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.