Exponents and the Scale of the Universe – a 21st Century math lesson

The other day in Algebra 1, I was talking about exponents and their connection to orders of magnitude and fractal structure, which I see as the fundamental concept here. Consider the powers of 3. Naturally, 3 is a triangle. Then 9 is simply a triangle of triangles – a “meta-triangle.” By the same thinking, 27 is a triangle of triangles of triangles. If you’re thinking like we were, maybe you’d call 27 a triangle of meta-triangles. Or is it a meta-triangle of triangles? Amazingly, it’s both!!!

Powers of Three

(Is it fair to call 27 a meta-meta-triangle?  Math up these comments.)

Anyhow, my students made the connection to Sierpinski’s triangle, and the whole “meta” conversation brought up Inception, video feedback, and other cultural references with which they are familiar, so the whole thing really connected to things they know. The exponent tells you how many levels deep the “inception” goes. Of course I talked about how 3^4=1x3x3x3x3, but these patterns are too rich to leave boiled down into something that simple.I hope you don’t think that’s too abstract or confusing for 8th graders.  These ideas are somewhat mind-boggling or perplexing, as Dan Meyer might say. They’re not impossibly difficult. They’re just weirdly fascinating and challenging. Ideas like that (and damn good problems) draw learners into the math world. The point here is that we had a chance to struggle with instantiated concepts, and by wrestling with things like meta-meta-triangles and by arguing about them, we did some math together. (Opinion: A mathematical conversation is math. Agree?)

If I tried to wrap the material up into a perfect little package that could be quickly delivered and easily understood or memorized, then there wouldn’t be any mathematics left at all.

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The base ten number system is built on exactly these ideas, and scientific notation exploits that beautifully, using the exponent to indicate the level of depth. This was where I hoped to lead the class, and I had some (slightly dry) worksheets at the ready, so I handed them out.

While they were working, I pulled up an AMAZING visualization called Scale of the Universe 2. (If you’ve never seen this before, I wouldn’t blame you if you stopped reading and just played with that for 40 minutes.) I was happy to report that it was designed and programmed by some high school kids in California, but I was even happier that the room started to fill with energy. I narrated our little viewing a bit, but I could hardly get a word in between their questions! Their questions. A marathon is that big? How many central parks is Angel Falls? What’s “total human height?”

I answered a few, which was almost too much fun, and then we started researching. We pulled up wikipedia and google and found the most amazing information. Even I learned an incredible amount, and for the rest of the period we were tapped into the magic of curiosity and learning in the information age. Sometimes we would do computations in our head, sometimes on the board, and sometimes we would just pull up WolframAlpha and had the computer solve it for us.  Arithmetic, paper, and pencil have their place, no doubt, but using a great tool isn’t a sin.  It’s a virtue.  Especially today, with the most incredible tools readily available.

I didn’t care that we stopped working on the scientific notation problems. We were using scientific notation right then to understand our world. It was OK that the sheets didn’t get filled in. We could do that later. This was going to leave a mark. I went next door, grabbed 10 netbooks, and put them in their hands. “Keep going. Whatever questions you have, answer them.”

I did it again with my 10th graders and again with my 5th graders. Amazingly, the Algebra 2 kids would ask me questions and not even realize they were staring at the screen of the most powerful information tool ever available. “Google that!” All of a sudden this was a lesson in using the internet. Whatever you find, whatever questions you answer, record it. Tomorrow we’ll share. “Where?  You could use Google Docs…”

If you believe that school is where we equip students for the world, is there any doubt they need these skills and tools in their kit?

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I’m telling this story for three reasons:

i) That’s cool math up there. Exponents and fractals are amazing, and they actually connect to my recent post, quite possibly the coolest problem I’ve ever come up with.

ii) To show how computers can help us meet the enormous demands of true interest. 100 years ago, when our school system was designed, I think this was literally impossible. Schools today, however, have an amazing opportunity if they can shift their mindset.

iii) I love when a class spins on a dime in the most unpredictable ways. It tells my students are awake. It tells me they are conscious and active, processing and guiding our work. I love it. I think of the transfer of power (not just information and technique) as a central goal of my teaching, so when stuff like Scale of the Universe 2 falls into my lap, my job can get very easy.

I’m so thankful to be at a school where we can follow our whims and simply study whatever is most fascinating.

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The “Lost in Recursion” Recursion

Math and Girls, Y’all!

Google loves women.

News to many (mostly delivered by Google’s home page), but today is International Women’s Day!

Fun fact: On March 8, 1917, Russian demonstrations marking International Women’s Day initiated the February Revolution! (Thanks to Bill Everdell, with whom I share my classroom, for that little tidbit.) More would know the holiday if we lived in Russia, of course. Even places from Afghanistan to Zambia have made this a national holiday.

Well, I’ve been thinking about gender equality for a long while, especially with regards to math, so I thought I’d share some thoughts.

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I would say we have a real problem with girls and math. I won’t site studies, but let me share some observations:

Graduate mathematics students are overwhelmingly male. This term, for instance, we have four women in Complex Analysis. Algebraic Topology has none.

Top scorers on math contests are routinely male. Why?

I teach a class of 8th graders that segregate themselves by gender every single day, without so much as a word about it. What is that?

At Saint Ann’s I created a course called “Algebra 2: Functions and Abstract Algebra.” In its first two years, only three girls have taken the class. Another two dropped the first day. Am I the problem?

We also now have a fine spread of one semester math electives for high schoolers. They’re buried here, but we offer incredible courses like Intro Topology, Non-Euclidean Geometry, Fractals and Chaos, and The Complex Plane. And yet, registrants are overwhelmingly male. My Complex Plane course hadn’t a single girl in it!

Even our incredible Mathematical Art Seminar (a group of more than 20 students) has only four girls!!! What is going on?

This is killing me.

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Especially, since I also teach 5th grade, where it is plain to see that my students are equally apt. Few will disagree that boys and girls look and behave differently (how else could we tell them apart?), but I don’t see even the slightest tendency to mathematical weakness in my female students. Some of the most delightfully playful, thoughtful, and powerfully-minded 11 year-old mathematicians I know are girls.

What gives? “What happens to girls in math class?” (I was asked that by a parent of two incredible girls this year.)

I don’t know…. I don’t have the answers…. I can’t fix the problem, on my own, but I’m sure that consciousness is the first step. (It almost always is.)

So I think about it everyday. How do I treat girls differently? Do I call on them as much? Do I expect the same from them? Do I talk to them the same way? Do I look at them the same way? Do I fear creeping them out? Does that differently shape the male-female teacher-student relationship? Is it not possible to have the same relationships with female students as I have with males?

It’s very easy for me question my actions, but extremely hard to know what it’s like for my female students, or even my male ones for that matter. I simply stay conscious and make every attempt, big or small, to encourage female mathematicians. We’ve made extra effort on Math Munch, for instance, to include stories about females in mathematics. Check out this and this.

The math question is only part of a much larger set of societal issues. As the spring clothes come out, think about what’s going on. What do we expect of our girls?

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A turning point for me came when I read Douglas Hofstadter’s incredible tome, Metamagical Themas. I greatly encourage you to read the section called “A Person Paper on Purity in Language.” In it, Hofstadter argues against our gender-based language habits, and by analogy (as usual) to racial language, reduces it to absurdity. Let me reiterate; This paper is totally worth reading. It’s had a great influence on me.

It’s the reason I feel completely awkward every time I hear or say, “you guys.” Sometimes I almost can’t stop myself, even when I’m speaking to a group of all girls. What is that about!? They’re not guys. Five years ago, I would have said, “it’s fine. Who cares? Even girls do it,” but now it actually makes me cringe.

And so, I have made a very small change.

I say “y’all.”

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My mom’s side of the family is from Tennessee and Kentucky, so I’ve heard y’all plenty of times. Younger me found y’all entirely repugnant. The plural “yous” is also an option, but unfortunately both often carry low-class connotations.

Nonetheless, I say y’all.

Saying it means confronting and denying a strange male-default. I take some pleasure in sounding a bit more like personal hero, Ben Folds, but I keep on, because I believe it’s right. In any case, it certainly sounds awkward at times, especially in private school NYC, but I couldn’t care less.

The more I say it, the more naturally it flows. Better yet, every time someone comments or questions me, I have the perfect opportunity to discuss gender equality!

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I’d LOVE to hear from readers about what you do to take on these issues. Please comment!

In any case, thanks for reading, y’all!!! Happy Women’s Day!