## Tuesday, September 22, 2009

### The Joy of Tutoring

Artemis* is 8. He arrived for his first tutoring session last week ready to learn more about trigonometry. He doesn't yet do algebra, and a few days ago said "I can't subtract", but trigonometry is what he's enamored of right now. (He can subtract, he just doesn't know how to use the standard algorithm, yet.) He's full of extremes like this. He was reading before he was 2, but had very little control of his body until recently.

For the past year he's been coming to the math salon I host, along with his parents and twin sister. The first time he came, he was so excited he just had to twirl around and get his whole body moving. (He reminds me of myself. When I'm really excited, I just have to wiggle and wag my tail.) He's still excited, but now he can be part of a group of people working together on a math problem. Part of his excitement during our tutoring session was that he got to have me all to himself. He snuggled up next to me on my sofa, and we dove in.

I started with the Pythagorean Theorem. He knew there were hundreds of proofs, but I don't think he'd really walked through one before. The proof I'm most familiar with involves a bit of algebra, and for him that was the complicated part. (Maybe he's ready for lots of heady stuff but not yet algebra? We'll see.) Just now I looked up proofs to try to find the one I used. Didn't get a good link for that one, but here are two I'll show him next Monday, both completely visual: One with the triangles hinged, the other with them sliding.

[In a previous post I mentioned mathematical holes that can cause students grief for years and years, like not learning your times tables in 3rd grade because you were out sick. I was unsure whether I wanted to say that because Artemis and others like him were in the back of my mind somewhere. When a student isn't expected to know things in a particular order, it's not too hard to work around them, and get to them later.]

I showed Artemis a few more basic geometry proofs, like the angles in a triangle adding to 180 degrees. In the middle of our one-hour lesson, he got so excited by it all he just had to move, so he took a 10-minute break on the trampoline. At the end of our lesson, I lent him Geometer's Sketchpad, Who Is Fourier?, and Mathematics: A Human Endeavor. He's been reading the Fourier book since then, and came in this week excited about one of the formulas he saw in it.

$sin(2x)+sin(3x)\neq sin(5x)$ [Ooh, this is my first time doing that. I like it! I used codecogs.]

It seemed to me that he was intrigued by the fact that sine isn't additive. So I played the mystery box (or Guess My Rule) game, where I have a function in mind, and he figures it out by giving me inputs to see what outputs I give him. It gave me a fun way to talk about functions, input, output, domain, range, etc. With each of the functions I used, I then drew a graph, and we looked at whether it would be additive. I talked about it as linearity.

He wanted to think about $y=sin(x)+cos(x)$, so I pulled out my TI calculator. I tried to keep chatting with him, but I found myself saying "Look!" a few times, and belatedly realized he was too entranced by the calculator to do anything else. So we looked at things like $y=sin(2x)+sin(3x)$, which I knew went with what he'd been reading in the Fourier book. I let him borrow the calculator, and later that day his mom went out and bought him one.

Next week he wants to take a walk and find math all around us. Sounds fun to me. (It took me a moment to let go of the notion that we had to do something more industrious.) ;^)

I am like a kid in a candy shop myself, getting to work with someone who loves math so much. It feels like jazz improv, taking his lead and doing a riff on it. Wow! I'll be taking on a few more students in the coming months. I wonder if any of the others will lead me as well as he does, so I can learn more about how to teach by following.

___
*He decided to use the pseudonym Artemis for my blog posts because he likes the Artemis Fowl books.