Sixth grade students’ emerging practices of data modelling

Abstract

We explore 11–12-year-old students’ emerging ideas of models and modelling as they engage in a data-modelling task involving inquiry based on data obtained from an experiment. We report on a design-based study in which students identified what and how to measure, decided how to structure and represent data, and made inferences and predictions based on data. Our focus was on the following: (1) the nature of the student-generated models and (2) how students evaluated the models. Data from written work generated by groups and transcripts of interviews were analysed using progressive focussing. The results showed that groups constructed models of actual data by paying attention to various aspects of distributions. We found a tendency to use differing criteria for evaluating the success of models. This data modelling process also fostered students’ making sense of key ideas, tools and procedures in statistics that are usually treated in isolation and without context in school mathematics. In particular, we identified how some students appeared to gain insights into how a ‘good’ statistical model might incorporate some properties that are invariant when the simulation is repeated for small and large sample sizes (signal) and other properties that are not sustained in the same way (noise).

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References

  1. Ainley, J., & Pratt, D. (2017). Computational modeling and children’s expressions of signal and noise. Statistics Education Research Journal, 16(2), 15–37.

    Google Scholar 

  2. Bakker, A., & Derry, J. (2011). Lessons from inferentialism for statistics education. Mathematical Thinking and Learning, 13(1–2), 5–26.

    Article  Google Scholar 

  3. Cobb, P. (2009). Individual and collective mathematical development: the case of statistical data analysis. Mathematical Thinking and Learning, 1(1), 5–43.

    Article  Google Scholar 

  4. Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Shauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.

    Article  Google Scholar 

  5. Common Core Standards Writing Team. (2013). Progressions for the Common Core State Standards in Mathematics (draft, July 4). High School, Modeling. Tucson: Institute for Mathematics and Education, University of Arizona.

    Google Scholar 

  6. English, L. D. (2010). Young children’s early modelling with data. Mathematics Education Research Journal, 22(2), 24–47.

    Article  Google Scholar 

  7. English, L. D., & Watson, J. (2017). Modelling with authentic data in sixth grade. ZDM Mathematics Education. https://doi.org/10.1007/s11858-017-0896-y.

    Article  Google Scholar 

  8. Fielding-Wells, F., & Makar, K. (2015). Inferring to a model: Using inquiry-based argumentation to challenge young children’s expectations of equally likely outcomes. In A. Zieffler & E. Fry (Eds.), Reasoning about uncertainty: Learning and teaching informal inferential reasoning (pp. 1–28). Minneapolis: Catalyst Press.

    Google Scholar 

  9. Garfield, J., & Ben-Zvi, D. (2008). Learning to reason about statistical models and modeling. In J. Garfield & D. Ben-Zvi (Eds.), Developing students’ statistical reasoning: Connecting research and teaching practice (pp. 143–163). Dordrecht: Springer.

    Chapter  Google Scholar 

  10. Gravemeijer, K. (2002). Emergent modeling as the basis for an instructional sequence on data analysis. In B. Phillips (Ed.), Proceedings of the sixth international conference on the teaching of statistics (ICOTS-6), Cape Town, South Africa, [CD-ROM]. Voorburg: International Statistical Institute.

  11. Hancock, C., Kaput, J., & Goldsmith, L. (1992). Authentic inquiry with data: Critical barriers to classroom implementation. Educational Psychologist, 27(3), 337–364.

    Article  Google Scholar 

  12. Konold, C., Harradine, A., & Kazak, S. (2007). Understanding distributions by modelling them. International Journal of Computers for Mathematical Learning, 12(3), 217–230.

    Article  Google Scholar 

  13. Konold, C., Higgins, T., Russell, S. J., & Khalil, K. (2015). Data seen through different lenses. Educational Studies in Mathematics, 88(3), 305–325.

    Article  Google Scholar 

  14. Konold, C., & Kazak, S. (2008). Reconnecting data and chance. Technology Innovations in Statistics Education, 2. http://repositories.cdlib.org/uclastat/cts/tise/vol2/iss1/art1. Accessed 21 Apr 2017.

  15. Konold, C., & Miller, C. D. (2011). TinkerPlots 2.0: Dynamic data exploration. Emeryville: Key Curriculum.

    Google Scholar 

  16. Konold, C., Robinson, A., Khalil, K., Pollatsek, A., Well, A. D., Wing, R., & Mayr, S. (2002). Students’ use of modal clumps to summarize data. In B. Phillips (Ed.), Proceedings of the sixth international conference on the teaching of statistics (ICOTS-6), Cape Town, South Africa, [CD-ROM]. Voorburg: International Statistical Institute.

  17. Lehrer, R., & Romberg, T. (1996). Exploring children’s data modelling. Cognition and Instruction, 14(1), 69–108.

    Article  Google Scholar 

  18. Lehrer, R., & Schauble, L. (2002). Investigating real data in the classroom: Expanding children’s understanding of math and science. New York: Teachers College Press.

    Google Scholar 

  19. Lehrer, R., & Schauble, L. (2004). Modeling variation through distribution. American Education Research Journal, 41(3), 635–679.

    Article  Google Scholar 

  20. Lesh, R., & Doerr, H. M. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning and problem solving. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 3–33). Mahwah: Lawrence Erlbaum Associates.

    Chapter  Google Scholar 

  21. Makar, K., & Rubin, A. (2009). A framework for thinking about informal statistical inference. Statistics Education Research Journal, 8(1), 82–105.

    Google Scholar 

  22. MEB (2018a). Matematik Öğretim Programı (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar). Ankara: TTKB.

    Google Scholar 

  23. MEB. (2018b). Matematik Öğretim Programı (9, 11 ve 12. Sınıflar) (p. 10). Ankara: TTKB.

    Google Scholar 

  24. Moore, D. S. (1990). Uncertainty. In L. Steen (Ed.), On the shoulders of giants: New approaches to numeracy (pp. 95–137). Washington, DC: National Academy Press.

    Google Scholar 

  25. Moore, D. S. (1999). Discussion: What shall we teach beginners? International Statistical Review, 67(3), 250–252.

    Article  Google Scholar 

  26. Noll, J., & Kirin, D. (2017). TinkerPlots model construction approaches for comparing two groups: Student perspectives. Statistics Education Research Journal, 16(2), 213–243.

    Google Scholar 

  27. Parlett, M., & Hamilton, D. (1972). Evaluation as illumination: A new approach to the study of innovatory programs, Occasional Paper no 9. University of Edinburgh, Centre for Research in the Educational Sciences, Edinburgh. (Retrieved from ERIC database (ED167634).

  28. Pratt, D., & Noss, R. (2010). Designing for mathematical abstraction. International Journal of Computers for Mathematical Learning, 15(2), 81–97.

    Article  Google Scholar 

  29. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.

    Article  Google Scholar 

  30. Tall, D., & Gray, E. (1991). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. The Journal for Research in Mathematics Education, 26(2), 115–141.

    Google Scholar 

  31. Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical inquiry. International Statistical Review, 67(3), 223–248.

    Article  Google Scholar 

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Acknowledgements

This work was supported by PAU-ADEP 2017KRM002-159. We would also like to acknowledge insightful and constructive feedback from all reviewers and editor of this manuscript, and the generous work of the students who participated in this study.

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Correspondence to Sibel Kazak.

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Kazak, S., Pratt, D. & Gökce, R. Sixth grade students’ emerging practices of data modelling. ZDM Mathematics Education 50, 1151–1163 (2018). https://doi.org/10.1007/s11858-018-0988-3

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Keywords

  • Data modelling
  • Distribution
  • Informal statistical inference
  • Middle school students