Last Thursday, I defended my thesis. The process was challenging, in that I have a tendency to be casual with certain details, and in this context I was called to task over each one of these. Most unexpected was being caught about a misplaced minus sign (not what you expect to be caught on in this context). Basically, I only cared if there was an arrow, but a member of my committee wanted to know whether it was pointing to the right or to the left.

It was a surprisingly difficult point.

In any case, I’m finished now, and I’m heading out of town on a brief vacation. Then I’m off to an annual conference in honor of R. L. Moore. Moore pioneered a technique of teaching math sometimes known as the Moore method, which is essentially a classroom setup that involves the students doing all the work: the professor provides a frame (some axioms and statements of theorems) and then the students have to prove virtually everything. It’s a great way to learn math. My math seminars at Swarthmore were usually taught in some variation of the Moore method, and when I got to graduate school I was shocked that we were back to boring old lecture.

Of course, the really hard work of math is always in the actual doing of math: the problem solving, and question asking. The question for the teacher is, how much class time should you spend on telling and showing students how to solve problems, and how much time letting them solve problems. It’s more time consuming to let them solve the problems, so you can “cover” much more material if you just lecture to your students. However, it’s very common to have students in this environment who feel like they understand but can’t really remember how to do stuff on their own. The gap between “knowing it in class” and “knowing it on the test” is vast, alas. Part of this is because we tell the students what to do but don’t devote class time and energy to helping them with the hardest part: doing it on their own. So a different approach which has the students working at the center may cover less material, but the students tend to understand it a lot better.

One variation on the Moore method that I’ve used a lot is the conceptual workshop (discussed in chapter 4 of Teaching with Your Mouth Shut, by my father, Don Finkel), where student groups work through a series of questions, which lead them to bigger ideas. I wrote a workshop for my differential equations class to lead them to invent a method to solve the simplest kind of 2nd order differential equations. In a purer view of the Moore method, you could just let students do all the work, and not break up the ideas into steps at all. Unfortunately, not every student is ready to attack a big problem right away. So the teacher must decide where to step in with a hint. Ideally, the students will be in that perfect blend of success and frustration where they stay engaged with the struggle. Too easy and there’s no point; too hard and students will give up.

It’ll be good to see what others are up to in the world of math teaching.