Intuitive Conceptions of Probability and the Development of Basic Math Skills
The idea of probabilities has been described as a “Janus-faced” concept, which can be thought of either in terms of frequencies or in terms of subjective confidence. This dualism contributes to debates about the nature of human rationality, and therefore
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Abstract The idea of probabilities has been described as a “Janus-faced” concept, which can be thought of either in terms of frequencies or in terms of subjective confidence. This dualism contributes to debates about the nature of human rationality, and therefore the pedagogical assumptions and goals of education. For this reason, the present chapter explores the evidence regarding how quantitative information is intuitively understood in the human mind over the course of elementary school education. Are particular interpretations of probability equally weighted or does one interpretation predominate as mathematical concepts are being acquired? We find multiple, converging lines of evidence that indicate a frequency interpretation of probabilistic information is developmentally primary and privileged. This has implications for mathematics education, even before the introduction of actual probabilities, in areas such as learning fractions and decimals. Educational practices should work to bootstrap from these privileged representations (rather than fight them) and built towards a more inclusive and comprehensive model of probability knowledge. We conclude that a fundamental issue is not just whether students think about probabilities as a frequentist or as a subjectivist, but rather how they recognize when to be one versus the other.
1 The Interpretation of Probabilities Mathematics may be the queen of the sciences, but this queen has a dirty little secret. Once in a while, she sneaks out and cavorts with the common people; arguing, gambling, and getting into fights. When she does this it is called probabilities. Far from the regal courtyards, the mathematics of probabilities developed as a way to evaluate what were good or bad gambles (Gigerenzer et al. 1989). She engaged in ferocious debates about what she was and how she should behave (Gigerenzer 1993). She behaves, in other words, in a very un-queenly manner. In all seriousness, the concept of probabilities in mathematics really does lead a metaphorical double life. On the one hand, there is a conception of probabilG.L. Brase (B) · S. Martinie · C. Castillo-Garsow Kansas State University, Manhattan, KS, 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_10, © Springer Science+Business Media Dordrecht 2014
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ities that involves using past events to statistically predict future outcomes. For example, a “30 % chance of rain” is based on a model of past weather events (It has rained on 3 out of the 10 previous days similar to this one). We will call this the frequentist concept of probabilities because it is, at its core, based on frequencies of past events. On the other hand, there is a conception of probabilities that involves degrees of belief. For example, a “1 % chance that the earth will be destroyed in 10 years” is based on a person’s subjective beliefs, not a history of past events. We will call this the subjectivist concept of pr
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