Inconsistencies in Students' Reasoning about Probability.

Subjects were asked to select from among four possible sequences the "most likely" to result from flipping a coin five times. Contrary to the results of Kahneman and Tversky (1972), the majority of subjects (72%) correctly answered that the sequences are equally likely to occur. This result suggests, as does performance on similar NAEP items, that most secondary school and college-age students view successive outcomes of a random process as independent. However, in a follow-up question, subjects were also asked to select the "least likely" result. Only half the subjects who had answered correctly responded again that the sequences were equally likely; the others selected one of the sequences as least likely. This result was replicated in a second study in which 20 subjects were interviewed as they solved the same problems. One account of these logically inconsistent responses is that subjects reason about the two questions from different perspectives. When asked to select the most likely outcome, some believe they are being asked to predict what actually will happen, and give the answer "equally likely" to indicate that all of the sequences are possible. This reasoning has been described by Konold (1989) as an "outcome approach" to uncertainty. This prediction scheme does not fit questions worded in terms of the least likely result, and thus some subjects select an incompatible answer based on "representativeness" (Kahneman & Tversky, 1972). These results suggest that the percentage of secondary school students who understand the concept of independence is much lower than the latest NAEP results would lead us to believe and, more generally, point to the difficulty of assessing conceptual understanding with multiple-choice items.

[1]  C. Konold,et al.  Informal Conceptions of Probability , 1989 .

[2]  D. Krantz,et al.  The effects of statistical training on thinking about everyday problems , 1986, Cognitive Psychology.

[3]  L. McDermott Research on conceptual understanding in mechanics , 1984 .

[4]  D. Krantz,et al.  The use of statistical heuristics in everyday inductive reasoning , 1983 .

[5]  Ruth Beyth-Marom,et al.  A curriculum to improve thinking under uncertainty , 1983 .

[6]  G. A. Miller,et al.  Book Review Nisbett, R. , & Ross, L.Human inference: Strategies and shortcomings of social judgment.Englewood Cliffs, N.J.: Prentice-Hall, 1980. , 1982 .

[7]  L. Ross,et al.  Human Inference: Strategies and Shortcomings of Social Judgment. , 1981 .

[8]  A. Tversky,et al.  The framing of decisions and the psychology of choice. , 1981, Science.

[9]  J. Michael Shaughnessy,et al.  Misconceptions of probability: An experiment with a small-group, activity-based, model building approach to introductory probability at the college level , 1977 .

[10]  A. Tversky,et al.  Judgment under Uncertainty: Heuristics and Biases , 1974, Science.

[11]  A. Tversky,et al.  Subjective Probability: A Judgment of Representativeness , 1972 .

[12]  A. Tversky,et al.  BELIEF IN THE LAW OF SMALL NUMBERS , 1971, Pediatrics.

[13]  I. Lakatos PROOFS AND REFUTATIONS (I)*† , 1963, The British Journal for the Philosophy of Science.

[14]  J. Shaughnessy Research in probability and statistics: Reflections and directions. , 1992 .

[15]  Douglas A. Grouws,et al.  Handbook of research on mathematics teaching and learning , 1992 .

[16]  Clifford Konold,et al.  Understanding Students’ Beliefs About Probability , 1991 .

[17]  Joan Garfield,et al.  Difficulties in Learning Basic Concepts in Probability and Statistics: Implications for Research. , 1988 .

[18]  Catherine A. Brown Secondary School Results for the Fourth NAEP Mathematics Assessment: Discrete Mathematics, Data Organization and Interpretation, Measurement, Number and Operations. , 1988 .

[19]  J. Minstrell Explaining the ’’at rest’’ condition of an object , 1982 .

[20]  B. Fischhoff,et al.  Cognitive Processes and Societal Risk Taking , 1976 .

[21]  A. Tversky,et al.  On the psychology of prediction , 1973 .