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Stochastic misconceptions of pre-service teachers.

Article Excerpt
Abstract

Analyzing explanations from 108 participants on a probability and statistics assessment, this study examined the use of two stochastic misconceptions by pre-service K-8 teachers. Representativeness heuristics were used more often when the problem was set in a social context rather than a traditional mathematical setting. Although some participants used representativeness to guide them to a correct answer, most participants were led astray by this heuristic. The majority of the participants did not realize that a conjunction is less probable than either of its constituent parts.

Introduction

According to Shaughnessy (1981) and others, probability is one of the mathematical topics for which misconceptions are highly likely. Shaughnessy (1992) also postulated that unless we deal with misconceptions of teachers, we cannot expect them to help their students. Therefore, this study examines specific misconceptions involving the representativeness heuristic and the conjunction fallacy of future K-8 teachers with the thought that a better understanding of these misconceptions will guide curricular reforms and lead future teachers to a solid background in probability and statistics.

Misconceptions are systematic, rather than careless, errors. Misconceptions can coexist with competing correct ideas in the minds of students unless they are directly confronted and remediated (Carpenter & Hiebert, 1992). Psychologists are interested in stochastic misconceptions because they "provide information about the processes underlying subjective judgments of probability" and "suggest limitations on the quality of human judgments" (Gavanski & Roskos-Ewoldsen, 1991, p. 181). The work of Amos Tversky and Daniel Kahneman showed that the same heuristics that help people make daily decisions under uncertainty also lead to "severe and systematic errors" that contradict established probability theory (1974, p. 1124).

One example is the representativeness heuristic for which people use the degree to which event A resembles class B to estimate the probability that event A belongs to class B (Tversky & Kahneman, 1974). Additionally, "people expect that a sequence of events generated by a random process will represent the essential characteristics of that process even when the sequence is short" (Tversky & Kahneman, 1974, p. 1125). For instance, the sequence of coin tosses HTHTTH appears to be more random (and thus more likely) than HHHTTT. Similarly, the sequence HHHTTT is perceived to be more likely than HHHHTH because the former appears to represent the fairness of the coin better (Tversky & Kahneman, 1974). However, in reality, all 64 outcomes are equally likely. Fast (1997) gave a similar question to secondary mathematics student teachers and found that a third of them failed to provide the correct answer. Tversky and Kahneman (1974) recognized people's belief that all samples were representative of the population regardless of sample size; specifically, people believed that approximately half the babies born in a hospital (regardless of the number of babies born in that hospital) would be male. This contradicts the law of large numbers which indicates that a large sample should be much more representative of the population (50% male) than a small sample. Fischbein and Schnarch (1997) used the same concept with 500 students from fifth grade through college. Only 1% of their participants chose the correct answer, while over 55% indicated that the probabilities were equivalent in both hospitals (Fischbein & Schnarch, 1997).

Tversky and Kahneman (1983) also explored the conjunction fallacy in which people do not realize that the probability of the conjunction of A and B is less than both the probability of A and the probability of B because the conjunction of A and B is contained in both A and B. A possible explanation for this is that people are judging representativeness rather than probability when this problem is placed in a context (Tversky & Kahneman, 1983). However, the fallacy still occurs (though at a lesser rate) when the problem lacks a context (Gavanski & Roskos-Ewoldsen, 1991). Evidence for the conjunction fallacy has been reported for students in many age groups ranging from second grade to college (Davidson, 1995;...

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