Date: Fri, 4 Oct 1996 From: Ray Mines Subject: Maybe I have found a break through. One of my students wrote \pi(\pi - 1)(\pi - 2) \dots (\pi - 3 + 1) on the board. So we turned to page 132 of Brualdi and looked at the expression r(r-1) \dots (r - k + 1). I asked them how many factors there are in this expression. Random answers from the class. I asked what factor came before the last one. More random noise. Finally one student said that the term before the end was (r - k + 2), and she was positive she was correct. I asked her how she knew she was correct. "I worked it out for k = 7" was her response. I said that works for 7 but does it work for 1,000,000? More talk some students did not understand the case k = 7. The student explained that carefully. Confusion about the \dots was rampant. Finally, with some help from me, we rewrote the above expression replacing the (r - k + 1) by (r - (k -1)). All at once everyone knew how many factors there are. Next we talked about the change is their position. In the beginning I could say that there were less than k factors or more than k factors and the students would have to go along with what I said because I was the authority. At the end they had the power to argue against me. I had lost my power to them because they understood the notation and could count the number of terms. They were amazed to see that they actually could control the conversation. They did not have to believe what I said. They understood what a proof is. We'll see what happens next week. This exercise works even better in an one-on-one situation. The students like having power over me.