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I believe the same applies to mathematicians doing arithmetic.

It’s a running joke among mathematicians that they’re bad with numbers. This confuses outsiders, like hearing surgeons plead clumsiness, or poets claim illiteracy, or Rick Astley confess that actually he *is* going to give you up and let you down, maybe even run around and desert you.

Does it come from some false modesty? A skewed sense of humor?

No, some mathematicians insist: it’s really true, we’re bad at arithmetic.

I’m choosing my words carefully: “mathematics” and “arithmetic” are not interchangeable. “Arithmetic” refers to calculations with numbers: 17.9 + 18.32, for example. “Mathematics,” meanwhile, is far broader: it tackles shape, structure, change, and all kinds of quantities.

The reality is that mathematicians aren’t professional arithmetic-doers, any more than musicians are professional players of scales.

I’ve heard mathematicians lament that their ability with arithmetic peaked sometime in grade school. That sounds overblown, but they’re probably not wrong.

As early as high school, specific numbers start taking a back seat to *patterns among numbers*. You stop working with 7 and 9 and 22, and start working with an *x* or an *n* that can refer to all of them at once. As you move into more abstract realms, your arithmetic gets rusty.

And the more math you study, the more extreme this gets.

When I began to teach 3D vectors two years ago, I realized I first had to teach it to myself, because I’d never actually learned it. My college courses skipped straight to “n-dimensional vectors.”

This is how mathematicians approach things: why discuss the 2D or 3D case when you can just climb the ladder of abstraction and cover all cases at once? Surely you can figure out the 3D specifics when you need them, right?

Well… maybe.

But going from abstract to concrete isn’t always as easy as you’d think.

I’m not a professional mathematician, but I’m proud to say I have all the bad habits of one. To wit: my students are often surprised at my clumsiness with arithmetic. Just today I casually said “40,000” when I meant “400,000.”

This happens a lot.

Now, are mathematicians *actually* that bad at arithmetic? Compared to engineers and accountants, perhaps. Compared to the average person on the street, of course not. The “bad at arithmetic” thing is probably overplayed. So why do mathematicians love to bring it up? Here’s one reply:

*Imagine you’re an artist, and people are convinced that your job consists of rolling and unrolling canvasses. That’s it.*

*Week after week, people ask: How fast you can roll a canvas? What’s the biggest canvas you’ve ever unrolled? Can you come over and unroll my canvas for me this weekend? And so on, and so on.*

*You keep trying to clarify – to talk about the actual *paint* you put *on* the canvas – but they don’t really get it. They just laugh and say, “Oh, you artists!”*

*Wouldn’t you start pretending to be bad at canvas-rolling, just to change the subject?*

That’s one answer. But I have to admit there’s another possibility:

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Last week, I visited my dad, who still gets the newspaper.

(For my younger readers: that’s a stack of cheap paper printed with a detailed description of yesterday.)

Anyway, for an ungrateful millennial like me, a print newspaper means one thing: puzzles.

Like Sudoku.

You already know the rules: nine rows, nine columns, and nine medium squares, each containing the digits 1 through 9. You’re given some; you fill in the rest. It looks something like this (by which I mean, “here’s an example lifted from the Wikipedia page”):

Now, I’m not much of a Sudoku player. (Crossword guy, to be honest.) But glancing at the puzzle, my dad and I got to wondering: How do they generate these puzzles?

We weren’t sure.

So we found a more tractable question: **What if you were a lazy Sudoku maker?**

That is, suppose you managed to generate a single Sudoku puzzle. (Or steal it from the Wikipedia page.) And suppose you wanted to make a few bucks selling collections of puzzles in airport bookshops. But there’s a catch: You’re not sure how to make more.

**How many “different” puzzles can you get from a single Sudoku?**

Well, let’s start with this: the numbers don’t actually matter.

For example, you could switch the 1’s and the 2’s. Nothing really changes. Every row still has a 1 and a 2. So does every column. So does every medium square.

The symbols in Sudoku are meaningless. It doesn’t matter *what* they are—numbers, letters, emoji. It just matters *where *they are.

Swapping 1’s and 2’s isn’t all we can do. You could also switch the 3’s and the 4’s. Or scramble the 8’s and 9’s. Or turn the 5’s into 6’s, the 6’s into 7’s, and the 7’s into 5’s.

There are a *lot* of ways to do this.

To see how many, let’s clean the slate, and turn them all into letters.

Now, for **a**, we have nine choices. Namely, it can be any digit.

Then, for **b**, we’ll have eight choices: any digit *except* the one claimed by a.

And for **c** we’ll have seven choices: any digit except the two already chosen.

Proceeding that way, we get **9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 **total possibilities. That’s also known as **9!**, and it’s big: to be precise, **362,880**.

Here’s an example, drawn at random from those nearly 400,000 possibilities:

We’ve already got a *lot* of puzzles, and we’ve only begun.

If you want another kind of change, you can switch the first and second column.

As with our number-shuffling above, the change is only superficial. Every column, every row, every medium square—they all still have the digits 1 through 9, precisely once each.

So we haven’t broken the puzzle, just reconfigured it slightly. That’s what we’re going for: costume changes that preserve the structure of the puzzle, while disguising it for human eyes.

How many puzzles does this kind of switcheroo give us?

Well, just focusing on the first three columns, we’ve got six possibilities:

We can also scramble the middle three columns, or the final three columns. That’s 6 x 6 x 6 possibilities so far.

Or, if we want, we can consider not just individual columns, but *bands* of columns, like this:

As above, these bands can be rearranged in six orders:

Thus, by rearranging these bands of columns, we’ve got 6 more possibilities, to multiply by our early 6 x 6 x 6. That’s a total of **6 ^{4}**, or

But of course, Sudoku is symmetrical! What we can do for columns, we can just as easily do for rows. So that gives *another* 1,296 rearrangements, any of which we can do in combination with any of the 1,296 above.

Multiply these together, and you’ve got **6 ^{8}**, which is

And remember: these rearrangements work for *every single one *of the **362,880 **“different” puzzles we generated earlier by resequencing the numbers.

So, are we done?

Not quite.

We can also *rotate* the puzzles 90^{o}.

Yeah, like that.

Obviously this puts the numbers sideways, but that’s a quick fix. And it gives us a totally new puzzle:

Other rotations are possible, too—we could go 180^{o}, or 90^{o} the other way—but these can be accomplished just as easily by rearranging rows and columns. (The same goes for various reflections.) That leaves us with just one relevant rotation.

This turns each puzzle above into 2 puzzles.

So, what are we left with?

That’s a lot.

Like, a *lot* a lot.

If your printer could print out one per second, it would take nearly 40,000 years to get them all. The paper would fill 14,000 stacks the height of Everest.

What we’re exploring here are the **symmetries** of Sudoku. They’re the transformations that don’t really transform, the changes that leave an object fundamentally the same.

Sudoku may be small—only 81 squares—but it has more than a trillion symmetries. All from a single puzzle.

Suffice it to say: That’s a lot of puzzle books.

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**“In mathematics you don’t understand things. You just get used to them.”**

On one level, this runs against everything I believe as a teacher. Mathematics should not be an intimidating collection of inscrutable methods! It should be a *timidating *collection of *scrutable* methods! We should accept nothing on authority. Everything in mathematics is there to be understood.

And I do believe that.

But I also know that mathematics is full of startles and shocks. I know that even the simplest objects can bury deep secrets.

Take the number line.

Boring, right? Nothing could be more prosaic.

Well, that’s only because you’ve gotten used to it. To my mind, the number line merits only one possible reaction:

It’s not the integers that spook me, although those are plenty weird.

To wit: although we distinguish numbers like “4986” vs. “4987,” not all human languages are so fussy. After all, does the difference between these numbers really *matter*? Are you good enough at counting to reliably tell one from another? The fact that we assign these distinct identities is a bit of a mind-popper.

No, what keeps me up at night is the stuff between the integers: the numbers a 12-year-old student of mine recently dubbed “the disintegers.”

When we draw a line connecting 0 to 1, what exactly are we drawing?

You might think we’re drawing a line of fractions. After all, fractions are like bedbugs: tiny and everywhere, living in every nook and crevice you can imagine.

And yet there are lots of numbers that *aren’t* fractions, perhaps the most famous being the square root of 2.

The square root of 2 can’t be written as a fraction (by which I mean, one whole number divided by another). Sure, you can get close – say, **665,857 / 470,832**, which is the square root of roughly **2.0000000000045**. But you can’t get the square root of 2 precisely.

Okay, so the disintegers aren’t all ratios of whole numbers. But that’s okay. Maybe, like the square root of 2 (which is the solution to the equation x^2 = 2), they’re just the solutions of equations. That’s not so bad, right?

We call such numbers – the solutions to simple polynomial equations – **algebraic**.

Unfortunately, some numbers escape even this widened circle.

There’s no nice algebraic equation that gives you pi or e. These numbers are called “transcendental.” For them, the best you can do is to describe a process by which a computer could, in theory, compute the number.

(For example, if you want to compute e, you can take a big number n, find 1 + 1/n, and then raise this sum to the nth power. It won’t give you e *precisely*, but the bigger the n you choose, the closer you’ll get to e.)

At this point, we’re forced to widen our conception of “number” again. Numbers, we shall say, are “things you can compute.” Maybe they’re not fractions. Maybe they’re not even the solutions to equations. But they are things that you could tell a computer to calculate.

Unfortunately… that’s not quite right.

There are non-computable numbers.

In fact, *almost everything on the number line is non-computable.*

What does this mean?

If you pick a random number between 0 and 1, you will arrive at a number that literally cannot be communicated or expressed. Sure, you can list the decimal digits, but if you ever stop, even after a trillion years, then you will have failed to specify your number – and there is no process by which you can tell a computer to carry on the job.

These are numbers that cannot be touched by the human mind.

And this is *most numbers*.

Well, you know what they say: In mathematics, you don’t understand things. You just get used to them.

FURTHER READING:

Fawn Nguyen taught a badass lesson where she posed kids the philosophical and mathematical challenge of **finding the square root of 7 on a number line**.

Simon Gregg calls noncomputable numbers “**the dark matter of the number world**” and points out that biology has its own version of the “most of what’s out there is beyond our imagination” problem.

Also worth reading is **Mr. Sock Monkey’s** comment below, which speaks for us all:

monkey still not sure of definition of word number or if number have existence independent of human mind. when monkey read definition of number by mr russell it make simple brain of monkey spin.

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A fair question: how did “i” get the name of “imaginary number”?

It seems harsh. In some sense, *all* numbers are imaginary. After all, is there really such a thing as negative numbers? You can’t have -2 friends, no matter how alienating your Facebook posts are.

Or what about the irrationals? If you take a 1-meter stick and mark it up into equal segments, then no matter how tiny and minute the divisions, you’ll never get an irrational length. Even if you go down to the atomic level. That’s kind of weird.

Heck, what about the natural numbers, like 7 and 15? Isn’t it a little weird to pretend that these exist? I mean, 7 *what*? Numbers are made for counting. How can you have a number without anything to enumerate?

So sure, I’ll grant that *i *is imaginary, but only insofar as every number is!

Of course, back in the day, mathematicians saw something fishy about these numbers. After all, they’re neither bigger than zero, nor smaller than zero, nor equal to zero. THey give negative results when squared. So you can’t blame mathematicians like Euler for using names like these:

- “imaginary”
- “impossible”
- “inconceivable”
- “fancied”

Clearly, we ought to envy the inhabitants of the nearby parallel universe where these are called “fancied” numbers. But I, for one, pity those poor souls in the universe where they are called “inconceivable,” which lacks the playful color of “imaginary.”

(The word “imaginary,” by the way, came from Descartes. Like many other great names, it began as a slur.)

It’s fun to try to think of other names for *i* and its multiples.

“Orthonumbers” or “orthogonal numbers” seems like a popular choice. It was the first that came to my mind, and I’m not alone. After all, they appear not *on* the number line, but perpendicular (or “orthogonal”) to it.

I also think we could call real numbers “posroots” (since they are the square roots of positive numbers) and imaginary numbers “negroots” (since they are the square roots of negative numbers).

(Bonus for “Guardians of the Galaxy” fans: you get to picture *i* saying “I am negroot.”)

Do names really matter? Maybe not. Mathematical creatures are whatever they are, no matter what we call them.

(A rose by any other name would have the same fractal dimension.)

Still, it’s hard not to want our names to reflect the symmetry and structure of the mathematical world.

Now, when it comes to imaginary numbers, you may say I’m a dreamer.

But I’m not the only one.

I hope someday, you’ll join us.

And the world will know *i* is “real” as 1.

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“Because two negatives make a positive!”

Then, if you listen carefully, you will hear something else: the low rumble of my teeth grinding together with tectonic force.

“Two negatives make a positive” is one of those math slogans that drives me crazy, because it is so pithy, so memorable, so easy to apply… while also being so vague and non-mathematical that I’m amazed students find it useful at all.

We can all think of many, many cases where two negatives *don’t* make a positive. Rain on your wedding day *plus* grand larceny on your wedding day does not make for a winning combination, despite what “two negatives make a positive” would suggest.

It’s not even true with negative numbers, where -10 + -30 does NOT equal +40 (although I have seen students claim that it does, citing “two negatives make a positive” as their justification).

In fact, that’s one of my major complaints with “two negatives make a positive”: it is such a swift, over-arching generalization that students wind up applying it in places where it doesn’t make much sense.

In fact, “two negatives make a positive” doesn’t really make much sense *anywhere*.

What does make sense is a slight variant, less catchy but far more true: “The opposite of the opposite is just the thing itself.”

What’s the opposite of “the opposite of happy”?

Well, “the opposite of happy” is sad.

So the opposite of *that* is “happy” again.

For adding and subtracting with negatives, I tend to favor a debt model.

For multiplying and dividing with negatives, I think a slightly more abstract approach is necessary – it’s all about the properties of multiplication.

Good mental models are more effective than mantras like “two negatives make a positive,” I believe. But even if they weren’t – even if the use of mantras led to error-free computation with negatives – I’d still favor the “mental model” approach. Learning new models engenders the kind of rich thinking that math class is supposed to be about; learning new mantras engenders the uncritical thinking of the cult-follower.

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*Tune in next week for more wacky hijinks as Young Gottfried tries to build a horizontal windmill!*

[NOTE: No, I am not actually doing a serialized webcomic on the adventures of a teenaged Gottfried Leibniz, c0-inventor of the calculus. You are welcome to do so yourself; I promise I won’t litigate.]

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So here are some ideas of other ways to slice, dice, and rearrange the mathematics currently taught in high schools. (And hey, maybe we’ll want to switch out an ingredient here or there, too.)

I’ll lay out four proposals.

First up…

This is an approach with a simple goal: Make Math Useful.

You’d begin with a condensed version of the Geometry/Algebra sequence (** Mathematical Reasoning**). We’d cut some of the dregs (compass-and-ruler constructions, circle theorems, conic sections, fussy technical proofs). Instead, you’d do a semester of geometric proof, focusing on the greatest hits, and a semester of algebra, focusing on expressing linear relationships.

Then, we’d get a year of probability and statistics (** Data Mathematics**). You’d learn about randomness, standard deviation, conditional probability, p-value, and how to build a spreadsheet. All that tasty, jobs-y goodness.

For junior year, you’d get a course that covers some traditional content (functions, graphing skills, exponential growth, derivatives) but with a focus on financial and economic applications (** Financial Mathematics**). How does interest work? How does debt work? Emphasizing the real-world relevance of these ideas wouldn’t have to undercut their theoretical beauty; rather, it would add to it.

And finally, you’d get a course in slightly more advanced topics relevant to the digital age (** Computer Mathematics**): first-order logic; expressing transformations with position vectors and matrices; binary; writing and understanding algorithms.

Yes, we lose some stuff. (Goodbye, arcsecant—and good riddance!) We even lose some good stuff. (I love teaching about limits in depth, but there’s no space for it here.) But I think we gain in coherence and purpose. Instead of organizing the curriculum around traditional divisions within mathematics, we organize them around goals that make actual sense for actual students.

Or does it worry you to blur the historic boundaries between subfields? Well, if traditional disciplinary structure is your thing, then you might go for…

Our second proposal:

Personally, I’d be jazzed to teach any of these courses.

But perhaps you’re not. Perhaps you’re a dreary Eeyore soul, and you’ve observed that this is a forbiddingly large number of different classes for a high school to teach.

Well, you conservative stickler, here’s a menu for you.

Our third proposal:

Wait—isn’t this what we have now?

More or less, yeah. But I’m picturing one big change.

Currently, there are two years of AP Calculus available. Lots of students take the first one as juniors, and the second as seniors, meaning that half of their high school math time is spent on calculus.

This stuns and baffles the Brits I teach with. They approach calculus very differently, scattering it throughout 11^{th} and 12^{th} grade. (You might even encounter easy bits of differentiation in 9^{th} and 10^{th}.) Some topics get cut entirely—for example, they never really tackle limits, and frankly, they don’t seem to miss them.

So my suggestion: ditch the second year of calculus.

This requires some cutting (goodbye, partial fractions and trig substitution). So why is it worth it?

Because ambitious college-bound students would still want two years of AP mathematics. And, without the impressive two years of calculus on the table, they’d turn to the alternative, a course that’s useful for citizenship and which everyone should probably be taking anyway:

Statistics.

Of course, if this modest proposal is too stodgy and incremental to get your blood flowing, I’ve got one final suggestion.

The fourth proposal:

Teachers like me urge each other to think in terms of verbs. “Education is not about helping them *know* stuff,” we say. “It’s about helping them *do* stuff.”

So let’s make the “doing” central.

Begin with lots of modeling activities. Explore how proportions, linear relationships, and trig ratios can help us express a whole host of real-world relationships. Do three-acts. Build inclinometers. Go wild.

Then, in 10^{th} grade, hit the disciplinary core of mathematics: proof. Build systematically from Euclid’s axioms (the roots of geometry) all the way to subtle proofs about planar figures and Platonic solids (the canopy of geometry). Hit up algebraic classics, too—the proof that root-2 is irrational, and that the primes never end. Get kids arguing, justifying, deploying logic.

Next, in 11^{th} grade, dive into the hot topic in mathematics: statistical analysis. The focus here is on describing the messy, multifaceted world we live in. How can probability capture the breadth of possibilities in life? How can statistics tease apart true claims from spurious ones?

Finally, in 12^{th} grade, reconquer the terrain where mathematics education currently spends most of its time: calculation. Explore the rules of calculus, contrasting the systematic completeness of differentiation rules with the scattershot, ad hoc methods that integration allows. Don’t just carry out algorithms; master the idea of an “algorithm” itself. Investigate numerical techniques like Newton’s Method. Write computer programs to execute rote steps so you don’t have to do it yourself.

Would any of my ideas be strictly better than the existing status quo? Maybe, maybe not. As I’ve written before, the devil’s in the details.

But it’s always fun to imagine a new recipe.

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