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Seeing chance: perceptual reasoning as an epistemic resource for grounding compound event spaces

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Abstract

The mathematics subject matter of probability is notoriously challenging, and in particular the content of random compound events. When students analyze experiments, they often omit to discern variations as distinct events, e.g., HT and TH in the case of flipping a pair of coins, and thus infer erroneous predictions. Educators have addressed this conceptual difficulty by engaging students in actual experiments whose outcomes contradict the erroneous predictions. Yet whereas empirical activities per se are crucial for any probability design, because they introduce the pivotal contents of randomness, variance, sample size, and relations among them, empirical activities may not be the unique or best means for students to accept the logic of combinatorial analysis. Instead, learners may avail of their own pre-analytic perceptual judgments of the random generator itself so as to arrive at predictions that agree rather than conflict with mathematical analysis. I support this view first by detailing its philosophical, theoretical, and didactical foundations and then by presenting empirical findings from a design-based research project. Twenty-eight students aged 9–11 participated in tutorial, task-based clinical interviews that utilized an innovative random generator. Their predictions were mathematically correct even though initially they did not discern variations. Students were then led to recognize the formal event space as a semiotic means of objectifying these presymbolic notions. I elaborate on the thesis via micro-ethnographic analysis of key episodes from a paradigmatic case study.

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Notes

  1. Strictly speaking, the physical marbles-scooping experiment is hypergeometric, not binomial, because as each marble is captured by a concavity in the scooper, there is one less of that color in the bin. However, the fairly minute ratio of the sample size (4) to the total number of marbles in the bin (hundreds) enables us to think of this experiment as quasi-binomial and, for all practical effects, as actually binomial. In this paper I do not expand on the computer-based simulations, because my thesis here pertains primarily to students’ perceptual judgments of the random generator itself and, in particular, how students coordinated these judgments with their guided perceptions of the event space. Moreover, my experimental design was such that students engaged the computer activities only after they had made sense of the event space, so that any observations of students grounding the actual outcome distribution in the event space are contaminated by the prior activities. That is, the study was designed as an experimental unit, not a comparison experiment.

  2. Square brackets communicate indexical information with respect to speech referents, which can be gleaned quite unequivocally from the agent’s gestures.

  3. An accompanying video clip of 2′15″ min duration can be viewed online at http://tinyurl.com/dor-tamar.

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Acknowledgments

The original research reported in this paper was conducted with the support of a National Academy of Education/Spencer Postdoctoral Fellowship. I wish to thank the ZDM Editor-in-Chief as well as three anonymous reviewers for very helpful critiques.

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    Dor Abrahamson

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Abrahamson, D. Seeing chance: perceptual reasoning as an epistemic resource for grounding compound event spaces. ZDM Mathematics Education 44, 869–881 (2012). https://doi.org/10.1007/s11858-012-0454-6

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