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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The title: *The SAT Changed Their Guessing Policy to Appear Fairer, But It’s Actually Less Fair.* “With the ACT pulling ahead in the admissions test Cola Wars,” I wrote, “I struggle to greet the SAT’s announced changes with anything but cynicism.”

I was halfway into the boxing ring when I realized I was on the wrong side of the fight.

This little fable is about the SAT’s “guessing penalty,” and while it’s a tale full of technicalities, I promise it’ll end with a moral. A moral so obvious, it’s surprising.

Or perhaps vice versa: so surprising, it’s obvious.

Although the term “guessing penalty” appears practically everywhere (including the *New York Times*), the College Board never uses it. And the College Board is right. There’s no way to penalize guessing, per se—after all, the SAT only sees right answers and wrong ones. They’ve got no way of knowing whether you arrived at your response by cold, logical deduction or by blind, stupid luck.

Instead, the current SAT penalizes *wrong *answers. You get 1 raw point for answering a question right, and lose ¼ of a point for answering it wrong. (A blank answer neither adds nor subtracts from your total.)

This system doesn’t aim to penalize guessing. It aims to neutralize it.

Let’s say you guess blindly on five questions. On average, you’ll get one right (+1) and the other four wrong (-¼ – ¼ – ¼ – ¼), for a net score of zero. Since the outcome is no different than if you’d simply left all five questions blank, guessing should neither help nor hurt you overall. It just adds a greater element of randomness to your score.

Why did the College Board ditch this longstanding system? For simplicity’s sake.

In their announcement, they described the old system as “Complex Scoring.” That’s fair. For their whole lives, students have taken tests with no distinction between blank answers and wrong answers—both are worth zero. Then, on the SAT, when the stakes are the highest, the rules of the game suddenly switch, and blank becomes (slightly) better than wrong.

This unfamiliar system might seem to confer an advantage on wealthy families. After all, they can afford tutors to explain the rule and its implications, while poorer students remain in the dark.

Hence, the new and cleaner system: One point for right answers. Period.

Unfortunately, this revision doesn’t make the test less gameable. It makes the test *appear* less gameable, while actually making it *more* so.

To see why, suppose that you and I both take a timed section with 20 questions. Being similarly skilled (and rather slow) students, we each get through question #10, finishing it with a few seconds to spare.

But then, I spend those final few moments working (naturally, it would seem) on problem #11. You, on the other hand, spend the remaining time randomly bubbling answers for #11-20.

Clever move on your part. Under the old rule, this would’ve merely added randomness to your score. But under the new rule, you’ll get credit for your right guesses and suffer no cost for your wrong ones. On average, it’ll boost your score by 2 points—a 20% improvement over mine.

You and I knew precisely the same number of actual answers, but you did significantly better, because you employed the right test-taking strategy.

Isn’t that precisely the scenario the new SAT is supposed to help avoid: rewarding students not for their knowledge of the content, but of the test?

But this is where I got it wrong, and where the SAT—to their credit—got it right.

The crucial fact is that, even though it’s perfectly possible to earn a solid score (say, a 600 per section) while answering only two-thirds of the questions, virtually no students take that approach. Out of every 25 questions, the typical student answers 24.

It’s strange—the system gives them no incentive to do this. They’d largely be better off focusing on 12 or 14 questions, making sure to get them right, rather than spreading themselves thin by hurrying to finish. Even so, students answer virtually every question, leaving almost no blanks.

The SAT’s revision, then, is just a response to the facts on the ground. Kids are *already* answering all the questions. In this landscape, penalizing wrong answers, on top of rewarding right ones, is redundant and confusing.

And here we come to the moral, the take-home message, the thing I learned from picking a losing rhetorical battle with the College Board, and then wising up.

It’s vacuous to call a test “good in theory.” For an assessment, the only thing that matters is practice. The measure of a test’s success lies in its interactions with the humans that take it.

When designing (or pondering the design of) a behemoth test like the SAT, it’s tempting to think of it as a cold, impersonal system of incentives. It’s easy to forget that it will land on the desk not of some abstract test-taking machine but of real, flesh-and-blood students.

As such, it must obey the imperative of every test: to respond to the needs and habits of the test-takers.

A test may appear transparent, fair, and perfectly organized to an outsider, but if it confuses the students and obscures their abilities, then it’s worthless. An ingenious bit of test-craft gets you nowhere if it asks students to fight against instincts that they simply can’t overcome.

So College Board, I apologize for the hate-letter I never sent. You were right.

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If people looked like bad drawings, he’d look like this:

He also taught me one of the most enduring lessons I’ve learned about economics, modeling, and the limits of theory to explain the social world.

But the lesson wasn’t about those things. Not explicitly. It was about the minimum wage.

A model is a simplified representation of reality. A helium atom is reality, but this little picture is a model, because it captures some key features while leaving out others.

The classical model of the minimum wage goes something like this. Start with a world that has no minimum wage. The market should stand at a nice equilibrium. If you’re working a job, then the wage you earn must be worth more to you than the time you spend. Otherwise, you’d quit.

Meanwhile, if you’re an employer, then the workers you hire must contribute more to your business than they cost in wages. Otherwise, you’d fire them.

For example, say Bob is working at McDonald’s for $4 per hour. It’s worth it to him—in fact, he’d work for just $3 per hour. And it’s worth it to McDonald’s, too—Bob contributes $6 per hour to their bottom line. So both Bob and McDonald’s benefit from the exchange.

But now the government institutes a minimum wage of $7. What happens?

Well, unfortunately, Bob is toast. He’s only worth $6/hour to McDonald’s, but they’re required to pay him $7 per hour, which is a bad tradeoff for them. So they fire Bob instead.

Bob is willing to sell his labor for $4 per hour. And McDonald’s is willing to buy it. But the government won’t let them. That minimum wage law dooms Bob—and anyone else unable to provide $7/hour of value to a business—to unemployment.

So goes the classical model. It predicts that raising the minimum wage will drive low-productivity workers into unemployment.

But Peter showed me another possibility.

You’ve heard of a **monopoly**, which occurs when there’s only one company selling a certain product. Similarly, a **monopsony **occurs when there’s only one company *buying* a certain product—in this case, people’s labor.

So, let’s suppose that McDonald’s is the only employer around, and each new employee provides them with $10/hour of value.

Anita is willing to work for just $5/hour, so McDonald’s happily hires her.

Biff is willing to work for $6/hour. McDonald’s would like to hire him, but if they do, they’ll have to raise Anita’s wage too. (They can’t pay a new hire more than an experienced veteran!) So hiring Biff really costs them $6/hour, *plus* the $1/hour raise for Anita. That’s a total of $7/hour, which is still worth it, so they hire Biff.

Carmen is willing to work for $7/hour. By the same logic as before, hiring her will cost McDonald’s $9/hour. Still worth it, so they do it.

Diego is willing to work for $8/hour. But hiring him costs McDonald’s $8/hour + $3/hour = $11/hour. That’s not worth it to them. So Diego stays home.

Again, Diego is willing to work for $8. And McDonald’s is willing to hire him for $8. But it doesn’t happen—not because of government interference, but because of the structure of the labor market itself.

Now, introduce a minimum wage of $9. As before, McDonald’s hires Anita, Biff, and Carmen. But now, hiring Diego only costs them $9, because they don’t need to raise their other employees’ salaries simultaneously. That’s a good tradeoff. So they do it.

In this model, a minimum wage doesn’t drive up unemployment. It actually drives it down.

As Peter finished, I began mentally picking apart this model. My classmates did the same aloud: “Is that monopsony a reasonable assumption?” “Are starting wages really tethered together in this way?” “Doesn’t this depend on the government magically setting the minimum wage in the sweet spot between worker productivity and workers’ willingness to sell their labor?”

Peter handled the questions amiably and fairly. He acknowledged the model’s flaws, affirmed its strengths, and elucidated its predictions.

“But what makes this better than the traditional model?” someone finally asked.

Peter blinked. “Nothing, of course,” he said. “Nothing intrinsic to the model.”

“Then why teach it to us?” someone else asked.

“Because when one model predicts A, and another model predicts B, then it’s time to set aside the models and go gather some data. You can have the world’s most elegant model, but if it can’t predict what happens in real life, then it’s just a pretty piece of mathematics.”

To me, it’s a lesson that reaches beyond introductory economics. It’s about dogma. It’s about resisting the rigid principles of any ideology that prompts us to dismiss the counsel of lived experience.

Not to get too political, but let’s take the Tea Party. My problem isn’t necessarily that I disagree with their small-government principles. It’s that they fling those principles outward like a javelin, into a reality they haven’t full investigated. Let’s deregulate the economy! Let’s revive nullification as a doctrine! Let’s default on the national debt! And let’s do this all with utter conviction, never mind that no one can possibly know for certain what will come from any of these untested policies!

The Tea Party is all model and no data.

In the real world—the one out there, with air and trees and seven billion real people—any cleanly-stated principle is bound to be, at best, a crude first-order approximation of wisdom. The world’s too subtle for easy answers. Principles, like all models, are simple; the world, full of data, is not. So we’d best roll up our sleeves and learn what we can.

Oh, by the way: the research suggests that raising the minimum wage has a borderline-significant (read: rather small) negative effect on employment, though some argue (and this is a more fun punchline) that it may have virtually no effect at all.

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C’mon, dude. You already get credit for the laws of motion, that cool apple story, and the tasty fig cookies. Let me have this.

-Leibniz

Dear Student,

I know it stings to fail a test that badly. But hey, silver lining: you’re so far into the area below the curve that you’re practically an integral.

-Teacher

Dear Student,

I’ll put it this way: You don’t seem to understand me yet, but you keep getting closer.

-Limits

Dear “Ordinary” Differential Equations,

How does it feel to know you’re not special?

-Partial Differential Equations

Dear “Partial” Differential Equations,

How does it feel to know you’re not complete?

-Ordinary Differential Equations

Dear Mean Value Theorem,

Please, show mercy! I beg you, have a heart!

Imploringly,

The Kind Value Theorem

Dear Local Maximum,

Mr. Big Shot, huh? You are just a mid-sized fish in a very big pond, my friend. Watch yourself.

-The Global Maximum

Dear Television Series,

Pssh. You call yourselves “series”? You’re finite! What fun is that?

-Geometric Series

Dear Related Rates Problems,

You guys are like snowflakes—each of you is totally unique! And if there are enough of you, it’s safer to call off school and stay home.

-Students

Dear Teacher,

I know you called this unit Optimization, but I’m feeling more like Pessimization right now.

-Struggling Student

Dear Global Minimum,

Whenever I feel down about myself, I just look at you and I think, “Hey, at least I’m not *that* guy.” So thanks for being the lowest of the low!

-Local Minimum

Dear Student,

To answer your question: Yes, I would say that the Fundamental Theorem of Calculus *is* something you should know for this calculus course. It’s very useful. Crucial, even. Heck, you might even call it “fundamental.”

-Your Teacher

Dear Teacher,

You misunderstood me! When I called the behavior at that point “discontinuous,” I meant it like, “Dis be continuous, mon.”

-Student

Dear Lindsay Lohan,

Now I can’t finish a calculus test without whispering (and then screaming) “THE LIMIT DOES NOT EXIST.” Thanks a lot.

-Student

Dear Physics,

If acceleration is the derivative of velocity, and velocity is the derivative of position, then what is position the derivative of? Did I just blow your mind?

-Stoner mathematician

Dear 1950s Alabama,

What’s your problem? Integration is easy—here, I’ll show you.

-Calculus student

Dear Product Rule and Chain Rule,

I finally tracked you down! Don’t deny it: You’re my parents, aren’t you?

-Quotient Rule

Dear Worker,

First, you endanger everybody on this crew by pulling that ladder away from the wall at a constant velocity. And now you want help creating a mathematical model for the consequences of your irresponsibility? Here’s a consequence: You’re fired.

-Your foreman

Dear Derivative,

You’re so inconsiderate. It’s like you don’t even notice I’m there!

-Constant term

Dear Shepherd,

I realize it saves fence if you use the river as one border of your pasture, but aren’t you worried about us falling into the water? I’m not sure we can swim!

-Your Sheep

Dear Polynomials,

Want to race? You can even have a head start.

-Exponentials

Dear Students,

Remember, doing a u-substitution is like the aftermath of a break-up. You go through and clear away all the signs of your ex, until there’s nothing left but you.

-Your teacher

Dear Teacher,

If the best student in our class maintains an A, and the worst student manages to raise his grade until it’s *also* an A, wouldn’t you have to give the rest of us A’s, based on the squeeze theorem?

-C student

Dear f(x) = 1782x^{2} + 1,

I’m sorry to say it, but you’ve got a huge concavity.

-Your Dentist

Dear Infinite Sequence,

Never mind. We thought somebody said “infinite sequins.”

-Costume Designers

Dear Teacher,

Why did you take off points for the question about finding a formula for dy/dx for the curve x^{2} + y^{2} = 16? I mean, sure, I didn’t actually write anything down, but I figured it was implicit.

-A student

Dear Monarchy,

Shall we use the guillotine? Or do you prefer that we revolve this rope around the axis of your neck? Vive la resistance!

-Solids of Violent Revolution

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