This is going to be another one of those unbalanced productions: the bulk of
it is another AWM Education column preview, but a recent local event was too
neat to leave unmentioned, so I will start with a brief description of that.
The event in question was a Brown Bag last Tuesday. It was run by Dylan Helliwell and David White -- two graduate students who had a bright idea and are doing bright things with it. Backing up ever so slightly: our department has been awarded an NSF VIGRE (Vertical InteGration of Research and Education) grant, one of whose objectives is to find students who might make good math majors and entice them into the field. Some months ago, Dylan and David saw the film "A Math Life," and realized that just seeing that could have a major impact on a potential mathematician. So they built a project around that idea, including ways to get students to see it, and other good, enticing things to offer them, and VIGRE is going to support that project. At Tuesday's Brown Bag they described their plans and then (after a technological cliff-hanger getting the DVD to do its thing) we all watched "A Math Life." I was planning on burbling a bit about the film, which I have now seen three times and would love to see another three, preferably with friends who regard "mathematician" and "weirdo" as synonymous. Then I discovered a web site where the description is a good deal more substantial than my offerings would be, so I'll give that instead: http://www.cs.dartmouth.edu/~rockmore/mathlife.html.
And now the AWM column: Statistics as a Fundamental Situation
Sometimes a return to something one worked on several years before produces a feeling of "Been there, done that, let's get on." Sometimes (alas!) it even produces a "Why did I ever think that was interesting??" Occasionally, though, returning with a new perspective unveils deeper aspects that had somehow remained out of sight before, and then the resonance can be really exciting. I have just had that experience, and while it is still resonating, I want to write about it -- all the layers at once, rash though the attempt may be.
First, some background: I was introduced to Probability as an undergraduate at Bryn Mawr, and became addicted on the spot. The addiction has taken various forms. First came a doctoral thesis on Stochastic Control Theory. Then, as my interests turned more and more to teaching, came the creation of a course for liberal arts majors which focused mainly on Probability. This led to the gradual discovery of the degree to which Probability requires a different kind of thinking from that to which students are accustomed, and how very challenging that thinking is. As I was discovering that, the K-12 world was in the process of decreeing that the teaching of Probability really ought to begin by Middle School at least - a decree that produced a considerable demand for me to do exactly what I most enjoyed doing: expanding my repertoire of ways to entice people to learn and enjoy elementary Probability. Eventually I added one more ambition: to have people not just enjoy Probability in the classroom, but recognize the ways in which it crops up constantly outside of the classroom, and be able to apply what they have learned from me after they have left my classroom. I'm still working on that one.
Meanwhile, in a different context, I became first intrigued and then deeply
involved with Didactique, a French Mathematics Education research program
founded in the late sixties by Guy Brousseau. The form that the involvement
eventually took was the collaboration with Brousseau himself on a series of
papers, simultaneously in his French and my English. The papers go back to
his early work and describe it from his current perspective, in a way that
clarifies it to people who are not part of the research program (a group
that includes all Anglophones). Well before we finished our first such
paper, we had agreed that the second would be about an experiment in the
teaching of Statistics that he had carried out decades before and been
pining to re-visit. I was delighted with the prospect and thoroughly enjoyed
the project. In due course "An Experiment in the Teaching of Statistics" was
published by the Journal of Mathematical Behavior [volume 20 #3, 2001], and
our collaboration turned to other areas.
The article might have remained in the category of things finished and gently gathering dust, but for an outside circumstance. For a couple of years I have been working on an "Invitation to Didactique" designed to provide English speakers (and readers) with a general introduction to the foundations of the field. While I was cogitating on which basic ideas to choose and how to present them, I had a wonderful conversation with Brousseau, who was feeling quite pleased about a seminar he had just given using the Experiment in Statistics as a basis for a discussion of the concept of Fundamental Situation. Since the area of Didactique on which the "Invitation" centers is the Theory of Situations, my ears went up with a whoosh, and I dived gleefully into his seminar notes. At first I thought I could simply translate the notes and use them directly, but I eventually had to face the fact that they were written to create a discussion amongst a bunch of people who have been working together for years, not to introduce anybody to anything. So I backed off, cleared my head, and revisited the scene, working to cut through to the central ideas that would make interesting connections for my readers. That's when things began to resonate.
The Theory of Situations is based on the hypothesis that a concept is best learned when the learner is put in circumstances in which there is a problem to be solved and the learner must invent the concept in order to solve the problem. Clearly this necessitates careful calibration and a lot of study, because the circumstances must be such that the invention is possible -- the learner must have the appropriate prior knowledge, as well as a sufficiently high level of motivation and self-reliance. Research in Didactique has explored many Situations for many concepts. Some are relatively compact, such as the Situation for Proof I described in the 10/03 Newsletter. The experiment in Statistics was longer, comprising 20 sessions of varying lengths carried out in the fourth grade classroom of Guy Brousseau's wife Nadine. The article owes much to the records she kept and her loving memory for her students' learning process.
Condensing mightily, the Situation centered around three opaque containers (first sacks, later bottles set up to match them), each known to contain 5 marbles, each marble being either white or black. The bottles had transparent caps, so that when they were upended one marble's color could be seen. The class was challenged to determine the contents of each bottle. After fruitless attempts to peer into the bottle, they settled on taking turns flipping the bottle and reporting on the color of the visible marble. This netted the information that each bottle had at least some of each color, but nothing more. Then one of the kids came up with the idea of doing sequences of five flips. After discovering, with some distress, that the same bottle could produce different sequences, they decided to list the sequences for each bottle. As the lists got longer but continued variable, one student came up with the idea of a terminating condition: if one combination appeared twice more than any other, it was the official content of that bottle. This idea was a big hit, and got them through two bottles. The third, though, gave an official winner of 3 whites and 2 blacks, despite the fact that 4 blacks and 1 white had turned up far more often than 4 whites and 1 black. Their concern over that led them to set up four bottles on their own, one with each of the interesting combinations, and start checking how they did on long sequences (no longer in sets of five). Ultimately the teacher brought in a computer that could be set to produce results for really long sequences of flips of a virtual bottle whose content could be varied, and the students came up with an informal but highly functional notion of convergence. They finished with a game where the computer's "bottle" was set up with a hidden distribution of a known number of marbles. Students were given a certain number of tokens with which to buy bunches of samples. When they felt "sure enough" they stopped buying samples and declared what they thought was in the bottle. If they were right, they doubled the number of tokens they had left. If they were wrong, they lost them all.
So that is the Situation, and a charming one it is. Now what does it mean for Brousseau to call it a Fundamental Situation for Statistics? In effect it is a bold claim that the students are creating for themselves not just the specific concepts the leap to the eye -- sampling, the Law of Large Numbers and hypothesis-testing, for instance -- but the central, key knowledge of Statistics itself. That's not a claim to be taken lightly. In fact, it is a claim that has no meaning unless you address the question: what is the key knowledge at the core of Statistics? It was cogitating on that that brought me back into contact with all I have observed in my years of teaching the basics of Probability. What people struggle with, and what those kids in the end had pretty clearly internalized, is the idea of reasoning about uncertainty, accepting the possibility of "knowing" an answer without ever being able absolutely to check it (the teacher never did open the bottles), and accepting that small samples may differ widely in ratio from what their Probability predicts. How many people with a lot of elementary Statistics under their belts still razz the Weather Bureau because some of the time that they say "80% probability of rain" it doesn't rain? And where would Las Vegas be if most people didn't believe that if the roulette wheel has hit red a lot of times it must be black's turn? The class acquired some vital knowledge all right. So I will accept the claim that the Situation is Fundamental, and lay claim myself to having deepened my own understanding in the process of studying it. Now I wonder what the probability is that I will be able to impart any of that new understanding to my students?