Unexpected Classroom Dynamics, and a plain linear algebra class

Originally Posted 01-25-2012

when teaching today, one each for dynamics and linear algebra.

First, dynamical systems. I wanted to give the students their first real taste of the interesting behavior that makes a chaotic dynamical system, so I gave them some stuff to do about the doubling map x \mapsto 2x \mod 1. I was trying to set up a conversation about the dangers of using a computer and just trusting the output without thinking about what the computer is really doing, so I asked the students to compute a bunch of orbits both “exactly” and by starting with numerical approximations to their inputs. Specifically, I asked the students to find the nature of the orbits of 1/k for k from 2 to 30. I didn’t expect anything deep to come out of this particular conversation, except maybe the students would notice something about the orbits of rational numbers. But, being well-trained in the mystic arts, some students noticed that there were almost patterns to the periods involved. This blew up my plan for the day in the most wonderful way. The best way to see what happens is to try it yourself and see the apparently simple pattern with just a few oddballs. And the oddballs aren’t some simple to describe phenomenon. So, we have a big task staring us in the face: “What is going on here?” The students asked it themselves, they made a couple of conjectures in class, and they seem pretty motivated. I expect we will be productively sidetracked for a couple of meetings. It is just wonderful.

Second, to follow up my good linear algebra activity, I had a very blah class today. I just couldn’t figure out how it wasn’t meeting my expectations or how to adjust it. I was preoccupied with this during my unattended office hour, and I finally came to a realization: I never set out my expectations. That is, in the flurry of last minute planning, I never took the time to set some basic learning goals for my linear algebra class. This is why I feel aimless. So, tomorrow I am going to spend some time fixing this problem. I think I will share my list directly with the students and ask them which ones they think they have met. We’ll set straight to work on ones they don’t feel are comfortable. I hope it resets and refocuses our study.

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