Secondary and university students' understanding of independence and conditional probability

František Mošna

doi:10.14712/18047106.3013

Vol. 14 No. 1 (2023): Scientia in educatione Research Reports

Secondary and university students' understanding of independence and conditional probability

František Mošna, ¹

Pedagogická fakulta UK

DOI: https://doi.org/10.14712/18047106.3013

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Mošna, F. (2023). Secondary and university students’ understanding of independence and conditional probability . Scientia in Educatione, 14(1), 15-26. https://doi.org/10.14712/18047106.3013

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Abstract

The primary objective of the study was to investigate the perceptions of Czech secondary and university students regarding independence in tasks involving multiple repetitions of random events and their understanding of conditional probability. The study employed a sample of 43 students, ranging in age from 15 to 23, who engaged in think-aloud interviews. The selection of eight tasks was based on existing literature.
A qualitative analysis of the interview transcripts established that students encountered difficulties comprehending the concepts of independence and conditional probability, irrespective of whether they had previously undertaken a university course on probability. Notably, certain misconceptions about independence only surfaced in more challenging tasks, wherein students relied more on their intuition than their acquired knowledge. The misconceptions primarily manifested when describing the random space. The research findings have significant educational implications, which are discussed.

Keywords Independence, conditional probability, intuitive perceptions, teaching of probability.

References

Albert, J.H. (2003). College students’ conceptions of probability. The American Statistician, 57(1), 37–45. https://doi.org/10.1198/0003130031063 Open DOI Search in Google Scholar
Batanero, C. (2014). Probability teaching and learning. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 491–496). Springer. https://doi.org/10.1007/978-94-007-4978-8 128 Open DOI Search in Google Scholar
Batanero, C. (2015). Understanding randomness: Challenges for research and teaching. In K. Krainer, & N. Vondrová (Eds.), Proceedings of the 9th Congress of European Society for Research in Mathematics Education (pp. 34–49). Charles University in Prague, Faculty of Education and ERME. Search in Google Scholar
Batanero, C., & Borovcnik, M. (2016). Statistics and probability in high school. Springer. https://link.springer.com/book/10.1007/978-94-6300-624-8 Open DOI Search in Google Scholar
Batanero, C., & Sanchéz, E. (2005). What is the nature of high school students’ conceptions and misconceptions about probability? In G. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 241–266). Springer. https://doi.org/10.1007/0-387-24530-8 11 Open DOI Search in Google Scholar
Dale, A. I. (1982). Bayes or Laplace? An examination of the origin and early applications of Bayes’ theorem. Archive for History of Exact Sciences, 27(1), 23–47. https://doi.org/10.1007/BF00348352 Open DOI Search in Google Scholar
de Moivre, A. (1756). The doctrine of chances. (3rd ed.). A. Millar. Search in Google Scholar
Díaz, C., & Batanero, C. (2009). University students’ knowledge and biases in conditional probability reasoning. International Electronic Journal of Mathematics Education, 4(3), 131–162. https://doi.org/10.29333/iejme/234 Open DOI Search in Google Scholar
Díaz, C., Batanero, C., & Contreras, J. M. (2010). Teaching independence and conditional probability. Boletin de Estadistica e Investigacion Operativa, 26(2), 149–162. Search in Google Scholar
Díaz, C., & de la Fuente, E. I. (2007). Assessing students’ difficulties with conditional probability and Bayesian reasoning. Journal on Mathematics Education, 2(3), 128–148. https://doi.org/10.29333/iejme/180 Open DOI Search in Google Scholar
Evans, B. (2007). Student attitudes, conceptions, and achievement in introductory undergraduate college statistics. The Mathematics Educator, 17(2), 24–30. Search in Google Scholar
Falk, R. (1989). Inference under uncertainty via conditional probabilities. In R. Morris (Ed.), Studies in mathematics education: The teaching of statistics (Vol. 7, pp. 175–184). UNESCO. Search in Google Scholar
Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children. D. Reidel Search in Google Scholar
Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic: Intuitively based misconceptions. Journal for Research in Mathematics Education, 28(1), 96–105. https://doi.org/10.2307/749665 Open DOI Search in Google Scholar
FEP (2007). Framework Education Programme for Secondary General Education (Grammar Schools). VUP. Search in Google Scholar
Gal, I. (2005). Towards “probability literacy” for all citizens: Building blocks and instructional dilemmas. InG.A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 39–63). Springer. https://doi.org/10.1007/0-387-24530-8 3 Open DOI Search in Google Scholar
Galavotti, M.C. (2017). The interpretation of probability: Still an open issue? Philosophies, 2(3), 20. https://doi.org/10.3390/philosophies2030020 Open DOI Search in Google Scholar
Graham, A. (2006). Developing thinking in statistics. SAGE Publications Inc. Search in Google Scholar
Humphreys, P. (1985). Why propensities cannot be probabilities. The Philosophical Review, 94(4), 557–570. https://doi.org/10.2307/2185246 Open DOI Search in Google Scholar
Jones, G.A., Langrall, C.W., & Mooney, E. S. (2007). Research in probability: Responding to classroom realities. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 909–955), Charlotte. Search in Google Scholar
Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6(1), 59–98. https://doi.org/10.1207/s1532690xci0601 3 Open DOI Search in Google Scholar
Konold, C., Pollatsek, A., Well, A., Lohmeier, J.H., & Lipson, A. (1993). Inconsistencies in students’ reasoning about probability. Journal for Research in Mathematics Education, 24(5), 393–414. https://doi.org/10.2307/749150 Open DOI Search in Google Scholar
Maher, A.A., & Sigley, R. (2014). Task-based interviews in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 579–582). Springer. https://doi.org/10.1007/978-94-007-4978-8 Open DOI Search in Google Scholar
Martignon, L., & Wassner, C. (2002). Teaching decision making and statistical thinking with natural frequencies. In B. Phillips (Ed.), Proceedings of the 6th International Conference on Teaching of Statistics [CD]. International Statistical Institute. Search in Google Scholar
Mccurdy, Ch. S. I. (1996). Humphrey’s paradox and the interpretation of inverse conditional propensities. Synthese, 108(1), 105–125. https://doi.org/10.1007/BF00414007 Open DOI Search in Google Scholar
Mooney, E. S., Langrall, C.W., & Hertel, J.T. (2014). A practitioner perspective on probabilistic thinking models and frameworks. In E. Chernoff, & B. Sriraman (Eds.), Probabilistic thinking. Advances in mathematics education (pp. 495–507). Springer. https://doi.org/10.1007/978-94-007-7155-0 27 Open DOI Search in Google Scholar
Mošna, F. (2022). Pojetí pravděpodobnosti a statistiky ve výuce. [Conceptions of probability and statistics in teaching.] Pedagogická fakulta, Univerzita Karlova. Search in Google Scholar
Nemirovsky, I., Giuliano, M., Pérez, S., Concari, S., Sacerdoti, A., & Alvarez, M. (2009). Students’ conceptions about probability and accuracy. The Montana Mathematics Enthusiast, 6(1–2), 41–46. https://doi.org/10.54870/1551-3440.1132 Open DOI Search in Google Scholar
Piaget, J., & Inhelder, B. (1951). La genése de I’idée de hasard chez I’enfant. Presses Universitaires de France. Search in Google Scholar
Rolka, K., & Bulmer, M. (2005). Picturing student beliefs in statistics. Zentralblatt für Didaktik der Mathematik, 37(5), 412–417. https://doi.org/10.1007/s11858-005-0030-4 Open DOI Search in Google Scholar
Roney, C. (2016). Independence of events, and errors in understanding it. Palgrave Communications, 2, 16050. https://doi.org/10.1057/palcomms.2016.50 Open DOI Search in Google Scholar
Salda˜na, J. (2015). The coding manual for qualitative researchers. Sage. Search in Google Scholar
Tanujaya, B., Prahmana, R.Ch. I., & Mumu, J. (2018). Designing learning activities on conditional probability. Journal of Physics: Conference Series, 1088, 012087. https://doi.org/10.1088/1742-6596/1088/1/012087 Open DOI Search in Google Scholar
Tarr, J. E., & Lannin, J.K. (2005). How can teachers build notions of conditional probability and independence? In G.A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 215–238). Springer. https://doi.org/10.1007/0-387-24530-8 10 Open DOI Search in Google Scholar
Tversky, A., & Kahneman, D. (1971). Belief in the law of small numbers. Psychological Review, 76, 105–110. https://doi.org/10.1037/h0031322 Open DOI Search in Google Scholar

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[1] Albert, J.H. (2003). College students’ conceptions of probability. The American Statistician, 57(1), 37–45. https://doi.org/10.1198/0003130031063 Open DOI Search in Google Scholar

[2] Batanero, C. (2014). Probability teaching and learning. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 491–496). Springer. https://doi.org/10.1007/978-94-007-4978-8 128 Open DOI Search in Google Scholar

[3] Batanero, C. (2015). Understanding randomness: Challenges for research and teaching. In K. Krainer, & N. Vondrová (Eds.), Proceedings of the 9th Congress of European Society for Research in Mathematics Education (pp. 34–49). Charles University in Prague, Faculty of Education and ERME. Search in Google Scholar

[4] Batanero, C., & Borovcnik, M. (2016). Statistics and probability in high school. Springer. https://link.springer.com/book/10.1007/978-94-6300-624-8 Open DOI Search in Google Scholar

[5] Batanero, C., & Sanchéz, E. (2005). What is the nature of high school students’ conceptions and misconceptions about probability? In G. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 241–266). Springer. https://doi.org/10.1007/0-387-24530-8 11 Open DOI Search in Google Scholar

[6] Dale, A. I. (1982). Bayes or Laplace? An examination of the origin and early applications of Bayes’ theorem. Archive for History of Exact Sciences, 27(1), 23–47. https://doi.org/10.1007/BF00348352 Open DOI Search in Google Scholar

[7] de Moivre, A. (1756). The doctrine of chances. (3rd ed.). A. Millar. Search in Google Scholar

[8] Díaz, C., & Batanero, C. (2009). University students’ knowledge and biases in conditional probability reasoning. International Electronic Journal of Mathematics Education, 4(3), 131–162. https://doi.org/10.29333/iejme/234 Open DOI Search in Google Scholar

[9] Díaz, C., Batanero, C., & Contreras, J. M. (2010). Teaching independence and conditional probability. Boletin de Estadistica e Investigacion Operativa, 26(2), 149–162. Search in Google Scholar

[10] Díaz, C., & de la Fuente, E. I. (2007). Assessing students’ difficulties with conditional probability and Bayesian reasoning. Journal on Mathematics Education, 2(3), 128–148. https://doi.org/10.29333/iejme/180 Open DOI Search in Google Scholar

[11] Evans, B. (2007). Student attitudes, conceptions, and achievement in introductory undergraduate college statistics. The Mathematics Educator, 17(2), 24–30. Search in Google Scholar

[12] Falk, R. (1989). Inference under uncertainty via conditional probabilities. In R. Morris (Ed.), Studies in mathematics education: The teaching of statistics (Vol. 7, pp. 175–184). UNESCO. Search in Google Scholar

[13] Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children. D. Reidel Search in Google Scholar

[14] Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic: Intuitively based misconceptions. Journal for Research in Mathematics Education, 28(1), 96–105. https://doi.org/10.2307/749665 Open DOI Search in Google Scholar

[15] FEP (2007). Framework Education Programme for Secondary General Education (Grammar Schools). VUP. Search in Google Scholar

[16] Gal, I. (2005). Towards “probability literacy” for all citizens: Building blocks and instructional dilemmas. InG.A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 39–63). Springer. https://doi.org/10.1007/0-387-24530-8 3 Open DOI Search in Google Scholar

[17] Galavotti, M.C. (2017). The interpretation of probability: Still an open issue? Philosophies, 2(3), 20. https://doi.org/10.3390/philosophies2030020 Open DOI Search in Google Scholar

[18] Graham, A. (2006). Developing thinking in statistics. SAGE Publications Inc. Search in Google Scholar

[19] Humphreys, P. (1985). Why propensities cannot be probabilities. The Philosophical Review, 94(4), 557–570. https://doi.org/10.2307/2185246 Open DOI Search in Google Scholar

[20] Jones, G.A., Langrall, C.W., & Mooney, E. S. (2007). Research in probability: Responding to classroom realities. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 909–955), Charlotte. Search in Google Scholar

[21] Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6(1), 59–98. https://doi.org/10.1207/s1532690xci0601 3 Open DOI Search in Google Scholar

[22] Konold, C., Pollatsek, A., Well, A., Lohmeier, J.H., & Lipson, A. (1993). Inconsistencies in students’ reasoning about probability. Journal for Research in Mathematics Education, 24(5), 393–414. https://doi.org/10.2307/749150 Open DOI Search in Google Scholar

[23] Maher, A.A., & Sigley, R. (2014). Task-based interviews in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 579–582). Springer. https://doi.org/10.1007/978-94-007-4978-8 Open DOI Search in Google Scholar

[24] Martignon, L., & Wassner, C. (2002). Teaching decision making and statistical thinking with natural frequencies. In B. Phillips (Ed.), Proceedings of the 6th International Conference on Teaching of Statistics [CD]. International Statistical Institute. Search in Google Scholar

[25] Mccurdy, Ch. S. I. (1996). Humphrey’s paradox and the interpretation of inverse conditional propensities. Synthese, 108(1), 105–125. https://doi.org/10.1007/BF00414007 Open DOI Search in Google Scholar

[26] Mooney, E. S., Langrall, C.W., & Hertel, J.T. (2014). A practitioner perspective on probabilistic thinking models and frameworks. In E. Chernoff, & B. Sriraman (Eds.), Probabilistic thinking. Advances in mathematics education (pp. 495–507). Springer. https://doi.org/10.1007/978-94-007-7155-0 27 Open DOI Search in Google Scholar

[27] Mošna, F. (2022). Pojetí pravděpodobnosti a statistiky ve výuce. [Conceptions of probability and statistics in teaching.] Pedagogická fakulta, Univerzita Karlova. Search in Google Scholar

[28] Nemirovsky, I., Giuliano, M., Pérez, S., Concari, S., Sacerdoti, A., & Alvarez, M. (2009). Students’ conceptions about probability and accuracy. The Montana Mathematics Enthusiast, 6(1–2), 41–46. https://doi.org/10.54870/1551-3440.1132 Open DOI Search in Google Scholar

[29] Piaget, J., & Inhelder, B. (1951). La genése de I’idée de hasard chez I’enfant. Presses Universitaires de France. Search in Google Scholar

[30] Rolka, K., & Bulmer, M. (2005). Picturing student beliefs in statistics. Zentralblatt für Didaktik der Mathematik, 37(5), 412–417. https://doi.org/10.1007/s11858-005-0030-4 Open DOI Search in Google Scholar

[31] Roney, C. (2016). Independence of events, and errors in understanding it. Palgrave Communications, 2, 16050. https://doi.org/10.1057/palcomms.2016.50 Open DOI Search in Google Scholar

[32] Salda˜na, J. (2015). The coding manual for qualitative researchers. Sage. Search in Google Scholar

[33] Tanujaya, B., Prahmana, R.Ch. I., & Mumu, J. (2018). Designing learning activities on conditional probability. Journal of Physics: Conference Series, 1088, 012087. https://doi.org/10.1088/1742-6596/1088/1/012087 Open DOI Search in Google Scholar

[34] Tarr, J. E., & Lannin, J.K. (2005). How can teachers build notions of conditional probability and independence? In G.A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 215–238). Springer. https://doi.org/10.1007/0-387-24530-8 10 Open DOI Search in Google Scholar

[35] Tversky, A., & Kahneman, D. (1971). Belief in the law of small numbers. Psychological Review, 76, 105–110. https://doi.org/10.1037/h0031322 Open DOI Search in Google Scholar

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