Math for Love

Confessions of a Converted Lecturer

Sun, 07 Mar 2010 21:20:00 -0800

This is an absolutely excellent talk, which I can’t recommend highly enough.

It’s not just that he talks about the problems with lecturing and the benefits of using other methods. What’s phenomenal about this is that he actually collects solid data, and scientifically argues his point. Anyone who’s interested in teaching, and especially those who are interested in teaching math or science should watch this talk.

It does make me consider my own teaching as well. I’ve long been an advocate of using other methods besides lecture, and have found ways to have my students really work stuff out for themselves in my differential equations class, for example. And yet, I haven’t really done as much as I could be doing in this context. I suppose the reason for this is that I feel like there’s no way to go over all the material that I have a curricular mandate to cover without lecturing. Also, the book we use is pretty weak. But now I wonder—surely there’s a better way.

I’m looking forward to the day when I can teach outside the auspices of any curriculum, and let the students’ learning take precedence over every other consideration.

A timely comic

Sat, 06 Mar 2010 00:57:25 -0800

A timely comic:

Given my recent talk and blog on the Collatz conjecture, it’s fun to see xkcd mock it here.

"In mathematics, the art of asking questions is more valuable than solving problems."

Tue, 02 Mar 2010 00:08:32 -0800

“In mathematics, the art of asking questions is more valuable than solving problems.”

- Georg Cantor

data tells stories if you present it right

Wed, 17 Feb 2010 08:37:00 -0800

data tells stories if you present it right:

For those who doubt the power of graphs to tell a story, I just received this graph from Obama’s group, Organizing for America. This is a very nice example of how to highlight the information that you find important, and use data to back up a version of events.

I’d be curious what the political response is to something like this. The conclusion seems indisputable when presented in this form. Unfortunately, we probably won’t see an actual discussion at a high level of context in the press. It’s more convenient for them to respond to data out of context (last month’s job numbers, say), and let everyone have their say.

The classic text on this topic is Tufte’s. He’s worth checking out.

3n + 1

Mon, 08 Feb 2010 02:02:55 -0800

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

Tue, 02 Feb 2010 12:25:22 -0800

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

Mon, 01 Feb 2010 13:55:37 -0800

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

Wed, 06 Jan 2010 16:27:34 -0800

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.

Massive Multiplayer Mathematics

Thu, 10 Dec 2009 11:36:05 -0800

Massive Multiplayer Mathematics

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

Sat, 24 Oct 2009 14:28:16 -0700

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

Thu, 08 Oct 2009 10:39:42 -0700

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

Wed, 09 Sep 2009 18:28:00 -0700

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

Sun, 23 Aug 2009 18:28:51 -0700

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

Tue, 14 Jul 2009 20:06:46 -0700

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

Sun, 07 Jun 2009 20:50:00 -0700

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.]

Teachers struggle in math

Wed, 03 Jun 2009 11:50:41 -0700

Teachers struggle in math:

This article hits on a disturbing (and common) thread: teachers of young children often have very little positive experience with math to pass on.

Math is the lie that makes us realize the truth.

Wed, 20 May 2009 13:05:23 -0700

Math is the lie that makes us realize the truth.:

A great guest column in the NYTimes on the mathematics of scale. That such a simple idea is so applicable always strikes me as profound.

What faith is required in mathematics

Wed, 20 May 2009 12:49:24 -0700

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

Thu, 23 Apr 2009 17:06:00 -0700

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.

Right.

The moment, again

Tue, 21 Apr 2009 16:51:02 -0700

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.