MIDDLE SCHOOL STUDENTS’ STATISTICAL REASONING ABOUT DISTRIBUTION IN THEIR STATISTICAL MODELING PROCESSES
DOI:
https://doi.org/10.52041/serj.v22i1.312Keywords:
Statistics education research, Distribution, Informal statistical inference, Statistical modeling, Statistical reasoning, TinkerPlotsAbstract
Statistical reasoning about a population through samples can be achieved by modeling the relationship between population and sample. One way to do this is to model real data situations in a technology-integrated environment. With this view, we aimed to investigate how middle school students formed distributions and examined their statistical modeling processes through the informal reasoning process within the Reasoning with Informal Statistical Models and Modeling (RISM) framework. The case study reported in this paper focuses on how the conjecture and data models students designed throughout three activities evolved and how their inclusion of a fundamentally probabilistic mechanism matured. Findings show the students approached the distribution probabilistically with their inferences during the modeling process and grasped the statistical concepts they would encounter at a more advanced level. Therefore, we claim that students shifted from understanding empirical distributions to understanding theoretical distributions.
References
Aridor, K., & Ben-Zvi, D. (2017). The co-emergence of aggregate and modelling reasoning. Statistics Education Research Journal, 16(2), 38–63. https://doi.org/10.52041/serj.v16i2.184
Arnold, P., & Pfannkuch, M. (2014). Describing distributions. In K. Makar, B. de Sousa, & R. Gould (Eds.), Sustainability in statistics education. Proceedings of the Ninth International Conference on Teaching Statistics (ICOTS9), Flagstaff, Arizona. https://icots.info/icots/9/proceedings/pdfs/ICOTS9_8G1_ARNOLD.pdf
Bakker, A. (2004a). Reasoning about shape as a pattern in variability. Statistics Education Research Journal, 3(2), 64–83. https://doi.org/10.52041/serj.v3i2.552
Bakker, A. (2004b). Design research in statistics education: On symbolizing and computer tools. [Doctoral dissertation, Universiteit Utrecht]. CD-B Press.
Bakker, A., & Derry, J. (2011). Lessons from inferentialism for statistics education. Mathematical Thinking and Learning, 13(1–2), 5–26. https://doi.org/10.1080/10986065.2011.538293
Bakker, A., & Gravemeijer, K. P. (2004). Learning to reason about distribution. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 147–168). Springer.
Ben-Zvi, D., & Amir, Y. (2005). How do primary school students begin to reason about distributions? In K. Makar (Ed.), Reasoning about distribution: A collection of current research studies. Proceedings of the Fourth International Research Forum on Statistical Reasoning, Thinking, and Literacy (SRTL-4), Brisbane, Australia.
Ben-Zvi, D., & Arcavi, A. (2001). Junior high school students’ construction of global views of data and data representations. Educational Studies in Mathematics, 45(1–3), 35–65. https://doi.org/10.1023/A:1013809201228
Biehler, R., Ben-Zvi, D., Bakker, A., & Makar, K. (2012). Technology for enhancing statistical reasoning at the school level. In M. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. Leung (Eds) Third international handbook of mathematics education (pp. 643–689). Springer. https://doi.org/10.1007/978-1-4614-4684-2_21
Braham, H. M., & Ben-Zvi, D. (2017). Students’ emergent articulations of statistical models and modeling in making informal statistical inferences. Statistics Education Research Journal, 16(2), 116–143. https://doi.org/10.52041/serj.v16i2.187
Brown, E. N., & Kass, R. E. (2009) What Is Statistics? The American Statistician, 63(2), 105-110. https://doi.org/10.1198/tast.2009.0019
Canada, D. (2004). Preservice teachers’ understanding of variation. [Doctoral dissertation, Portland State University].
Chance, B., Ben-Zvi, D., Garfield, J., & Medina, E. (2007). The role of technology in improving student learning of statistics. Technology Innovations in Statistics Education, 1(1). https://doi.org/10.5070/T511000026
Cobb, P., McClain, K., & Gravemeijer, K. (2003). Learning about statistical covariation. Cognition and instruction, 21(1), 1–78. https://doi.org/10.1207/S1532690XCI2101_1
Creswell, J. W., & Poth, C. N. (2016). Qualitative inquiry and research design: Choosing among five approaches. SAGE Publications.
delMas, G., Garfield, J., Ooms, A., & Chance, B. (2007). Assessing students’ conceptual understanding after a first course in statistics. Statistics Education Research Journal, 6(2), 28–58. https://doi.org/10.52041/serj.v6i2.483
Dvir, M., & Ben-Zvi, D. (2018). The role of model comparison in young learners’ reasoning with statistical models and modeling. ZDM Mathematics Education, 50(7), 1183–1196. https://doi.org/10.1007/s11858-018-0987-4
Dvir, M., & Ben-Zvi, D. (2019). Repurpose and extend: Making a model statistical. In U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Eleventh Congress of the European Society for Research in Mathematics Education (CERME11) (pp. 922–929). Utrecht, The Netherlands.
Dvir, M., & Ben-Zvi, D. (2023). The double-edged sword of conjecturing. Mathematical Thinking and Learning, 25(2), 153–176. https://doi.org/10.1080/10986065.2021.1940427
Garfield, J., & Ben-Zvi, D. (2005). A framework for teaching and assessing reasoning about variability. Statistics Education Research Journal, 4(1), 92–99. https://doi.org/10.52041/serj.v4i1.527
Garfield, J., & Ben-Zvi, D. (2008). Developing students’ statistical reasoning: Connecting research and teaching practice. Springer.
Garfield, J. B., Ben-Zvi, D., Chance, B., Medina, E., Roseth, C., & Zieffler, A. (2008). Learning to reason about variability. In J. Garfield & D. Ben-Zvi (Eds.), Developing students’ statistical reasoning (pp. 201–214). Springer.
Hammerman, J., & Rubin, A. (2004). Strategies for managing statistical complexity with new software tools. Statistics Education Research Journal, 3(2), 17–41. https://doi.org/10.52041/serj.v3i2.546
Kazak, S., Fujita, T., & Wegerif, R. (2014). Year six students’ reasoning about random “bunny hops” through the use of TinkerPlots and peer-to-peer dialogic interactions. In K. Makar, B. de Sousa, & R. Gould (Eds.), Sustainability in statistics education. Proceedings of the Ninth International Conference on Teaching Statistics (ICOTS9), Flagstaff, Arizona
Konold, C., Harradine, A., & Kazak, S. (2007). Understanding distributions by modeling them. International Journal of Computers for Mathematical Learning, 12(3), 217–230. https://doi.org/10.1007/s10758-007-9123-1
Konold, C., & Higgins, T. (2003). Reasoning about data. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 193–215). National Council of Teachers of Mathematics.
Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33(4), 259–289. https://doi.org/10.5951/jresematheduc.33.4.0259
Lehrer, R., & Schauble, L. (2004). Modeling natural variation through distribution. American Educational Research Journal, 41(3), 635–679. https://doi.org/10.3102/00028312041003635
Makar, K., Bakker, A., & Ben-Zvi, D. (2011). The reasoning behind informal statistical inference. Mathematical Thinking and Learning, 13(1), 152–173. https://doi.org/10.1080/10986065.2011.538301
Makar, K., & Confrey, J. (2002). Comparing two distributions: Investigating secondary teachers’ statistical thinking. In B. Philips (Ed.), Developing a statistically literate society. Proceedings of the Sixth International Conference on Teaching Statistics (ICOTS-6), Cape Town, South Africa.
Makar, K., & Confrey, J. (2005). “Variation-talk”: Articulating meaning in statistics. Statistics Education Research Journal, 4(1), 27–54. https://doi.org/10.52041/serj.v4i1.524
Manor, H., & Ben-Zvi, D. (2017). Students’ emergent articulations of statistical models and modeling in making informal statistical inferences. Statistics Education Research Journal, 16(2), 116–143. https://doi.org/10.52041/serj.v16i2.187
Mills, J. D. (2002). Using computer simulation methods to teach statistics: A review of the literature. Journal of Statistics Education, 10(1). http://www.amstat.org/publications/jse/v10n1/mills.html
Moore, D. S. (1997). New pedagogy and new content: The case of statistics. International Statistical Review, 65(2), 123–137.
Noll, J., & Shaughnessy, J. M. (2012). Aspects of students' reasoning about variation in empirical sampling distributions. Journal for Research in Mathematics Education, 43(5), 509–556. https://doi.org/10.5951/jresematheduc.43.5.0509
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.
Patton, M. Q. (2014). Qualitative evaluation and research methods (4th ed.). SAGE Publications.
Pfannkuch, M. (2006). Comparing box plot distributions: A teacher’s reasoning. Statistics Education Research Journal, 5(2), 27–45. https://doi.org/10.52041/serj.v5i2.498
Pfannkuch, M., & Reading, C. E. (2006). Reasoning about distribution: A complex process. Statistics Education Research Journal 5(2), 4–9. https://doi.org/10.52041/serj.v5i2.496
Pfannkuch, M., Ben-Zvi, D., & Budgett, S. (2018). Innovations in statistical modeling to connect data, chance and context. ZDM Mathematics Education, 50(7), 1113–1123. https://doi.org/10.1007/s11858-018-0989-2
Reading, C. E., & Reid, J. (2006). An emerging hierarchy of reasoning about distribution: From a variation perspective. Statistics Education Research Journal, 5(2), 46–68. https://doi.org/10.52041/serj.v5i2.500
Rossman, A., & Chance, B. (2005). Workshop statistics student toolkit. Key College Publishing.
Rumsey, D. J. (2002). Statistical literacy as a goal for introductory statistics courses. Journal of Statistics Education, 10(3). https://doi.org/10.1080/10691898.2002.11910678
Scheaffer, R. L., Gnanadesikan, M., Watkins, A., & Witmer, J. (2004). Activity-based statistics: Student guide (2nd edition). Key College Publishing.
Shaughnessy, J. M. (2019). Recommendations about the big ideas in statistics education: A retrospective from curriculum and research. Cuadernos de Investigación y Formación, 18, 44–58.
Watson, J. (2005). Developing an awareness of distribution. In K. Makar (Eds.), Reasoning about distribution: A collection of current research studies. Proceedings of the Fourth International Research Forum on Statistical Reasoning, Thinking, and Literacy, Brisbane, Australia.
Watson, J. M. (2009). The influence of variation and expectation on the developing awareness of distribution. Statistics Education Research Journal, 8(1), 32–61. https://doi.org/10.52041/serj.v8i1.456
Watson, J. M., & Moritz, J. B. (2000). Developing concepts of sampling. Journal for Research in Mathematics Education, 31(1), 44–70. https://doi.org/10.2307/749819
Wild, C. (2006). The concept of distribution. Statistics Education Research Journal, 5(2), 10–26. https://doi.org/10.52041/serj.v5i2.497
Wild, C. J., Pfannkuch, M., Regan, M., & Horton, N. J. (2011). Towards more accessible conceptions of statistical inference (with discussion). Journal of the Royal Statistical Society Series A: Statistics in Society, 174, 247–295. https://doi.org/10.1111/j.1467-985X.2010.00678.x
