Finding a good question

I just met with my advisor, Sandor Kovacs. Meeting with him is great, and I walk out excited to get to work. No exception today, but my work feels particularly slippery at the moment.

The reason is that I don’t yet have a good question. I’m narrowing in on it, but it keeps eluding me. There’s a truism in math that finding a good question is half the work. I estimate that locating mine is going to take me pretty much this entire academic year.

I’ll be teaching a course this summer called Turtles All the Way Down, in which I hope to teach, among other things, the art of asking questions. Finding your question, in or out of mathematics, can inspire your creative work for years. Yet we devote so little time to learning how to do it well.

The fun thing about math is that the classic questions which begin math usually feel like they should be easy. Doubling the square is a good example. If I give you a length and tell you to double it, it’s really easy to do it (just take two of those lengths and put them together). Likewise, if I give you a square (imagine a one inch by one inch square if you need a specific example), and tell you to double that—by which I mean double the area it take up—it seems like it should be just as easy: just double the sides of the square. But then you have a two inch by two inch square, and that’s four times as big as a one inch by one inch square (draw it!). So how do you actually get the square that’s twice as big?

A seemingly innocuous question. The answer isn’t terribly hard, but it’s far from obvious, and requires some insight. Once you see it, though, there’s that aha! feeling. (Plato brings it up in the Meno). It also points in a very interesting direction. In fact, if we follow this question as fully and honestly as we can, we end up travelling the path that leads to modern mathematics. My old colleague Paul Lockhart once remarked, after a very bright second grader he knew cried in frustration when he was trying to work out an implication of this problem, that these were the tears where mathematics begin.

Off to work!

Origami Article

Link: Origami Article

“I realized origami was geometry without numbers. I fell in love.”

—Rona Gurkewitz, from the article.

I’ve seen some pretty exceptional origami in my life, from mathematicians. What’s inspiring about origami from a mathematical point of view is that it starts as a craft—the art of paper folding—and questions transform it into mathematics. I’ve been thinking about this a lot lately. Questions are at the heart of mathematics.

So what, specifically, is the question here? Probably something like: “What is it possible to make with a square sheet of paper?” Here’s Rona again:

“It’s playing with shapes and seeing what is going to happen.”

And that’s the next step—an inspiring question leads to play, and this play (serious play, as Keith Johnstone might say) leads us to a better understanding of what is possible.

The Best and Worst Jobs in the U.S. – WSJ.com

Link: The Best and Worst Jobs in the U.S. – WSJ.com

“It’s a lot more than just some boring subject that everybody has to take in school… It’s the science of problem-solving.”

— Ms. Courter, a research mathematician in the article below.

I’m not linking to this article to make lumberjacks feel bad. But this is a small part of a body of evidence that there’s something in mathematics that can bring great joy into our lives. You can’t see what it is from this article; you generally can’t see what it is when people write about math. I honestly think you have to start doing mathematics to feel it. I’ll try to put more on this site that might lead you to feel it.

Note that that top six jobs are mathematically heavy. Part of the reason matematician is such a great job is that there’s a tremendous amount of intellectual freedom. Sadly, most of us experience the opposite when we take math courses in school. Who here associates mathematics with freedom or choice? And yet, making choices is at the heart of mathematics.

The reason to do math is for love

This blog is my effort to change the way we talk and think about math. We all agree math is important; we debate furiously about the best curricula to use in schools. But sadly, the activity of mathematics—and the joy, beauty, and wonder it brings—is absent from almost all of our lives.

I’m going to do my best to show you what this subject is really about. I love math, and I believe it to be one of our inheritences as human beings, another rich form of expression like music, art, or speech. I’ll try to show you what it looks like in its uncorrupted form, and what it feels like when you do it. I’ll be posting videos, interviews, musings, essays, and news stories.

Have courage! Having math in your life will change the way you see everything.