Page 211 - Contributed Paper Session (CPS) - Volume 8

CPS2254 Dan C.
formal mathematical analysis, was beyond the purview of the middle school
students.

2. Methodology
The main reason “falling raindrops” was used as the context for the task
instead of “falling snowflakes” is because the project initially took place with
12 students in a city Tanzania, and again at a later date with 21 students in a
city in Vietnam (both places where snow was generally unfamiliar). The
students had some basic skills in probability and statistics: For example, they
could make simple graphs of data, and talk about distributions of data in terms
of centers and ranges. Also, they could discuss likelihoods and compute
probabilities for simple one-stage events.
Initially, when presented with the question of “Where might the first
sixteen drops land?” as described in the previous section, students made
marks on a 4 x 4 grid and also wrote down why they held that view. Whole-
group discussion ensued, with student opinions ranging from a more
deterministic approach (i.e. expressing that each of the sixteen tiles should
contain a raindrop in the center of each tile) to more of random approach (i.e.
the raindrops should look like less of a discernible pattern). The nature of the
discussion had similar types of thinking as reported in similar results from
other researchers (Engel & Sedlmeier, 2005; Green, 1982). Some students
wondered how it was possible to make any prediction since “anything can
happen” or “rain can fall anywhere”, while others mused about how factors
like wind might influence the results.
As we transitioned to the question of “How could we model this idea?”,
students were very creative. Among the ideas were finding a way to “splatter”
water over a grid, or other (more viscous) liquids that were easier to record a
single drop. Eventually students turned to other methods like tossing coins,
blocks, and even “confetti” they made from shredded newspaper (the latter
actually gave a strong impression of falling snow). Some students went up to
a 2nd - floor balcony to distribute their “raindrops” (many of which missed the
grid entirely), tossing things out into an alley or hall, and others used the
height of a desk, chair, or simply standing up in a room over a grid. We allowed
for all kinds of different materials and different sizes of grids (as long as they
comprised sixteen squares in a 4 x 4 array), with the only requirement being
that students felt their modelling technique was “as unpredictable as rain”. All
of the physical experimentation was photographed and videotaped for further
reflection.
Once sixteen token “raindrops” had landed somewhere on the 4 x 4 grid
of their choice, we did provide uniform pages of identical 4 x 4 grids on paper
where they could record their results, carefully marking on the recording paper
what their physical model showed. We then hung the recording papers all

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