The founding year: 1859.

The elevation: 8463 feet.

The population: 118 people.

And at the bottom—oh, the glorious bottom—these three numbers have been added together, yielding a total of 10,440. You can check it yourself: 10,440 is exactly correct. The arithmetic is flawless.

It’s perfectly right… and profoundly wrong. It’s a memorable token of a common mathematical mistake: carrying out an operation without investigating its meaning.

I could easily spin out 1000 words bagging on this poor sign-maker. But I’m not going to. (For one thing, there’s a chance the error was a deliberate joke, and even if it wasn’t, there’s enough ridicule out there for bad math.) Instead, I want to argue the opposite.

This error isn’t brainless, stupid, or contemptible. Rather, in several ways, the Gold Hill error is a uniquely sophisticated and modern one.

Hop in a time machine. Tour the range of human societies. You’ll find that most consider the idea of “mindless” computation paradoxical, nonsensical—a contradiction in terms.

That’s because, for most of human history, computation was really hard.

As recently as 1000 years ago, simple multiplication—the stuff we teach to 9-year-olds now—required expert equipment. Determining a square root was the special provenance of academics and well-trained merchants.

Back then, “simple” math didn’t feel so simple.

Now, of course, we’ve got calculators, iPhones, and—most importantly—the Hindu-Arabic numeral system. This way of writing numbers, familiar and obvious to us today, arrived as a revelation. It’s far easier to compute with our friendly numerals than it was with, say, Roman numerals.

For most of our history as mathematical thinkers, the Gold Hill error would’ve been tremendously unlikely. Computations took too much effort for you to waste time performing a silly one!

Math and science teachers chide students for “forgetting their units.” Omitting units is like leaving a number naked and unclothed. Do you mean 7 feet? 7 meters? 7 miles? Or 7 jars of jelly?

That’s the basic problem with the Gold Hill sign. The three numbers have very different units (years, people, and feet). These can’t be added.

But for many mathematical cultures, “forgetting” the units would have been unimaginable, because numbers (and operations) were inextricably tied to physical representations.

That is: numbers *needed* units.

Take square numbers. Why is multiplying a number by itself known as “squaring”? Simple: it’s equivalent to finding the area of a square (with the original number as a side length).

What about “cubing”? Likewise, that’s just finding the volume of a cube (with the given side length).

For Greek mathematicians, a single number represented a length. The product of two numbers stood for an area. And the product of three numbers represented a volume.

What about the product of four numbers? It’d be nonsensical, impossible. In Euclid’s famous geometry book, he carefully avoids ever multiplying four numbers together. For him, it would have violated common sense, gone against his geometric conceptualization of number.

What’s the point of all this? It’s that our very ability to conceive of numbers without units is a pretty remarkable abstraction. The Greeks would find our notion of numbers, even as practiced by little kids, a bit eerie and ethereal. Their vision of math was beautiful but immensely concrete, closely tied to geometric and visual meanings.

And so to a Greek, the Gold Hill sign would be… well, like Greek.

Most of the math that we teach to our students comes from long, long ago. Integers and fractions have been studied since prehistory. All of our geometry was known to the Greeks. Simple algebra dates back to medieval Arabia.

But one particular branch of mathematics is incredibly new—almost shockingly modern in comparison to the lumbering dinosaurs with which it cohabits our curriculum. That young, upstart branch?

Statistics.

Although it’s often used as a synonym for “number,” the word “statistic” in fact has a specific and little-discussed meaning. It refers to a single number that summarizes a whole sample. For example, if I grab five people off the street, ask them how annoyed they are (on a 1-to-10 scale) at being grabbed by me, and then average those numbers, that average is a statistic.

The thing about statistics is that a lot of numbers go into them. Just as honeybees need acre upon acre of flowers to produce a single jar of honey, statisticians need lots of raw data to produce a simple summary.

But where past societies lacked for data, we’ve got a surplus. Heck, it’s 2015; we’re practically swimming in it. If you clicked on this post through Facebook, Google, or Twitter, then your click is another data point in some massive collection in Silicon Valley. We’re so immersed in numbers—drowning in them, really—that we *need* statistics. We need summaries.

We need totals.

Now, I don’t mean to suggest that finding a total is a new idea. The number at the bottom of the Gold Hill sign is a simple sum. It’s the result of a basic addition operation—a computation as old as time.

But the attitude that produced the Gold Hill error—the practice of summarizing data immediately after presenting it—is a new development in our mathematical culture, a strange necessity created by our glut of information.

So yes, it’s easy to see the Gold Hill sign as an emblem of everything that’s wrong with our mathematical culture. We push students into mindless computations and unnecessary abstraction, and the result, too often, is gibberish.

But if you look deeper, you can see the Gold Hill sign as a bizarre but encouraging artifact of the advances we’ve made in mathematics. We’ve developed (and propagated) technologies for computing with ease. We’ve built (and popularized) an impressively abstract concept of number. And we’ve grown so prolific at gathering and processing data that we practically do it in our sleep.

The Gold Hill sign is still wrong. But it’s wrong for fascinating reasons.

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I submitted the requisite form, giving all of them A’s. My chairman was indignant. “How can they all be A’s?” he asked. “Is this some kind of joke?”

I said, no, it wasn’t a joke, but that the more I got to know each student, the more he seemed to me distinctive. My A was not some attempt to affirm a spurious equality but rather an acknowledgment of the uniqueness of each student. I felt that a student could not be reduced to a number or a test, any more than a patient could. How could I judge students without seeing them in a variety of situations, how they stood on the ungradable qualities of empathy, concern, responsibility, judgment?

Eventually, I was no longer asked to grade my students.

Dr. Sacks is a neurologist. His expertise ranges so far and wide (he has written on autism, Tourette’s, migraines, colorblindness, sign language, musical hallucinations) that the word “specialization” no longer fits.

Now, I’m a teacher, not a doctor. But reading Sacks’ autobiography, I’m struck by how teachers and doctors both feel a crucial tension, confronting the same fundamental choice in how to define our professional selves. Am I a narrow specialist, applying my expertise to address a specific need of the pupil or patient?

Or am I generalist, embracing the full complexity and interconnectedness of the human before me?

Sacks embeds his answer in practically every paragraph. His purpose, always, is the health and flourishing of the human being. This doesn’t mean jettisoning professionalism. Insofar as this is a debate, Sacks refuses to take a “side.” Instead, through his work and writing, he marries the clinical and the human, the scientific and the spiritual, the pragmatic and the poetic. His prose bubbles and eddies with insight and compassion, rich currents of thought cascading in long sentences that layer adjectives like river sediments, each deposit deepening and shifting the fluid organic whole.

Sacks does not choose between being *scientist* and *humanist*. Instead, he weds the roles together, merging them into one.

How does this process begin? For Sacks, it starts with listening to his patients. He lends not only an analytic eye but a sympathetic ear. Sacks grew up with three brothers; two became doctors, but the third, Michael, was troubled by schizophrenia. Over time, Michael found he could confide only in his younger brother Oliver:

[Michael] had begun to think of… the entire medical profession as determined to devalue or “medicalize” everything he thought and did, especially if it had any hint of mysticism, for they would see it as an intimation of psychosis. But I was still his little brother, just twelve years old, not yet a medicalizer, and able to listen sensitively and sympathetically to anything he said, even if I could not fully understand it.

Much of neuroscience aims to reduce our minds to mechanisms. We seek the brain region responsible for X, the neurotransmitter that underlies Y, the biological mechanism for Z. (Never mind if X, Y, and Z happen to be our most precious expressions of self.) Shining in this way, the light of neurology is bright and deadly. It explains and sterilizes us, turns us into labeled anatomical diagrams.

But Sacks resists this “medicalization.” People are people: gloriously unique, irreducibly complex. They are not bundles of symptoms. He gazes into the deepest, most mystical parts of the human psyche, and—far from extinguishing the living mystery of experience, of selfhood—Sacks’ science embraces and nourishes it.

So it must be with teachers. Recent decades have given us a wealth (you might say a clutter) of tools and checklists. Our shelves overflow with tables of state standards, banks of test items, standardized assessments. Like doctors, we’ve got a plethora of diagnostics at our fingertips.

But we must not “medicalize” our students. These tools begin a conversation; they do not end it. To learn any subject—math, history, Spanish, even neurology—is to undertake a fabulous and singular journey, to exercise your humanity at its highest level.

That’s not on the checklists; rather, it’s what the checklists are there for.

To be sure, specialization has its benefits. As a math teacher, my job is quite specifically to help my students learn mathematics, not to cook their breakfast or to counsel them on romance. (Neither my cooking nor my counsel would do them any favors.) For Sacks, this goes double: a neurologist is nothing if not an exquisitely trained specialist.

But when Sacks began working at a headache clinic, he found that his patients’ needs forced him outside of this mindset:

[Seeing patients] gave me a feeling of what seemed wrong with American medicine, that it consisted more and more of specialists. There were fewer and fewer primary care physicians, the base of the pyramid…. I found myself feeling not like a super-specialist in migraine but like the general practitioner these patients should have seen to begin with. I felt it my business, my responsibility, to enquire about every aspect of their lives.

Later, of his patients at Beth Abraham hospital, he writes:

I lived next door to the hospital and sometimes spent twelve or fifteen hours a day with them. They were welcome to visit me; some of the more active ones would come over to my place for a cup of cocoa on Sunday mornings.

This might seem to threaten the idea of clinical distance. In patient/doctor and pupil/teacher relationships, we are rightly afraid of impropriety. So we grow to fear attachment, enmeshment—any sort of excess sympathy—as a sign that lines are being crossed, balances disturbed. Instead, we stay cool. Removed.

“Professional.”

But to Sacks, propriety is not at odds with intimacy. In fact, the whole purpose of professionalism is to create space for a clear and purposeful connectedness. He writes of his own psychoanalyst:

I still see Dr. Shengold twice a week, as I have been doing for almost fifty years. We maintain the proprieties—he is always “Dr. Shengold” and I am always “Dr. Sacks”—but it is because the proprieties are there that there can be such freedom of communication. And this is something I also feel with my own patients. They can tell me things, and I can ask things, which would be impermissible in ordinary social intercourse.

Reading Sacks, I’m always moved by the extraordinary empathy he lends to every patient. In those crippled by illness or disorder, he finds hidden strengths, adaptive resilience. In those dismissed by others—as psychologically diminished, intellectually null, or beyond treatment—he finds untold complexity and richness. He writes:

I find every patient I see, everywhere, vividly alive, interesting and rewarding; I have never seen a patient who didn’t teach me something new, or stir in me new feelings and new trains of thought.

I wish, desperately, that I could say the same of every student I’ve taught. But the day is only so long; owing sympathy and attention to all of my students, I inevitably shortchange most of them. They each face unique puzzles and struggles, carry with them distinctive strengths and personal motives—but hell if I manage to discover half of these stories before they graduate and move on.

I’ll never understand my students as deeply as Sacks understands his patients. Still, it’s an ideal I can strive towards.

As it is, my “knowledge” of my students is often statistical. I know how they’ve performed on homework, quizzes, and tests. I can predict how they’ll fare on the high-stakes exams that will circumscribe their opportunities for the future. Our educational system is increasingly a statistical one.

In medicine, statistics are also ascendant, and in a footnote (it’s always a footnote, with Sacks—the notes for *Awakenings* originally ran to twice the length of the text itself) Sacks relays one memorable anecdote on the topic:

“How many patients do you have on L-dopa?” he asked me.

“Three, sir,” I replied eagerly.

“Gee, Oliver,” Labe said, “I have three hundred patients on L-dopa.”

“Yes, but I learn a hundred times as much about each patient as you do,” I replied, stung by his sarcasm.

Sacks doesn’t deny the value of statistics. (“All sorts of generalizations are made possible by dealing with populations,” he remarks.) But he insists that they are not enough. Alongside the cold and towering arsenal of data, he supplies an utterly necessary complement: a renewed humanism, a depth of caring and compassion, a creative receptiveness to the variety of human strengths and experiences. He doesn’t rebel against the statistical regime; he simply denies that it can substitute for older and more personal forms of wisdom.

That’s the balance I seek in my teaching. The word *essential* springs to mind, in its two distinct senses: the newer one of “very important,” and the older one of “capturing the essence.” Test scores, statistics, student data—these things are very important, but they are not the essence.

Tests are only stethoscopes. They are not the heartbeat itself.

Sacks is dying of cancer. In a February essay, he wrote, “It is up to me now to choose how to live out the months that remain to me.” When he passes, the world will lose an extraordinary being, a form of life every bit as distinctive and wonderful as the patients to whom he lent his sympathy, his Sunday afternoons, and—perhaps most lasting of all—his words.

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Dear Benedict Carey,

I very much enjoyed your book *How We Learn*. It blends the vast and varied harvest of research on learning into something light, flavorful, and nutritious. A psych-berry smoothie, if you will. It’s a lovely summer read for a math teacher like me.

But I’m also a blogger—which is to say, a cave-dwelling troll, forever grumping and griping. And so I’d like to dive into your chapter on practice (“Being Mixed Up: Interleaving as an Aid to Comprehension”). In it, you purport to remain impartial in “the math wars,” but it’s my view that you come down distinctly on one side.

It’s towards the end of the chapter that you hit the culture war in mathematics. On the one hand (you explain), there are top-down **progressives**, who urge children “to think independently rather than practice procedures by rote.” On the other, you’ve got bottom-up **conservatives**, who put their faith in “the old ways, in using drills as building blocks.” All fair enough.

Then, however, you land on a rockier claim:

This clash over math was (and is) about philosophies, and in math of all subjects it is results that matter, not theories.

Maybe so. But it’s far from obvious *which* results matter. That depends on what philosophy you subscribe to.

For example, one teacher might care a lot about standard textbook questions like these:

Another might scowl and spit upon those, preferring more open-ended questions:

Others might embrace all these questions. Or despise them all.

The disagreement isn’t just about the most effective path to excellence. It’s about what “excellence” even means.

At one extreme, you can view math as a fixed and specific body of knowledge, a pre-existing library of tools, techniques, and numerical skills. Or, at the other extreme, you can view mathematics as a frame of mind, more about habits of thought and broad problem-solving approaches. Like most teachers, I fall somewhere in the complicated middle. I have my own views about which tasks are meaningful, and which are rubbish—which are math, and which are mush.

In the end, it’s silly to say that “results” matter and “theories” don’t. Our choice of the former is inextricable from the latter.

Next, you dive into a study (promising that it consists of “real math” and implying that it ought to transcend the debate between mathematical progressives and conservatives). In the experiment, kids learn rules for computing the numbers of faces, edges, corners, and angles that a prism has, given the number of sides in its base:

Unfortunately, this is a task that many progressives would reject outright.

Here, students must recognize a code word (*face*, *edge*, *corner*, or *angle*) and match it with an arithmetic operation (*+ 2*, *x 3*, *x 2*, or *x 6*). To me, this represents a narrow and overemphasized aspect of mathematics: signal-triggered computation. You characterize it as high-level (“We are not only discriminating between the locks to be cracked; we are connecting each lock with the right key”) but I suspect I’m not alone in considering it rather drab and rote, not to mention irrelevant to the primary purpose of mathematics education.

What’s the alternative?

Well, first off, mathematics is about framing questions. So what is this “prism” thing we’re discussing—not to mention “faces” and “edges”? What definitions are we using? What properties are we focusing on? How are these 3D objects similar to (and different from) 2D objects? Are there borderline cases that are hard to categorize? For example, which of the following shapes would count as “prisms”? (And are there any that *aren’t* prisms, but still follow the rules?)

Now, as for those rules: sure, it’s nice to use them, but can we explain *why* they’re true? Could we have uncovered them for ourselves? Can we test the boundaries of their applicability, rooting out exceptions, or proving to our satisfaction that no exceptions exist?

These are the mathematical activities I care most about.

Framing questions.

Reasoning.

Contrasting.

Extending.

Explaining.

Understanding.

I don’t mean to hate on the study. Yes, I find the specific task pretty silly, but I buy the essential conclusion that interleaved practice (where you mix all four types of questions together) is more effective than blocked practice (where you practice each type in isolation). Insofar as my teaching is conservative and skill-focused (which it sometimes is—certain skills, I believe, demand automaticity), that’s useful guidance.

But I worry about your presenting this as a consensus vision of “real” math. Many of us believe that math is about far more than memorizing and applying procedures, just as we believe that a song consists of more than its bass line. It’s a shame to see someone trying to transcend the debate over the aims of the math curriculum, but then focusing entirely on how to accomplish the aims championed by just one “side.” It’s if someone said, “I won’t take a position on whether aliens exist. Now, here’s a method for contacting extraterrestrials that’s 39% more effective.”

I should be clear: This matter occupies only a few pages of what is, to be clear, a fun and practical book, from the story of Winston Churchill failing his classics exam to your delicious images of the creative process (“For me, new thoughts seem to float to the surface only when fully cooked, one or two at a time, like dumplings in a simmering pot”). So I thank you for the enjoyable book you’ve written.

In any case, I hope I’ve been able to offer a more detailed (if not entirely novel) perspective from the muddy trenches of these ongoing “math wars.”

Best,

Ben Orlin

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(see posts 1 and 2)

Somehow, I suspect I wouldn’t survive long on the frontier.

Drop me in the American West, circa 1850, and I fear my math-blogging and bad-drawing skills might not carry me far. I need indoor plumbing. I need the rule of law. I need chain coffee shops. I’m not cut out for the frontier.

And yet the frontier is exactly where I found myself the other day, when I came across this formula in the wonderful *Penguin Book of Curious and Interesting Numbers*, by David Wells:

I decided to play around with this product a bit. After all, what are products for, if not playing around?

(Go ahead and play with your Apple products. I’ll play with my infinite ones. We’ll see who has more fun.)

I felt like there should be an easier way to write this expression, exploiting the repetition of factors, so I gave it a shot, and created this:

Then my brain exploded and the universe dissolved around me, because I had just punched logic in the face, and it had punched me back.

The left side of that equation is π/2. It’s roughly 1.57.

The right side of that equation, however, is a product of many numbers—*all of them below 1.*

What happens when you multiply two numbers smaller than 1? You get *another* number smaller than 1.

How the heck could that equal 1.57?

Somehow, law had broken down. I was stranded on the frontier, with no cavalry or sheriff to come save me. This was a dispute I’d have to resolve myself, by hook, by crook, or by showdown at high noon.

At this point, I faced three grim possibilities:

- The cool book where I found this formula was wrong.
- Logic was wrong.
- I was wrong.

Pitting my own abilities against (1) those of David Wells and (2) the tensile strength of the fabric of the cosmos, it became pretty clear where the error lay. So I set out to find where I’d gone wrong.

Going back to the original equation, I came up with an alternative way to rewrite it:

This one was clearly greater than 1. (After all, it was an infinitely long product, with each of its factors greater than 1.) Checking a few terms in Excel verified that it was approaching π/2.

Then I found another way to rewrite the original:

This one starts at 2. Then it gets smaller and smaller, by tinier and tinier adjustments. Once again, it seemed to approach π/2.

So what had I done wrong in my original approach?

Soon enough I realized it. I’d made a mistake that seemed perfectly innocuous at the time but had, in fact, contained the seed of my own annihilation. I’d made a move that doesn’t matter with *finite* products, but which is forbidden with *infinite* products.

I’d forgotten to multiply by 1.

It perhaps goes without saying, but when you’re working with infinities, you need to be careful. Notation that’s usually harmless can explode like dynamite, making rubble of everything.

Infinity is the frontier. Law breaks down.

Back home, in the realm of finite products, multiplying by 1 makes no difference. 3 x 2 is the same as 3 x 2 x 1 (or as 1 x 3 x 2 x 1 x 1, for that matter). Finite products—that is, lists of factors that eventually end—can be rearranged to your heart’s content.

Not so with infinite products. Just look what else I can do, if I allow myself to play around with 1’s:

Throwing in extra 1’s allows me to rearrange the numerators and denominators virtually however I please. It seems I can make this product as large or as small as I want!

I should also confess that steps like the second one above – where I take a long product of fractions, and turn it into a single fraction with a long product on top and another long product on bottom – are a little dubious in this setting. With finite products? Fine. With infinite ones? Not so much.

In short: the frontier is a strange place.

I’m working here with products, but the same dangers apply with sums. Take this one, called the *alternating harmonic series*:

This sequence starts at 1.

Then it jumps backwards to ½.

Then it jumps forwards to 5/6.

Then it jumps backwards to 7/12.

Then it jumps forwards to 47/60.

As you go, the sum keeps jumping forwards and backwards, forwards and backwards… but by smaller and smaller steps. Looking carefully, you can see that it’s hovering around a certain destination—a place it will never in our lifetimes reach, but which it *approaches*.

That point is smaller than 1.

It’s bigger than ½.

It’s smaller than 5/6.

It’s bigger than 7/12.

In fact, it turns out to be roughly 0.693, or more precisely, ln(2).

So far, so good. But what if I rearrange the terms, like this?

We start at 1.

Then we add something positive.

Then we add something *else* positive.

And so on.

Clearly, the answer has to be larger than 1—which 0.693 is decidedly not. So I’ve changed the result sum, simply by rearranging the terms.

That’s not supposed to happen with addition! 3 + 4 is the same as 4 + 3. Making your family members stand in a different order shouldn’t make your family bigger, smaller, smarter, or taller—it’s the same family, isn’t it?

But with infinite things to add, the order in which you arrange them turns out to matter a great deal. In fact, this sequence can be rearranged to form literally any number you like.

Why do we explore this mathematical frontier? What drives us to abandon the safe coastal cities that we’ve always known, and throw ourselves into the unknown interior of the mathematical continent? Aren’t we afraid? Mightn’t we get eaten by bears, swallowed by rapids, or tangled in paradoxes? Why do we risk it?

Simple: because humans are explorers.

It’s as true of mathematicians as it is of Lewis and Clarke. We want to chart new landscapes, to find fresh challenges, to go where our old ways don’t necessarily help or hold true. We enjoy the thrill of surviving on our wits alone.

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(see also part 1)

A few weeks ago, the webcomic Saturday Morning Breakfast Cereal posted a cartoon about the harmonic series.

(Obviously it’s a mistake to post an actual cartoonist’s work alongside my own second-grade-quality scrawl, but hey, maybe I’ll benefit from a math humor cheerleader effect.)

Now, what is the harmonic series? It’s this:

The sum never stops. It goes on forever and ever. Lovely, yes, but does it—in any meaningful sense—“equal” anything?

That is to say: Does it level out at some value?

Or does it just keep rising, forever and ever, eventually exceeding a million, then a billion, then a trillion, and so on, surpassing any ceiling or limit we might imagine?

Perhaps your first thought is this: “You silly man; you’ve already answered your own question. You said the sum goes on forever. So it must be infinite.”

Not so fast.

Take a look at this series, for example:

We start at 1. The next term brings us halfway to 2. The next term brings us halfway *again* to 2. The next term brings us halfway *AGAIN* to 2.

And so on.

This sum is “infinite” in the sense that it goes on forever, with no final term. But it’s finite in the sense that no matter how many numbers you add, you’ll never exceed 2. Sure, you’ll get close—achingly, painfully, infinitesimally close—but you’ll never surpass it. We say that the sum “converges” to 2.

Now, back to our original sum, the harmonic series. What should it equal?

Well… let’s look at some running totals as we go along.

Where is it going to settle down?

It isn’t.

To be clear, this total grows *incredibly* slowly. In fact, the word “slowly” fails to evoke its agonizingly incremental pace. Instead, imagine that you’re waiting in line at the DMV.

And there’s only one employee.

And that employee is one of those **talking trees** from Lord of the Rings.

And he’s stoned.

And the line includes all 7 billion people on earth.

That line you’re in? It moves like ball lightning compared with the growth of this series.

And yet… this series never settles down. This sum eventually exceeds a million, then a billion, on its way to the stars. In mathematical language: the series *diverges*.

This is all weird enough. But now we get to the truly strange part of the whole ordeal, the truth that prompted the inimitable** Zach Weinersmith** of SMBC to build a punchline around it:

If you throw out the numbers with 9’s in them, the series is small enough to converge.

“Nonsense, you gullible old toad!” you are perhaps shouting to your screen. “Why should throwing out the numbers with 9’s make such a difference? We’ve still got *all the other numbers*!”

Again, I say: not so fast. You’re making a classic mistake. When you think of “numbers,” you’re only picturing *little* numbers.

“No I’m not!” you may say. “I’m thinking of big numbers. Huge numbers. Like 9 million, or 47 billion, or 228 trillion.”

Exactly my point: small numbers.

You see, the longer a number gets, the harder it is to avoid a 9. Every time you add a digit, you add a new opportunity for a 9. That might not feel like a big danger—after all, those 9’s will pop up only 10% of the time. But look what happens:

It’s as if, every time you type a digit, there’s a 10% chance that a giant numeral “9” falls from the sky and crushes you. For short numbers, with just 2 or 3 digits, you’re not very worried. The chances are in your favor.

But now you begin to type a 100-digit number. How do you like your chances? Sure, that giant “9” probably won’t fall on *this* digit… nor on *this* digit… but how long do you think your luck will last? Eventually you’ll get unlucky, and that “9” will come plummeting from the sky.

In the long run, *most* numbers have 9’s in them. Virtually all of them, in fact. So if you throw out these numbers from the harmonic series, it’s no surprise that it now converges. You’ve thrown out almost the entire series!

Want to know the weirdest part? The same logic applies to a longer sequence of digits. Say, 999.

At first, most numbers won’t have this sequence. But picture a billion-digit number. Just writing this number out—in size 8 font, double-siding the printing to save trees—takes a stack of paper as tall as a house.

Surely *somewhere* in there the number is bound to have the digits “999” in that order, right?

The same is true of *every* equally big number. And MOST numbers are this big! After all, there are only finitely many *smaller* numbers, and infinitely many *bigger* numbers. So, again, we’re throwing out almost the entire series.

Thus, the harmonic series *also* converges if you throw out all the numbers that include the digits “999.”

Perhaps you can see where this is going. (If so, you may be experiencing vertigo and/or nausea; this is normal.) The logic above works for *any string of digits* you can possibly think of. Even, say, a sequence of a million 9’s in a row.

So, in conclusion, the harmonic series *diverges*. It eventually outgrows any ceiling you’d put on it.

But simply throw out the numbers that happen to include a string of a million 9’s in a row… and suddenly the series *converges*. It plateaus. There’s some value that it will never surpass.

Somehow, by excluding only those numbers with a million 9’s in a row, you’ve changed the nature of the series.

In the words of Weinersmith:

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**A.**The threat of Grexit**B.**The bittersweet knowledge that someday, when all of this has passed, we’ll have fewer opportunities to use the amazing word “Grexit”**C.**People thinking functions are linear when they’re SO NOT LINEAR**D.**Other (e.g., cat bites)

If you answered C, then congratulations! You are probably a teacher of math students ages 13 to 20, and we all share in your pain.

For everyone else (including you poor cat-bitten D folk), what are we talking about? We’re talking about errors like these (warning—mathematical profanity ahead):

What’s wrong with these statements? Well… everything.

I don’t blame the kids, of course. These errors are a natural—perhaps even inevitable—byproduct of the way we teach mathematics.

Every rule in secondary school math is actually a statement about objects—shapes, numbers, whatever. But they don’t *feel *like that to students. They feel like purely symbolic manipulations, rules for how x’s and y’s move around a page, not for how actual things relate to each other. When you see rules that way, it’s easy to fall prey to certain systematic errors.

Why is this particular error so tempting? Perhaps because it closely resembles a *real* rule students have learned:

This is called the Distributive Law, and it’s actually a deep fact about the numbers, an essential link between addition and multiplication.

You can discover it through numerical examples:

Or you can understand it with arrays:

Or you can use my colleague’s wonderful phrasing and think of an expression in parentheses as a “mathematical bag”:

The distributive law is crucial. It underlies most of what we do in algebra, such as factoring and “gathering like terms.” So what’s the problem?

It’s that students don’t learn the distributive law as a fact about *numbers*. They learn it as a fact about *parentheses*.

And unfortunately, mathematicians use parentheses in two very different ways: first, to group numbers; and second, to designate the inputs of a function.

The notation “f(x)” doesn’t mean “f times x.” It means “what I get when I put x into the function.”

But our faithful symbol-pushers don’t always catch the distinction. They see parentheses, and think, “Oh yeah! That rule I learned should apply here.”

What’s the solution? Here’s my current game plan:

**Teach the distributive law more carefully**. Draw pictures. Work examples. Talk about “bags.” Make sure they understand the meaning behind this symbolism.**Teach function notation much more carefully**. Give them the chance to practice it. Think like Dan Meyer and seek activities that create the intellectual need for function notation.**Keep stamping out the “everything is linear” error when it crops up**. Like the common cold, it’ll probably never be entirely eradicated, but good mathematical hygiene should reduce its prevalence.

What’s the only thing giving me pause here, making me doubt my nice, tidy theory of this error? In two words: Jordan Ellenberg.

In *How Not to Be Wrong*, he catalogs a variety of cases where people falsely assume “all curves are lines.” His examples are never just symbolic. Each time, people are messing up on a conceptual level. They’re not just pushing x’s and y’s around a page, but genuinely believing their own wrong statements.

So is my approach off-base? Are people actually acting on false beliefs about functions, or (as I’ve posited) are they failing to think about them as “functions” at all?

I’d love to hear what you think. And if you’ve got solutions for the Greek economy or cat bites, well, I’m all ears.

*UPDATED*: edited to correct the fact that I do not know the difference between numbers and letters. Oh well, like I always say, “C strikes and you’re out.”

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If there’s one thing about math that people love—and to make it through the average day, I have to believe there’s at *least* one—it’s infinity.

Throw the word into a math lesson, and ears perk up. *Infinity? Did he say infinity? *It’s like a distant celebrity, the subject of endless gossip and rumor. “I heard infinity isn’t even a number!” “Only the universe is *really* infinite.” “My last teacher said infinity times two is the same as infinity.” “I can use infinity to prove that 1 = 0!”

Infinity is a sound too high for our ears, a light too bright for our eyes, a taste so sweet that it would tear through our tongues like acid. Basically, it’s mathematical Mountain Dew.

Tellingly, all of our words for infinity define it by what it isn’t. *Infinite*: not finite. *Unlimited*: not limited. *Boundless*: without bounds. It’s hard to articulate what infinity *does*, so we settle for naming what it *doesn’t*: end. Infinity is the Anansi of mathematics, a trickster spider weaving baffling webs of paradox and contradiction.

Take this example: which has more numbers, List A or List B?

Well, they’re both continue forever, so they’re both infinite. But the first list includes every number on the second (2, 4, 6, 8…), and lots more as well (1, 3, 5, 7…). Thus, clearly, List A is the bigger group of numbers. Right?

Wrong. The lists are exactly the same size.

To see why, just go through list A and double every number on it, like so:

Now, doubling each number shouldn’t affect the size of the list. We haven’t added any items, or taken any way. It’s like taking a group of people and giving them fake beards—you don’t create or destroy any humans in the process.

And yet… without changing its size, we’ve somehow turned list A into list B. That can only mean that the two lists were the same size to begin with.

What’s happening here? Throughout our lives, we’ve relied on a simple rule for comparing the sizes of groups: *If one group has all the members of another, plus some others, then it must be bigger.* For example, if your list of Facebook friends includes all of mine, *plus *you’re friends with Kanye West, then you have more friends than I do (and probably cooler weekend plans, too).

And that’s perfectly true, for finite groups. But infinite groups… those are another matter.

Infinity shreds our intuitions. The startling truth is that all infinite lists are the same size, regardless of whether one happens to include the other.

The mathematician David Hilbert captured this paradoxical truth with the parable of Hilbert’s Hotel. Imagine there’s an infinite hotel, in which every room is currently occupied. (There’s a very popular convention in town, apparently.)

You show up, asking for a room. “Sorry, they’re all occupied,” says the innkeeper—who’s probably a little frazzled, what with the infinite calls for room service coming in.

“Yes,” you say, “but just ask everyone to move down one room, and there will be space.”

It’s true. The person in Room 1 goes to Room 2. The person in Room 2 goes to Room 3. The person in Room 136,004 goes to Room 136,005. And so on. Everybody has someplace to go.

Now, whether it’s good manners to show up at a hotel and demand that a literally infinite number of people suffer a minor inconvenience just so you can avoid trying the Best Western down the road—that’s another matter. You’re probably a bad person for making this request. But you’re a fine mathematician.

Can we do this without inconveniencing an infinite number of people? Unfortunately, no. For example, if you take Room 1, and send that guy packing, he’s going to have the same problem as you: every room is now occupied. As long as you’re shuffling a finite number of rooms, normal math applies, and somebody will be left out. Your infinite problem demands an infinite solution.

It gets crazier. What if you’re coming straight from the Infinite Man March, and you’ve got an infinite line of new friends in tow? Can the hotel hold you all now?

Impressively: Yes.

This time, each current guest needs to look at their room number, double it, and move to that doubled room. So the chap in Room 1 moves to Room 2. The gal in Room 2 moves to Room 4. The fellow in Room 3003 moves to room 6006. And so on.

At this point, which rooms are filled? Only the evens. So you and your infinity friends fill the odd rooms, and voila! Problem solved.

(Unless you’re the bellhop in charge of setting out the continental breakfast; then, the problems have just begun.)

At this point, infinity is perhaps beginning to feel like a squishy, flexible, silly-putty sort of idea. Infinite lists? They’re the same size. Infinity plus one? Still the same size. Infinity plus another infinity? Yup, still the same size. It raises the provocative question: Are *all* infinities the same size? Tune in next week for the answer.

Oh, who am I kidding? The answer is no. Some infinities are *much* bigger than others.

Tune in next week for *why*.

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**Logarithms**

**Algebraic Thinking**

**Infintesimals**

**Dress Codes at Mobius High School**

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What’s the difference? Well, in the UK, a “rubber” is an eraser, and in the US…

Oh! You mean for “grades.” Well, **in the US, “grades” are given by teachers**. They aim to assess the quality of your work throughout an entire term. **In the UK, your “grades”—the ones that matter to universities—are your scores on a handful of high-stakes exams**.

In the US, the scores you get from your teachers form the bulk of your permanent academic record; in the UK, those scores don’t even appear.

In my experience, both American and British educators react with utter horror to the opposite system. And, weirdly, many objections are mirror images of one another.

**Objection #1: Your System Drowns Students in Testing**

*The American shock*: “Wait… so years of work come down to a single test? Doesn’t that put tectonic pressure on the kids, and give them a distorted, blinkered view of education as nothing but glorified test prep? How do you convince them there’s more to learning than test scores?”

*The British horror*: “Wait… so you’re giving meaningful tests *every few weeks*? Doesn’t that put kids under constant pressure, the perpetual threat of judgment—25 hours of testing, spread throughout the year, instead of just 4 hours at the very end? How do they ever find time to learn?”

*My take*: Both accusations ring true, because both systems are testing-driven. True, the word “test” carries different connotations—in the US, it’s less definitive and final, more a landmark signifying the end of a topic. (Many teachers allow you to raise your score with corrections or retakes.) But ask the students in either country, and they’ll tell you: the tests are what count.

**Objection #2: Your System Misjudges Student Achievement**

*The American cry*: “There’s so much a test can’t capture! You’re trying to judge years of work by a single day—and not only that, but you’re employing a narrow, time-pressured measure of achievement. Your universities are basing their decisions on a woefully incomplete picture.”

*The British lament*: “So your grades are just in the hands of… whoever happens to teach you? What if my teacher is particularly stingy, or overly generous? How do you ensure fairness across the system? Your universities are basing their decisions on whims and opinions, not real data.”

*My take*: Neither accusation holds water here.

Yes, the British tests are imperfect. But I find they’re a hell of a lot better than most of their American cousins. There’s no multiple-choice to be found; every question requires a written answer. Also, they assess specific curricula and skills—far more akin to AP tests than the nebulous, intimidatingly vague SATs.

As for the American system: yes, Brits, some teachers are arbitrary, inconsistent graders. But each kid’s transcript includes dozens of grades. Some may be unfairly high; some, unfairly low; some (probably most) about right. Average it all together, and you’re likely to get a pretty fair assessment of student achievement.

How can such different systems give equally valid results? Well, teachers, think back to your last round of exams. Did any students bomb after doing well all year? Did anyone ace the exam after tanking all year? Or—as always happened with my classes—did your final exam scores show a shockingly clean alignment with exactly what you’d seen from students all year long?

Neither system is perfect. But both are perfectly functional.

** **

**Objection #3: Your System Warps the Student-Teacher Relationship**

*The American diatribe*: “In your system, teachers are powerless. They’re glorified test prep tutors, with no opportunity to inspire, or to share their love for the subject. All that matters is the standardized test. How can they leave any impression on the students’ lives? How can they forge any meaningful bond?”

*The British harangue*: “In your system, teachers are despots. They write and grade tests themselves, with no external check. Won’t this allow them to bully the poor students, whose academic futures they singly control? And wouldn’t this turn students into ghoulish hyenas or terrible sycophants, gnawing and sucking at their teachers for any extra points they can get?”

*My take*: Partial truths on both sides here.

First of all: Yes, Brits, that’s exactly what happens in the US, and yes, it’s awful. Students feel cheated by teachers, and teachers feel hounded by students—not always, but often enough to cause headaches. I’m not sure this is the fault of our grading system, though. Instead, I blame our litigious instincts. We’re all salesmen and lawyers at heart. If there’s an advantage to be had, we’ll press for it, no matter what the institutional framework. “Deference to authority” is not the American way.

Second: Yes, Americans, I don’t think the Brits quite share our image of the “inspirational teacher.” Just compare the websites of Teach for America (US) and Teach First (UK), organizations with nearly identical missions of sending bright, ambitious young people to teach in the nation’s neediest schools.

The British version begins: “How much you achieve in life should not be determined by how much your parents earn.”

The American version begins: “Change and be changed.”

Whatever your feelings on such soaring rhetoric, it’s clear that Britain has a slightly narrower vision of a teacher’s role. Teachers aren’t expected to enmesh themselves in students’ lives—and perhaps it’s the students’ loss. But again, I don’t think this is a question of grading systems. It’s culture. The British are not a terribly sentimental people: the “stiff upper lip” remains a cultural ideal (albeit a fading one). “Change and be changed,” with its mushy American hug-it-out emotionality, is a simply un-British notion.

So, yes, American and British teachers relate differently to their students. But I don’t think it’s the fault of the grading systems.

** **

**The Actual Difference: Singular vs. Plural**

So, to recap, I’ve covered three mutual objections that Americans and Brits might levy against each other:

- “Your system over-emphasizes testing!” Yes, but so does yours.
- “Your system misjudges student achievement!” Not really.
- “Your system warps the student-teacher relationship!” From your perspective, perhaps; but it’s an issue of culture, not grading system.

Where does that leave us? Can these two systems, which feel so fundamentally different, actually be equivalent—even interchangeable?

Of course not! Don’t be silly.

To grasp the essence of the difference, look no further than the names of the countries. Both are “United,” but one is a singular “Kingdom,” and the other is a plural patchwork of fifty distinct “States.”

Brits expect standardization and nationalization. They expect their country to act as a single unit, evenhanded and fair. Their entire college application system, for example, runs through a centralized hub. You can apply to precisely six schools, including either Oxford or Cambridge, but not both. If you’re an American thinking, “How strange—that’s like if you had to pick between Harvard and Yale!” then you don’t realize the half of it. It’s more like if you could only apply to one school in the entire Ivy League.

In the UK, everyone surrenders minor personal preferences here or there for the sake of cohesion. They don’t think twice about it. It’s what “United Kingdom” means to them.

America is different. We remain individualists, frontiersmen at heart, and we expect local control. When teachers find themselves forced to deliver scripted curricula, following every step of a prescribed course, it almost always leads to revolts. We expect to write our own tests and grade them however we like, serving our own private visions of how our subjects should be taught. We take a diversity of approaches for granted. “Of *course* your algebra class wasn’t precisely the same as mine; we went to different schools!”

I’m not trying to take sides on Common Core here; in my view, it represents a fairly modest degree of nationalization. But the bitter opposition against it speaks to our powerful American belief that education is a local matter.

It’s strange being an American in the British system. I often find myself objecting to the quirks of how British exams are scored. “If a question is posed in degrees, an answer in radians gets no credit? That’s so silly!” “Saying a probability is ‘1/6’ is fine, but ‘1 in 6’ receives no points? That’s so unfair!”

(Both these examples are from last Friday. This happens all the damn time.)

In moments like this, my colleagues don’t know what to make of me. Following such conventions is second nature for them. It takes no effort. It’s as if I said to them, “What’s that stupid symbol you guys use for 7? I’ve always drawn a little devil face instead! It’s mathematically equivalent!” *Well, okay Ben, that’s fine, except you can’t teach that to your students because no one will have any idea what the hell they’re talking about.*

I stumble over my own convictions, my ingrained belief that I shouldn’t have to bow to anyone else’s approach. Teaching is full of a thousand arbitrary decisions, forks in the road where either direction is really fine. I’m accustomed to picking my own path, according to my own taste. It’s strange, now, needing to pay careful attention to the road signs, lest I steer my students against the flow of traffic.

Which system is better? I don’t think there’s a definitive answer, any more than it matters whether a country drives on the left side of the road or the right. What I’m learning, though, is this: Whatever country you’re in, you’d better make sure you’re driving on the same side as everyone else.

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In February, I was throwing together a geometry test for my 12-year-olds. I wanted a standard angle-chasing problem, but – and here’s the trick – I’m lazy. So I grabbed a Google image result, checked that I could do it in my head, and pasted it into the document.

But when I started writing up an answer key, I ran into a wall. Wait… *how* did I solve this last time? I trotted out all the standard techniques. They weren’t enough. A rung of the logical ladder seemed to have vanished overnight, and now I was stuck, grasping at air.

Eventually, some colleagues and I solved it by with industrial-strength tools: the Law of Sines, the Law of Cosines, and some fairly sophisticated algebra. Yet the problem looked so elementary, I felt sure that our approach was overkill—that we were bashing down the door, whereas a defter hand could simply pick the lock.

Apparently, I’d actually chosen a famous gem of recreational mathematics, born in 1922 from the mind of Robert Langley, and known since as “Langley’s Adventitious Angles.”

And, as I suspected, the niftiest solution requires no trigonometry or algebra, just a single ingenious move: construct a line here, at a 20^{o} angle to the base.

It’s a masterstroke. It pops the lid right off of the problem. And, at least to me, it’s utterly un-guessable. If you had a thousand monkeys with a thousand typewriters and a thousand protractors, you’d get full verses of Shakespeare long before any of our furry friends stumbled upon this solution.

Sometimes the general techniques fail, and you need a sneaky trick. That’s life.

To me, a “trick” is a disposable insight, a funny little key that opens one particular closet door. And a “technique” is something more useful: a skeleton key, fitting the lock for a whole hallway of rooms.

But is the difference always so clear?

There’s a certain algebraic move that I first encountered in high school, which you might call “multiplying by the conjugate.”

You use it here, to simplify an expression with square roots:

And here, with complex numbers:

And here, again with square roots, to solve a limit (without recourse to the heavy machinery of L’Hopital’s Rule):

And here, to boil down a trigonometric expression:

At first glance, it feels like a trick: a little too slick, a little too cute. It’s hard to imagine it blossoming into a general technique. But it eventually proves useful in a shocking variety of situations, cutting across the whole secondary mathematics curriculum. This funny little key, which barely looks like it should open a single door, turns out to open hundreds.

Ultimately, I can’t find any categorical difference between a trick and a technique. They’re both problem-solving innovations, lying along a continuum that runs from “almost never useful” to “useful all the darn time.” A technique is simply a trick that went viral, and a trick is simply a technique that fizzled out after a single use.

It makes me wonder how to define the job of the mathematician (and of her little spiritual cousin, the mathematics student). Is it to master existing techniques? To cook up new tricks from scratch? Or to recognize how the tricks of today can be nurtured and expanded into the techniques of tomorrow?

Yes to all of that. But of all the aspects of the mathematician’s job, I think the ability to birth truly new ideas is overrated. Leaf through the pages of history, from Pythagoras to the present, and you’ll find a stunning supply of tricks, techniques, and everything in between. In the five millennia we’ve been working on this, humanity has accrued a remarkable wealth of mathematical ideas. Yes, we want our fresh young minds to make new deposits into the bank of our knowledge; but we also want them to know how to access the existing funds.

I would never, in a million years, come up with the trick to solve Langley’s Adventitious Angles. But I don’t need to. Someone else already has.

By the way, as promised, here’s the rest of the solution to Langley’s Adventitious Angles:

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