A Symphony with an Irrational Time Signature

a weekly roundup of cartoons, links, and things to make your eardrums bleed

2017.9.11 zeno's persuasion

This cartoon draws inspiration from the tireless work of Julia Galef, the patron saint of being patient in internet arguments. Recently, she has been compiling lists of “unpopular ideas” (about political systems, social norms,  and criminal justice).

Even better, she offers this list of reasons to engage in internet arguments, even when you know that neither of you is likely to change your mind: Continue reading


There Is No Perfect Teacher (Just a Bunch of Great Ones)

I’ve taught at two schools in my career.

The first, in California, had a dozen teachers in total. I adored my colleagues, but we each had our own domain. I handled Trig, Precalc, Calculus, and Stats. Soon I fell into unquestioned habits, built on assumptions I didn’t know I was making.

My second school, in England, had a dozen *math* teachers. It was as if, after years of playing bass in my empty garage, I had suddenly been recruited into an actual band.

My first winter, I took myself on a tour of the department, observing a lesson from each of my new colleagues. I came away convinced that there’s no one way to teach mathematics, that our methods are necessarily as diverse as our goals.

Take two very different teachers: Simon and Tom.

Whereas I’m always fretting about students who can execute procedures without understanding them, Simon wastes no such worrying; he simply weaves the two together far better than I do. His notes at the board model clear and disciplined thinking, and he gives written comments every bit as careful and analytical as the work he expects. (He is also one of the two most competitive sportsmen I have ever met.)


Tom couldn’t be more different. At the front of the room, he’s less a lecturer than a provocateur: the mischief-maker in chief, whose highest goal is to create space for mathematical exploration.  That means open-ended problems, and multiple days spent wrestling with a single provocative question. He incites debates, sets traps, and shines a spotlight on the students’ own thinking. (He is also one of the two most competitive sportsmen I have ever met.)


But the diversity in math teaching runs deeper than the Simons vs. the Toms, traditionalists vs. progressives.

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Three Rules for Tackling a World-Famous Math Problem

While I was listening to Sir Andrew Wiles speak, I saw the writer next to me had jotted three adjectives on his pad: “Calm. Elegant. Precise.”

Astute as those three words are, they miss the basic strangeness of Wiles’ life story. For all his calm, elegance, and precision, the guy is also a unicorn, a sasquatch, a one-of-a-kind creature from the pages of myth. He is, if you will, a walking oxymoron.

He is a celebrity mathematician.


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Is Algebra Just a Series of Footnotes to the Distributive Property?

Last month, I was helping some 6th-graders prepare for their final exam when it became clear that their teacher had utterly failed them. “You poor souls!” I said. “Orphaned by intellectual negligence!”

They blinked hopefully, as children do.

“Who was it?” I demanded. “Who taught you – or should I say, failed to teach you?”

“Um…” they hesitated. “You?”


“Well, whoever it was,” I said, “he denied you the chance to experience the beauty and centrality of the distributive property. For this, he shall have my undying scorn.”

The philosopher Alfred North Whitehead once described European philosophy as “a series of footnotes to Plato.” Twist his arm enough, and perhaps he’d have consented to describe algebra as “a series of footnotes to the distributive property.”


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The Three Phases of the Mathematical Life

This autumn, I got the chance to ask a few questions of Ngô Bảu Châu.

If your jaw is not on the floor, it’s because (A) you’ve spent shockingly little time browsing the list of Fields Medal winners, and (B) you’re not Vietnamese.

A helpful Vietnamese journalist I met explained to me that Châu is “the biggest celebrity in Vietnam.” Châu won his Fields Medal in 2010 for proving—hands inside the vehicle, please, because this is a wild ride—a key relationship between “orbital integrals on a reductive group over a local field” and “stable orbital integrals on its endoscopic groups.”

In Vietnam, that relationship is apparently the one sizzling on tabloid covers.


Châu is not your prototypical superstar. Even in Vietnam, apparently, he is a cryptic figure; not a chatty TV celebrity, but a silent legend. At the press conference where I met him, at the Heidelberg Laureate Forum, he gave some journalists terse one-sentence answers. Not because he was being standoffish, but because a mathematician like Châu never proves in ten lines what he can prove in just one.

I didn’t know what to ask him. I’m not a research algebraist and have never been mistaken for one. So I asked about his education, his youth in Vietnam, his mathematical coming of age.

How does Ngô Bảu Châu get to be Ngô Bảu Châu? Continue reading

The Joy of Slightly Fishy Proofs

I’ve got a proof stuck in my head.

It comes from a man as brilliant as he is grumpy. Last week in the staff room, one of my all-time favorite colleagues busted out a dozen-line proof of a world-famous theorem. Though I doubt it’s original to him, I shall dub it Deeley’s Ditty in his honor:


This argument is sneaky like a thief. I admire but don’t wholly trust it, because it solves a differential equation by separation of variables, a technique I still regard as black magic.

What I can’t deny is that it’s about 17 million times shorter than the proof I usually show to students, which works via Taylor series:


You might call this one Taylor’s Opus. It builds like a symphony, with distinct movements, powerful motifs, and a grand finale. It takes familiar objects (trig and exponential functions) and leaves them transformed. It’s rich, challenging, and complete.

It’s also as slow as an aircraft carrier making a three-point turn.


Here we have two proofs. The first is a nifty shortcut, persuasive but not explanatory. Watching it unfold, I’m unsure whether I’m witnessing a scientific demo or a magic trick.

The second is an elaborate work of architecture. It explains but perhaps overwhelms. If an argument is a connector between one idea and another, then for some students, this will feel like building the George Washington Bridge to span a creek.

The question it prompts, to me, is: What do we want from a proof?

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