Levels of Probabilistic Reasoning of High School Students About Binomial Problems
In this chapter, some aspects of the process in which students come to know and use the binomial probability formula are described. In the context of a common high school probability and statistics course, a test of eight problems was designed to explore
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bstract In this chapter, some aspects of the process in which students come to know and use the binomial probability formula are described. In the context of a common high school probability and statistics course, a test of eight problems was designed to explore the performance of students in binomial situations. To investigate the influence of instruction to overcome some common cognitive bias or their persistency, the first three problems are formulated in a way that may induce bias. Each one is structurally equivalent to another problem phrased to avoid any bias that was included in the test. Also, the second and third problems were administered before and after the course to assess the changes produced by instruction. A hierarchy of reasoning, designed in a previous study, was adapted and used to classify the answers of the students in different levels of reasoning. The classification of these answers points out that the components of knowledge, the classical definition of probability, the rule of product of probabilities, combinations, and the binomial probability formula, are indicators of transitions between levels. The influence of the phrasing of the problems is strong before instruction, but weak after it.
1 Introduction The binomial distribution is one of the most important discrete probability distributions. It is part of the high school syllabus and also of the first college course of statistics. The probability content of high school statistics courses includes probability distributions of elementary random variables (at least binomial and normal), mathematical expectation and variance. As Jones et al. comment, “The major new conceptual developments at this level are the inclusion of random variable and probability distributions” (Jones et al. 2007, p. 914). On the other side, it has been found that people respond to certain kind of probability problems influenced by some cognitive bias, like representativeness, availability, the illusion of linearity, the supposition of equiprobability, etc. (Fischbein and Schnarch 1997; Batanero and Sánchez 2005). Students respond to several situations E. Sánchez (B) · P.R. Landín Cinvestav-IPN, México, DF, Mexico e-mail: [email protected] E.J. Chernoff, B. Sriraman (eds.), Probabilistic Thinking, Advances in Mathematics Education, DOI 10.1007/978-94-007-7155-0_31, © Springer Science+Business Media Dordrecht 2014
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based on heuristic of representativeness when it is ineffective in the given cases, and that behavior seems to be persistent even after students have learned some techniques of probability which would help them avoid it (Shaughnessy 1992). However, despite the importance to clarify this observation by Shaughnessy, there is no research report that describes and analyzes how students who have learned some techniques of probability contend with problems that may influence them to respond based on cognitive bias. Are the answers of these students similar to those of students who have not studied probability? However, f
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