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P.S. Thanks to the Birmingham Uni postdoc who told me the “shoelaces” line!

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**Case Study #2: Quadratics.**

**Case Study #3: Equation.**

**Case Study #4: Doubling a fraction.**

**Case Study #5: Comparing functions.**

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Like, how do you factorize a quadratic? How to you differentiate a cubic? How do you solve a system of simultaneous linear equations? How do you poach an egg?

(Apparently you need a gentle whirlpool to get the egg moving. Whirlpools: the unsung hero of the breakfast table.)

Why are they so skilled at *how*?* *It’s because students like procedures. They like certainty, clarity, the feeling that you know exactly what to do at every moment.

But they struggle with *why*. And – even more basically – they struggle with *what*.

For example…

I find that questions like this elicit one of two responses from students. Either this:

Or this:

These aren’t questions students are accustomed to answering in math class. In history, perhaps, where they have to write IDs of historical figures and events; or even in science, where they have to understand each component’s role in a theory.

But not in math. We math teachers tend to ask lots of *how* questions, and not so many *what* questions.

If you ask me, that’s sort of sad. They’re experts in *how*, and they can’t even tell you what the *how* is for.

And in this case, it turns out, there’s a pretty satisfying answer.

First, note that quadratics are much more complicated and interesting than their simple flat-brained cousins, the linears:

And then, note that quadratics are much *simpler* than their roller-coaster contortionist siblings, the cubics, quartics, and other high-degree polynomials:

To me, this is the appeal of quadratics. As degree-2 polynomials, they occupy a sweet spot between the dull degree-1’s, and the intimidating, intractable degree-3’s.

Just as Goldilocks sought the perfect bed (not too hard, not too soft) and the perfect porridge (not too hot, not too cold), so the mathematician seeks the perfect polynomial. Not too hard, not too easy. Not too complex, not too simple.

Just about right.

Of course, this line of reasoning is open to an obvious attack. *Okay,* a disgruntled student might say, *you’ve convinced me that, if I’m going to study polynomials, I ought to focus on quadratics first.*

*But why should I study polynomials to begin with?*

The answer to that is trickier, I think. You might as well ask this:

This question has as many different answers as mathematics has teachers. Some like to focus on the applications of math. Some argue it’s all bout the beauty. Some just say, “Because,” and then sigh, because it’s been a long day.

But for me, it’s about thinking.

And the role that the quadratic plays in polynomials… well, that’s exactly the role that mathematics plays human thought.

In every walk of life, humans need to reason.So of course, they can learn these intellectual skills in other places. You don’t *need* math. But gosh, does math make it easier!

You can learn to taxonomize in biology, by considering the classification of organisms. But your taxonomies will never be perfect, because life doesn’t fit into neat little boxes. (I’m looking at you, protists.)

Life doesn’t… but math does.

Or you can learn to dissect arguments in civics. But emotions will flare. It’ll be tough to agree on premises. And even if you do, words like “justice,” “freedom,” and “common good” are subject to fuzzy interpretations and subtle misunderstandings. All words are like that: a little vague, tricky to pin down.

Except in math.

Logic shows up everywhere. But in math, it’s the whole game. Math isolates the operations of logic and reason so that we can master them.

In short: math is the playground of reason.

*This post is hastily adapted from a talk I gave yesterday at University of Birmingham, titled Death to the Quadratic Formula (or, Long Live the Quadratic Formula). Thanks to Dave Smith and the IMA for the invitation!*

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- Burn down a used bookstore.
- Sponsor legislation banning the use of peanut butter in desserts.
- Offhandedly mention that you think baby chimpanzees aren’t that cute.

- Or, if you want to take the easy route, just bust out this quote from the great mathematician G.H. Hardy:

The blood…

The boiling…

Oh, the boiling of the blood…

Try as I might, I can’t hate Hardy. He wrote wonderful textbooks, proselytized on the beauty of mathematics, and did other good deeds.

But I can hate this view, this toxic meme, which I believe is latent in our stereotypes of mathematics: this belief that generating new mathematical ideas is man’s highest calling, while wallowing in old ideas is grunt-work fit only for mules, washouts, and the dim bulbs we call teachers.

Hardy didn’t invent this idea. He simply gave it voice.

So on behalf of the mules, the washouts, and the dim bulbs we call teachers, I wish to mount a defense of the art of explanation.

For my first witness, I call to the stand… Euclid of Alexandria!

Now, Euclid was not the first Greek scholar to tackle geometry. Far from it. Centuries of brilliant folks had left their signatures by the time he arrived on the scene. So why is he known today as the founder of mathematical proof? Why is traditional school geometry often known as “Euclidean”? Why am I name-dropping him 2500 years after his death?

It’s because he wrote the book.

It’s because he consolidated the ideas.

It’s because he *explained*.

Before Euclid, geometry lay scattered in pieces, like a collection of jewels thrown to the sands. The ideas were there. The proofs were there. But they lacked organization, structure, coherence.

Euclid unified geometry into a clear, logical system. He laid out simple assumptions (called axioms) and traced every geometric truth, no matter how remote or sophisticated, back to these axioms. His project brought together the work of diverse mathematicians into a single, coherent whole.

Euclid transformed mathematics not by creating new ideas, but by elucidating the connections between the existing ones.

That is: by explaining.

Not bad for a second-rate mind, right?

Euclid’s not the only one to change the world through mathematical explanation. Leonardo of Pisa (best known by his posthumous nickname “Fibonacci”) brought the modern numeral system to new lands.

Before Leonardo, Europe sweated it out with Roman numerals: clunky, slow, inefficient for computation. But in the shipyards of Algiers, the young Leonardo learned a better system. These numerals—1, 2, 3, 4, 5, 6, 7, 8, 9, and the curious 0—were born in India, perfected in Arabia, and now transmitted, through Leonardo, to Europe. They caught on among the merchant class. And eventually, they swept the entire world.

Leonardo re-explained arithmetic to a whole continent. In the process, he nudged history towards its modern, globalized language of number.

Score another one for the second-rate minds.

It was Albert Einstein—another dim bulb, clearly—who said, “If you can’t explain it simply, you don’t understand it well enough.” But it doesn’t require Einstein to see the wisdom here.

Every schoolchild knows that explaining something to a friend helps you master it yourself.

Every instructor (from primary school to grad school) has felt how teaching something can help snap into place the loose fragments of your own understanding.

And every researcher has witnessed how the effort to write or speak about your ideas winds up purifying and clarifying them—like boiling off excess water to make smoother, creamier sauce.

Explanation isn’t just good for other people. It’s good for the explainer, too.

Hardy was a proud, contributing member of the mathematical research community. So I find it curious that he so blindly and guilelessly separated the “research” from the “community.”

Now, to research mathematics is to develop new ideas. This adds to our library of collective wisdom. That’s obviously a very cool thing.

But when you teach new minds, or write accessible books—i.e., when you explain—you’re doing something equally cool.

You’re *staffing* that library.

If we all shared Hardy’s stated priorities, we’d simply be stuffing the shelves with new and largely unread books. No one would be editing or consolidating this knowledge. No one would be coordinating our efforts. No one would be recruiting new scholars into the library.

Hardy’s advice is self-defeating. It creates a research community that’s all research and no community.

It creates a library full of great books and with no readers to read them.

We ought to be glad, then, that Hardy saw past his own bad advice—that he wrote books like *A Mathematician’s Apology* and his other accessible texts, in spite of his grudging attitude towards such work.

Maybe you think I’m exaggerating about mathematicians’ antipathy towards explanation. Surely this pro-research, anti-teaching, anti-anything-that-smells-like-teaching prejudice can’t be all *that* pervasive and damaging. Can it?

In that case, I introduce my final exhibit: the ABC Conjecture.

First proposed in 1985, the ABC Conjecture is one of the great unproved statements in mathematics. It’s a powerful claim about number theory, and if it’s true, it has many deep repercussions.

And, in August 2012, it was proved.

Well… maybe.

We’re not sure.

The mathematician with the proof is a fellow named Shinichi Mochizuki. He spent decades developing his theory, which spans more than 500 dense pages, full of novel notation and new conceptual machinery. The work is so inventive that—unfortunately—no one has been able to check it.

His individual triumph of problem-solving has been swamped by a collective failure of explanation. Mochizuki’s proof sits there, unexplained, not budging, like an undigested meal caught in the throat of a snake.

We’d better get some second-rate minds over to help.

Right, Hardy?

Of course, my complaint isn’t really with Hardy.

My complaint isn’t even with the casual elitism of the academy. (Trying to purge the self-importance and sense of intellectual superiority from the research enterprise is like trying to purge the sex references from rock ‘n roll. It probably can’t be done, and you wouldn’t like the result if you did.)

My complaint is against the twisted belief that you should care deeply about ideas, but resent having to share them.

My complaint is against the self-defeating effort to separate research itself from the ecosystem of activities that support, sustain, and justify it.

My complaint is against anyone who thinks explanation is a lesser art.

And most of all, my complaint is against people who don’t think baby chimpanzees are cute.

I mean, come on, guys. Open your eyes!

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