But what does it MEAN?

For a variety of reasons, I ended up having a student teacher observe my calculus class today.

I started the class with the following problem (ripped from the NCTM website):

A stack of oranges forms a pyramid with a rectangular base measuring 5 × 8 oranges. Each orange above level 1 rests in a pocket formed by 4 oranges on the level below it. A single row of oranges completes the stack. How many oranges are in the stack?

Before collecting papers, I asked if anyone thought they had the answer, then chose one student, and asked him to share his PROCESS, all the way through to the answer (ie, don't start with the answer, but rather, work up to the answer). He did, and I took notes on the board of what he was saying. So, EVERYONE got a chance to cheat a little, by copying down what I wrote on the board-- but they still had to finish the arithmatic, since I didn't finish out the solution.

After class, the student teacher who was observing got to talking with the other student teacher about that problem, just as a "what did you see in the class you observed." Since I was right there, I pulled out the student responses, and pointed out a few things that the student responses showed:

1. There were far more students responding with the correct response than the student teacher had expected (he figured about half, it was actually 18 of 24).
2. Of the correct responses, only about half showed evidence that convinced me that they really had a solid lock on the problem. The remainder had some sign that they had at least grasped the gist of the explanation given in class (specifically, they copied down the method, and finished the arithmatic).

What was interesting to me was the difficulty I had in relating to the student teachers that I wasn't convinced. In one case, a student had drawn the bottom (8x5) rectangle, labeled the sides thereof, added the 40 circles within it, and even drawn sketched lines of the pyramidal shape of the pile. To the side were the multiplication expressions that I had written on the board, evaluated and summed to find the correct answer. I contended that it was not clear, from that evidence, what the student was really thinking.

Since they seemed unconvinced, I changed gears a little: I contend that it is generally difficult to really understand what a student's conception is. The faculty at Northwestern really conviced me of this, particularly with the (relatively well-known) "Summer/Winter" problem (ask someone to explain why it's cold in the winter and hot in the summer; most people, regardless of education and intelligence, will first latch on to the elliptical orbit of the earth around the sun...)

So, I gave the example of one student who called me over to check his solution. "It's 97." He said. "Close, but you're a little short," I responded. He paused just a few seconds, grinned, and said, "Ah. It's 100."

The question I posed to the student teachers: What was that student thinking?