Teaching with the Socratic Method

Link: Teaching with the Socratic Method

A professor of mine once said, speaking about his education, “If they had actually taught us anything, we’d all be geniuses!” And indeed, when I talk to people who hate/fear math, a bad teacher is sure to come up in the conversation somewhere.

This conversation with a group of third graders about base 2 is a lovely counterpoint to what many of us experienced in our own educations. Read the question and answer section (at least—the whole thing is good), and see if he doesn’t make you think differently about numbers, and education.

In teaching as in math, there are many options.

Hilbert’s Hotel

Link: Hilbert’s Hotel

I love this Waylay comic about Hilbert’s Hotel. She does a beautiful job of explaining the paradox succinctly, and of tapping into the sense of human (nonmathematician) frustration with the whole deal at the same time. It’s an old favorite of mine.

There’s more on Hilbert’s Hotel at wikipedia. The classic version is: a hotel has an infinite number of rooms, and a person is staying in each room. A new guest arrives and is given a room. How is this possible?

Here’s a harder version: the infinite hotel across the street has a power outage, and sends all their (infinite) guests over to Hilbert’s hotel for the night. How do you give all of them rooms as well?

Sketch of my math dance piece

It’s called “What I was working on today.”

I come out and aggressively challenge the audience:

“I’m Dan, and I’m a math student PhD, and tonight one of you is going to come up here and solve a math problem, because you’re not dumb. You—you look nervous. This is easy—everyone can do this. You—” etc.

Then I step back and relax.

“When I tell people I do math, sometimes I think that’s what they expect me to do. I’ve got news—doing math and being smart have nothing to do with each other. There are a lot of dumb mathematicians in the world and a lot of very smart people who don’t do math at all.

“I’m going to tell you a story… A teacher once gave his class an assignment to keep them busy: add up the numbers 1+2+3+4+5+6+7+8+9 and keep adding until you get to 100. The teacher figured this would be a good way to keep all the kids busy, and the kids set to work, adding 1+2+3+4 and it was a waste of time for all of them. Because they weren’t making their own choices, they were doing it the way the teacher told them. And you know what—it was boring for them, and they all they all got it wrong. But one student made a choice that was his own: instead of adding in the teacher’s order, he thought, what if I add the biggest and smallest numbers together? So he added the 1 to the 100, and got 101, and set it aside. Then he put the 2 and the 99 together and got 101, and set it aside; then he added 3 to 98 and got 101 and put it aside; and added 4 to 97 and got 101 and put it aside; and he realized that if he paired the numbers this way he would always get 101. How many pairs? 100 numbers means 50 pairs, so that’s fifty sets of 101. That’s just a multiplication problem! Fifty times 101. He could do that, and he did: the answer was 5050, and he was the only one to get it right, and in doing so he discovered a beautiful and powerful pattern.

This is what doing math feels like: you make choices and discover beautiful patterns. I can’t give you the full feeling of what it’s like to be a mathematician, but I think it starts with asking a question. Somewhere in there you all have questions about math that haven’t been answered. Today, I want to give you the opportunity to ask them, and if I can, I’ll answer them. My brain is open.”

And then I take questions. After I answer a few, I say “I’m going to show you what I was working on today.” Then I move into a dance where I try to relate physically what my current work looks like and feels like.

Notes from my group: I need to practice more on fielding the questions so I can make ANY question sound really interesting, and take it in cool directions. Second, they liked my dancing, and thought I could expand it from a brief taste to fuller arc.

It’s pretty exciting stuff. Afterwards people came up and talked to me about math in their lives. I need to start getting interviews and lectures on this blog.

A math op-ed, circa 1996

Link: A math op-ed, circa 1996

By Suzanne Sutton. Here’s a quote:

It is among the greatest ironies of education that a subject so graceful and elegant, so able to inspire and bolster confidence, and so useful for living a joyous and effective life, should be presented in a manner that strips it of its substance and glory, and leaves students feeling bludgeoned and inept, convinced they “stink at math”, unaware of its beauty, or their own precious abilities.

Another meeting with my advisor

First of all, the math dance piece went quite well (at least, I had some positive feedback). In part of it I ask for questions from the audience, and I’m not sure I answered them all as well as I could have, but I was able to spiral into and beyond the question “What is the quadratic formula” to explain that evidence of the quadratic formula has been unearthed dating back to 1000 B.C. and beyond, and all of the interesting questions it leads to. More on this particular subject later on.

I met with my advisor today, for the first time in a while. Meeting with him has been uniformly productive of late. He was pleased by my recent little result, and suggested some new avenues to explore. I’m starting to get into a better and better place with my question: I can draw analogies to previous work and have a sense of what should (or might) be true with what I’m working on.

More on this later too.

Math Dance Tonight

I’ll be premiering my math dance piece tonight when my dance group, Stimulate Dance performs. Very exciting. I also get to do my sibling piece, which I know will be okay—it’s a very likeable piece. The math piece feels like it’s a little more ambitious. Can I get people to think about mathematics during an art performance? Can I argue, in essence, that math belongs in an artistic context?

I’ll try my best.

This is actually a fractal picture! There’s a gallery of them here. Gorgeous.

Where does Pac-Man live?

Here’s a question I’ve always liked: what kind of world does Pac-Man on?

We know we live on a sphere, or course, or something close to it. But we often imagine our world on a rectangular map. What are the rules? Well, if you go out the left side, you come back on the right side. If you go to the top or bottom, you arrive a single point (the north or south pole). So following the same logic, what are the rules in Pac-Man’s world? If he goes out the left side or his world, he comes back in the right side. But—if he goes out the top, he comes back in the bottom. Our map represents a sphere. What shape is he living on?

You could also think of it like this: take a stretchy square of paper, glue the left side to the right side and the top to the bottom. What do you get?

It turns out that this shape is of great mathematical interest. See if you can figure it out. If you get stuck, there’s a nice video here.

The nightmare

The other day a friend related a math nightmare. She was in a prison-like compound. People guarded the exits, and they wouldn’t let her out until she solved a calculus problem. She couldn’t ever do it, and her teeth started to fall out.

Apparently this had been a recurring nightmare for her through high school. I wonder how many people have had dreams like this about math. Or any dreams where math played a starring role.