A student of mine recently was interested in finding out if a selection technique another teacher uses to choose a random "volunteer" was in fact fair. One of the teachers in my school uses a simple finger game to choose a student who then has to be the first person to do their presentation. Each student puts up between 0 and 3 fingers, the total number of fingers is found, and then the teacher chooses a person (usually someone in the corner of the room) and starts counting out students, until the total number of fingers up is reached. This is supposed to be a random way of selecting a student, my student wanted to verify that this is in fact true.
He did a bunch of research on probability theory, learned about tree diagrams, conditional probability, and a few other more advanced probability techniques, all with the aim of understanding the notion of random selection. He came to me for some help, and with about 30 minutes of discussion, we outlines a method for solution, with the idea in mind that in fact the random selection technique is not fair. It turns out that in the two player version of this game, odd numbers come up more often than even numbers, out of 16 possible outcomes, odd numbers come up 8 times, even numbers 8 times, but one of those even numbers is 0, which if it comes up has to be discarded and the process restarted. If there was agreement that one was going to add one to the final answer, and include 0 as a possible response, perhaps this game would be fair.
He also observed that each possible outcome for each student doesn’t happen with equal frequency because of human selection bias, and moreover that some students will tend to always choose the same number. Given this additional information, it seems clear that this technique is far from random. The student plans on writing up his analysis of this idea and presenting it to the teacher to open a dialogue about his practice.
Students need freedom to explore these kinds of ideas. At the end of the year we will be doing a unit on probability, and I have agreed that if the student writes up an analysis of this problem and submits it, I’ll use that analysis as his assessment for our unit on probability, regardless of whatever other projects I have planned. His project definitely fits within the sphere of probability and is more advanced than what I plan on covering, so why not use it? Clearly he is demonstrating his understanding of the concept. If we provided this amount of flexibility in all of our courses, it seems clear to me that students will benefit.