Problem of the Week (or, ask another question)

Link: Problem of the Week (or, ask another question)

My old school, Saint Ann’s, in Brooklyn Heights, has an ongoing problem of the week competition for its middle schoolers. The problems tend to be fun, and require a little more thinking (and explanation) than most standard math problems, and the teachers have fun brainstorming new ones.

This one is pretty cute, and I think you can solve it if you want to. On the front of asking good questions though, I think there are some pretty interesting followup questions you could ask about this problem. The main one is, what happens to the ball after it hits the right side of the table? It will keep bouncing around the table, but will it ever go into the corners? Will it ever hit its starting position again?

Following these questions up, you can start to vary the size of the table and the angle and placement from which you shoot the ball in the first place. I suspect there’s a whole theory of how the ball bounces around the table that is elegant and satisfying. You just have to follow the initial question with a few more of your own.

Anyone else have any thoughts on good questions to ask about the billiard ball bouncing problem in the link?

After Tutoring

Just had a fun little tutoring session with a new student. (How could it surprise you that I tutor?) She strikes me, happily, as one of the many people who have a decent aptitude for math but, sadly, don’t like it much and don’t think they’re any good as it. How much I enjoy tutoring shocks me sometimes. The last problem got me genuinely excited: a neat geometric picture involving a stationary circle with another circle beside it shrinking away to nothing. You make a line from the top of the first circle to the place where the two circles intersect. The question, basically, is where does that line intersect the horizontal line through the center of both circles? We both looked at the picture and both of us thought, intuitively, that it should go off infinitely down the line.

But how to prove it? I suggested translating the problem from geometric to mathematical language. It involved quite a few steps, and wasn’t at all obvious how to do it at first, but she advanced from one to the next with little prompting from me, and we managed to state the whole thing as an algebra problem, where the answer would be given by a limit. Kind of astonishing that it’s possible to translate pictures into mathematics like this. We ran the limit, and with a little more work, found that the answer was: 4. A very finite number.

I have to say, this is one of the great things about math: you get surprised. Where did our intuition go wrong? What did the 4 have to do with the geometric picture (it was a graph, I should mention). I was immediately seized by a desire to start messing with the initial conditions to try to understand what made it be that number, and not another. And why not infinity? Something was wrong with our intuition, it seems. It’s a powerful thing to refine your intuition, and very satisfying.

The work of translating from one situation to another is also very key to the art of mathematics. Many people assume that all you need to do is solve problems, but before you can solve a problem you usually need to look at it from different perspectives. This algebraic/geometric correspondence is one of the richest in mathematics, in my opinion. It’s these connections between fields where the true insight lies.

A little example: have you ever noticed that as you go from one square number to another (1, 4, 9, 16, 25, 36, 49, etc.) you’re jumping an odd number each time (1, 3, 5, 7, 9, 13, etc.)? Looking at that arithmetically isn’t too enlightening: what does going from 3X3 from 4X4 have to do with the number 7?

But if you make it geometric, and think about a 3 by 3 array of dots going to a 4 by 4 array of dots, it becomes clear how to add those seven dots (in an L shape around the outside). Think about it a little more, and it becomes clear why it should be seven, and why it should always be odd.

And before you’ve known it, you have a proof of the purely arithmetic fact that the sum of the first n odd numbers is n^2. You can figure out that 1+3+5+7+9+…+91+93+95+97+99 = 2500 other ways (and I encourage it!), but the argument you get from playing with the square arrays sure is sweet.

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. –

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

“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.