This is one of the hardest days of the year for me. Classes start tomorrow. I'm nervous, although there's nothing in particular to be nervous about. I always claim that being nervous is just a sign of professional pride, but that's probably just rationalization. There's not much I can do to prepare for class. I just need to...wait.
My first day of classes, I always try to have something active for the students to do. I was always bored by professors who came in and read us the syllabus, and I always resented to wasted time when I did it. So I quit doing it years ago.
Tomorrow, I have two classes. In Linear Algebra, I'll set them to work on the "Hippos Problem," which is a nice little problem that I picked up in graduate school. It's a fairly straightforward problem, solvable with high school algebra. The catch is, there are two solutions, and which one you get depends on how you set up the equations. If things go right, I'll have some groups get one solution, and other groups get the other. Then we have to backtrack to figure out why there are two solutions, and whether there might be three or more. The moral of the story: The calculator can do the math, but you have to be smarter than the calculator.
In Modern Algebra, I will have them generate the Cayley tables for some simple permutation groups. (I know, many of you didn't understand that. Bear with me.) What it is is a very brief introduction to the mathematical structure that will take up most of our time in the course. They can do it using manipulatives. They take a little isoceles triangle and twist it around and put it down on top of itself, and see how the corners match up. It's a nice exercise, that I stole from a book by Ellen Maycock Parker.
On Tuesday, I have Elementary Stats. I stole a great opening day exercise from Allen Rossman and Beth Chance. I have stickers made up with some survey questions -- How old are you? Do you prefer Coke or Pepsi? How old do you think the professor is? I stick them to their backs, and they go around the room and answer the questions on each other's backs. Then they have to try to guess what the question on their own back is. It's a lot of fun. It would be quicker, of course, to just hand out a survey, but this gets them up and moving, and provides a little spark of interest.
The common theme, you will notice, is that I didn't come up with any of these exercises myself. I have, from time to time, tried to develop my own in-class activities, but without exception, my best activities were invented by someone else. It's a good thing that I don't have to pay royalties on classroom activities, or my classes would cost me a lot of money.