Abstract
The purpose of this article is to contribute to the research investigating the use of logical fallacies, in particular the fallacy of composition, to account for normatively incorrect responses given by prospective teachers to relative probability comparisons. Our results respond to certain assumptions made regarding research on relative probability comparisons of coin flip sequences, which have suggested that participants were actually comparing events rather than sequences, and demonstrates that even when presented with events, the majority of respondents still give normatively incorrect responses. As with all research in this area, abductive reasoning is employed to substantiate our claim that the fallacy of composition is the most probable explanation of respondents reasoning.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abrahamson, D. (2009). Orchestrating semiotic leaps from tacit to cultural quantitative reasoning: The case of anticipating experimental outcomes of a quasi-binomial random generator. Cognition and Instruction, 27(3), 175–224.
Batanero, C., & Serrano, L. (1999). The meaning of randomness for secondary school students. Journal for Research in Mathematics Education, 30(5), 558–567.
Batanero, C., Chernoff, E., Engel, J., Lee, H., & Sánchez, E. (2016). Research on teaching and learning probability. ICME-13 Topical Surveys. New York: Springer.
Borovcnik, M., & Bentz, H. (1991). Empirical research in understanding probability. In R. Kapadia & M. Borovcnik (Eds.), Chance encounters: Probability in education (pp. 73–106). Dordrecht, The Netherlands: Kluwer.
Chernoff, E. J. (2009). Sample space partitions: An investigative lens. Journal of Mathematical Behavior, 28(1), 19–29.
Chernoff, E. J. (2012a). Logically fallacious relative likelihood comparisons: The fallacy of composition [Special issue: National year of mathematics]. Experiments in Education, 40(4), 77–84.
Chernoff, E. J. (2012b). Recognizing revisitation of the representativeness heuristic: An analysis of answer key attributes [Themed issue: Probability in reasoning about data and risk]. ZDM Mathematics Education, 44(7), 941–952.
Chernoff, E. J., & Mamolo, A. (2015). Unasked but answered: Comparing the relative probabilities of coin flip sequence attributes. Canadian Journal of Science, Mathematics and Technology Education, 15(2), 186–202.
Chernoff, E. J., & Russell, G. L. (2012). The fallacy of composition: Prospective mathematics teachers’ use of logical fallacies. Canadian Journal of Science, Mathematics and Technology Education, 12(3), 259–271.
Evans, J. S. B. T., & Pollard, P. (1981). Statistical judgment: A further test of the representativeness heuristic. Acta Psychologica, 51, 91–103.
Falk, R. (1981). The perception of randomness. In C. Laborde (Ed.), Proceedings of the Fifth International Conference for the Psychology of Mathematics Education (pp. 222–229). University of Grenoble.
Garfield, J., & Ahlgren, A. (1988). Difficulties in learning basic concepts in probability and statistics: Implications for research. Journal for Research in Mathematics Education, 19(1), 44–63.
Jones, G. A., & Thornton, C. A. (2005). An overview of research into the teaching and learning of probability. In G. A. Jones (Ed.), Exploring probability in school (pp. 65–92). New York, NY: Springer.
Jones, G. A., Langrall, C. W., & Mooney, E. S. (2007). Research in probability: Responding to classroom realities. In F. Lester (Ed.), Handbook of research on mathematics teaching and learning (2nd ed., Vol. 2, pp. 909–955). Greenwich, CT: Information Age Publishing and National Council of Teachers of Mathematics.
Kahnemann, D., & Frederick, S. (2002). Representativeness revisited: Attribute substitution in intuitive judgment. In T. Gilovich, D. Griffin, & D. Kahnemann (Eds.), Heuristics and biases: The psychology of intuitive judgment (pp. 49–81). New York: Cambridge University Press.
Kahneman, D., & Tversky, A. (1972). Subjective probability: A judgement of representativeness. Cognitive Psychology, 3(3), 430–454.
Kahneman, D., Slovic, P., & Tversky, A. (1982). Judgment under uncertainty: Heuristics and biases. Cambridge, England: Cambridge University Press.
Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6(1), 59–98.
Konold, C., Pollatsek, A., Well, A., Lohmeier, J., & Lipson, A. (1993). Inconsistencies in students’ reasoning about probability. Journal for Research in Mathematics Education, 24(5), 392–414.
Kunda, Z., & Nisbett, R. E. (1986). Prediction and partial understanding of the law of large numbers. Journal of Experimental Social Psychology, 22(4), 339–354.
Lecoutre, M. P. (1992). Cognitive models and problem spaces in “purely random” situations. Educational Studies in Mathematics, 23(6), 557–568.
Lipton, P. (1991). Inference to best explanation. New York, NY: Routledge.
Peirce, C. S. (1931). Principles of philosophy. In C. Hartshorne & P. Weiss (Eds.), Collected papers of Charles Sanders Peirce (Vol. 1). Cambridge, MA: Harvard University Press.
Shaughnessy, J. M. (1977). Misconceptions of probability: An experiment with a small-group, activity-based, model building approach to introductory probability at the college level. Educational Studies in Mathematics, 8(3), 295–316.
Shaughnessy, J. M. (1992). Research in probability and statistics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 465–494). New York: Macmillan.
Stohl, H. (2005). Probability in teacher education and development. In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 345–366). New York: Springer.
Tversky, A., & Kahneman, D. (1971). Belief in the law of small numbers. Psychological Bulletin, 76, 105–110.
Tversky, A., & Kahneman, D. (1974). Judgement under uncertainty: Heuristics and biases. Science, 185(4157), 1124–1131.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Chernoff, E.J., Vashchyshyn, I., Neufeld, H. (2018). Comparing the Relative Probabilities of Events. In: Batanero, C., Chernoff, E. (eds) Teaching and Learning Stochastics. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-72871-1_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-72871-1_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72870-4
Online ISBN: 978-3-319-72871-1
eBook Packages: EducationEducation (R0)