Among the goals of CCML (Creating a Community of Mathematics Learners) are
Some of these goals have been addressed since the project got
under way in autumn, 1996; others are on the agenda for the upcoming
years.
One other goal, however, has a separate status: an element which
is essential for there to be any serious impact from the project
is that its participants share an experience which is intellectually
stimulating, mathematically challenging and pedagogically thought-provoking.
Such an experience cannot be tucked between the folds of an ongoing
teaching year, because it demands far too much focus and energy
of its participants. This, then, became the goal of the summers.
We planned for a one week institute each summer, to be offered
at two different times so as to accommodate the personal schedules
of the participants.
The question of the topics for the two turned out to be easily
resolved. In the course of planning the project we ran a survey
of the teachers which included a request for information about
mathematical areas on which they would most like to work. An impressively
solid majority requested probability and geometry. This was especially
pleasing to us, because they were exactly the areas from which
we had felt the teachers would most benefit (no, we did not
rig the survey!) The two share the property of being important
areas of mathematics which have been drastically under-represented
in the school curriculum for many years and are now being reinserted
into it. Mathematically, this is an excellent development, but
it is pretty rough on teachers in whose own school experience
those topics played a very minor role.
We chose, not quite randomly, to do the probability in 1997 and
the geometry in 1998. Without difficulty, we arrived at the general
structure for each day: mornings were to be spent in completely
homogenized groups (mixed districts, mixed levels, mixed degrees
of confidence), working purely and directly on mathematical issues.
In the afternoons the participants would be grouped in such a
way that people teaching in the same building would be in the
same group, and to a large extent districts would be concentrated.
The work would be taken from middle school curriculum materials,
and each afternoon's work would (as much as possible) deal with
the same mathematical issues as the morning's, but at a middle
school level.
It was in making the next decisions that the planning process
became intellectually stimulating, mathematically challenging
and pedagogically thought-provoking. What about probability made
us all agree that it is valuable and important? What aspects
of it are, in fact, key? What are the distracters that lead people
astray in learning it? How can it be made accessible with no
trace of trivialization? And, as we delved a little deeper, what
is there about it that so frequently inspires a fear that is
close to panic?
Many hours of really interesting discussion later, we agreed that the central topics were
Arching over these was a property of probability which we felt
a compelling need to emphasize: doing probability requires almost
no mathematical hardware . A tremendous number of interesting
and highly applicable problems can be solved bare-handed. Even
when counting arguments are involved, most can be derived on the
spot. A lot of emphasis is needed on this point, because one
of the most common results of a half-understood course is the
impression that probability exists as a corollary of combinatorics,
and anyone not fully conversant with permutations and combinations
might as well not tackle probability at all.
And along with all of these was the need to balance an emphasis
on context (any real life decision has a 90% probability of having
a probabilistic basis) and manageability (it's a lot easier for
the intuition to deal with a flipped coin than a 50% chance of
rain).
From this abundance of challenges and constraints there emerged
the series of days described below. Most were identical for the
two workshops, though the workshop in June had the advantage
of novelty and the one in July that of a few replacements for
bits that just didn't accomplish what they were intended to.
Day 1: We spent the first few minutes on a general discussion of Probability: its meaning, their reaction to it, why it is in the middle school curriculum, where it turns up in life in general and in their lives in particular. Then we launched them into a worksheet with a sequence of questions about pulling marbles out of a bag. The sequence has the property that the first question can be (and has been) done with fourth graders, and the last is a challenge for math majors. By the end of the allotted time, all groups were well into the sequence, and a couple were working away at the last question. The rest of the morning we spent on a sequence of questions about the rolling of two dice (being sure to give them actual dice to solidify their discussions.) The sequence was designed to raise their consciousness about issues of independence and conditional probabilities, though there was no fanfare about either one.
We finished the morning by describing the Monty Hall Game Show
problem and assigning it for homework.
The afternoon activity came from STEM materials. It set up an
intuition-jolting example in the context of drawing for a lucky
bean, then provided both materials to model the situation and
questions to guide discussion of it.
Day 2: Using the context of genetics, we plunged them into really heavy-duty conditional probabilities. The central issue with which they had to grapple was the fact that if one member of a couple has blue eyes and the other brown, then each successive brown-eyed baby that the couple produces increases the probability that the brown-eyed parent is heterozygous, and hence that the next baby will have brown eyes. Understanding this involves dealing with the difference between change in an actual outcome and change in our estimate of its probability, which is a subtle issue indeed. It generated extremely lively discussion in every section. When the time seemed ripe, we followed it up with a physical model of the same situation: each pair of participants was given a paper bag containing two coins, one of them fair and the other two headed. They were to draw one out without examining it, estimate the probability that a flip would turn up heads and then flip it. If it came up tails, they put it back in the bag and started over. If it came up heads, they re-estimated the probability of getting heads on the next flip, then re-flipped. This continued until either tails came up or else five heads in a row did. For many, the genetics situation became a good deal clearer after they carried out this modeling.
We finished the morning with a collection of conditional probability problems.
The afternoon activity from the Connected Mathematics project
involved mazes, and provoked a wide variety of mathematical effort,
ranging from trees to lengthy discussions.
Day 3 was devoted to the Expected Value. Starting with
a problem which asked them to invent the idea, by requesting them
to figure out what might be expected in the long run in a repeated
situation, we then took them through a sequence of problems where
they had to take the basic idea and extend it bit by bit, notably
to situations not involving money.
The afternoon session looked at Expected Value as introduced
in the Connected Mathematics curriculum, with a context of basketball
and a number of nice ways to model it.
Day 4 began with a look at a quite different mathematical aspect-that of theoretical versus experimental probability. We progressed from ordinary dice through loaded dice (foam rubber dice with map pins stuck in them in a recognizable pattern) to "weird dice", which were triangular prisms with various knobs and troughs-decided not regular. Participants had no trouble distinguishing between their types of probability, and the resulting discussion produced a number of excellent insights.
In sections where time permitted, we also presented the "Cereal
Box Problem": if Wheaticrunch cereal boxes offer colored
pens in six different colors, with each box containing a single
pen in a randomly selected color, and if you are determined to
get one of each color, how many boxes of Wheaticrunch do you expect
to have to consume?
The afternoon activities, mostly from the STEM materials, involved
modeling a baseball game using sundry dice, and extended to issues
of theoretical versus experimental probabilities. We also took
the opportunity to bring in some issues of assessment, using as
an example a task suggested in the Mathematics in Context curriculum
which one of our facilitators had kindly offered to carry out
in her classroom before the end of the year, so that we had actual
student responses to look at.
Day 5 extended our studies into the field of Statistics.
It also represented something of a change of pace, in that the
morning's activities were all ones which could be taken directly
back into a classroom. The key concept aimed at was that increasing
sample size improves the accuracy of estimates, and hence leads
more often to correct decisions. Participants carried out experiments
with M&M's, modeled a mark-recapture fish counting process
with goldfish crackers (and were treated to a lively talk by one
of the facilitators about a running of the same experiment with
genuine fish at a local elementary school), and estimated the
percentage of earth covered by water using the Monte Carlo method
and some inflated globes.
The afternoon began with a highly accessible assessment activity
with yet more M&M's. This we followed up with a short general
de-briefing and a final community activity: participants displayed
in the assembly room the T-shirt and/or poster probability designs
they had put together during a couple of allocated chunks of time.
After a little while of munching cookies, unwinding and admiring
each other's work, they gathered for a brief salvo of information
and farewell, then headed out with a well-earned sense of satisfaction
and a feeling (made clear in their written evaluations) that they
came out knowing a lot more probability than they went in with.
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