3n + 1

This weekend I gave a middle school version of a keynote lecture at the local Mathcounts tournament, as a group of 5th to 8th graders finished slices of pizza and waited for the contest results to be posted. The talk actually went quite well. It was my first ever powerpoint, and I must admit there is some real power to including video in a presentation.

I talked about the 3n+1 problem, a lovely little unsolved problem that is incredibly simple to relate, but virtually impossible to get anywhere with. It goes like this: say you pick a number, and generate a sequence based on the following two rules:

1. If your number is odd, multiply it by 3 and add 1.

2. If your number is even, divide it by 2.

Then you repeat that process with your new number. So for example, if you start with 5, you get the sequence

5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1 …

and the 4, 2, 1 loop repeats.

Now the absolutely astonishing thing is that no matter what number you start with, you always seem to come back to 1 eventually. But nobody knows if it’s true for ALL numbers, and no one can prove it. Try it… even some simple numbers can take a long time, but they all get to one eventually.

The nature of these unsolved problems, especially when their statement is so simple, makes me want to know their answers. Why does this happen? Does it even happen, for sure, always? I need to know why, why multiplying by 3 and adding 1 has this strange behavior.

Incidentally, I tried multiplying by 3 and subtracting 1 and didn’t get anything like this. Only about 34% of the numbers I tried came back to 1 in that case.

If I chose a and n in some sufficiently random way, and built some an+1 sequence in a sensible manner, what is the chance the sequence generated by those choices would return to 1?

Sometime I feel we don’t even understand how multiplication and addition are truly related, at the deepest level.

And no one can satisfy my desire to know. We live with mysteries.

Steve Strogatz explains math for the Times

Link: Steve Strogatz explains math for the Times

I seem to be falling further behind, as my opinion that the public at large is ready for actual mathematics in their lives is adopted by more and more capable people. The latest is Steve Strogatz, who will be explaining math, from simple topics (counting) to, apparently, very complicated (??). I’m curious to see how he manages. His first article generated a very positive result, if the comments are any judge.

How math is written

Link: How math is written

Abstruse Goose, the mathy-est comic there is (even beating out xkcd.com) has this great look at how math proofs are written.

I, meanwhile am in the middle of writing my thesis, and I just discovered a much quicker proof for a result that was pretty long before. So, should I replace the long version with the short, or put them both in? Is it more useful to know that I had a long, twisted version of things before I happened on this slicker path? Or just I just present the pared down version like everyone else tends to?

Planning inspiration judiciously

I have begun teaching two, not one, but two sections of differential equations this quarter, and immediately, the classes are different from each other. In one, the students contribute, respond, emote; in the other, I feel like I’m facing a mute wall. This is natural, of course, and not anything to worry about, but what does occur to me is that what works with one won’t work for the other.

For example, I was thinking of what I might say to the first class about the complex numbers. Just a teaser—I won’t teach them in earnest for a few weeks, when we need them. But still, it’s such a natural question to ask: if we invent complex numbers to have a place for the square root of one (which we call the imaginary number i), what is the square root of i? Such a natural question. And then, I think of Slaughterhouse 5, and the aliens description of our 3-dimensional life as comparable to riding a roller coaster with only a tiny pinhole to see out of. That’s exactly what the real numbers feel like once you’ve gone to the complex. They’re just a tiny slice of the whole picture, and once you know what the whole picture is, and how beautiful it is, you can’t imagine living without that knowledge.

This is what I thought of telling my students. But maybe just the one class. There has to be a naturalness to it, improvising off the script, and I have to talk to each of these classes in the best way for them.

My response to the question: What could be done to make math seem less uncool?

Link: My response to the question: What could be done to make math seem less uncool?

Here are a few suggestions:

1. Take student questions seriously. In my experience, students aren’t born disliking math, and probably everyone is naturally interested in the subject. But once they’re taught that it has no relevance to their lives, and that there are no questions to answer, just insipid “problems” to solve, they learn to hate it. In my experience, the questions students ask are often the most interesting (and historically relevant) anyway. “Is infinity plus one the same as infinity” is actually a deep and mathematically useful question to ask. What does it mean for infinite sets to be “equal” in size? These questions take you interesting places, and if you’re a clever teacher, it’s not hard to take student questions seriously and still cover the curriculum you need to cover.

2. Mention great results and unsolved problems. So many people think that there’s nothing left to do in math. In reality, we know so little it’s shocking. We have a quadratic formula for degree 2 polynomials. We have a cubic formula for degree 3, and a quartic formula for degree 4. Quintic formula? Don’t have one. Ditto for every degree above 5. Is it even possible to find it? No. How do we know? How is it possible to prove anything is impossible without testing all the (infinite) cases?

3. Teach math along with it’s history. Isn’t math more interesting if we learn that the field of probability started with two mathematicians gambling? Or that the person who proved that there can be no quintic (or higher) formula died in a duel, after having written a letter the night before that solved the historic problem and gave birth to three new fields of mathematics?

These are all illustrations of the point which others have already mentioned: if students are invited to have a personal, relevant engagement with a living subject, they’ll find it interesting; if they’re forced to memorize a bunch of rules for no reason that have no usefulness anyway, they’ll find it boring and stupid.

(For more, check out this fluther question about the excellent essay A Mathematician’s Lament.)

Gelfand Passes

Link: Gelfand Passes

The mathematician Israel Gelfand died today. His obituary contrasts the direction of his work—fundamental and tool developing—with his advisor’s, which was more ambitious and difficult to follow. More and more, I’m becoming convinced that spanning fields rather than pushing into the stratosphere of a single one, is the more valuable contribution, if you have the potential to do both.

Though let’s not fool ourselves. Clearly Gelfand was deep in his work. It’s important not to use “broadness” as an excuse for lack of depth.

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!