The role of decision making and resources in group solutions of a problem involving Bayes' formula

Gloria Stillman

doi:10.7146/nomad.v10i1.147155

Authors

Gloria Stillman

DOI:

https://doi.org/10.7146/nomad.v10i1.147155

Abstract

This study was concerned with the decision making behaviour of students at the upper secondary level of schooling. Transcripts of verbal reports produced by student pairs solving a probability problem involving Bayes' Formula were analysed using Schoenfeld's protocol parsing scheme. The students' monitoring and decision making were significant contributors to how solutions evolved but so too were particular words in the problem statement. Both the quality of control behaviour and students' resources contributed to the success or otherwise of their solution. Resource-related factors included misinterpretation of the problem, misreading of the problem, failure to make the connections between probability rules and their conditions of use, and confusion between compound and conditional probability.

References

Adibnia, A., & Putt, I. J. (1998). Teaching problem solving to year 6 students: A new approach. Mathematics Education Research Journal, 10(3), 42-58. https://doi.org/10.1007/BF03217057

Barkatsas, A.N., & Hunting, R. (1996). A review of recent research on cognitive, metacognitive and affective aspects of problem solving. Nordisk Matematik Didaktik (Nordic Studies in Mathematics Education), 4(4), 7-30.

Falk, R. (1988). Conditional probabilities: insights and difficulties. In R. Davidson & J. Swift (Eds.), Proceedings of the Second International Conference on Teaching Statistics. Victoria, BC: University of Victoria.

Galbraith, P., Pemberton, M., & Haines, C. (1996). Testing to a purpose: assessing the mathematical knowledge of entering undergraduates. In P. Clarkson (Ed.), Technology in mathematics education, Proceedings of Nineteenth Annual Conference of the Mathematics Education Research Group of Australasia (pp.215-220). Melbourne: MERGA.

Geiger, V., & Galbraith, P. (1998). Developing a diagnostic framework for evaluating student approaches to applied mathematics problems. International Journal of Mathematical Education in Science and Technology, 29 (4), 533-559. https://doi.org/10.1080/0020739980290406

Gigerenzer, G. (1996). On narrow norms and vague heuristics: A reply to Kahneman and Tversky (1996). Psychological Review, 103(3), 592-596. https://doi.org/10.1037/0033-295X.103.3.592

Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102(4), 684- 704. https://doi.org/10.1037/0033-295X.102.4.684

Goos, M., & Galbraith, P. (1996). Do it this way! Metacognitive strategies in collaborative mathematical problem solving. Educational Studies in Mathematics, 30, 229-260. https://doi.org/10.1007/BF00304567

Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: an introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics, (pp.1-27). Hillsdale, NJ: Lawrence Erlbaum.

Ichikawa, S. (1989). The role of isomorphic schematic representation in the comprehension of counter intuitive Bayesian problems. Journal of Mathematical Behaviour, 8, 269-281.

Kahneman, D., & Tversky, A. (1996). On the reality of cognitive illusions. Psychological Review, 103(3), 582-591. https://doi.org/10.1037/0033-295X.103.3.582

Lester, F., Garofalo, J., & Kroll, D. (1989). The role of metacognition in mathematical problem solving: A study of two grade seven classes. Final report to the national Science Foundation of NSF project MDR85-50346.

Nathan, M. J., Kintsch, W. & Young, E. (1992). A theory of algebra-word- problem comprehension and its implications for the design of learning environments. Cognition and Instruction, 9(4), 329-389. https://doi.org/10.1207/s1532690xci0904_2

Peard, R. (1996). Problems with probability. In P. C. Clarkson (Ed.), Proceedings of the 19th Annual Conference of the Mathematics Education Research Group of Australasia: Technology in Mathematics Education (pp. 437- 444). Melbourne: MERGA.

Pollatsek, A., Well, A. D., Konold, C., Hardiman, P., & Cobb, G. (1987). Understanding conditional probabilities. Organizational Behaviour and Human Decision Processes, 40, 255-269. https://doi.org/10.1016/0749-5978(87)90015-X

Schoenfeld, A. (1985a). Making sense of 'out loud' problem solving protocols. The Journal of Mathematical Behaviour, 4, 171-191.

Schoenfeld, A. (1985b). Mathematical problem solving. Orlando, FL: Academic Press.

Schoenfeld, A. (1987). Confessions of an accidental theorist. For the Learning of Mathematics, 7(1), 30-38.

Schoenfeld, A. (1989). Teaching mathematical thinking and problem solving. In L. B. Resnick & B. L. Klopfer (Eds.), Toward the thinking curriculum: Current cognitive research (pp.83-103). Washington, DC: American Society for Curriculum Development.

Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp.334-370). New York: Macmillan.

Scott, N. (1994). Profiles of some non-routine problem solving episodes. In G. Bell, B. Wright, N. Leeson & J. Geake (Eds.), Challenges in Mathematics Education: Constraints on Construction, Proceedings of the Seventeenth Annual Conference of the Mathematics Education Research Group of Australasia, Vol. 2, (pp. 531-538). Lismore: MERGA.

Smith, N., Wood, L. N., Gillies, R., & Perrett, G. (1994). Analysis of student performance in statistics. In G. Bell, B. Wright, N. Leeson, & J. Geake (Eds.), Challenges in Mathematics Education: Constraints on Construction, Proceedings of the 17th Annual Conference of the Mathematics Education Research Group of Australasia, Vol. 2, (pp.539-543). Lismore: MERGA.

Stillman, G. (1998). Task context and applications at the senior secondary level. In C. Kanes, M. Goos, & E. Warren (Eds.), Teaching Mathematics in New Times Proceedings of the Twenty First Annual Conference of the Mathematics Education Research Group of Australasia, Vol. 2, (pp.564-571). Gold Coast: MERGA.

Stillman, G. A., & Galbraith, P. L. (1998). Applying mathematics with real world connections: Metacognitive characteristics of secondary students. Educational Studies in Mathematics, 36(2), 157-195. https://doi.org/10.1023/A:1003246329257

Tversky, A., & Kahneman, D. (1980). Causal schemes in judgements under uncertainty. In Fishbein, M. F. (Ed.), Progress in social psychology. Hillsdale, NJ: Erlbaum.

Walpole, R. E., & Myers, R.H. (1978). Probability and statistics for engineers and scientists (2nd ed.). New York: Collier Macmillan. https://doi.org/10.2307/2530629

Wilson, J., & Clarke, D. (2004). Towards the modelling of mathematical metacognition. Mathematics Education Research Journal, 16(2), 25-48. https://doi.org/10.1007/BF03217394