Advertisement

ZDM

, Volume 50, Issue 7, pp 1139–1150 | Cite as

Statistical modelling and repeatable structures: purpose, process and prediction

  • Katie MakarEmail author
  • Sue Allmond
Original Article

Abstract

Children have limited exposure to statistical concepts and processes, yet researchers have highlighted multiple benefits of experiences in which they design and/or engage informally with statistical modelling. A study was conducted with a classroom in which students developed and utilised data-based models to respond to the inquiry question, Which origami animal jumps the furthest? The students used hat plots and box plots in Tinkerplots to make sense of variability in comparing distributions of their data and to support them to write justified conclusions of their findings. The study relied on classroom video and student artefacts to analyse aspects of the students’ modelling experiences which exposed them to powerful statistical ideas, such as key repeatable structures and dispositions in statistics. Three principles—purpose, process and prediction—are highlighted as ways in which the problem context, statistical structures and inquiry dispositions and cycle extended students’ opportunities to reason in sophisticated ways appropriate for their age. The research question under investigation was, How can an emphasis on purpose, process and prediction be implemented to support children’s statistical modelling? The principles illustrated in the study may provide a simple framework for teachers and researchers to develop statistical modelling practices and norms at the school level.

Keywords

Statistics education Informal statistical inference Statistical modelling Repeatable structures 

Notes

Acknowledgements

We gratefully acknowledge funding for this work from the Australian Research Council (ARC DP120100690, ARC DP170101993) and our research assistants Janine and Ali who transcribed these data and assisted with preliminary analysis.

References

  1. Ainley, J., Pratt, D., & Hansen, A. (2006). Connecting engagement and focus in pedagogic task design. British Educational Research Journal, 32(1), 23–38.  https://doi.org/10.1080/01411920500401971.CrossRefGoogle Scholar
  2. Allmond, S., & Makar, K. (2014). From hat plots to box plots in Tinkerplots: Supporting students to write conclusions which account for variability in data. In K. Makar, B. de Sousa & R. Gould (Eds.), Proceedings of the 9th International Conference on Teaching Statistics (pp. 1–6). International Association for Statistical Education.Google Scholar
  3. Bakker, A., & Derry, J. (2011). Lessons from inferentialism for statistics education. Mathematical Thinking and Learning, 13(1–2), 5–26.CrossRefGoogle Scholar
  4. Barbosa, J. C. (2006). Mathematical modelling in classroom: A socio-critical and discursive perspective. ZDM, 38(3), 293–301.  https://doi.org/10.1007/BF02652812.CrossRefGoogle Scholar
  5. Biehler, R., Frischemeier, D., & Podworny, S. (2017). Editorial: Reasoning about models and modelling in the context of informal statistical inference. Statistics Education Research Journal, 16(2), 8–12.Google Scholar
  6. Doerr, H., delMas, R., & Makar, K. (2017). A modeling approach to the development of students’ informal inferential reasoning. Statistics Education Research Journal, 16(2), 86–115.Google Scholar
  7. Doerr, H. M., & English, L. D. (2003). A modeling perspective on students’ mathematical reasoning about data. Journal for Research in Mathematics Education, 34(2), 110–136.  https://doi.org/10.2307/30034902.CrossRefGoogle Scholar
  8. English, L. D. (2012). Data modeling with first-grade students. Educational Studies in Mathematics, 81(1), 15–30.  https://doi.org/10.1007/s10649-011-9377-3.CrossRefGoogle Scholar
  9. Fielding-Wells, J. (2014). Developing argumentation in mathematics: The role of evidence and context. Unpublished doctoral dissertation, The University of Queensland.Google Scholar
  10. Fielding-Wells, J., & 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. 1–27). Minneapolis: Catalyst Press.Google Scholar
  11. Flyvbjerg, B. (2006). Five misunderstandings about case-study research. Qualitative Inquiry, 12(2), 219–245.CrossRefGoogle Scholar
  12. Font, V., Godino, J. D., & D’Amore, B. (2007). An onto-semiotic approach to representations in mathematics education. For the Learning of Mathematics, 27(2), 2–7, 14.Google Scholar
  13. Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1–3), 111–129.CrossRefGoogle Scholar
  14. Hancock, C., Kaput, J. J., & Goldsmith, L. T. (1992). Authentic inquiry with data: Critical barriers to classroom implementation. Educational Psychologist, 27(3), 337–364.CrossRefGoogle Scholar
  15. Hestenes, D. (2010). Modeling theory for math and science education. In R. Lesh, P. L. Galbraith, C. R. Haines & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies: ICTMA 13 (pp. 13–41). New York: Springer.CrossRefGoogle Scholar
  16. Konold, C. (2002). Hat plots? Unpublished manuscript. University of Massachusetts, Amherst.Google Scholar
  17. Konold, C., Finzer, W., & Kreetong, K. (2017). Modeling as a core component of structuring data. Statistics Education Research Journal, 16(2), 191–212.Google Scholar
  18. Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29–63.CrossRefGoogle Scholar
  19. Lehrer, R., & English, L. (2018). Introducing children to modeling variability. In D. Ben-Zvi, K. Makar & J. Garfield (Eds.), International handbook of research in statistics education (pp. 229–260). New York: Springer.CrossRefGoogle Scholar
  20. Lehrer, R., Jones, R. S., & Kim, M. J. (2014). Model-based informal inference. In K. Makar, B. de Sousa, & R. Gould (Eds.), Proceedings of the Ninth International Conference on Teaching Statistics (ICOTS9). Dordrecht, the Netherlands: ISI.Google Scholar
  21. Lehrer, R., Kim, M. J., & Jones, R. S. (2011). Developing conceptions of statistics by designing measures of distribution. ZDM – The International Journal on Mathematics Education, 43(5), 723–736.  https://doi.org/10.1007/s11858-011-0347-0.CrossRefGoogle Scholar
  22. Lehrer, R., & Schauble, L. (2010). What kind of explanation is a model? In M. K. Stein & L. Kucan (Eds.), Instructional explanation in the disciplines (pp. 9–22). New York: Springer.CrossRefGoogle Scholar
  23. Lesh, R. A., & Doerr, H. M. (Eds.). (2003). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  24. Makar, K. (2014). Young children’s explorations of average through informal inferential reasoning. Educational Studies in Mathematics, 86(1), 61–78.  https://doi.org/10.1007/s10649-013-9526-y.CrossRefGoogle Scholar
  25. Makar, K., & Rubin, A. (2009). A framework for thinking about informal statistical inference. Statistics Education Research Journal, 8(1), 82–105. Retrieved from http://iase-web.org/documents/SERJ/SERJ8(1)_Makar_Rubin.pdf.
  26. Makar, K., & Rubin, A. (2018). Learning about statistical inference. In D. Ben-Zvi, K. Makar & J. Garfield (Eds.), International handbook of research in statistics education (pp. 261–297). New York: Springer.CrossRefGoogle Scholar
  27. Peters, S. (2014). Developing understanding of statistical variation: Secondary statistics teachers’ perceptions and recollections of learning factors. Journal of Mathematics Teacher Education, 17, 539–582.  https://doi.org/10.1007/s10857-013-9242-7.CrossRefGoogle Scholar
  28. 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
  29. Powell, A. B., Francisco, J. M., & Maher, C. A. (2003). An analytical model for studying the development of learners’ mathematical ideas and reasoning using videotape data. Journal of Mathematical Behavior, 22, 405–435.  https://doi.org/10.1016/j.jmathb.2003.09.002.CrossRefGoogle Scholar
  30. Pratt, D., & Ainley, J. (2014). Chance Re-encounters: ‘Computers in Probability Education’ revisited. In T. Wassong et al. (Eds.), MitWerkzeugen Mathematik und Stochastik lernen—Using tools for learning mathematics and statistics. New York: Springer.Google Scholar
  31. Pratt, D., Johnston-Wilder, P., Ainley, J., & Mason, J. (2008). Local and global thinking in statistical inference. Statistics Education Research Journal, 7(2), 107–129. Retrieved from http://iase-web.org/documents/SERJ/SERJ7(2)_Pratt.pdf.
  32. Shaughnessy, J. M. (2007). Research on statistics learning and reasoning. In Second handbook of research on mathematics teaching and learning (pp. 957–1010). Reston, VA: NCTM.Google Scholar
  33. Shaughnessy, J. M., & Pfannkuch, M. (2002). How faithful is Old Faithful? Statistical thinking: A story of variation and prediction. Mathematics Teacher, 95(4), 252–259.Google Scholar
  34. Wells, J., Makar, K., & Allmond, S. (2012). Evidence triangle (Poster). Brisbane: The University of Queensland. Retrieved from http://www.mathsinquiry.com.
  35. Wild, C. (2017). Modelling: Connecting the worlds. Keynote presentation at the 10th International Forum for Research on Statistical Reasoning, Thinking and Literacy (SRTL10), 2–8 July 2017, Rotorua, New Zealand.Google Scholar
  36. Wild, C., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223–265.CrossRefGoogle Scholar
  37. Zapata-Cardona, L. (this issue). Students’ construction and use of models: A socio-critical perspective. ZDM Mathematics Education.Google Scholar

Copyright information

© FIZ Karlsruhe 2018

Authors and Affiliations

  1. 1.School of Education, Social Sciences BldgThe University of QueenslandBrisbaneAustralia

Personalised recommendations