The Real World as a Trick Question: Mathematical Modeling, Knowledge, and Assessment

UCB Committee on Research: Junior Faculty Research Grant, 2006-7 [$6k]

This research direction, part of the Seeing Chance project, consisted of conducting and analyzing probability-related clinical interviews with college students majoring in statistics to explore issues of intuitive reasoning.

Abrahamson, D. (2007, April). The real world as a trick question: Undergraduate statistics majors’ construction-based modeling of probability. Paper presented at the annual meeting of the American Education Research Association, Chicago, IL.

ABSTRACT: 24 undergraduate/graduate students enrolled in mathematical programs participated in one-to-one interviews as part of a design-based research study of the cognition of probability. The students were asked to estimate outcome distributions of a very simple randomness generator consisting of an exposed bin full of marbles, half green and half blue, and a scooper—a 2-by-2 array of concavities—for drawing out exactly four marbles from the mix. This array formation (4-block) featured also in combinatorial-analysis materials and computer-based simulations of the probability experiment. Central to the design is the combinations tower, an assembly of the 16 unique outcomes in the form of a 1:4:6:4:1 “picto-barchart,” i.e., with the outcomes themselves, not just stark columns as in regular histograms. All students said that the relatively most common experimental outcome should have 2 green and 2 blue marbles, but only 10 students initiated combinatorial analysis as a means of warranting their guess, of whom only 4 conducted it successfully. For all students, the combinations tower constituted a context for coordinating between the sample space of the stochastic device and distributions of actual outcomes in experiments with this device. I argue for the utility of guided, situated problem solving for the learning and consolidation of probability concepts.