Rethinking Probability Education: Perceptual Judgment as Epistemic Resource
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 outcomes, e.g., HT and TH in the case o
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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 outcomes, 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 pedagogical foundations and then 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. Along the way, I explain the design-based research methodology, highlighting how it enables researchers to spin thwarted predictions into new theory of learning.
A few years ago, I was kicked out of a party. I had told the hostess that I study people’s intuition for probability, and she wanted to hear more. So I did what I usually do in such situations: I asked her the two-kids riddle. Here’s a rough transcription of our dialogue. Dor: I have two kids. One of them is a girl. What’s the sex of my other kid? D. Abrahamson (B) University of California at Berkeley, 4649 Tolman Hall, Berkeley, CA 94720-1670, USA e-mail: [email protected] E.J. Chernoff, B. Sriraman (eds.), Probabilistic Thinking, Advances in Mathematics Education, DOI 10.1007/978-94-007-7155-0_13, © Springer Science+Business Media Dordrecht 2014
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Hostess: I don’t get it. It’s just, like, 50–50—it’s the same chance of getting a boy or a girl. What does your other kid have to do with it? They’re totally independent events! Dor: Indeed they are independent events, but together they make a compound event. There’s double the chance that my other kid is a boy than a girl. Hostess: Well excuse me, but that just doesn’t make any sense at all. How could this possibly be true? And I resent your patronizing tone. Dor: You see, two-kid families are either Girl–Girl, Girl–Boy, Boy–Girl, or Boy–Boy. That’s all the possibilities. Now, we know that one of
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