Math for Love

Mar 7

Confessions of a Converted Lecturer

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.

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Mar 6

A timely comic

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

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Mar 2

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

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Feb 17

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.

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Feb 8

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.

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Feb 2

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.

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Feb 1

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?

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Jan 6

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.

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Dec 10

Massive Multiplayer Mathematics

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Oct 24

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.)

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I'm a mathematician. This is my blog. It's about math, creativity, culture, education, and beauty. It's for everyone.

3n + 1

Planning inspiration judiciously

I'm a mathematician. This is my blog.
It's about math, creativity, culture, education, and beauty.
It's for everyone.