This Monday, there was no energy in the room. Students were less engaged and responsive, and one was even sleeping. (After class, someone suggested that the Super Bowl might have been part of our problem. I never would have thought of that...) I left class depressed. It doesn't matter how much I like this material, if I can't get the students interested.
On Wednesday, I came in early and asked questions of the students who were there. "Do you like this class?", "What's working for you?", "What isn't?" We usually sit in a circle, but I left them as they were so I could keep asking my questions. As more students arrived, they joined in the conversation. And the desks stayed in rows.
A number of them said they'd like more lecture, so I jumped into the problem we'd been considering on Monday. R's conjecture was that if the are more odd vertices than even in a graph it will be impossible to trace a path over every edge just once, and if there are more even vertices than odd, this will be possible. M had an alternate conjecture, that if there are more than 2 odd vertices, it will be impossible, and with 2 or less it will be possible.
We had an example with 4 odd vertices, so I asked if we could get an example with 3 odd vertices. That turned out to be the perfect question (for that group at that moment). They worked alone, with a partner, or with a small group. They came to the board to show their examples. I said, "This is how mathematicians work. We look for examples, and after enough failure, we try to use our failure to prove it's impossible. If we have trouble with that, we might go back, with new thoughts from the proof attempt, to looking for an example."
So we tried to prove it was impossible. Our attempts gave us plenty of chances to use the language of graphs (edges, vertices, degree), that they are just learning. One student said it might be easier to show that it's impossible to have just 1 odd vertex. I liked that. I said, "One good strategy is to start with a simpler case. Another is to generalize. Maybe we could try to prove that any odd number of odd vertices is impossible."
We weren't getting there quickly, and some students were still looking for an example. S came to the board to show his example. It turned out that he'd counted the edges wrong at some of the vertices. He said sorry, and I talked about how we want to celebrate our mistakes. He came up to the board later, so at least he's willing to take the risk, although he might not be celebrating yet.
M made up a new notation for keeping track of how many even and odd vertices we had. I loved that.
V said something like "We have to notice what doesn't matter." (I think his wording was better than this, but I don't have it now...) I loved what he said, wrote it on the board, and asked people to put it in their notes. If they're going to take notes, I want their notes to be on problem-solving strategies as much as on the content.
Another student, maybe R, suggested adding up the vertex degrees. There was some discussion, and it seemed like they'd pulled together everything they needed. So I said I thought it might be time to write down what they'd come up with. I did that, with lots of questions to them, and lots of unsolicited suggestions from them. We struggled with defining degree of a vertex.
We had an example on the board with a loop, so defining degree as 'the number of edges attached to that vertex' seemed wrong. (The lower vertex in this drawing has degree 3, but only has two edges attached to it.) I meant to check our text, but didn't manage that during class. Her definition is no better. We settled on: The degree of a vertex is the number of edge endings at that vertex. I loved that we were dealing with the issue of why definitions need to be precise.
I then wrote what I thought I had heard from a student: The total degree is the sum of all the vertex degrees. But I see now that I should have gotten this step from the students. I think a bunch of them suddenly saw it after I wrote that. It would have been better if I'd asked the student who had offered it to help me with the wording.
This whole process took over an hour. If I had done more of the 'work', it would have used up 5-10 minutes of class time, and the students would have believed they understood it. It would have seemed pretty obvious to most of them. And yet, they wouldn't have learned nearly as much.
Here's our proof:
I thought after class about Bob Kaplan's list of ways he attempts to 'disappear'. Yesterday I felt like I achieved that to some degree. I mentioned that after class to a student, and he said students would have talked longer if I hadn't stood up. Yep, I still have a ways to go.
And yet, it was standing up at the beginning of class (instead of sitting in the circle with them) that got the energy level up. Interesting...