On Friday I realized—yet again—that my too-clever-for-their-own-good students were finding ways to answer questions without understanding the ideas.

Rather than reckon with the concept of slope, they were memorizing a complex rule:

That’s all true, so far as it goes, but it’s as opaque and sinister as the tax code.

“Math is supposed to make sense!” I told them, and in my flailing to explain why, I found myself reaching for my favorite rhetorical tool: the overly-detailed analogy.

So, to see what math class is like for memorization-driven students, imagine that you’re a household robot.

Your job is to clean the dishes.

Now, you don’t know anything about food, meals, or human culinary practices—you’re a robot, after all. All you know is what you’re told. And your only wish in life is to follow your instructions, and to carry out the right procedures.

You’re thrilled when a nice family takes you home, teaches you how to recognize dishes as “dirty,” “clean,” and “dry,” and then lays out your simple rules to obey:

Of course, you don’t know what dishes are for (decoration? sport?). You don’t know how they get dirty (carelessness?). You don’t know the point of cleaning them (don’t they keep getting dirty?). And you don’t know why they belong in the cupboards afterwards (they’re hidden when clean and displayed when dirty?). You just know that these are the steps.

And you follow steps. You’re a good robot.

On your first day, you notice some dry dishes out on the table. You start stacking them in the cupboard, just like you’ve been told, when your owner starts screaming. “What are you doing, you dumb machine? I just set the table a minute ago!”

You’re confused. Weren’t you just following the rule that says ‘Stack the dry dishes’?

Soon, the rules are amended:

Okay. Weird, and contradictory, but whatever. One exception isn’t so hard to learn.

A little later, you notice the plates getting dirty. There’s all kinds of lumpy organic stuff piling up on them. So you grab them and start cleaning.

“NO!” you hear. “STUPID ROBOT! That was our lasagna!”

What’s all this “lasagna” talk? Isn’t this organic dirt on the dishes? Aren’t dirty dishes supposed to be cleaned? Isn’t that, like, *your entire job*?

Soon, your first rule is amended:

“Wait until the large dirt particles are gone?” How paradoxical. You’re supposed to focus your cleaning on the *less* dirty dishes, waiting for the *really *dirty dishes to magically clean themselves halfway?

Well, no matter. Strange and inconsistent rules are still rules, and—being a good robot—you follow rules.

But the next day, they’re yelling at you again. “Why didn’t you clean that plate? You left it out overnight!”

You look at it. It’s still got lots of dirt on it. Didn’t they say not to clean it if it’s got such big particles left?

A moment later, you’re programmed with a *new* first rule:

You’re starting to get anxious now. What once seemed a simple scheme of rules has grown rather complicated. You could really use some encouragement for your efforts, and so you leap at the first chance to put this newly modified rule into action. The removal of particles on one dish has stopped, so you begin to clean it.

But no, only more shouting—“I was just pausing to take a sip of water! I wasn’t done!”—and a new, even more complicated rule:

It all seems cruelly arbitrary. This once-straightforward task (“clean the dirty dishes”) has become a nightmare cobweb of exceptions and contradictions, in which you must calibrate your behaviors precisely based on bizarre conditions and inscrutable requirements.

Ladies and gentlemen, welcome to math class.

What’s missing, of course, is the crucial understanding of what it’s all for. It turns out that you can’t succeed as a dishwasher until you understand why dishes get dirty in the first place.

“The point of dishes,” the robot needs to be told, “is to hold humans’ food while they eat it. When they’re done eating, that’s when the dishes need to be cleaned.”

Every rule – even the craziest, most arbitrary mandate – has a reason rooted in this essential purpose. (*Why leave the dishes with big particles? Because the person is still eating!*) And so it is in math class. If you understand slope not as “that list of steps I’m supposed to follow” but as “a rate of change,” things start making more sense. *(Why is it the ratio of the coefficients? Because, look what happens when x increases by 1!*)

You get to work a lot less, and think a lot more.

Now, conceptual understanding alone isn’t enough, any more than procedures alone are enough. You must connect the two, tracing how the rules emerge from the concepts. Only then can you learn to apply procedures flexibly, and to anticipate exceptions. Only then will you get the pat on the back that every robot craves.

With my students on Friday, I garbled the whole analogy. I tend to do that.

But there’s a simple takeaway. Even if you don’t care about understanding for its own sake; even if you’re indifferent to the beauty and deeper logic of mathematics; even if you care only about test results and right answers; even then, you should remember that the “how” is rooted in the “why,” and you’re unlikely to master methods if you disregard their reasons.

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This September, I gave my 7^{th}-graders an elegant little problem about a 12-step staircase. You’re climbing from the bottom to the top, using combinations of single and double steps. The question is, how many ways can you do this?

I was stunned when some of my students offered answers almost immediately. “145!” one screamed, as if he had just gotten bingo. “Am I right?”

“Whoa, that was fast!” I said. “Why 145?”

“12 times 12, plus 1!” he announced. “Am I right?”

“But…” I hesitated. “But why 12 times 12? Why plus 1? Are we just doing random computations that sound like fun?”

He listened to my questioning with the same patience you’d give a friend’s mediocre guitar solo. Then he launched right back into his chorus: “So,” he said, “am I right?”

To him, at that moment, “doing math” meant “making a number smoothie.” You take the numbers in front of you, throw them all into the blender, and mash the “pulse” button until you get something.

The funny thing is, in our classes, this often *works*. You see a thick block of text; you pick out the numbers; you run them through the formula; and voila, you’ve got a solution, no thinking required!

It’s like I’m trying to teach you to make smoothies, and I always start by laying out exactly the ingredients you need. “Here’s a banana, a cup of strawberries, a cup of milk, and a cup of vanilla yogurt. Can you make a smoothie?”

Of course you can!

But what happens when I throw extra ingredients in front of you? “Here’s a banana, an onion, a cup of strawberries, paprika, dry pasta, cooked pasta, milk, vegetable oil, a cup of vanilla yogurt, and toothpaste. Can you make a smoothie?”

Maybe you can. Or maybe you’ll conjure up a foul paprika pasta paste.

The number smoothie is a classic way to avoid thinking. It is the blind mashing of a button. Whereas real mathematics is thoughtful, selective, and carefully considered, the number smoothie is precisely the opposite: indiscriminate, wanton, thoughtless.

It doesn’t mean you should never use formulas, any more than you should swear off smoothies. I’ve written before about the need to understand where formulas come from, but even after you’ve done that, a formula needn’t be a lifeless tool.

Learning a formula doesn’t need to be the terminus of thought.

Instead, try something like this:

For the blue, pink, and purple triangles, you’ve got too much information. It’s the mathematical equivalent of needing to sort out the onions and toothpaste from the real ingredients. For the green, you don’t have *enough* information—the mathematical equivalent of needing to root through the fridge to find the missing strawberries.

Or you could ask a question like this:

Now, there are no ingredients at all! You’ve got to go grocery shopping all by yourself. It’s a newfound freedom (and, for many students, a new level of cognitive challenge).

Or, you could ask questions like this:

Whereas before you just blended yourself a puree and called it quits, these questions demand a more sophisticated kind of cooking. Now, using the formula is just one small step along the path.

Or how about this:

You’ve been given an ingredient that *looks* like a strawberry, but isn’t. You’d better know your stuff if you want to recognize this poisonous imposter-berry before it’s too late.

Occasionally, we teachers grow frustrated with our formula-thirsty students. (Okay, more like “often” or “weekly.”) Sometimes, we even denounce formulas altogether, deriding them as “brainless plug-and-chug” or “not real math.”

Of course, that’s going too far. The intelligent use of formulas is an important part of mathematics. But we’re right about one thing: there’s a lot more to formulas than just throwing numbers into a blender.

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“I’m planning a series of posts,” I told my dad the other day as he drove me home from the airport. “The title is *How to Avoid Thinking in Math Class*.”

Before I could get any further, he rubber-stamped the idea. “That sounds great! I always tell people, the point of school is to help you not to think.”

It’s a good thing he was the one behind the wheel, because if it were me, I’d have slammed the brakes and spat my latte all over the windshield.

“Really?” I sputtered. “I meant the exact opposite. I was going to catalogue all the silly ways that kids avoid doing real math. You know: blindly carrying out operations; hating word problems; worshipping right answers; focusing on procedures instead of ideas. I was going to take inventory of all those cognitive shortcuts and how they impede learning.”

“Well, sure,” he said. “But they can’t think *all* the time.”

“Well, isn’t thinking the whole point of math education?” I countered. “Math class is like a gymnasium for thought. You work out daily with a trainer who knows how to stretch your limits. You watch your mental muscles build. You see the payoff of intellectual risk. You learn that just as people can grow faster and more agile, they can grow smarter, too.”

“Yeah,” he mused, “that’s part of it. But that’s not the whole thing.”

“Why not?”

“Because thinking is *hard*.”

I had to admit, I knew what he meant. Just a few hours before, on the plane, I’d been reading a book when I realized that my eyes were just gliding over the words in a mindless imitation of reading. Meanwhile, my attention roamed and drifted. I wondered which sandwiches can and cannot be improved by bacon.

“Sure, sometimes we focus deeply,” my dad continued as he absentmindedly accelerated to make a yellow light. “But our natural state is autopilot.

“It’s like he says in that book,” my dad added. This is my dad’s usual level of specificity; he once described his favorite television character as, and I quote: “Oh, you know. Him. The guy.” Luckily, this time I knew which book he meant.

“*Thinking, Fast and Slow*,” I said.

“Right. By Khina… Khena…”

“Daniel Kahneman,” I said. My dad nodded.

Kahneman (a psychologist and Nobel laureate) describes human thought as governed by two systems. The first is fast and automatic. It employs shortcuts, heuristics, and dogmas—quick-and-dirty rules that save our mental strength. The second system is slow and deliberate. It musters focus, frames scenarios, tests assumptions, and reconciles contradictions—all those cognitive activities that exhaust students and excite teachers.

“We can only do that deep thinking on special occasions,” my dad said. “It’s just too tiring to focus intensely all the time. It’d be like sprinting everywhere instead of walking.

“That’s why the goal of school has to be automaticity,” my dad concluded. The Sunday morning roads were empty, and we’d nearly made it home. “Take learning your times tables. You’ve got to know them cold so that you can go on to finding common denominators, or reasoning about algebraic functions, or whatever. You need each task to become automatic before you can move onto the next intellectual step.”

“I guess you’re right,” I said. “But how do you reach automaticity? Isn’t the only way to get there by thinking deeply? Don’t you need to wrestle with the ideas before you can make them automatic?”

He drove quietly for a moment. “Yeah.”

“So then,” I said, “you’re saying that the purpose of school is to make you think hard now, so you won’t have to think later.”

(All this, of course, to allow you to think even harder.)

“Yeah,” he said, signaling a turn and pulling into the garage. “That’s learning, pretty much.”

And that, in a nutshell, is what I hope to write about over the next month.

In teaching math, I’ve come across a whole taxonomy of insidious strategies for avoiding thinking. Albeit for understandable reasons, kids employ an arsenal of time-tested ways to short-circuit the learning process, to jump to right answers and good test scores without putting in the cognitive heavy lifting. I hope to classify and illustrate these academic maladies: their symptoms, their root causes, and (with any luck) their cures.

But, as my dad points out, it’s not always bad to avoid thinking. It’s often healthy, even necessary. So I also hope to highlight the good heuristics, the sensible shortcuts, and the wisdom of seeing math class as an effort to program your autopilot.

I hope to make posts on the following topics (subject to abrupt change, regional blackout, second thoughts, and personal whim):

*Number Smoothies*. Throw the numbers in a blender, and press “puree.”*Dish-Washing Robots*. When thinking less means working more.*Word Quarantine*. Do math with no word problems! Then, paint with no pigments, and breathe with no air.*Garden of Doubt*. Watering the seeds of uncertainty.*Sharpening the Axe*. The wisdom of Abe Lincoln, math teacher.*Little Waldens*. As Thoreau advised: “Simplify, simplify, simplify!”*Church of the Right Answer*. Where do my students worship?

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Like Roald Dahl and Catherine-Zeta Jones, the equals sign was born in Wales.

It was 1557—not that long ago, in the scheme of things. Just a few years before the birth of Shakespeare. In fact, the Danish prince and the Scottish king captivated the public long before their humble Welsh neighbor reached wide renown.

The early equals sign was a lovely but ungainly thing, a long pair of parallels that its inventor called Gemowe Lines:

Over the centuries, this stilt-legged creature shortened into the compact and tidy symbol we know today.

And before that? Well, mathematicians simply spelled out equalities with the phrase “is equal to.”

10 is equal to 7 + 3.

8 x 9 is equal to 72.

And of course, a^{2} + b^{2} is equal to c^{2}.

The equals sign offered a way to avoid the tedious repetition of these words. Or, as Robert Recorde, the father of the symbol, put it: *to auoide the tedioufe repetition of thefe woords*.

An equals sign, then, is a verb. It’s the mathematical equivalent of “to be”—just as common, just as concise, and just as powerful.

But that’s not what kids see, is it?

To them, ‘equals’ means something other than ‘equals.’

In their arithmetic years, kids almost always encounter equal signs in a single, limited context: to call for the result of an operation. They fill their days with questions like this:

They get so used to statements of the form *[number] [operation] [number] = [answer] *that they’re a little creeped out by statements like this:

And totally deceived by statements like this:

Asked what goes in the blank, kids choose 9, because 4 + 5 = 9.

Or, when asked to perform multiple operations—start with 7, multiply by 5, subtract 9, and divide by 2—I often find my students writing streams of ungrammatical gibberish like this:

Of course, only the last of those equal signs makes any sense. The other two are bald lies. 7 x 5 doesn’t equal 35 – 9, and 35 – 9 doesn’t equal 26 / 2.

Poor old Robert Recorde would hang his head in sorrow.

Luckily, there’s a simple visualization of equality that can brush aside many of these misconceptions in one forceful sweep.

An equation is a statement of balance.

See the two sides? I’ve got different weights on each but the total is the same.

I can take away the same weight—say, 15 pounds—from each side, and they’ll still be equal. They’re not the same as they were *before*, but still the same as *each other*.

Similarly, I could add 5 pounds to each side, and they’d still be equal. Or I could double each side. Or halve each side.

So long as I do the same thing to each side, they’ll still be equal.

That familiar mantra—“Do the same thing to both sides of the equation”—is not an arbitrary dictate, cooked up by the Algebraic Rules and Regulations Committee in some air-conditioned boardroom. It’s a simple fact, which I capture in this rhyme:

*If two things are equal
then do what you will
to both things at once;
they’ll be equal still.*

Other symbols make sense in this light, too. The “>” symbol means “the thing on the left weighs more.”

The “<” symbol means the opposite.

And the “≠” symbol means “these two things aren’t equal, although I’m not telling you which one is bigger.”

I’ve had students ask me whether we can switch the sides of an equation, as if they need to consult the Bylaws of Algebra in some dusty legal library before making such a move. But understood with balance statements, it’s obviously true.

My younger students can mostly solve linear equations in x. But they do so by wordless numerical intuition (“I just *knew* x had to be 7”) or by blindly executed procedures (“I subtracted 11, then I divided by 2, but I don’t know why that works”). With balances in mind, suddenly it all makes sense:

Weirdly, my school’s textbooks teach “solving for y in terms of x” as a fundamentally different problem than merely “solving for x.” But in this light, they’re virtually identical.

The visualization can offer insight to older students, too. Students often solve simultaneous equations by “adding the two equations.”

Of course, you can’t really “add equations.” That’s nonsense. What you *can* do is add the same thing to both sides of one equation.

And then, on the left, instead of “52” we write something that’s equal to 52:

I don’t mean to say that this visualization is a magic pill, a cure for all misconceptions and fevers. No single key can unlock every door in mathematics. For that, you need flexible thinking, creativity, a healthy faith in your own abilities and a healthy skepticism of your own results.

Still, thinking properly about the equals sign sure helps.

I’ll confess: moving to the UK has heighted my affection for Wales. Sure, the English gave us a global language, the industrial revolution, and soccer. But the Welsh have gifted our planet a humble little symbol that compresses into two quick pen-strokes the far-reaching idea of equality itself.

*Further Reading, for the Curious*

FROM THE COMMENTS (i.e., the superior shadow-blog existing just below the surface of mine):

*Nevin objects to my objection*: “There are so many mathematical terms and symbols that are already overloaded to have multiple meanings, that **[this use of the equals sign] really isn’t hurting anything**.” *The problem, to me is that** students are not consciously adopting a different convention. Instead, they’re doing something unconventional, believing it’s conventional, and potentially missing an important concept in the process.*

*John Cowan points out, sensibly: “I think they use = that way because that’s what a calculator does; its = button means ‘Compute the (possibly intermediate) answer.'”*

*Howard worries about over-emphasizing notational issues*: “It seems to me that there is an insane desire to make stuff look like math and **force the mathematical language down like the production of foie gras**“

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**Law: **In the beginning, God gave His creatures free will, wisely limiting His own liability for any damage they might cause.

**Computer Science: **God threw something together under a 7-day deadline. He’s still debugging.

**History: **God wrote the Bible, which claims that the heavens and earth were created by God. This is exactly why you can’t always trust primary sources.

**Literary Theory: **After creating the world, God left scant evidence of His existence, as a deliberate exploration of the problematic nature of authorship.

**Political Theory: **When God created the world, He made sure to favor incumbents, being one Himself.

**Economics: **God created us in His own image, as rational consumers. But as sinners, we strayed.

**Physics: **God modeled the universe on the card game Mao: There are lots of strange rules and He refuses to explain any of them.

**Chemistry: **On the second day, God created entropy, to make sure the universe would turn itself off if He accidentally left it running.

**Psychology: **God said “Let there be light,” but what did He *mean* by that?

**Political Science: **In the beginning, God created the heavens and earth: pork-barrel construction projects that greatly benefited His district.

**Medicine: **In the beginning, God created a great clinical trial, although He hasn’t told us yet whether mankind received the treatment or the placebo.

**Accounting: **On the sixth day, God created man, whom he tasked with conducting a proper audit of His other creations.

**Finance: **God invested His creatures with life, and has received only a middling return on investment.

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This year, I’m teaching younger students than I’ve ever taught before. These guys are 11 and 12. They’re newer than iPods. They watched YouTube before they learned to read.

And so, instead of derivatives and arctangents, I find myself pondering more elemental ideas. Stuff I haven’t thought about in ages. Decimals. Perimeters. Rounding.

And most of all: Multiplication.

It’s dawning on me what a rich, complex idea multiplication is. It’s basic, but it isn’t easy. So many of the troubles that rattle and unsettle older students (factorization, square roots, compound fractions, etc.) can be traced back to a shaky foundation in this humble operation.

What’s so subtle about multiplication? Well, rather than just tell you, I’ll try to show you, by using a simple visualization of what it means to multiply.

Multiplication is making an array.

If you think of multiplication as “repeated addition”—that is, 5 x 3 as 5 + 5 + 5, then great! That’s the same thing.

And if you think of multiplication as “groups”—that is, 5 x 3 means 5 groups of 3, or 3 groups of 5, then even more great! That’s *also* the same thing.

Now, what’s the benefit of this visual model? Ah, where to begin! Without it, you’re multiplying from behind a blindfold. Tear that cloth from your eyes, and begin to see!

Take the *distributive property*, a seemingly opaque bit of symbolism that says a(b+c) = ab + ac. Its misuse haunts algebra teachers’ nightmares. But it’s no mystery—just a simple fact about adding two arrays together.

Or what about the *prime numbers*, those invisible atoms of the mathematical world? Well, under this view, they’re as pleasingly tactile as moss on a stone.

First, *composites* (the opposite of primes) are numbers that can form arrays.

And primes? They’re numbers that *can’t* form arrays.

By the way, why isn’t 1 prime? (It’s a fact that confounds students, and “because we want prime factorization to be unique” isn’t a satisfying answer.) Well, visually, it doesn’t quite fit either pattern. It can’t form an array of smaller numbers, so it’s not composite. But it isn’t an awkward misfit like the primes. It’s just an indivisible little unit. So it’s neither prime nor composite.

Ever heard the word “commutative”? It means you can switch the sequencing of an operation without changing the outcome. For example 5 x 3 = 3 x 5. And this makes perfect visual sense for multiplication, because rotating the array shouldn’t change its size.

We can dig deeper. A “factor” or “divisor” of a number is a side length you can use for its array.

And a “common factor” for two numbers is a width that they share. It lets you combine them into one larger array.

Similarly, a “multiple” of a number is a larger array that contains multiple copies of it.

And a “common multiple” of two numbers is an array inside which they can each fit repeatedly, like tiles.

On a simpler level, why is it called *squaring* a number? Because it makes a square!

We can take this further. Replace your dots with squares.

Suddenly, we’re talking about *area*. We’re looking at the formula for the area of a rectangle, from which we can derive all other basic area formulas.

This way, we can even multiply fractions! Multiplying 4/5 x 3/5 splits a unit square into 25 pieces, and picks 12 of them—hence, 12/25!

I find mixed success when inflicting drawings like these on my students. A few Y9’s are so excited they practically leap out of their seats. Others squint skeptically. “So you want us to draw a picture every time we multiply?” they ask. “Isn’t that a waste of time?”

“Yes,” I tell them. “It is.”

“Do *you* ever use these?” they ask.

“No,” I say. “I don’t.” Fluency means taking simple steps mentally, without resorting to fingers or scratch paper. A kid who can’t use the distributive property in his sleep is as doomed as a kid who doesn’t know 9 x 4 = 36.

“So why do these pictures matter?”

They’re an early-stage ingredient. When you’re first developing a concept, they’re something you throw into the batter. Later, when you think about multiplication, you’ll rarely need to draw these pictures. Their flavor will have suffused your understanding, like vanilla extract baked into your cookies.

But trying to add this visual understanding later is often fruitless. You can’t pour vanilla extract over the top of your finished cookies. The concept is done baking.

For now, I’m enjoying my days with the young’n’s, these kids who are younger than the Euro. It’s never been so fun feeling so old.

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**Mathematical Slogan****Real Slogan**

**Mathematical Slogan** **Real Slogan**

**Real Slogan**

**Mathematical Slogan**

**Real Slogan**

**Mathematical Slogan**

**Real Slogan**

**Mathematical Slogan**

**Real Slogan**

**Mathematical Slogan**

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Once there weren’t numbers,

and life was cold and sad.

You might say “I’ve got lots of stuff!”

but not how much you had.

You could gather flowers,

but you couldn’t count them up.

You could ask for chocolate milk,

but not a “second” cup.

And though their eyes could see just fine,

the people still were blind.

They held things in their arms and hands,

but never in their minds.

Then a number sprouted up!

No one knows quite how.

It just appeared,

and people cheered,

“We’ve got a number now!”

It glistened in the morning dew,

and sparkled in the sun.

It stood up straight and proud, and cried,

“Hello! I’m number one.”

And after that, the people saw the world a different way.

They might come home and tell their friends,

“I saw one cow today!”

“One sandwich!” they could tell the cook.

“One song!” they’d ask the singer.

“One glass of water,” they’d request,

while holding up one finger.

But if they wanted *more* to drink,

well, that was not as fun.

They’d say, “One more. One more. One more,”

until the thirst was done.

Finally, one day, One cried out,

“This simply will not do!

I’m not enough! You need my friend!”

And out stepped number two.

The people cheered!

Two waved and grinned,

and said, “Oh, mercy me!

Let’s have *my* friend come join us now!”

And out stepped number three.

Three just blushed and introduced

its friend, the number four.

Each number brought another friend,

and more, and more, and more!

By the time the sun had set,

the numbers filled the air.

They stretched into the evening sky—

Numbers everywhere!

The world had changed that very day:

The people now could *count*.

Instead of saying, “Look! Some sheep!”

They’d state the right amount.

“Eleven children in the class.”

“Twenty boats at sea.”

“The ants upon the ground below?

There’s seven-hundred three.”

Things were good.

Yes, things were great!

Except one thing was not.

Two friends had baked themselves a cake,

then said, “It seems we’re caught.

We want to share this cake we’ve made.

We want to split it fair.

But what amount should we each get?

We’re simply not aware.”

No one knew just how much cake

to give to both the friends.

They argued till their throats were sore,

and patience hit its end.

Then the numbers happened by,

and broke into a laugh.

“You don’t need us!” the numbers cried.

“You need our friend one-half!”

Indeed, they did—and quickly, they resolved their cake transaction.

“One-half! What *are* you?” people asked.

And it said, “I’m a fraction!”

And then there sprouted from the ground,

from crannies and crevasses,

fractions by the millions,

fractions by the masses!

“What a world of numbers!” people cried,

and danced,

and sang.

They thought their tale had ended with a satisfying bang.

They thought they knew the numbers.

Every uncle, aunt, and kid:

they thought they knew the numbers!

They really thought they did.

For years, they’ve kept on coming, though,

emerging through the mist.

There’s too many to fathom.

There’s too many to list.

They come in different sizes,

and they come in different signs.

There’s negatives,

and radicals,

and 1-4-1-5-9’s.

Some of them are perfect,

and some of them are square,

and some are just irrational,

like clumps of unbrushed hair.

Some are made of many parts,

and some are elemental.

Some are real,

and some are not,

and some are transcendental.

Some are large as universes,

others small as seeds.

You plant a single number

and the others sprout like weeds.

And even now—yes, even now!—on strange and special days,

numbers help the people see the world in different ways.

Once there weren’t numbers,

but that was long ago.

And now they cover everything,

like freshly fallen snow.

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