Fostering Probabilistic Reasoning Away from Fallacies: Natural Information Formats and Interaction between School Levels

Francisco Vargas; Tommaso Benincasa; Giuseppe Cian; Laura Martignon

doi:10.29333/iejme/5716

Full Text (PDF)

Francisco Vargas ¹ ² ^* , Tommaso Benincasa ¹ ³, Giuseppe Cian ³, Laura Martignon ²

More Detail

¹ Universidad el Bosque, COLOMBIA² Ludwigsburg University of Education, GERMANY³ Leonardo da Vinci School, COLOMBIA^* Corresponding Author

Abstract

The article reports an empirical study on the introduction of elementary probabilistic concepts in school, focusing on tasks related to the psychological tradition of heuristics and biases. The concepts involved were studied using an extensional natural frequencies approach. We describe the school intervention conducted in an interaction across different school levels (5th and 9th grades) with the aim of promoting motivation and cooperation thereby strengthening learning. The different tests were assessed both qualitatively (based on argumentation analyses) and quantitatively. The results provide further evidence on the diversity of obstacles tied to probabilistic notions. More importantly, they exhibit an overall improvement in performance of students at both levels. This work confirms the efficacy of natural frequencies in eliciting the intended interpretation of probabilistic tasks and suggests that an appropriate interaction between different scholastic levels can be implemented as a fruitful learning arrangement.

Keywords

probabilistic thinking
natural frequencies
conjunction fallacy
cooperative learning
learning through teaching

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, Volume 14, Issue 2, May 2019, 303-330

https://doi.org/10.29333/iejme/5716

Publication date: 17 Feb 2019

Article Views: 3272

Article Downloads: 2099

Open Access References How to cite this article

References

Brousseau, G. (2006). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970–1990 (Vol. 19). Springer Science Business Media.
Bruner, J. S. (1966). Toward a theory of instruction, Cambridge, Mass.: Belkapp Press.
Chernoff, E., & Sriraman, B. (2014). Probabilistic thinking. AMC, 10, 12. https://doi.org/10.1007/978-94-007-7155-0
Chevallard, Y., & Bosch, M. (2014). Didactic transposition in mathematics education. In Encyclopedia of mathematics education (pp. 170-174). Springer Netherlands. https://doi.org/10.1007/978-94-007-4978-8_48
D’Amore, B. (1999). Elementi di didattica della matematica. Bologna: Pitagora.
Diaz, C., & Batanero, C. (2009). University students’ knowledge and biases in conditional probability reasoning. International Electronic Journal of Mathematics Education, 4(3), 131-162.
Diaz, C., & De La Fuente, I. (2007). Assessing students’ difficulties with conditional probability and Bayesian reasoning. International Electronic Journal of Mathematics Education, 2(3), 128-148.
Duval, R. (2017). Understanding the Mathematical Way of Thinking–The Registers of Semiotic Representations. Springer International Publishing. https://doi.org/10.1007/978-3-319-56910-9
Engel, J., & Sedlmeier, P. (2005). On middle-school students’ comprehension of randomness and chance variability in data. ZDM, 37(3), 168-177.
Falk, R. (1986). Conditional probabilities: insights and difficulties. In Proceedings of the Second International Conference on Teaching Statistics (pp. 292-297).
Fiedler, K. (1988). The dependence of the conjunction fallacy on subtle linguistic factors. Psychological research, 50(2), 123-129. https://doi.org/10.1007/BF00309212
Fischbein, E., & Gazit, A. (1984). Does the teaching of probability improve probabilistic intuitions? Educational studies in mathematics, 15(1), 1-24. https://doi.org/10.1007/BF00380436
Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic, intuitively based misconceptions. Journal for research in mathematics education, 96-105. https://doi.org/10.2307/749665
Galesic, M., Garcia-Retamero, R., & Gigerenzer, G. (2009). Using icon arrays to communicate medical risks: overcoming low numeracy. Health Psychology, 28(2), 210. https://doi.org/10.1037/a0014474
Gigerenzer, G. (2008). Rationality for mortals: How people cope with uncertainty. Oxford University Press.
Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological review, 102(4), 684. https://doi.org/10.1037/0033-295X.102.4.684
Goldin, G. A., Epstein, Y. M., Schorr, R. Y., & Warner, L. B. (2011). Beliefs and engagement structures: Behind the affective dimension of mathematical learning. ZDM, 43(4), 547. https://doi.org/10.1007/s11858-011-0348-z
Gould, S. J. (1992). Bully for Brontosaurus: Reflections in natural history, New York: Norton.
Gras, R., & Totohasina, A. (1995). Conceptions d’élèves sur la notion de probabilité conditionnelle révélées par une méthode d’analyse des données: Implication-similarité-corrélation. Educational Studies in Mathematics, 28(4), 337-363. https://doi.org/10.1007/BF01274078
Grice, H. P. (1975). Logic and conversation, in Cole, P. and Morgan, J. L. (eds.), Syntax and Semantics 3: Speech acts (pp. 41-58), New York: Academic Press.
Harris, J. R. (2011). The nurture assumption: Why children turn out the way they do. Simon and Schuster.
Hertwig, R. (1995). Why Dr. Gould’s homunculus doesn’t think like Dr. Gould: The “conjunction fallacy” reconsidered. Konstanz: Hartung-Gorre.
Hertwig, R., & Gigerenzer, G. (1999). The ‘conjunction fallacy’ revisited: How intelligent inferences look like reasoning errors. Journal of behavioral decision making, 12(4), 275. https://doi.org/10.1002/(SICI)1099-0771(199912)12:4<275::AID-BDM323>3.0.CO;2-M
Hertwig, R., Benz, B., & Krauss, S. (2008). The conjunction fallacy and the many meanings of and. Cognition, 108(3), 740-753. https://doi.org/10.1016/j.cognition.2008.06.008
Hintikka, J. (2004). A fallacious fallacy? Synthese, 140, 25–35. https://doi.org/10.1023/B:SYNT.0000029938.17953.10
Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence: An essay on the construction of formal operational structures. New York: Basic Books. https://doi.org/10.1037/10034-000
Kahneman, D. (2011). Thinking, fast and slow. Macmillan.
Kahneman, D., Slovic, P., & Tversky, A. (1982). Judgement under uncertainty: heuristics and biases. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511809477
Kleiter, G. D. (1994). Natural sampling: Rationality without base rates. In Contributions to mathematical psychology, psychometrics, and methodology (pp. 375-388). Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4308-3_27
Kurz‐Milcke, E., Gigerenzer, G., & Martignon, L. (2008). Transparency in risk communication. Annals of the New York Academy of Sciences, 1128(1), 18-28. https://doi.org/10.1196/annals.1399.004
Martignon, L. (2014). Fostering children’s probabilistic reasoning and first elements of risk evaluation. In Probabilistic Thinking (pp. 149-160). Springer Netherlands. https://doi.org/10.1007/978-94-007-7155-0_9
Martignon, L., & Krauss, S. (2009). Hands-on activities for fourth graders: A tool box for decision-making and reckoning with risk. International Electronic Journal of Mathematics Education, 227–258
Massini, J. (2018). How to foster the coordination of intensional and extensional reasoning in young children Analyzing the Conjunction Fallacy in 4th class., Master thesis im Masterstudiengang (M.A.): Paedagogischen Hochschule Ludwigsburg.
Meder, B., & Gigerenzer, G. (2014). Statistical thinking: no one left behind. Probabilistic Thinking (pp. 127-148). Springer Netherlands. https://doi.org/10.1007/978-94-007-7155-0_8
Multmeier, J. (2012). Representations facilitate Bayesian reasoning: computational facilitation and ecological design revisited (Doctoral dissertation, Freie Universität Berlin).
Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children. Norton.
Saldanha, L., & Liu, Y. (2014). Challenges of developing coherent probabilistic reasoning: rethinking randomness and probability from a stochastic perspective. In Probabilistic Thinking (pp. 367-396). Springer Netherlands. https://doi.org/10.1007/978-94-007-7155-0_20
Slavin, R. E., Hurley, E. A., & Chamberlain, A. (2003). Cooperative learning and achievement: Theory and research. Handbook of psychology, 177-198. https://doi.org/10.1002/0471264385.wei0709
Stenning, K., & van Lambalgen, M. (2008). Human Reasoning and Cognitive Science. Cambridge, MA: MIT Press. https://doi.org/10.7551/mitpress/7964.001.0001
Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. science, 185(4157), 1124-1131.
Tversky, A., & Kahneman, D. (1983): Extensional versus intuitive reasoning: the conjunction fallacy in probability judgment. Psychological Review, 90, 293-315. https://doi.org/10.1037/0033-295X.90.4.293
Wassner, C., Martignon, L., & Biehler, R. (2004). Bayesianisches Denken in der Schule. Unterrichtswissenschaft, 32(1), 58-96.
Zhu, L., & Gigerenzer, G. (2006). Children can solve Bayesian problems: The role of representation in mental computation. Cognition, 98(3), 287-308. https://doi.org/10.1016/j.cognition.2004.12.003

How to cite this article

APA

Vargas, F., Benincasa, T., Cian, G., & Martignon, L. (2019). Fostering Probabilistic Reasoning Away from Fallacies: Natural Information Formats and Interaction between School Levels. International Electronic Journal of Mathematics Education, 14(2), 303-330. https://doi.org/10.29333/iejme/5716

Vancouver

Vargas F, Benincasa T, Cian G, Martignon L. Fostering Probabilistic Reasoning Away from Fallacies: Natural Information Formats and Interaction between School Levels. INT ELECT J MATH ED. 2019;14(2):303-30. https://doi.org/10.29333/iejme/5716

AMA

Vargas F, Benincasa T, Cian G, Martignon L. Fostering Probabilistic Reasoning Away from Fallacies: Natural Information Formats and Interaction between School Levels. INT ELECT J MATH ED. 2019;14(2), 303-330. https://doi.org/10.29333/iejme/5716

Chicago

Vargas, Francisco, Tommaso Benincasa, Giuseppe Cian, and Laura Martignon. "Fostering Probabilistic Reasoning Away from Fallacies: Natural Information Formats and Interaction between School Levels". International Electronic Journal of Mathematics Education 2019 14 no. 2 (2019): 303-330. https://doi.org/10.29333/iejme/5716

Harvard

Vargas, F., Benincasa, T., Cian, G., and Martignon, L. (2019). Fostering Probabilistic Reasoning Away from Fallacies: Natural Information Formats and Interaction between School Levels. International Electronic Journal of Mathematics Education, 14(2), pp. 303-330. https://doi.org/10.29333/iejme/5716

MLA

Vargas, Francisco et al. "Fostering Probabilistic Reasoning Away from Fallacies: Natural Information Formats and Interaction between School Levels". International Electronic Journal of Mathematics Education, vol. 14, no. 2, 2019, pp. 303-330. https://doi.org/10.29333/iejme/5716