Last year, I got the high school math teacher’s version of a wish on a magic lamp: a chance to ask a question of the world’s most famous mathematician.
Andrew Wiles gained his fame by solving a nearly 400-year-old problem: Fermat’s Last Theorem. The same puzzle had captivated Wiles as a child and inspired him to pursue mathematics. His solution touched off a mathematical craze in a culture where “mathematical craze” is an oxymoron. Wiles found himself the subject of books, radio programs, TV documentaries—the biggest mathematical celebrity of the last half-century.
In a month, I’ll be returning to Heidelberg, Germany. I’ll interview young researchers, ask a few questions of Fields Medalists, and save my deepest inquiries for the German chocolate cake. It’s the kind of absurd opportunity I had no reason to believe my career would afford when I began teaching math in 2009.
This has me thinking about a brief conversation I had last year with John Hopcroft, one of the honored laureates in Heidelberg.
You could be forgiven for thinking that John Hopcroft’s impressive career has followed a preordained trajectory. Bachelor’s, PhD, professorship. Stanford, Princeton, Cornell. Textbook author; National Science Board appointee; Turing Award winner. A well-groomed C.V. born from strategic calculations, right?
“A sequence of strange events that happened,” summarizes Hopcroft.
Hopcroft describes his career in terms of chance encounters and curiosities pursued. “I’ve never really planned things,” he says. “I’ve just been lucky.”
I’ve been having a little argument with five-years-ago me. The question is this:
Five-years-ago me? Throw his drink in your face. He’d tell you that rote thinking is the bane of his working days, that deep understanding is the whole point of learning mathematics. He’d tell you: No black boxes, ever.
Today-me is less convinced. Don’t working mathematicians, from the ground floor all the way up to Andrew Wiles, sometimes use black boxes? Isn’t it common sense that sometimes you need to use tools that you can’t build for yourself?
I’m still wary of equipping students with black boxes, but these days I’m willing to do it, so long as three conditions are met. I hesitate to share this crude checklist, knowing my colleagues out there in the profession will have wiser ways to frame the tradeoffs. (After all, aren’t checklists too binary, too black-and-white, for an idea as elusive and shaded as “understanding”?)
Nevertheless, my checklist goes something like this:
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
Thomas Mann once said, “A writer is someone for whom writing is more difficult than it is for other people.”
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.
If the Food Network has taught me one thing, it’s that how you plate a meal matters almost as much as what you’re serving.
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.
This is an approach with a simple goal: Make Math Useful. Continue reading
Sometimes students say precisely what they meant. “I don’t understand the question” means they don’t understand the question. “This is too hard” means it’s really too hard.
But sometimes, it takes a little translating…
Half of my classroom conversations go like this.
Student: “I don’t get the question.”
Me: [longwinded, exhaustive explanation of what the question is asking]
Student: “Yeah, I knew that. But I don’t get the question.
Me: “Oh. This is one of those conversations.”