Induction to Induction

23 June 2006

So we had the choice to try either an inductive lesson, a cooperative groups activity or a folding paper exercise. As much as I like folding paper, I decided to try another inductive lesson.

So, I was teaching solving one and two-step inequalities. We had already covered up through equations with only variables in them, so the solving part wasn't a problem. I just needed the class to realize that when you multiply by a negative on both sides of the inequality the inequality sign flips.

Generally, I set up a comparison. Side by side I had a list of equations: x >1, x>2, x>3 (labeled 1a, 2a, 3a) and their opposites (labeled 1b, 2b, 3b). I set up a table that gave x values and asked the students if each value satisfied the inequality. From those values, I asked the students to help me graph each on the board. By comparing the graphs from the a's and the b's, I asked the class to create a rule that would generalized this property. I asked what was different about the a's and the b's (eventually eeking it out of them that the only difference was that if you multiplied/divided one you'd get the other, but with the inequality flipped). It was really great that many actually did begin to see this rule.

I love this kind of lesson. I feel like the majority of my lessons may not be strictly inductive, but my most comfortable mode of instruction uses the parts of inductive teaching that encourage problem-solving, but, for expediency's sake, provide a bit more direction. Mentor Moe calls it Socratic and I'm not opposed to that label.

Speaking of Mentor Moe, I agreed with his two major comments. On the one hand, it was amazingly satisfying to see two of the kids on the right, much much quieter side of the room actively involved. Not only were they actively involved, they were the engine behind the induction. KR came up with the rule and TW, normally frustrated and somewhat bitter, was positive and engaged. On the flip side, the normally active students were somewhat less active. In fact, I'm sure the whole class did not follow the inductive logic.

Two ways to improve: 1) clearly and repeatedly state the problem that is driving the induction. In addition, from the beginning stress that we are making a rule and continue to come back to both the problem that necessitate the rule and the progress toward the rule itself. It's really not that difference from writing a good paper. Start off with a clear problem and continually link back to that problem in each paragraph. 2) Use an overhead or make a handout. I originally planned a handout, but I wanted to try it without one. But, I think all the erasing I had to do on the board was too cumbersome and wasteful. An overhead (which our classroom inexplicably lacks) would have been perfect. Just throw the template up there and reuse reuse reuse.

I'm planning on having a highly inductive/Socratic classroom. it might be interesting to have a couple of inductive sessions and then some practice workshops to hammer out skills where the class works in groups or alone on just dozens of examples. I think that would be a really interesting way to combine overall concepts, skills, inductive teaching and cooperative learning within a good classroom routine.

I guess we'll find out.