ARGUMENT STRUCTURES OF RESPONSES TO A CONTEXTUALLY PROVOCATIVE HOSPITAL PROBLEM VARIANT
DOI:
https://doi.org/10.52041/serj.v24i1.764Keywords:
Statistics education research, Argumentation, Context, Empirical Law of Large Numbers, Probability, StatisticsAbstract
Numerous variants of Kahneman and Tversky’s (1972) hospital problem have been used to investigate intuitions about the Empirical Law of Large Numbers (eLLN). A largely separate line of research has focused on interactions between context knowledge and statistical reasoning. The present study merges these two lines of research by analyzing tertiary students’ reasoning about a hospital problem variant set in a provocative context from their academic major. Participants’ reasoning structures were diagrammed and compared against one another. Some responses closely matched an anticipated argument structure, and others differed along dimensions such as syllogistic structure, types of justifications offered, and task interpretation. Results of the study illustrate the importance of going beyond the metric of participants’ success rate choosing the intended sample when doing research with contextually provocative hospital problem variants. Analyses of responses to such variants can be enhanced by examining the depth of eLLN intuition they reflect, their underlying syllogistic structures, and the extent to which application of the eLLN is qualified as needed in a given context.
References
Arnold, P., & Franklin, C. (2021). What makes a good statistical question? Journal of Statistics and Data Science Education, 29(1), 122–130. https://doi.org/10.1080/26939169.2021.1877582
Bar-Eli, M., Avugos, S., & Raab, M. (2006). Twenty years of ‘hot hand’ research: Review and critique. Psychology of Sport and Exercise, 7(6), 525–553. https://doi.org/10.1016/j.psychsport.2006.03.001
Bargagliotti, A., Franklin, C., Arnold, P., Gould, R., Johnson, S., Perez, L., & Spangler, D. A. (2020). Pre-K–12 guidelines for assessment and instruction in statistics education II (GAISE II): A framework for statistics and data science education. American Statistical Association.
Bar-Hillel, M. (1979). The role of sample size in sample evaluation. Organizational Behavior and Human Performance, 24(2), 245–257. https://doi.org/10.1016/0030-5073(79)90028-X
Ben-Zvi, D., & Aridor-Berger, K. (2016). Children’s wonder how to wander between data and context. In D. Ben-Zvi & K. Makar. (Eds.), The teaching and learning of statistics: International perspectives (pp. 25–36). Springer. https://doi.org/10.1007/978-3-319-23470-0_3
Ben-Zvi, D., Aridor, K., Makar, K., & Bakker, A. (2012). Students’ emergent articulations of uncertainty while making informal statistical inferences. ZDM Mathematics Education, 44(7), 913–925. https://doi.org/10.1007/s11858-012-0420-3
Brown, E. C. (2019). Growing certain: Students’ mechanistic reasoning about the empirical law of large numbers (Publication No. 13895270) [Doctoral dissertation, University of Minnesota]. ProQuest LLC.
Chazan, D., Sela, H., & Herbst, P. (2012). Is the role of equations in the doing of word problems in school algebra changing? Initial indications from teacher study groups. Cognition and Instruction, 30(1), 1–38. https://doi.org/10.1080/07370008.2011.636593
Cobb, G. W., & Moore, D. S. (1997). Mathematics, statistics, and teaching. The American Mathematical Monthly, 104(9), 801–823. https://doi.org/10.1080/00029890.1997.11990723
Cornish, F., Gillespie, A., & Zittoun, T. (2013). Collaborative analysis of qualitative data. In U. Flick (Ed.), The SAGE handbook of qualitative data analysis (pp. 79–93). Sage.
delMas, R. C. (2004). A comparison of mathematical and statistical reasoning. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 79–95). Kluwer. https://doi.org/10.1007/1-4020-2278-6_4
Eldh, A. C., Årestedt, L., & Berterö, C. (2020). Quotations in qualitative studies: Reflections on constituents, custom, and purpose. International Journal of Qualitative Methods, 19. https://doi.org/10.1177/1609406920969268
Evans, J., & Dusoir, A. E. (1977). Proportionality and sample size as factors in intuitive statistical judgement. Acta Psychologica, 41(2–3), 129–137. https://doi.org/10.1016/0001-6918(77)90030-0
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/10.5951/jresematheduc.28.1.0096
Gerbic, P., & Stacey, E. (2005). A purposive approach to content analysis: Designing analytical frameworks. The Internet and Higher Education, 8(1), 45–59. https://doi.org/10.1016/j.iheduc.2004.12.003
Gil, E., & Ben-Zvi, D. (2011). Explanations and context in the emergence of students’ informal inferential reasoning. Mathematical Thinking and Learning, 13(1–2), 87–108. https://doi.org/10.1080/10986065.2011.538295
Groth, R. E., & Choi, Y. (2023). A method for assessing students’ interpretations of contextualized data. Educational Studies in Mathematics, 114(1), 17–34. https://doi.org/10.1007/s10649-023-10234-z
Henriques, A., & Oliveira, H. (2016). Students’ expressions of uncertainty in making informal inference when engaged in a statistical investigation using TinkerPlots. Statistics Education Research Journal, 15(2), 62–80. https://doi.org/10.52041/serj.v15i2.241
Kahneman, D. (2011). Thinking, fast and slow. Farrar, Straus, and Giroux.
Kahneman, D., & Klein, G. (2009). Conditions for intuitive expertise: A failure to disagree. American Psychologist, 64(6), 515–526. https://doi.org/10.1037/a0016755
Kahneman, D., & Tversky, A. (1972). Subjective probability: A judgment of representativeness. Cognitive Psychology, 3(3), 430–454. https://doi.org/10.1016/0010-0285(72)90016-3
Keith, W., & Beard, D. (2008). Toulmin’s rhetorical logic: What’s the warrant for warrants? Philosophy & Rhetoric, 41(1), 22–50. https://www.jstor.org/stable/25655298
Knipping, C., & Reid, D. (2015). Reconstructing argumentation structures: A perspective on proving processes in secondary mathematics classroom interactions. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education: Examples of methodology and methods (pp. 75–101). Springer. https://doi.org/10.1007/978-94-017-9181-6_4
Langrall, C. W., Nisbet, S., Mooney, E., & Jansem, S. (2011). The role of context expertise when comparing data. Mathematical Thinking and Learning, 13(1–2), 47–67. https://doi.org/10.1080/10986065.2011.538620
Lem, S. (2015). The intuitiveness of the law of large numbers. ZDM Mathematics Education, 47(5), 783–792. https://doi.org/10.1007/s11858-015-0676-5
Lem, S., Van Dooren, W., Gillard, E., & Verschaffel, L. (2011). Sample size neglect problems: A critical analysis. Studia Psychologica, 53(2), 123–135.
Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic inquiry. Sage Publications.
Madden, S. (2011). Statistically, technologically, and contextually provocative tasks: Supporting teachers’ informal inferential reasoning. Mathematical Thinking and Learning, 13(1–2), 109–131. https://doi.org/10.1080/10986065.2011.539078
Makar, K., & Confrey, J. (2004). Secondary teachers’ reasoning about comparing two groups. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 353–373). Kluwer.
Masnick, A. M., Klahr, D., & Morris, B. J. (2007). Separating signal from noise: Children’s understanding of error and variability in experimental outcomes. In M. C. Lovett & P. Shah (Eds.), Thinking with data (pp. 3–26). Lawrence Erlbaum Associates.
Maxara, C., & Biehler, R. (2010). Students’ understanding and reasoning about sample size and the law of large numbers after a computer-intensive introductory course on stochastics. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. Proceedings of the Eighth International Conference on Teaching Statistics (ICOTS8, July 2010), Ljubljana, Slovenia. International Statistical Institute. https://iase-web.org/documents/papers/icots8/ICOTS8_3C2_MAXARA.pdf?1402524969
Menary, R. (2007). Writing as thinking. Language Sciences, 29(5), 621–632. https://doi.org/10.1016/j.langsci.2007.01.005
Miles, M. B., Huberman, A. M., & Saldaña, J. (2019). Qualitative data analysis: A methods sourcebook (4th ed.). Sage.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics.
Nilsson, P., Schindler, M., & Bakker, A. (2018). The nature and use of theories in statistics education. In D. Ben-Zvi, K. Makar, & J. Garfield (Eds.), International handbook of research in statistics education (pp. 359–386). Springer. https://doi.org/10.1007/978-3-319-66195-7_11
Pfannkuch, M. (2011). The role of context in developing informal statistical inferential reasoning: A classroom study. Mathematical Thinking and Learning, 13(1–2), 27–46. https://doi.org/10.1080/10986065.2011.538302
Reagan, R. (1989). Variations on a seminal demonstration of people’s insensitivity to sample size. Organizational Behavior and Human Decision Processes, 43(1), 52–57. https://doi.org/10.1016/0749-5978(89)90057-5
Rossman, A., Chance, B., & Medina, E. (2006). Some important comparisons between statistics and mathematics, and why teachers should care. In G. F. Burrill & P. C. Elliot (Eds.), Thinking and reasoning with data and chance: Sixty-eighth yearbook of the National Council of Teachers of Mathematics (pp. 323–333). National Council of Teachers of Mathematics.
Rubel, L. (2009). Middle and high school students’ thinking about effects of sample size: An in and out of school perspective. In S. L. Swars, D. W. Stinson & S. Lemons-Smith (Eds.), Proceedings of the 31st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 636–643). Georgia State University.
Sedlmeier, P. (1999). Improving statistical reasoning: Theoretical models and practical implications. Lawrence Erlbaum Associates.
Sedlmeier, P., & Gigerenzer, G. (1997). Intuitions about sample size: The empirical law of large numbers. Journal of Behavioral Decision Making, 10(1), 33–51. https://doi.org/10.1002/(SICI)1099-0771(199703)10:1<33::AID-BDM244>3.0.CO;2-6
Shaughnessy, J. M., & Pfannkuch, M. (2002). How faithful is Old Faithful? Statistical thinking: A story of variation and predication. Mathematics Teacher, 95(4), 252–259. https://doi.org/10.5951/MT.95.4.0252
Sherlock-Shangraw, R. (2013). Creating inclusive youth sports environments with the Universal Design for learning. Journal of Physical Education, Recreation, & Dance, 84(2), 40–46. https://doi.org/10.1080/07303084.2013.757191
Society of Health and Physical Educators. (2013). National standards for K-12 physical education.
Sommerhoff, D., Weixler, S., & Hamedinger, C. (2023). Sensitivity to sample size in the context of the empirical law of large numbers: Comparing the effectiveness of three approaches to support early secondary school students. Journal for Didactics of Mathematics, 44(1), 233–267. https://doi.org/10.1007/s13138-022-00213-x
Stanovich, K.E. (2018). Miserliness in human cognition: The interaction of detection, override and mindware. Thinking & Reasoning, 24(4), 423–444. https://doi.org/10.1080/13546783.2018.1459314
Tabor, J., & Franklin, C. (2019). Statistical reasoning in sports (2nd ed.). Bedford, Freeman, and Worth High School Publishers.
Tirosh, D., & Stavy, R. (2000). How students (mis-)understand science and mathematics: Intuitive rules. Teachers College Press.
Toulmin, S. (1958). The uses of argument. Cambridge University Press.
Toulmin, S. (2003). The uses of argument (updated edition). Cambridge University Press.
Warren, J. E. (2010). Taming the warrant in Toulmin’s model of argument. The English Journal, 99(6), 41–46. https://www.jstor.org/stable/20787665
Weixler, S., Sommerhoff, D., & Ufer, S. (2019). The empirical law of large numbers and the hospital problem: Systematic investigation of the impact of multiple task and person characteristics. Educational Studies in Mathematics, 100(1), 61–83. https://doi.org/10.1007/s10649-018-9856-x
Well, A. D., Pollatsek, A., & Boyce, S. J. (1990). Understanding the effects of sample size on the variability of the mean. Organizational Behavior and Human Decision Processes, 47(2), 289–312. https://doi.org/10.1016/0749-5978(90)90040-G
Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223–248. https://doi.org/10.1111/j.1751-5823.1999.tb00442.x
