanding to know whether an idea will be “on the exam” before they deign to learn it. But let’s be honest. From right answers come good grades, and from good grades come all worldly prizes and pleasures.

Students worship right answers because that god delivers.

I’m not fond of this reality. But most of the time, I simply work within the confines of the grade-driven system.

I write the best tests I can. I mix in posters, projects, and presentations. I dabble in standards-based grading. I give second and third chances to succeed. I sing the praises of learning, in the best tenor I can muster. And at day’s end, when my job demands that I sort and label my students, I try to do so with sympathy and transparency.

In short, I’m constantly nudging students to think more deeply, but I never really challenge the dogmas of the Church of the Right Answer. I’m a good, rule-abiding cop, in a city where the rules are sometimes grossly unfair.

That doesn’t always satisfy me. Some days I don’t want to nibble at the edges. I crave a more radical assault: a reformation, a new religion.

Some days, I want a Church of Learning.

Now, I don’t have 95 theses ready to nail on the wall. It’s easy to critique a system, and damn hard to build a better one. You want a magic plan for education, a foolproof blueprint for helping the next generation to flourish? Me too! Send me the link when you find it.

In the meantime, I’ll tell you the four theses I’ve got, and let you supply the other 91.

Think of all the people you know who are curious and thoughtful. People who ask real questions and listen to the answers. People who read weird, well-written stuff, and then tell you about it. They’re not so scarce, are they? In many places, they’re numerous and thriving. The soil is not as barren as it seems.

I’ve seen it often. I’ll meet a student who knows everything about quadratics: how to factorize them, how to graph them, how to complete the square. And, like most students burdened with such knowledge, they view quadratics with pure contempt.

Then, with a little coaching, they’ll suddenly see what a quadratic expression *is*. They glimpse the idea beneath it all. And all of those useless fragments of procedural knowledge finally snap together, swift and tight, into a whole and gorgeous understanding.

The idea pops like a firework, and the kid lights up like a sky.

Real learning touches something deep inside people. It endows a sense of power and purpose and wonder that you’ll never get from regurgitating facts and executing shortcuts. Once you start thinking, you want to keep thinking.

This is a theology that doesn’t require a steeple. We don’t have to march on Washington, or take a scalpel to the face of education. We’re talking about the simple conviction that ideas are worth celebrating. Such as love of ideas can take hold as… well…

As an idea.

The Church of Learning doesn’t require that I reject the economic structures around me. I don’t need to drop out of school, eat from dumpsters, and refuse to clip my nails. Sure, the school-to-workforce pipeline doesn’t always honor the things I care about. But that’s okay. My material life need not dictate the terms of my intellectual life.

I can be a capitalist, and still be curious.

(In fact, I haven’t subtitled my memoirs yet, but dibs on “The Curious Capitalist.”)

In a zero-sum game, every winner needs a loser. I can’t gain a dollar unless you forfeit one. I can’t triumph unless you fail.

It’s pretty much how we’ve built our educational system. Students compete for finite resources: admissions spots at elite schools, A’s in a tough class, titles like “valedictorian” and “cum laude.” If no one is failing, we sometimes worry that we’re doing something wrong—perhaps setting our standards too low.

*Somebody* has to fail, right?

To some degree, this continues after school. Adults compete for jobs, for customers, for parking spots. But—and I can’t scream this loudly or often enough—

*Life isn’t just competition!*

Wherever my students go—business, academia, government, Mars—they’ll reckon with problems. They’ll create things. Their hands will get good and dirty. And if they topple the challenges, fix what’s broken, and imagine wonderful new things that everybody needed and that nobody ever realized, then the world will get better.

Really.

A better world is not like a fancy college. The spots aren’t limited. The more we populate the planet with curious, thoughtful, talented people, the better it becomes.

There’s a word for this idea: “education.”

When my students leave, it won’t particularly matter what grades I bestowed, or what tests they aced, or what shortcuts to the right answers they could and couldn’t find. What will matter are their abilities.

Creativity.

Grit.

Ingenuity.

Their wealth of knowledge.

Their curiosity, independence, and capacity to learn new things.

Those skills can be developed. It just takes time and effort. And that’s what teachers are for: to help make it happen.

Like my students, I have good days and bad. They and I are jostled by the same storms, pulled by the same tides. Sometimes I scold them for worshipping at the Church of the Right Answer, and sometimes I play back the tape and discover that I’ve been preaching its sermons myself. I try to track the harm and the good that I do in the classroom, measure the one against the other. It isn’t easy. There are no right answers.

But of course, life’s not about right answers.

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*Thanks for reading! This is the sixth (and final!) in a series of posts about How to Avoid Thinking in Math Class. See also parts 1, 2, 3, 4, and 5.*

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(See Also Parts 1, 2, 3, and 4)

Sometimes I fantasize about making scarecrows of myself.

They’d wear jackets, ties, and expressions of thoughtful patience. I’d scatter them around my classroom—maybe even one every desk (if scarecrow manufacturers happen to give bulk discounts). And they’d work wonders for my students, because a lot of the time, the students don’t actually need me.

They just think they do.

This idea goes back to my first year teaching, when one student would come to me after school, homework in hand. “I need help,” he’d say.

“Sure. What question?”

“All of them.”

It sounded grim. But when pressed, he could explain perfectly how to tackle every problem. His understanding was solid. “So what’s the issue?” I’d ask.

He’d shake his head solemnly. “I lack confidence.”

That autumn, we developed a peculiar routine. While I worked at my big wooden desk, he’d perch on the edge, quietly and independently doing his homework. He refused to sit at an adjacent desk of his own. He wanted to be extra close, with his paper in my field of vision—never mind that I wasn’t providing a shred of assistance, or even watching him work. It reassured him simply to have an expert close at hand.

He just needed a scarecrow.

The fact is, uncertainty is hard. It scares us. I myself run from it, without meaning to, or even realizing what drives my behavior. Faced with a hard task, I procrastinate: fleeing to Facebook, turning to Twitter, or suddenly “remembering” that I need to re-shuffle my computer files from 2007.

Sometimes, it seems I’ll do anything to avoid wallowing in my own uncertainty.

As for students, it can be frightening to start a math problem. You don’t know quite where it will lead. Will my approach be fruitful? Will it falter? Where do I even begin?

But unlike my desk-perching student, most kids don’t recognize that one rope holding them back is fear of the unknown. They just hesitate: too afraid to leap without a net, but never bothering to go in search of a net for themselves.

Sometimes they write nothing, paralyzed by not knowing. They won’t commit pencil to paper until they’re 100% sure.

Or, if they *do* write something, they immediately check answers in the back of the book, before inspecting or reflecting. Craving quick approval, they cut short their own learning.

Often, they just push the task aside, idly waiting for a resolution, too shy or un-self-aware even to ask for help.

Other times, they ask. And ask. *And ask*. They’re hoping I’ll slice the task into bite-sized pieces for them, sparing them the trouble of actually grappling with the unknown.

In all these cases, students are refusing to engage with their uncertainty. But if you’re uncomfortable with doubt, you’ll never break through to the other side. You’ll never have a “Eureka!” moment or an intellectual “Aha!” You’ll never… well… *learn*. After all, if you can’t bear to face the unknown, how will you ever come to know it?

I find that my desk-percher has it right. At times like these, the mere presence of an expert can supply the confidence you’re lacking.

“Just write down what you know.”

“Hey, give it a shot.”

“Well, how do *you* think we can solve this problem?”

Platitudes like these—despite being as vague and vacuous as any pop lyric—actually succeed in urging students forward in their problem-solving. I find myself saying such things constantly. In those moments, the kids don’t really need me. They just need a nudge. A pull-string teacher, with a few pre-programmed slogans, would suffice.

To be fair—and in defense of my employability—that’s not *always* the case. Many days, kids are legitimately stuck. And what seems on the surface like a generic suggestion (“Try a simpler case, like when there’s only one person”) or a pro forma question (“How do we actually *define* a circle?”) can actually point towards the key idea. A mannequin-me couldn’t give that kind of targeted feedback and encouragement.

Still, I’m surprised how often a scarecrow is all it takes.

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(See Also Parts 1, 2, and 3)

This speaks more to my naiveté as a first-year teacher than anything else, but I was shocked to find how fervently my students despised the things they called “word problems.”

“I hate these! What is this, an English lesson?”

“Can’t we do regular math?”

“Why are there *words* in *math class*?”

Their chorus: *I’m okay with math, except word problems.*

They treated “word problems” as some exotic and poisonous breed. These had nothing to do with the main thrust of mathematics, which was apparently to chug through computations and arrive at clean numerical solutions.

I was mystified—which is to say, clueless. Why all this word-problem hatred?

To me, the very phrase “word problems” sounds bizarre. It’s like “food meals” or “page novels.” Of *course *meals have food. Of *course* novels have pages.

And of *course* math problems have words!

One half of mathematics is “pure math.” It deals with abstract ideas, pursued for their own sake. It demands clear definitions and logical arguments. In short, it’s chock full of words. You don’t have trust me on this: take a look at a random research paper in pure math. There are more sentences than equations.

The other half of mathematics is “applied math.” It aims to solve real-life problems, working its fingers into fields as diverse as science, finance, design, and government. If you want to communicate about the quantitative problems in these fields (let alone solve them), well, you’d better get ready to use some words.

Word problems aren’t an invasive species. They’re the whole biosphere.

My first year, I taught students who had come through a tough, testing-driven middle school. They had bought into a simple vision of education: school is work. To them, cranking through math problems was like hammering nails or folding laundry—not necessarily a barrel of laughs, but satisfying in a familiar, workaday way. You’ve got a task. You do it. Then it’s done.

Willing to work hard? Yes; admirably.

Willing to think hard? Not necessarily.

I suspect many students share this rather joyless, overcast view of education. They may not love math, but at least they know what to expect. Word problems violate that contract; they interrupt the flow.

To see why, take an example:

Now, I look at this and think, “Ah, a problem about traveling objects.” This conjures up a whole mental map of ideas:

Brushing aside the clutter, you’ve got this basic structure to my knowledge:

The concept forms the foundation. Everyday words, mathematical symbols, whatever—those are just different languages for talking about the underlying idea. With this kind of understanding, solving the problem is pretty straightforward:

But if I’m a typical word-problem-hating student, I look at things a little differently. My knowledge is structured like this:

For a student like me, “doing math” means learning which symbols trigger which procedures. For example, the symbols “4 + 2x = 18” triggers the procedure “18 take away 4, then divide by 2,” giving an answer of 7.

These connections between symbol and procedure are mostly arbitrary. I take away 4 because… well, because that’s what I’m supposed to do. There’s no deeper reason.

And uh-oh… here come word problems.

My old triggers (the symbols “x” and “+”) are nowhere to be seen, so I don’t know which procedures to execute anymore. I need to start from scratch, learning a whole new set of cues.

And even worse, these new triggers aren’t clear and unambiguous like the old ones. They’re subtle and tenuous. They’re embedded in that messy human creation called language—full as it is of paraphrase, omission, and implication.

My teacher thinks that these “word problems” are a lot like the numerical ones, but to me, they’re a whole new breed. All my old knowledge feels useless in this unfamiliar verbal swamp.

Is it any surprise I’d rather stick to what I know?

As I wrote last week, procedural thinking is often useful (even crucial!). But it cannot replace real understanding, the kind of conceptual thinking will keep bearing fruit. Moreover, success demands connecting the two, linking concepts to procedures, ensuring always that the *how* is rooted in the *why*.

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