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Visualizing Chance: Tackling Conditional Probability Misconceptions

  • Stephanie BudgettEmail author
  • Maxine Pfannkuch
Chapter
Part of the ICME-13 Monographs book series (ICME13Mo)

Abstract

Probabilistic reasoning is essential for operating sensibly and optimally in the 21st century. However, research suggests that students have many difficulties in understanding conditional probabilities and that Bayesian-type problems are replete with misconceptions such as the base rate fallacy and confusion of the inverse. Using a dynamic pachinkogram, a visual representation of the traditional probability tree, we explore six undergraduate probability students’ reasoning processes as they interact with this tool. Initial findings suggest that in simulating a screening situation, the ability to vary the branch widths of the pachinkogram may have the potential to convey the impact of the base rate. Furthermore, we conjecture that the representation afforded by the pachinkogram may help to clarify the distinction between probabilities with inverted conditions.

Keywords

Bayesian-type problems Conditional probability Dynamic visualizations 

Notes

Acknowledgements

This work is supported by a grant from the Teaching and Learning Research Initiative (http://www.tlri.org.nz/).

References

  1. Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215–241.CrossRefGoogle Scholar
  2. Arnold, P., Pfannkuch, M., Wild, C., Regan, M., & Budgett, S. (2011). Enhancing students’ inferential reasoning: From hands-on to “movies”. Journal of Statistics Education, 19(2), 1–32. Retrieved from http://www.amstat.org/publications/jse/v19n2/pfannkuch.pdf.
  3. Bakker, A. (2004). Reasoning about shape as a pattern in variability. Statistics Education Research Journal, 3(2), 64–83.Google Scholar
  4. Bar-Hillel, M. (1980). The base rate fallacy in probability judgments. Acta Psychologica, 44, 211–233.CrossRefGoogle Scholar
  5. Batanero, C., Chernoff, E., Engel, J., Lee, H., & Sánchez, E. (2016). Research on teaching and learning probability. In Proceedings of Topic Study Group 14 at the 13th International Conference on Mathematics Education (ICME), Hamburg, Germany (pp. 1–33).  https://doi.org/10.1007/978-3-319-31625-3_1.Google Scholar
  6. Bea, W. (1995). Stochastisches denken [Statistical reasoning]. Frankfurt am Main, Germany: Peter Lang.Google Scholar
  7. Biehler, R. (1991). Computers in probability education. In R. Kapadia & M. Borovnick (Eds.), Chance encounters: Probability in education (pp. 169–211). Boston, MA: Kluwer Academic Publishers.CrossRefGoogle Scholar
  8. Binder, K., Krauss, S., & Bruckmaier, G. (2015). Effects of visualizing statistical information—An empirical study on tree diagrams and 2 × 2 tables. Frontiers in Psychology, 6(1186).  https://doi.org/10.3389/fpsyg.2015.01186.
  9. Böcherer-Linder, K., Eichler, A., & Vogel, M. (2016). The impact of visualization on understanding conditional probabilities. In Proceedings of the 13th International Congress on Mathematical Education,Hamburg (pp. 1–4). Retrieved from http://iase-web.org/documents/papers/icme13/ICME13_S1_Boechererlinder.pdf.
  10. Borovnick, M. (2011). Strengthening the role of probability within statistics curricula. In C. Batanero, G. Burrill, & C. Reading (Eds.), Teaching statistics in school mathematics—Challenges for teaching and teacher education: A joint ICMI/IASE study: The 18th ICMI study (pp. 71–83). New York, NY: Springer.Google Scholar
  11. Brase, G. L. (2009). Pictorial representations in statistical reasoning. Applied Cognitive Psychology, 23, 369–381.  https://doi.org/10.1002/acp.1460.CrossRefGoogle Scholar
  12. Brase, G. L. (2014). The power of representation and interpretation: Doubling statistical reasoning performance with icons and frequentist interpretation of ambiguous numbers. Journal of Cognitive Psychology, 26(1), 81–97.  https://doi.org/10.1080/20445911.2013.861840.CrossRefGoogle Scholar
  13. Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101.CrossRefGoogle Scholar
  14. Budgett, S., Pfannkuch, M., Regan, M., & Wild, C. J. (2013). Dynamic visualizations and the randomization test. Technology Innovations in Statistics Education, 7(2), 1–21. Retrieved from http://escholarship.org/uc/item/9dg6h7wb.
  15. Chernoff, E. J., & Sriraman, B. (Eds.). (2014). Probabilistic thinking: Presenting plural perspectives. Dordrecht, The Netherlands: Springer.  https://doi.org/10.1007/978-94-007-7155-0.Google Scholar
  16. Clark, J., & Paivio, A. (1991). Dual coding theory and education. Educational Psychology Review, 3(3), 149–210.CrossRefGoogle Scholar
  17. Cobb, G. (2007). One possible frame for thinking about experiential learning. International Statistical Review, 75(3), 336–347.CrossRefGoogle Scholar
  18. Coppell, K. J., Mann, J. I., Williams, S. M., Jo, E., Drury, P. L., Miller, J., et al. (2013). Prevalence of diagnosed and undiagnosed diabetes and prediabetes in New Zealand: Findings from the 2008:2009 Adult Nutrition Survey. The New Zealand Medical Journal, 126(1370), 23–43.Google Scholar
  19. Finger, R., & Bisantz, A. M. (2002). Utilizing graphical formats to convey uncertainty in a decision-making task. Theoretical Issues in Ergonomics Science, 3(1), 1–25.  https://doi.org/10.1080/14639220110110324.CrossRefGoogle Scholar
  20. Garcia-Retamero, R., & Hoffrage, U. (2013). Visual representation of statistical information improves diagnostic inferences in doctors and patients. Social Science and Medicine, 83, 27–33.CrossRefGoogle Scholar
  21. Garfield, J., delMas, R., & Zieffler, A. (2012). Developing statistical modelers and thinkers in an introductory, tertiary-level statistics course. ZDM—International Journal on Mathematics Education, 44(7), 883–898.CrossRefGoogle Scholar
  22. Gigerenzer, G. (2014). Risk savvy: How to make good decisions. New York, NY: Viking.Google Scholar
  23. Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Bulletin, 102, 684–704.Google Scholar
  24. Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin, S. (2007). Helping doctors and patients make sense of health statistics. Psychological Science in the Public Interest, 8, 53–96.CrossRefGoogle Scholar
  25. Gigerenzer, G., Hoffrage, U., & Ehert, A. (1998). AIDS counseling for low-risk clients. AIDS Care, 10, 197–211.  https://doi.org/10.1080/09540129850124451.CrossRefGoogle Scholar
  26. Greer, B., & Mukhopadhyay, S. (2005). Teaching and learning the mathematization of uncertainty: Historical, cultural, social and political contexts. In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 297–324). New York, NY: Kluwer/Springer Academic Publishers.CrossRefGoogle Scholar
  27. Griffiths, T. L., & Tenenbaum, J. B. (2006). Optimal predictions in everyday cognition. Psychological Science, 17, 767–773.  https://doi.org/10.1111/j.1467-9280.2006.01780.x.CrossRefGoogle Scholar
  28. Hoffrage, U., Hafenbrädl, S., & Bouquet, C. (2015). Natural frequencies facilitate diagnostic inferences of managers. Frontiers in Psychology, 6(642), 1–11.  https://doi.org/10.3389/fpsyg.2015.00642.CrossRefGoogle Scholar
  29. Kahneman, D. (2011). Thinking, fast and slow. New York, NY: Allen Lane.Google Scholar
  30. Kahneman, D., & Tversky, A. (1973). On the psychology of prediction. Psychological Review, 80, 237–251.CrossRefGoogle Scholar
  31. Koehler, J. J. (1996). The base rate fallacy reconsidered: Descriptive, normative and methodological challenges. Behavioral and Brain Sciences, 19, 1–17.  https://doi.org/10.1017/S0140525X00041157.CrossRefGoogle Scholar
  32. Konold, C., & Kazak, S. (2008). Reconnecting data and chance. Technology Innovations in Statistics Education, 2(1). Retrieved from http://escholarship.org/uc/item/38p7c94v.
  33. Lane, D. M., & Peres, S. C. (2006). Interactive simulations in the teaching of statistics: Promise and pitfalls. In B. Phillips (Ed.), Proceedings of the Seventh International Conference on Teaching Statistics, Cape Town, South Africa. Voorburg, The Netherlands: International Statistical Institute.Google Scholar
  34. Lefevre, R. J., Pfautz, J., & Jones, K. (2005). Weather forecast uncertainty management and display. In Proceedings of the 21st International Conference on Interactive Information Processing Systems (UPS) for Meteorology, Oceanography, and Hydrology, San Diego, CA. Retrieved from https://ams.confex.com/ams/pdfpapers/82400.pdf.
  35. Makar, K., & Confrey, J. (2005). “Variation-Talk”: Articulating meaning in statistics. Statistics Education Research Journal, 4(1), 27–54.Google Scholar
  36. Mandel, D. R. (2015). Instruction in information structuring improves Bayesian judgment in intelligence analysis. Frontiers in Psychology, 6(387), 1–12.  https://doi.org/10.3389/fpsyg.2015.00387.CrossRefGoogle Scholar
  37. Mayer, R. E. (2009). Multimedia learning. New York: Cambridge University Press.CrossRefGoogle Scholar
  38. Mayer, R. E. (2010). Unique contributions of eye-tracking research to the study of learning graphics. Learning and Instruction, 20, 167–171.  https://doi.org/10.1016/j.learninstruc.2009.02.012.CrossRefGoogle Scholar
  39. Moore, D. (1997). Probability and statistics in the core curriculum. In J. Dossey (Ed.), Confronting the core curriculum (pp. 93–98). Washington, DC: Mathematical Association of America.Google Scholar
  40. Nance, D. A., & Morris, S. B. (2005). Juror understanding of DNA evidence: An empirical assessment of presentation formats for trace evidence with a relatively small random-match probability. Journal of Legal Studies, 34, 395–444.  https://doi.org/10.1086/428020.CrossRefGoogle Scholar
  41. Neumann, D. L., Hood, M., & Neumann, M. M. (2013). Using real-life data when teaching statistics: Student perceptions of this strategy in an introductory statistics course. Statistics Education Research Journal, 12(2), 59–70. Retrieved from https://iase-web.org/documents/SERJ/SERJ12(2)_Neumann.pdf.
  42. Paling, J. (2003). Strategies to help patients understand risks. British Medical Journal, 327, 745–748.  https://doi.org/10.1136/bmj.327.7417.745.CrossRefGoogle Scholar
  43. Pfannkuch, M., & Budgett, S. (2016a). Reasoning from an Eikosogram: An exploratory study. International Journal of Research in Undergraduate Mathematics Education, 1–28.  https://doi.org/10.1007/s40753-016-0043-0.CrossRefGoogle Scholar
  44. Pfannkuch, M., & Budgett, S. (2016b). Markov processes: Exploring the use of dynamic visualizations to enhance student understanding. Journal of Statistics Education, 24(2), 63–73.  https://doi.org/10.1080/10691898.2016.1207404.CrossRefGoogle Scholar
  45. Pfannkuch, M., Budgett, S., & Arnold, P. (2015). Experiment-to-causation inference: Understanding causality in a probabilistic setting. In A. Zieffler & E. Fry (Eds.), Reasoning about uncertainty: Learning and teaching informal inferential reasoning (pp. 95–127). Minneapolis, MN: Catalyst Press.Google Scholar
  46. Pfannkuch, M., Budgett, S., Fewster, R., Fitch, M., Pattenwise, S., Wild, C., et al. (2016). Probability modelling and thinking: What can we learn from practice? Statistics Education Research Journal, 11–37. Retrieved from http://iase-web.org/documents/SERJ/SERJ15(2)_Pfannkuch.pdf.
  47. Pfannkuch, M., Seber, G. A., & Wild, C. J. (2002). Probability with less pain. Teaching Statistics, 24(1), 24–30.CrossRefGoogle Scholar
  48. Pouget, A., Beck, J. M., Ma, W. J., & Latham, P. E. (2013). Probabilistic brains: Knowns and unknowns. Nature Neuroscience, 16, 1170–1178.  https://doi.org/10.1038/nn.3495.CrossRefGoogle Scholar
  49. Sacristan, A., Calder, N., Rojano, T., Santos-Trigo, M., Friedlander, A., & Meissner, H. (2010). The influence and shaping of digital technologies on the learning—and learning trajectories—of mathematical concepts. In C. Hoyles, & J. Lagrange (Eds.), Mathematics education and technology—Rethinking the terrain: The 17th ICMI Study (pp. 179–226). New York, NY: Springer.Google Scholar
  50. Schoenfeld, A. (2007). Method. In F. Lester (Ed.), Second handbook of research on the teaching and learning of mathematics (pp. 96–107). Charlotte, NC: Information Age Publishers.Google Scholar
  51. Sedlmeier, P., & Gigerenzer, G. (2001). Teaching Bayesian reasoning in less than two hours. Journal of Experimental Psychology: General, 3, 380–400.  https://doi.org/10.1037//0096-3445.130.3.380.CrossRefGoogle Scholar
  52. Shaughnessy, M. (2007). Research on statistics learning and reasoning. In F. Lester (Ed.), Second handbook of research on the teaching and learning of mathematics (Vol. 2, pp. 957–1009). Charlotte, NC: Information Age Publishers.Google Scholar
  53. Sirota, M., Vallée-Tourangeau, G., Vallée-Tourangeau, F., & Juanchich, M. (2015). On Bayesian problem-solving: Helping Bayesians solve simple Bayesian word problems. Frontiers in Psychology, 6(1141), 1–4.  https://doi.org/10.3389/fpsyg.2015.01141.CrossRefGoogle Scholar
  54. Sloman, S. A., Over, D. E., Slovak, L., & Stibel, J. M. (2003). Frequency illusions and other fallacies. Organizational Behavior and Human Decision Processes, 91, 296–309.CrossRefGoogle Scholar
  55. Spiegelhalter, D. J. (n.d.). Screening tests. Retrieved from Understanding Uncertainty: https://understandinguncertainty.org/screening.
  56. Sturm, A., & Eichler, A. (2014). Students’ beliefs about the benefit of statistical knowledge when perceiving information through daily media. In K. Makar, B. de Sousa, & R. Gould (Eds.), Proceedings of the Ninth International Conference on Teaching Statistics (ICOTS9), Flagstaff, Arizona, USA. Voorburg, The Netherlands: International Statistical Institute.Google Scholar
  57. Téglás, E., Vul, E., Girotto, V., Gonzalez, M., Tenenbaum, J. B., & Bonatti, L. L. (2011). Pure reasoning in 12-month-old infants as probabilistic inference. Science, 1054–1059.  https://doi.org/10.1126/science.1196404.CrossRefGoogle Scholar
  58. Teigen, K. H., & Keren, G. (2007). Waiting for the bus: When base-rates refuse to be neglected. Cognition, 103, 337–357.  https://doi.org/10.1016/j.cognition.2006.03.007.CrossRefGoogle Scholar
  59. Thomas, M. O. (2008). Conceptual representations and versatile mathematical thinking. In Proceedings of the Tenth International Congress in Mathematics Education, Copenhagen, Denmark (pp. 1–18).Google Scholar
  60. Villejoubert, G., & Mandel, D. R. (2002). The inverse fallacy: An account of deviations from Bayes theorem and the additivity principle. Memory & Cognition, 30, 171–178.  https://doi.org/10.3758/BF03195278.CrossRefGoogle Scholar
  61. Ware, C. (2008). Visual thinking for design. Burlington, MA: Morgan Kaufmann Publishers.Google Scholar
  62. Watson, J. M., & Callingham, R. (2014). Two-way tables: Issues at the heart of statistics and probability for students and teachers. Mathematical Thinking and Learning, 16(4), 254–284.  https://doi.org/10.1080/10986065.2014.953019.CrossRefGoogle Scholar
  63. Wolfe, C. R. (1995). Information seeking on Bayesian conditional probability problems: A fuzzy-trace theory. Journal of Behavioral Decision Making, 8, 85–108.CrossRefGoogle Scholar
  64. Zikmund-Fisher, B. J., Witteman, H. O., Dickson, M., Fuhrel-Forbis, A., Khan, V. C., Exe, N. L., et al. (2014). Blocks, ovals, or people? Icon type affects risk perceptions and recall of pictographs. Medical Decision Making, 34, 443–453.  https://doi.org/10.1177/0272989X13511706.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of StatisticsThe University of AucklandAucklandNew Zealand

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