Math for Love

Forgetting and learning

One of the remarkable things about learning math is that steps forward in understanding require a kind of forgetting. Everything always looks simple in retrospect; it’s letting go of your biases that prevent you from learning that is difficult. There are great examples of this in the book Ender’s Game: the main character goes into zero gravity for the first time and realizes that he can let go of his conception of “down.” Simple, but difficult.

I’ve often wondered about why this is so often the case. After all, learning seems constructive in nature, but forgetting and letting go of bias is deconstructive, and seems like it’s the opposite process. Today I had another thought on the topic though. I’ve long suspected that much learning involves a refinement of vision and categorization. Toddlers sometimes go through phases of calling all animals “doggies,” for example, because their categories aren’t sufficiently refined. (I heard one story of a child passing a field and point to the “moo doggies” there.) But this means that we learn through differentiating more deeply between things. In other words, we forget that they seemed alike. They’re actually different.

Of course, so much of math is about seeing similarities between things that look totally dissimilar. In effect, both skills are crucial: we need to disconnect ideas from each other (forget our biases) in order to connect them up to each other in new ways. Learning seems to involve the constructive and deconstructive in equal measure.

Meanwhile, I’m getting pretty hopeful about my thesis. I think I may actually be on the verge of calculating the number of Fourier-Mukai partners of certain K3 surfaces. Just have to go over (and over) the details.

Progress and Simplicity

I had a little progress on my thesis work, recently. Essentially, I was able to prove what form a composition of transformations would take in the most general case. I had a hunch (and a hope) that it would be the simplest thing I could think of: given two transformations defined by two numbers, the composition of them should be defined by the product of those numbers. And indeed, that’s the way it is, in the simplest case.

This is one of the reasons I like math: things sometimes work out like they’re supposed to. If you study biology, it seems like everything is a hopeless mess; any process in the body is affected by every other process, so you can barely ever get a clear look. In math, there’s this magical way that things end up being surprisingly simple when you look at them the right way. That’s what seems to be happening with my work: a complicated transformation involving geometric structures connected via an abstract algebraic process and the whole thing boils down to multiplication.

For the moment. I have a feeling the subsequent cases will be a bit rougher.

Turtles all the Way Down

I’ve been terribly delinquent about posting here recently.

I’ve just finished my second week of my summer course, Turtles All the Way Down, which I’ve been teaching through the Robinson Center at the University of Washington. I have to say, it’s been a thrilling and tiring ride so far. We meet from 9-2:20, Monday, Tuesday, and Thurdsay, which adds up to more than fifteen hours a week in class. The students, each of whom just finished seventh, eighth, or ninth grade, are a pretty impressive bunch.

My goal in the class has been to get the kids asking their own questions. I would say they’re on a roll at this point. Many of the issues that get raised in class are deeply mathematically substantial. What’s also fun is that I get to learn from my students and my TA.

Two of the gems from the course:

1. There are “bigger” infinites than the infinity we think of. We’ve discussed the two smallest in class so far. My students are clamoring to know if there are more (there are), and what they look like.

2. There’s a more general way to think about dimension that allows you to measure the dimension of fractals as being nonnatural numbers. We’ve seen fractals with dimension close to 1.77, for example. This was actually new to me, and pretty cool.

I’ll try to update with more highlights soon!

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 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.