Monday, October 23, 2006

The probability is rapidly approaching zero

I've been teaching a class for students planning to be elementary and middle school teachers. Recently, we've been learning about probability. It's a very hands-on class; we meet in a room with nice big hexagonal tables instead of standard desks and do lots of activities. There are two big cabinets filled with wonderful manipulatives. (Note for non-math-education sorts: Manipulatives are physical objects used to teach mathematical concepts. They include items like plastic chips or other counters, different shaped tiles and blocks, and a variety of interesting and creative tools people have come up with. They're a lot of fun to play with.)

Naturally, I try to lead students to discover important results, then hilight and review the results. I'm however realizing that conducting experiments in probability with students who are not so good at following directions is often a recipe for disaster.

A typical day consists of me giving them an experiment to conduct and record results from. (Perhaps rolling dice or flipping coins.) I explain and briefly demonstrate what they are going to do, then set them to collect data. When they are done, I smile like the Buddha and ask them what they found.

They tell me.

Then I say, "You got what? How--wait, what exactly did you do? That can't be right..." Then we proceed to go over the procedure until I find the mistake.

Actually, after the first experiment, I had way to much faith in my students. I recorded their results on the board, looked them over, and declared that this was a very unusual outcome. Something you would only expect to see maybe one in a million times, but nonetheless possible. But after all, I explained, that doesn't mean it can never happen. That was when a student asked "Wasn't that how we were supposed to do it?" And that was when I discovered we had not had a one-in-a-million event, but actually what turned out to be closer to a nine-times-out-of-ten event*: They had written down what they thought should happen instead of what actually happened. This tends to produce results that can politely be termed "wonky".

So new problem: What is the probability that my students will understand and follow all of my directions?
* Granted, Terry Pratchett does point out that one-in-a-million chances do come up 9 times out of 10.

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