Page 216 - Contributed Paper Session (CPS) - Volume 8

CPS2254 Dan C.
At this stage in the analysis of students’ written responses, a rubric is being
developed to better affix a quantitative measure to the degree of
improvement in students’ use of variability in their reasoning. For example,
Engel & Sedlmeier (2005) ascribed the “Novice” label to students whose
predictions included between one and three empty squares, and the label of
“Expert” to those between four and eight empty squares. They then structured
numeric scores based on those labels (and also the lower levels of
“Deterministic” and “Moderately Deterministic”). So far, an initial qualitative
examination of responses shows a general increase in student confidence in
making predictions, with a markedly stronger emphasis on trying to balance
variability with expected values.

4. Discussion and Conclusion
Among the surprising results of the project so far has been the new
avenues for questions that came from looking at the data generated by
Fathom. A good example was when students asked the waittime question
“How many trials would we expect before we hit exactly 6 empty squares?”
This question seemed natural enough, given that one student after another
might do a trial and not have that particular result. Or it might happen on the
first try.
Some students did have bit of prior knowledge about an expected value
for wait time as the reciprocal of the underlying probability, although it wasn’t
phrased that way. For instance, they might expect to roll a die six times to hit
a “4”. But again, there is variability to consider. In the context of the “falling
raindrops” task, students could see that six empty squares had a high
likelihood, say 0.342 for example. They then wonder if in fact 1/0.342 ≈ 2.92
might mean that “three trials” ought to be reasonable to hit exactly six empty
squares. We then turned to Fathom to see if that in fact “three” was a
reasonable answer for the above question on wait-time.
Perhaps the most intriguing question had to do with the probability of a
square having a particular nonzero number of raindrops. They surmised a
correlation between “number of empty squares” and “maximum number of
raindrops in a square”, but it turned out to be a challenging question to
determine a specific probability for a given nonzero number of raindrops. For
instance, “What’s the likelihood any given trial will have 6 raindrops as a
maximum on a square?” was a question that arose. Certainly, we could look at
our original experimental data – the paper grids up around the room – and
compute that experimental probability. But getting Fathom to “keep track” of
how many trials had exactly six raindrops on a square (and no more than six)
was complicated for us.
Instead, it was very easy to have Fathom run trials until the number of
raindrops was six or greater. So, we changed the question to “What’s the

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