Yesterday the Maths 1M students handed in an assignment question that asked them to prove a property of triangles using a vector-based argument. It’s not my job to help students do their assignment questions per se, but it is my job to help them learn skills to solve any future problem. This kind of problem, I find, is really hard to learn generalisable skills from.
For most students you really need to be there every step of the way as you try to solve the problem together, so that at the end you can look over what happened and figure out the sorts of things that made it possible to come up with a proof today. Students need to be hear the sorts of general self-questions you ask to help progress your thinking, even if they don’t lead anywhere straight away, and they need to see the dead-end paths you went down only to come back and go a different way.
The big problem is that if you’ve already seen the proof of this 15 times this week, it’s very very easy to guide students down a particular path that they could never possibly think of by themselves first go. It’s very easy to ask specific leading questions rather than general questions that might not lead anywhere. It’s very easy to push them away from the dead-end paths towards something that will give a result more quickly. You want to avoid doing that as much as possible, and the only way I know to do that is to pretend you haven’t seen the solution.
You’re going to have to pretend that you really don’t know how to do it and you really are just figuring it out with them today, and pretend to be surprised that something turned out nicely, and pretend to be frustrated when things don’t. It’s a real art and it takes a lot of practice and a lot of energy to pull it off.
I was very pleased the other day when I did pull it off. I was helping some students with this proof, and I said and did all the right things, including the dead-ends and everything.
After these students were happy with what we’d achieved and had a nice moral about problem-solving to take away, I turned to my other side to help the student who had been sitting there patiently. He had a whole different kind of proof to work on (mathematical induction), and I started as I often do by looking up the definition and writing that down, then saying “Now I’m not sure if this is going to help yet”. He responded to this by saying, “I don’t think I’ll ever believe you again when you say that.”
You see, I had helped him with the geometry proof only a couple of days before, and he had patiently sat there listening to the deja vu of me go through all the same things I went through with him. I looked him in the eye at that point and he said, “That was very impressive.” And he meant it. It’s nice when someone appreciates your craft.