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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It was the end of our first day on limits—a deep and slippery concept, the engine of calculus—when Melanie exclaimed, “Wait. Shouldn’t that limit be 4, not 6?”

Nope—it was 6. Melanie’s error suggested that she’d missed the lesson’s most basic truth, an idea that the class had spent the day paraphrasing, analyzing, and shouting in chorus. Talking one-on-one, I could have coached her through the misconception. But hers was a public declaration, in front of the whole room.

Even before the words had left Melanie’s mouth, I could hear the groan welling up among the students, murmured ridicule and the slapping of foreheads soon to follow. They all knew it. She didn’t. From Melanie’s blushing, you could read her self-esteem falling like a mercury thermometer.

And so I found myself confronting one of the teacher’s daily puzzles: what do you say when a student is wrong?

In the classroom—and beyond it—dealing with mistakes is delicate and crucial territory. At one extreme, we can deride errors, ridicule them, chase them out of town with a pitchfork—to the dismay of the poor kids who were trying their best.

At the other extreme, we risk coddling wrong statements, embracing them as “perspectives,” waiting in vain for the mistaken to see the light, and in the meantime allowing confusion to reign.

It’s no easy balance to strike, exposing falsehoods without steamrolling egos. To solve the predicament means tackling a deeper issue: what kind of dialogue, exactly, are we having? Which is primary: ideas, or the people voicing them?

Most of our daily conversations are centered on people. Consider the meandering path of dinner table chatter. It drifts from topic to topic, following tangents, pursuing loose connections. Questions are posed and never answered. Thoughts dissipate, half-articulated. A conversation, after all, is about sharing each other’s company. It’s a meal, not an interrogation.

Academic discussions differ. We’re hashing out ideas, each person pitching in to advance the conversation, not necessarily tabulating whose insight is whose. I say A; you say B; and we both think of C simultaneously. It’d be counterproductive to insist that your ideas deserve special status simply because they’re yours. The truth is indifferent to its speakers. Reason doesn’t notice who gives it voice.

In an academic dialogue, ideas ought to matter more than egos.

The challenge for teachers is that we’re having both conversations at once. Our students matter as people—we care about their thoughts, their feelings, their ambitions and fears. But ideas matter, too—we want to spread truth, to help our students master bodies of fact that past generations have painstakingly uncovered (and that textbook authors have clumsily compiled).

So how do we address wrong ideas, without trampling the people who espouse them? Here are four time-tested techniques that I’ve stolen from teachers far nimbler than I.

“That’s exactly the mistake I made when I first learned this material.” Or: “That’s a tricky point. In last year’s class, more than half of them made that error on the quiz.” Or the simplest: “That’s a really natural mistake to make.”

“It sounds like you’re generalizing from the pattern we saw earlier. But I’m not so sure those same results will apply here.” Even the most egregious mistakes have their roots in some identifiable thought process, and tracing that process helps students recognize that their *ideas* are not *them*. A train of thought can derail, and it doesn’t make the thinker any less worthy or intelligent.

“Hmm… go ahead and check that calculation again, then get back to me.” Private mistakes sting far less than public failures.

“Remember, mistakes are the essence of learning. When someone commits an error, we respectfully help them understand it, so that everybody can learn from it.” At its best, conversation is a beautiful thing—collaborative thinking, a shared brain. But students need to be taught the rules of the game.

With proper preparation, a class will rarely respond with the hostility they showed Melanie that day. But sometimes even a friendly pack can turn on one of its own.

“I’m glad you brought that up, Melanie,” I shouted over the groans. “It goes to show how *tricky* this transition is. You’ve all spent the last three years approaching graphs one way. This new perspective is the conceptual core of calculus, and the mental shift will take some time.”

The groans faded. Melanie nodded.

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Stop leering and asking me “What’s your sine?” I’ve already told you, it’s ½. Why don’t you ask me about my hobbies or something?

-π/6

Dear Secant,

Ooh, this isn’t totally related, but what you just said reminds me of a great story…

-Tangent

Dear SOHCAHTOA,

You would be really useful if I had a better grasp on phonetics. Is it SAHCOATOH? SOACOHTAH? Help me out here.

-Bad Spellers

Dear 3-4-5 Triangle and 5-12-13 Triangle,

Hey, can I play? How come you guys never pick me to hang out with?

-8-15-17 triangle

Dear Quadrant 1,

I’m closing the wormhole to Quadrant 3. The threat posed by the Dominion is simply too great.

-Captain Sisko, Deep Space Sine

Ladies and Gentlemen of the Jury,

Let’s concede, for the sake of argument, that sin(A)/a = sin(B)/b. If so, what does it prove? Nothing! My client is innocent.

-The Lawyer of Sines

Dear Teacher,

Don’t worry, I totally understand simple harmonic motion. “Simple” means dumb, “harmonic” means music, and “motion” is moving around. So simple harmonic motion is just bad dancing. I’m the master of that!

-Student

Dear Sine,

We can’t keep fighting each other like this. We just wind up right back where we started. Truce?

-Arcsine

Dear Radians,

I don’t care if you’ve got more critical acclaim. I’m more popular among audiences.

-Degrees

Dear Students,

Just when I thought things were going so well… you see “sin(45^{o})/sin(30^{o}),” and you cancel out “sin” to get 1.5? Back to square one, guys.

-Teacher

Dear Teacher,

Sorry for blurting out that answer so quickly. I guess I’ve got an itchy Trig finger.

-Student

Dear Trig Classes,

Oh, I see. You spend weeks drawing beautiful portraits of sine and cosine, but when it comes time to draw me, you’re “too busy” and “behind schedule”? How convenient.

-Cosecant

Dear Trigonometry, if That Is Your Real Name,

I’m going to have to ask you to wait here. We need to verify your Identity.

-Homeland Security

Dear Classmate,

Shhh… don’t tell the teacher, but for the test, I snuck a bunch of light waves into the room. That should make it easy to remember what sine curves look like. You can look at them too, as long as you don’t snitch.

-Your Classmate

Dear Mom,

I don’t think I can go to school today. I have too many questions about who I am, and what my purpose is in math class. Basically, I’m having a trig identity crisis.

-Your Kid

Dear Student,

Look, I know you want more points, but when I asked you to state DeMoivre’s theorem, you just wrote “DeMoivre’s De Man!!!” and drew a picture of a French guy in a beret holding a baguette. I think 3 out of 10 is *more* than generous.

-Your Teacher

Dear Numbers,

You’re so concrete. I don’t see how you could ever be useful in the real world.

-Mathematicians

Dear Teacher,

You want me to “evaluate” this expression? Okay—it’s muddled, poorly lit, and unengaging. I give it 1 star out of 4. Are you happy now?

-Student

Dear Teacher,

Aw, you’re letting us bring a cheat sheet to the test? Way to take all the fun out of cheating. Now I don’t even want to.

-Student

Dear Sin(2x) and 2sin(x),

Wait… you two guys are different? Please don’t think I’m racist, but I thought you were the same thing.

-A student

Dear “Scientific” Calculators,

What’s so scientific about having less functionality than a slide-rule?

-Graphing calculators

Dear Graphing Calculators,

Wow, you only cost $112 each? That’s just $109 more than the Wolfram Alpha app that does all the same things, with a better interface. What a bargain!

-Scientific calculators

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It’s not all wrong, I’m sure. But it makes me wonder: How well do we actually *know* the classroom? Before we start drastic renovations, we should make sure we’ve got a clear view of the structure that’s already in place. And I’m not sure we do.

The story of the classroom is devilishly hard to tell.

Not long ago, I wrote a piece for *The Atlantic* on the difficulty of being honest about the classroom. While writing, I asked my sister (a teacher and instructional coach), “Do you find that the teachers who *talk* a good game might not always *walk* a good one?”

“Definitely,” she said. “They’re not totally uncorrelated, but they’re different skills. Just because you can describe a classroom well doesn’t mean you can run one—and vice versa.”

It’s a dichotomy I think about often. I teach, and I tell stories, and those are very different things.

This matters because our stories—streamlined and simplified as they are—almost always carry more weight in memory and decision-making than the full, undigested truth. Those technologies I hear about aren’t trying to remake the actual classroom; they’re trying to remake the slightly distorted version we talk about.

We thus run a risk: letting the classroom as discussed and the classroom as practiced drift apart, like continents with an oceanic gap spreading wide between them, until each shore is barely visible from the other.

How does this happen? I see three main challenges to telling the true story of the classroom.

Most stories have a protagonist: *Hunger Games* has Katniss, *Star Wars* has Luke Skywalker, and the Civil War has Abraham Lincoln. Heroes are fun. They’re simple. Their posters look great on dorm room walls.

But now, step into that teeming multicellular organism we call a classroom. You’ll see groups of students talking; a teacher circling quickly from student to student, the interactions brief but potent; crossover between groups; notes and texts and funny faces sent across the classroom; isolated kids bored in the back row; in short, a whole mess of complex relationships.

You’ve got 25 or 30 protagonists to pick from, each with equal claim on the title.

So much is happening at once that our tellings are inevitably selective. To capture the whole classroom, you’d need to go *Game of Thrones* on it: write a dozen books of epic length, leaping from one character to the next, with seven different scenes unfolding at any given moment.

A story is an incident. “Backpacking, I looked across the ravine, and I saw a mother bear and two cubs”—that’s a story.

“Backpacking, I typically bring a pair of flip-flops to wear around the campsite”—that’s not a story, it’s a dull fact about a routine, and if we’re chatting at a party, then you’re pointing to your empty cup and telling me how it’s been great talking but you’ve got to go get a refill now.

Stories are fun. Routines are dull.

The problem is that, in the classroom, the routine often *is* the story.

Inevitably, the 180 days in the school year blur together. If a moment stands out as a good story, that’s often because it’s atypical and unrepresentative. You don’t remember the 179 days that Annie sat there, yawning and fidgeting; you remember the one day that she raised her hand so eagerly the momentum knocked her out of her chair.

The most satisfying stories to tell can actually paint a totally misleading portrait.

The lifeblood of a classroom runs not through one-time incidents, but through the endlessly repeated ones. Habits. Rules. Enforcement of those rules. Class customs. Student culture. Key lessons driven home week after week. It’s a slow grind, and its essence is often lost when we try to tell stories of isolated moments.

The first rule of narrative is, “Show, don’t tell.”

Oops—broke it! Let me try again, a little more colorfully this time.

I’ve seen active, engaged classrooms full of kids learning nothing valuable.

I’ve seen quiet, somewhat dull-looking classrooms where kids soaked up important and enduring lessons.

The kind of stuff that’s necessary for compelling storytelling—sights, sounds, smells—doesn’t always point towards the *right* story. The crucial activity in a classroom is the invisible, inaudible, odorless shifting of gears that happens inside the students’ skulls. What matters is their inner psychology—what they’re learning, and what they’re not. Short of first-person narration, there’s no good way to access that.

“So what do we do about all this?” I asked my sister.

“Keep looking carefully at the classroom,” she told me. “Keep measuring our stories against the reality, as best we can see it.”

Every minute in every class, things go wrong. Distracted chatter. Bullying whispers, out of earshot. Misconceptions unchecked. Seconds wasted. No one runs a perfect classroom; teaching is a job that affords us so much to think about, so much to consider, that we can attend to only a few issues at a time.

But that’s the beauty of the classroom, too. It’s relentless and unflinching. The classroom isn’t just your best days. It’s all of them. Teaching isn’t like basketball, with constant timeouts and stoppages and pauses. It’s more like soccer football, with the clock always running.

I try to heed the stories I tell. In moments of despair, I recount my triumphs. In moments of confidence, I dare to face my shortcomings. And when I tell others—friends, family, strangers—about my teaching, I do my feeble best to indicate the thousand threads of the tapestry, knowing all the while that no story I tell is ever quite complete.

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