Wednesday, February 09, 2005


According to Piaget (as quoted here), learning probability requires formal operational thinking. Only 35% of high school graduates reason at the formal operational level. What is the probability that half of the 26 students in my pre-service teacher class reason at the formal operational level? (You may assume that my students are randomly selected from the high school graduates in my state -- a flawed assumption.)

(Assuming I did the Bernoulli trial correctly -- the answer is 0.27. Edited: I was off by a factor of 6 when I computed 26-choose-13 by hand; the correct probability that exactly half of my students are at this level is 0.045. It is left as an exercise to the reader to compute P(at least half) are at this level.)

Today I'm giving a test on probability to the pre-service teachers, and they will do badly. I will need to buy a box of Kleenex to deal with this -- that's how badly they're going to do. Based on the only secondary-methods book that I've read, when you know that all your students are going to do badly, then you shouldn't give the test. But I have to for two reasons: First, I said on the syllabus that there would be a test on February 9, and in my department, the syllabus trumps all. Secondly, I need to send a strong message to (most of) my students that the same coasting techniques that have allowed them to get Cs in previous math classes are not going to be effective here. That they need to work hard each and every day and devote this semester to learning math.

Several students are not doing the homework. I do homework spot-checks (my theory being that this population should be responsible enough to do their homework with only small reminders from me), and many of them either don't hand in the sample problems or else copy answers from the back of the book. I've had three students come to my office hours for the first time this week -- just a day or two before the test. Some of them have only thought about math/probability during the eleven 50-minute class periods we've had so far. (Yes, after eleven classes, I'm pretty sure that they still can't do the suite of flip-three-coin and roll-two-dice problems.)

This is a recipe for disaster when you realize that probability requires formal operational thinking.

Further compounding the problem, while I know what the problem is, I don't know how to best teach my class to address it. I've had my students work on difficult problems in class. They hated it. Hated, hated, hated it. They would rather watch me solve difficult problems. I've done some problems at the board. ("I can understand the problems when you do them, but I can't do them on the homework.") Really what they want is a secret formula: if a problem is about choosing marbles from a bag, then you always do this; if it's about rolling dice, then you always do that. Everyone who can do these hard problems must know the secret formula (how else could we do the problems?) -- and we're not telling!

So I've been unable to communicate that these problems are different -- that the usual strategies of memorizing and naive pattern matching are ineffective. And I haven't motivated them to change their ways and grapple with the problems thoughtfully instead of flailing about seeking The Answer. I'm morally obligated to give an exam that tests the material at the level expected by the course (200-level).

And today they are going to fail.