# Misconceptions of probability: An experiment with a small-group, activity-based, model building approach to introductory probability at the college level

- 605 Downloads
- 49 Citations

## Summary

The day-by-day observations made by the experimenter during one of the activity-based classes indicate that college students *can* learn to discover some elementary probability models and formulas for themselves while working on probability experiments in small groups. Furthermore, the effects of sample size upon measures of central tendency and variability can be learned by students working on activities such as those developed for the experimental course in this study. Making guesses for the probability of events and checking guesses with a hand-held calculator seems to help college students to be more cautious about probability estimates, and helps to make them aware about some of their own misconceptions about probability. Small-group problem solving, keeping a log of all class work and activities, and investigating the misuses of statistics all appeared to have a positive effect upon college students' attitudes towards mathematics, as indicated in the questionnaires filled out by the subjects in the experimental sections.

The present results of this study support the hypotheses of Kahneman and Tversky [10, 12] which claimed that combinatorially naive college students rely upon availability and representativeness to estimate the likelihood of events. Kahneman and Tversky were skeptical about the possibility of helping students to overcome their reliance upon availability and representativeness. The results on the posttest in this study suggest that the manner in which college students learn probability makes a difference in their ability to overcome misconceptions that arise from availability and representativeness. Mere exposure to probability concepts is not sufficient to overcome certain misconceptions of probability. Fischbein [9] notes that the “synthesis between the necessary and the possible-which is the basis of probabilistic thinking-does not in fact take place spontaneously ....” He claims that science education emphasizes only the deterministic aspect, and neglects the study of uncertainty. Thus peoples' intuition of probabilistic thinking is distorted by science education's emphasis on the necessary, and neglect of the possible. This experiment suggests that the course methodology and the teaching model used in an elementary probability course can help develop peoples' intuition for probabilistic thinking. A course in which students carry out experiments, work through activities to build their own probability models, and discover counting principles for themselves can help students to overcome their misconceptions about probability, and can help restore the synthesis between the necessary and the possible which is essential to probabilistic thinking.

## Keywords

College Student Science Education Probability Model Model Building Probability Estimate## Preview

Unable to display preview. Download preview PDF.

## Bibliography

- [1]Tversky, A. and Kahneman, D., ‘Belief in the Law of Small Numbers’,
*Psychological Bulletin***76**(1971), 105–110.Google Scholar - [2]Shaughnessy, J. M., ‘A Clinical Investigation of College Students’ Reliance Upon the Heuristics of Availability and Representativeness in Estimating the Likelihood of Probabilistic Events’, Unpublished Doctoral Dissertation, (1976).Google Scholar
- [3]
- [4]Cohen, J.,
*Chance, Skill, and Luck: The Psychology of Guessing and Gambling*, Baltimore, Penguin Books, 1960.Google Scholar - [5]Cohen, J. and Hansel, M.,
*Risk and Gambling*, New York, Philosophical Libraries Incorporated, 1956.Google Scholar - [6]Edwards, W., ‘Conservatism in Human Information Processing’, in B., Kleinmutz (ed.),
*Formal Representations of Human Judgment*, New York, Wiley, 1968.Google Scholar - [7]Fischbein, E., Bărbat, I., and Mînzat, I., ‘Intuitions primaires et intuitions second-aires dans l'initiation aux probabilités’,
*Educational Studies in Mathematics***4**(1971), 264–280.Google Scholar - [8]Fischbein, E.,
*The Intuitive Sources of Probabilistic Thinking In Children*, Dordrect-Boston, D. Reidel, 1975.Google Scholar - [9]Fischbein, E., ‘Probabilistic Thinking in Children and Adolescents’, in R. Bechauf (ed.),
*Forschung zum Prozess des Mathematiklernens*, Institut für Didaktik der Mathematik der Universität Bielefeld, 1976, 23–42.Google Scholar - [10]Kahneman, D. and Tversky, A., “Subjective Probability: A Judgment of Representativeness’,
*Cognitive Psychology***3**(1972), 430–454.Google Scholar - [11]Kahneman, D. and Tversky, A., ‘On the Psychology of Prediction’,
*Psychological Review***80**(1973), 237–251.Google Scholar - [12]Kahneman, D. and Tversky, A., ‘Availability: A Heuristic for Judgeing Frequency and Probability’,
*Cognitive Psychology***5**(1973), 207–232.Google Scholar - [13]Kahneman, D. and Tversky, A., ‘Judgement Under Uncertainty: Heuristics and Biases’
*Science***185**(1974), 1124–1131.Google Scholar - [14]
- [15]Mosteller, F., Kruskal, W. H., Link, R. F., Peiters, R. S., and Rising, G. R.,
*Statistics By Example*, Reading, Addison-Wesley, 1973.Google Scholar - [16]Mosteller, F.,
*Fifty Challenging Problems in Probability With Solutions*, Reading, Addison-Wesley, 1962.Google Scholar - [17]Huff, D.,
*How to Lie with Statistics*, London, W. W. Norton, 1954.Google Scholar - [18]Weiss, N. A., and Yoseloff, M. L.,
*Finite Mathematics*, New York, Worth, 1975.Google Scholar - [19]Pollak, H., ‘On Some Problems of Teaching Applications of Mathematics’,
*Educational Studies in Mathematics***1**(1968), 24–30.Google Scholar - [20]Thompson, M., ‘Models, Problems and Applications of Mathematics’, Unpublished Pre-conference paper for a Conference on Topical Resource Books for Mathematics Teachers, Eugene Oregon, 1974.Google Scholar
- [21]Klamkin, M., ‘On the Teaching of Mathematics so at to be Useful’,
*Educational Studies in Mathematics***1**(1968), 126–160.Google Scholar - [22]Fitzgerald, W., ‘The Role of Mathematics in a Comprehensive Problem Solving Curriculum in Secondary Schools’,
*School Science and Mathematics*(1975), 39–47.Google Scholar - [23]Freudenthal, H., ‘Why to Teach Mathematics so as to be Useful’,
*Educational Studies in Mathematics***1**(1968), 3–8.Google Scholar