9/22/08: Lesson plans are tough

Last week I mentioned that George is about one day ahead of his lesson plans in Stats, since he's never taught the class before. While this isn't the most convenient thing for me in trying to plan what I want to talk about, after preparing my talks for today I have a whole new respect for just how hard it can be to make a lesson plan. Especially when you have not one class but two different classes each day. I mentioned in my last post that I wanted to try to do two talks, one for Stats and one for Algebra. Stats is about to do normal distributions (aka the bell curve, with the 68-95-99.7 rule), so I knew I could make a talk out of that, but algebra was actually much harder. How do you come up with something interesting for Algebra I?

Stats: The Normal Distribution

The normal distribution is created when you have a series of random events that add together. A simple example, and one we used in class today, was of flipping a coin ten times and counting how many heads you got. We should expect this to follow a normal distribution from 0 to 10 with a mean of 5. I brought a sack of pennies and had each student flip a coin ten times and call out their answers to me, and then I input them into an array in MATLAB and printed out a histogram really quickly. However, there are only about 15 people in each class, counting myself and George. That makes for a pretty poor normal distribution; it's not symmetric, the peaks aren't clear or are in the wrong places, etc. I asked the students what was wrong, why, for example, no one had gotten 7 heads. Is it impossible to get 7 heads out of 10? They knew that of course it's possible, and realized that if we had more coin-flippers we could fill in the data better. Planning for this, I wrote up a script in MATLAB last night (very simple, like 5 lines) to simulate a user-input number of coin flips over a user-input number of trials. In one stroke, I could show them what the distribution looked like with a hundred, or a thousand, or ten thousand coin flippers. Out popped the bell curve we all know and love, still bumpy at a hundred but nice at a thousand and perfect at ten thousand. The kids were pretty impressed with the rapid graphs, which was gratifying.

Then I showed another type of random event that generates a normal distribution, this time the random walk or the "drunkard's walk," as it was explained to me when I first learned about it. Basically, imagine a drunken sailor staggering along the dock. For every step forward, he is equally likely to stagger left or right. If you try to find how far to the left or right of his original point he is after some set number of steps, it follows a normal distribution. I got a lot of laughs demonstrating the stagger, Captain Jack Sparrow-style, down the center of the classroom. I wrote up a MATLAB script for this, too, simulating any number of sailors in a 50-step random walk and outputting the results to a histogram. Again we observed that with only 50 sailors, the distribution isn't clear. At 500 and 5000 it's better, and at 50000 it's almost perfect. For a wow factor I ran a simulation of 5 million random walkers last night, which took about half an hour on my laptop, and saved the resulting histogram. This paid off when one of the girls in class (I need to get better with their names! A personal copy of the seating charts is a must) asked what it looked like with a million tries.

After we did all this, I showed them the usual image of the bell curve with its 68-95-99.7 breakdown, and I think it helped to recognize the shape and what it meant since we'd just seen so many examples of it with our coin flips and random walks. I talked about things in nature that follow this distribution, like particle velocities in the air. I also talked about things we scale onto a bell curve, like test scores in classes and especially standardized tests, like the ACT. After seeing how random events build up a normal distribution, I got a lot of furrowed brows when I had them think about what that meant for using it as an intelligence distribution—trying to say how smart you are as if there were a series of points in your life where you had a random chance of either gaining a chunk of intelligence or not, like the flip of a coin. I finished by linking the topic to Brownian motion, so I explained what that is and showed a pretty sweet video I found online of a physics demo lab device that shows a bunch of tiny ball bearings hitting a hockey-puck sized disc and moving it erratically back and forth.

Algebra: Fun with x, Dividing by Zero and Ramanujan

About a week and a half ago I saw a play called A Disappearing Number at the Power Center, by a theater company called Complicite. The play was based on G.H. Hardy's A Mathematician's Apology, which was the story of his collaboration with the self-taught Indian savant Srinivasa Ramanujan. The play is heavily scattered with mathematical ideas, but the theater company does a remarkable job making the concepts understandable to a largely non-mathematical audience. I decided to use some of Complicite's tricks and some of my own to make my talk for Algebra.

I started with a riddle: Pick a number, and number. Now add five. Now double the number. Subtract four from what remains. Divide by two, and finally subtract your original number. The answer, assuming you did the math correctly, is always three. It's a verbal trip through an algebraic process, but to an untrained audience it is as good as magic. This brought gasps from the audience in the play, and it got some whoops and cries of "no way!" in the algebra class too. I went through the algebra on the overhead, and as we went I explained that this is the power of algebra. I can try these steps for each number individually, but there's always the question of whether there's some number I missed that doesn't give an answer of three. When I choose x as my number instead, it proves the process for any number I can think of, big or small.

With this example of the power of x in hand, we then went from another direction. I went through an old fallacy on the overhead, using x and dividing by zero under the guise of x to prove something nonsensical, that 1 = 2. No one was able to figure out where we went wrong, so I went back to a story from Ramanujan in school, a story of boys and fish that I picked up from the play. If I have ten boys and ten fish, says the teacher, how many fish does each boy get? One, of course. What about a hundred boys and a hundred fish? It's the same. In fact for any number of boys with the same number of fish, each boy always gets one fish – a number divided by itself is always one! Then the clever Ramanujan asks, "What if I have no boys and no fish? Does each boy still get a fish?" Or, is 0 / 0 = 1? This is a really deep question, and I went through some idea of how we could try plotting the function 1/x to get an idea of what happens with division by zero. Ultimately, I explained, mathematicians punted and called it "undefined." I tried to link this paradox for them with the limitations between math and the real world. I can always take things, like boys and fishes, and use mathematics to describe them, just as I can take those random numbers we picked and substitute in x to prove it for all cases. But the reverse, trying to apply something nonphysical, like zero boys each getting a fish, doesn't make sense, the same way that the things we do with x's don't always come out quite right, like in the case of dividing by x=0 to get 1 = 2.

This lecture was a stretch for a lot of the students. The concepts of infinity are really fuzzy at this stage for them, so thinking about plus and minus infinity is a trippy thing. I think I had a hair-blown moment when I told them to think about how, while there are an infinity of numbers 0,1,2,3,4, etc., and an infinity 0,-1,-2,-3,-4, there is also an infinity of numbers between any other two numbers, like between 2 and 3, or 2.2 and 2.3, or 2.22 and 2.23, and so ad infinitum. This had the side effect of leaving some of them pleasantly subdued when George took over. I lost many during the discussion though. This was perhaps an overly ambitious talk for a freshman class, but they seemed to enjoy it better than going over their homework.