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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But the fact is that, aside from being the butt of cheap jokes, mathematicians and scientists don’t share all that much in common.

And you can tell that from the way they look at each other’s fields.

When it comes to research, **scientists view mathematics the way a handyman views a toolkit**. To a scientist, math is a way of solving problems, as practical as a step-ladder or a roll of duct tape.

Want to describe an object falling to earth? Draw up a quadratic equation!

Want to investigate a rate of change? Take a derivative!

Want to model an electromagnetic attraction? Bring out the vector fields!

When people extol the “real-world” benefits of mathematics, they’re talking about moments like this, when a scientist employs quantitative techniques to analyze the world around us.

Meanwhile, **mathematics draws on science the way an artist draws on a muse**. Science reveals a real-world phenomenon—and the mathematician asks, how can we make this abstract? How can we generalize?

Take the idea of spatial dimension. Most of your ordinary objects—microwaves, teapots, housecats—are three-dimensional. That means they can be measured in three directions—length, width, and height.

Thus, any point in our three-dimensional world can be summarized with three numbers—call them, *x*, *y*, and *z*.

But the mathematician pushes further. What if we had four numbers? Or five? Or *n*? What would words like “volume” and “distance” come to mean? Can we imagine a six-dimensional sphere, and if so, what in blazes *is* it? Which of our 3D intuitions will carry over into higher dimensions, and which will break down?

For mathematicians, physical reality—that is, scientific reality—is nothing more or less than a source of inspiration. And like any inspired artist, mathematicians feel free to extrapolate and invent, to ask “What if?” and “How else?” and “Couldn’t we pretend…?” It doesn’t matter whether any of this higher-dimensional stuff really exists—it’s still a marvelous stroll to take your brain on.

Mathematics and science, then, aren’t like two members of the same species. They’re like two entirely different animals, sharing a lovely symbiosis. Math is like a parasite-eating bird, perched on the rhino of scientific reality.

Math gets nourished. Science solves a problem. Everybody wins.

The bird and rhino don’t share much in common, but they make a heck of a team.

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…when you run into a college classmate who dropped out after suffering from health issues. You always meant to write a nice, sympathetic letter of support, but it never crested to the top of your to-do list, and now your long silence seems callous. The classmate sees you, looks away, then marches right up to you and asks, point-blank, the question you’ve dreaded for years:

“Why,” your classmate spits, “does raising both sides of an algebraic equation to an even power potentially introduce extraneous solutions?”

…you’ve already skimmed every worthwhile article in the newspaper. You completed the crossword, the Sudoku, even the word jumble. Grudgingly, you turn to the paper’s last remaining puzzle: the Partial Differential Equation of the Day.

A difficult man, largely neglected by his own children, he has come to embrace you as one of his few points of contact beyond the nursing home. Now, in his waning hours, he mistakes you for his estranged son, Bradley, and delivers a tearful apology. “I was a terrible father,” he says. “Forgive me, Brad.”

“I do,” you say.

“If you truly forgive me,” he wheezes, “then prove it. Tell me the quadratic formula, one last time.”

…your desktop breaks down. It won’t even boot up. A new computer (lasting an expected five years) will be $800, or alternatively, you can diagnose the problem for $70, and repairs (which would extend the computer’s lifespan an expected three years) could cost anywhere from $50 to $600, uniformly distributed.

Now, it may feel impractical as a replacement for a desktop, but look how *pretty* that tablet over there is. Look how *shiny*. Don’t you want a pretty, shiny tablet instead? Don’t you? Don’t you?

…by the frat-boy culture at your office, you find yourself in left field, dreading any ball hit your way. Suddenly, a pop-up: it should probably be yours, but instead, the shortstop and centerfielder converge on it, and collide—hard. The shortstop is out cold, blood oozing from a temple.

“Quick!” the centerfielder shouts. “Is anyone a doctor, nurse, or topologist?”

Silence follows.

“C’mon!” he screams. “We need someone to treat this wound, diagnose it, or prove its invariance under diffeomorphism!”

…and you’re craving potato chips. Still, you want to minimize the caloric damage, so you’re standing in the 24-hour liquor store, comparing brands. Lay’s nutrition label claims it has 200 calories per serving and 6 servings per bag, while the generic brand has 600isin(11π/6) calories per serving and 5i servings per bag.

“Looks like these chips have some *complex* carbohydrates!” you chortle. The guy behind the register smiles appreciatively.

…transcending corporeal existence. A four-dimensional hypercube and a six-dimensional analog of a torus are gossiping about a good friend of yours, an equiangular polygon undergoing a dilation. “Heh, I’d like to project into *that* plane!” the hypercube says, and they both laugh with cruel mirth.

Do you say something, or just keep walking?

There’s a math section on the GRE.

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*New Law***: **For every action, there shall be a vicious and disproportionate reaction, followed by many months of inaction.

** Existing Law: **Planets orbit the sun in ellipses.

*New Law***: **Planets orbit the sun in a series of strange curlicues, to be redistricted whenever a new party takes power.

** Existing Law: **Objects in motion shall tend to remain in motion, unless acted upon by an outside force.

*New Law***: **Objects in office shall tend to remain in office, even if challenged in the primary by an outside force.

** Existing Law: **The universe is expanding.

*New Law***: **If the universe is indeed expanding, then we shall see to it that the national debt keeps pace.

*Existing Law***: **No temperature can ever reach absolute zero.

*New Law***: **No temperature can reach absolute zero anytime in the next three months. We shall revisit the issue after such time has elapsed.

*Existing Law***: **Heisenberg’s Uncertainty Principle: The more precisely you know the momentum of a particle, the less precisely you may know its position.

*New Law***: **The more precisely you know a politician’s ambitions, the less precisely you may know his positions.

*Existing Law***: **Perpetual motion machines are impossible.

*New Law***: **We are proud to award a $75 billion federal contract to Perpetual Motion Machines Inc. (We assure you that their generous campaign contributions had no effect on our decision whatsoever.)

*Existing Law***: **Natural selection drives evolution; that is, organisms better suited to survival and reproduction will tend to survive and reproduce more than their competitors.

*New Law***: **Natural selection drives the economy; by the way, we’re loaning Wall Street half a trillion dollars, hope everyone’s cool with that.

*New Law*: *[The value of E is currently vacant, as the Senate has filibustered all seven values that the president has nominated.]*

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So I gave them a survey, asking them to complete the following six sentences.

- Mathematics
**is**… - Mathematics
**is not**… - Mathematics is
**useful**for… - Mathematics is
**useless**for… - My
**favorite thing**about mathematics is… - My
**least favorite thing**about mathematics is…

Their answers spanned the whole spectrum of attitudes, from rapture to resignation, from joyful to jilted. (Apologies for the alliteration; I can’t contain it. Clearly.) Here’s a small sampling selection smattering collection of their answers, in all their silly wit and strange variety.

…playing with numbers.

…a long word for “maths.”

…the best!

…the story of a few people who decided to take these regular numbers and perform cheap tricks with them.

…fractions, pi, Graham’s number, Einstein, and sweaty classrooms.

…boring.

…compulsory.

…quite hard.

…fun and easy.

…a language and an art.

…confusing.

…what makes the world go ‘round. Literally.

…potentially interesting as well as potentially tedious.

…one of the most important tools one can use in life.

…a load of complicated numbers and operations.

…strangely entertaining.

…an ice cream flavor.

…a way to fight crime.

…like any other subject.

…very fun.

…everyone’s cup of tea.

…easy.

…easy.

…easy.

…easy. [*You get the idea. This was a popular one.*]

…very useful.

…about imagination.

…a subject to take lightly.

…an elephant.

…a debating subject, as most of the time there is one answer.

…just one thing.

…a complete waste of time.

…entirely boring.

…boring. If it is, then the teacher made it so.

…taught in a way that makes it fun or enjoyable.

…a modern concept; it has been around for a long time.

…easy to understand; there are many areas I am still struggling to get my head around, but I will eventually.

…always a simple answer (or any answer we can comprehend).

…everything.

…nothing.

…tax evasion.

…cooks.

…mining ore from caverns.

…banking, buses, trains.

…thinking outside the box.

…increasing our understanding.

…buying stuff.

…organizing the financial system.

…NASA’s operations.

…bringing order to chaos; making complicated things simple.

…maths homework.

…choosing sandwich spreads.

…writing epic poetry.

…being Kate Middleton.

…saving you from a tragic blimp crash.

…dancing.

…making friends.

…creativity, because maths is very logical.

…chickens, and other non-enlightened, non-sentient farm animals.

…emotions.

…imagination and dreams.

…artists.

…playing with my dog.

…rapping about blight and urban degradation.

…day-dreaming, cloud-spotting.

…working at McDonald’s.

…dividing numbers by another or doing any other simple problem, because technology is easily accessible and we can use calculators.

…the challenge.

…getting the answer right.

…solving problems, because it is like solving a murder case.

…magic squares.

…happy numbers.

…the complete order and chaos of it at the same time.

…the fact that I’m good at it.

…the teachers.

…you’re either right or wrong.

…you never have to give your opinion.

…the logic and sense.

…that it is a universal language.

…how it can be used for almost anything in life.

…how even if something appears complex, it can be solved step by step easily.

…the history of maths.

…doing very hard calculations for no good reason.

…a calculator.

…tricky word problems and impossible equations.

…strange theories that don’t make any sense.

…hard work.

…when I use a compass that is loose and it slips and makes my diagram inaccurate.

…showing your work even when you know the answer.

…the questions.

…not being able to understand something and feeling frustrated.

…that it is not always presented in a fun way and can be at times very complex and discouraging.

…tests.

…that the teachers are not as inspiring as other subjects, probably because they’re also fed up.

…the fact that in order for me to grasp mathematics, it takes a long time.

…explaining proofs.

…silly diagrams.

…when I spend ages on a problem, then realize I’ve made a mistake really early on.

…the fact that, after primary school, they decided to ruin it by diluting all those lovely numbers with letters.

…there’s no leg room in the maths classrooms.

…getting the answer wrong.

…the long tedious sums and the unsolvable ones.

…when you have to repeat the same sort of single thing over and over again like a mindless robot.

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