One out of three

I recently posed one of my favorite questions of all time to a student I’m tutoring. He’s something of a natural, and got it remarkably quickly. I personally spent hours on it the first time I heard it, and then years to understand all the subtlety of it. And yet, it’s possible that a group of kids with no mathematical training might be able to solve it out of necessity on the playground.

So here’s the question: you have a fair coin, meaning it’s equally likely to come up heads or tails. How can you design a game that involves only flipping the coin, that you have a one out of three chance of winning?

Every time I pose this, someone ends up coming up with a solution I’ve never thought of before. I’d love to see if any readers have new solutions for me.

[Note: I posted this question here. You can check out people’s responses.]

What faith is required in mathematics

Imagine I took a small handful of pennies and showed them to you clumped together, then spread the same collection out into a wider array; you know, without having to think about it, that the pennies are worth the same amount either way. It’s the same group of pennies, after all. To children, this isn’t obvious. The psychologist Piaget noticed that many of the the “obvious” things we know are not obvious to children until they encounter them in certain ways at certain ages. Moreover, there seems to be something irreducible about the leap in brain development. One day, a child can’t see that the pennies are worth the same amount, the next day it’s obvious, and he can’t imagine that he ever didn’t know.

As we grow up, we have less and less of these Piagetian type leaps. Learning becomes something more incremental. If we want to learn a piece of music, we practice it, and watch ourselves know it better, in tiny increments. The practice leads to our better understanding something.

But there are still Piagetian leaps that take place, I would argue, and they’re quite thrilling when they happen. I remember taking real analysis in college: essentially most of the course is centered on a techincal point of drawing a series of increasingly smaller circles around points in order to define a very precise notion of what “continuity” means. I remember it being baffling at the time. But by the time I was taking the following course, it felt so obvious that it was almost like second nature. I couldn’t imagine not understanding the principle.

I’ve discussed before how the nature of these Piagetian leaps forward make math hard to teach: the teacher puts the problem forward and feels like it must be easy to see, because he can’t imagine being unable to see it; the student can’t see how to begin thinking about it, because in some way, you can’t see the breakthough until, magically, you do. What’s astonishing, though, is how this continues as long as you’re studying math. I’m looking at incredibly abstract notions, abstractions piling on abstractions. There are moments when the whole idea escapes me completely. There is also the almost constant feeling like I’ll never be able to see what it is I want to see, or prove what it is I want to prove. It’s impossible to imagine knowing (and knowing naturally) what it is I don’t know now.

This is where faith is required in mathematics. I have to keep throwing myself at what I don’t understand. When I don’t understand what I don’t understand, I have to keep searching: it’s like hunting around an obstacle course for the right brick wall to throw yourself against. But even though I can’t see progress at times, I have to assume that something is happening inside my head. I’ve repeated the process of total confusion yielding, suddenly, to total understanding so many times that I have to believe that it will keep happening.

Keeping this kind of faith is challenging sometimes. Not knowing is such a vast feeling. Even more, the creative insight is so slippery that many mathematicians, I think, have an anxiety in them akin to writers: do I have another good idea in me?

Back to work, back to work… I will continue to believe that something is building inside me, though I cannot see it.

The meaning of the search

Link: The meaning of the search

This beautiful essay in the last edition of Wired magazine gave a really nice explanation of why science (and math) is worth doing, even if we don’t get the answers we’re looking for. Worth the read.

On a related note, I discovered an erroneous assumption I made a while back that invalidates the last four months’ results. Still reeling from it a tiny bit, but the main thing is… it’s the search.


The moment, again

I got some requests to clarify what I meant when I said that this was a moment for mathematics in our culture. Let me explain.

For the first time ever, math is starting to be hip. When I was in high school I couldn’t imagine math being respected by those cool, we-do-stupid-dangerous-things people. Perhaps they still don’t respect it. But starting 12(!) years ago with Good Will Hunting, and proceeding through movies, plays, and TV shows like Proof, A Beautiful Mind, Numbers, not to mention countless popular math books, math has started to be cool in a way it never was.

At the same time, math education became central in the endless education debates. Earlier it was phonics vs “whole language,” but the argument centered on language in the past. More recently, math (i.e. the “math wars”) has had center stage in education.

I think the moment is passing now, but I still think there’s some ripeness. The opportunity to put a different idea about what mathematics feels like is still open.

I hope so anyway.

Free time to work

I have the quarter off from teaching, which means it’s the perfect time to get some serious work done on my thesis. The fact that the work that it most important in my life is also the least urgent—I could do nothing for six months and no one would know or care—means I have to be especially disciplined with myself. To that end, I’m putting aside big tracts of time that are to be devoted to my work.

I feel like I’ve finally started to get the good work habits and general resolve in place. Becoming disciplined in this way was one of my original goals when I came to graduate school.

Unfortunately, the work that I’m looking at right now is in danger of devolving into long strings of algebraic manipulation with no clear end in sight. There’s a difference between devloting time to working and working well. With math as with so many things, the best flow opens up when you’re on a certain kind of a roll. Sadly, there seems to be so much to interrupt said roll in mathematics: so many slippery details that get away from you.

I had a professor once say that you only do an hour of work in graduate school: it just needs two years of preparation to make it happen.

Beauty Through Mathematics

Link: Beauty Through Mathematics

Here’s another group out to show the world a vision of beauty through mathematics: Imaginary. Computers have made wonderful things possible in the realm of visualization. It is so labor intensive to get pictures this good without them.

Also, on the topic of mathematics having it’s moment in the world—Germany just named 2008 the Year of Mathematics. Every dog has its day.

My favorite symbol

Earlier this year a middle schooler emailed me to ask me what my favorite mathematical symbol was. I didn’t have a good answer, because I’d never thought about it at length, and I ended up giving him the old standby answer: 0. The apparent contradictory nature of having something stand for nothing kept humanity from developing it for some time, and its existence allows numbers to be written in the modern base 10 (or base anything) form. Books have been written about the innovation, like this one, or this one.

Honestly, though, I don’t have much of a personal connection to zero. The symbol I find really compelling, though, is the arrow. Let me explain why.

First of all, it’s everywhere. I can’t think of a mathematical subfield that doesn’t involve drawing arrows (each with their own precise mathematical meaning). We call them morphisms, mappings, functions, edges of directed grahps, etc., but it seems like they’re always present. The idea of moving your problem from here to there is central in mathematics.

Second, I love the implication of movement. There’s a line about chess, that “the threat is more powerful than the move.” Understanding how to look at a chess board as a place where forces are exerting themselves represents a leap forward in one’s ability with the game. Similarly, feeling a mathematical situation as wanting to be represented elsewehre is a key part of a developed (or developing) mathematical intuition.

Third, it’s powerful. It’s shocking how much information is carried in arrowed diagrams. They allow you to ignore clutter and focus just on what is essential. Category theory is a dramatic example of this—everything is reduced to objects and the arrows between them.

So, to that middle schooler I led astray with an out-of-date, impersonal answer, I hope you find this post.