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?