Students’ Aggregate Reasoning with Covariation

  • Keren AridorEmail author
  • Dani Ben-Zvi
Part of the ICME-13 Monographs book series (ICME13Mo)


Helping students interpret and evaluate the relations between two variables is challenging. This chapter examines how students’ aggregate reasoning with covariation (ARwC) emerged while they modeled a real phenomenon and drew informal statistical inferences in an inquiry-based learning environment using TinkerPlotsTM. We focus in this illustrative case study on the emergent ARwC of two fifth-graders (aged 11) involved in statistical data analysis and modelling activities and in growing samples investigations. We elucidate four aspects of the students’ articulations of ARwC as they explored the relations between two variables in a small real sample and constructed and improved a model of the predicted relations in the population. We finally discuss implications and limitations of the results. This article contributes to the study of young students’ aggregate reasoning and the role of models in developing such reasoning.


Aggregate reasoning Exploratory data analysis Growing samples Informal statistical inference Reasoning with covariation Statistical modelling 



This research was supported by the University of Haifa and the I-CORE Program of the Planning and Budgeting Committee and the Israel Science Foundation grant 1716/12. We deeply thank the Cool-Connections research group who participated in the Connections project 2015, and in data analysis sessions of this research.


  1. Ainley, J., Nardi, E., & Pratt, D. (2000). The construction of meanings for trend in active graphing. International Journal of Computers for Mathematical Learning, 5(2), 85–114.CrossRefGoogle Scholar
  2. Ainley, J., Pratt, D., & Hansen, A. (2006). Connecting engagement and focus in pedagogic task design. British Educational Research Journal, 32(1), 23–38.CrossRefGoogle Scholar
  3. Bakker, A. (2004). Design research in statistics education: On symbolizing and computer tools (A Ph.D. Thesis). Utrecht, The Netherlands: CD Beta Press.Google Scholar
  4. Bakker, A., Biehler, R., & Konold, C. (2004). Should young students learn about boxplots? In G. Burrill & M. Camden (Eds.), Curricular development in statistics education, IASE 2004 Roundtable on Curricular Issues in Statistics Education, Lund Sweden. Voorburg, The Netherlands: International Statistics Institute.Google Scholar
  5. Bakker, A., & Gravemeijer, K. P. E. (2004). Learning to reason about distributions. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 147–168). Dordrecht, The Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar
  6. Bakker, A., & Hoffmann, M. (2005). Diagrammatic reasoning as the basis for developing concepts: A semiotic analysis of students’ learning about statistical distribution. Educational Studies in Mathematics, 60, 333–358.CrossRefGoogle Scholar
  7. Batanero, C., Estepa, A., & Godino, J. D. (1997). Evolution of students’ understanding of statistical association in a computer based teaching environment. In J. B. Garfield & G. Burrill (Eds.), Research on the role of technology in teaching and learning statistics (pp. 191–205). Voorburg, The Netherlands: International Statistical Institute.Google Scholar
  8. 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.CrossRefGoogle Scholar
  9. Ben-Zvi, D., Aridor, K., Makar, K., & Bakker, A. (2012). Students’ emergent articulations of uncertainty while making informal statistical inferences. ZDM—The International Journal on Mathematics Education, 44(7), 913–925.CrossRefGoogle Scholar
  10. Ben-Zvi, D., Gravemeijer, K., & Ainley, J. (2018). Design of statistics learning environments. In D. Ben-Zvi., K. Makar & J. Garfield (Eds.), International handbook of research in statistics education. Springer international handbooks of education (pp. 473–502). Cham: Springer.Google Scholar
  11. Biehler, R., Ben-Zvi, D., Bakker, A., & Makar, K. (2013). Technology for enhancing statistical reasoning at the school level. In M. A. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. Leung (Eds.), Third international handbook of mathematics education (pp. 643–690). Berlin: Springer.CrossRefGoogle Scholar
  12. Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events. Journal for Research in Mathematics Education, 33(5), 352–378.CrossRefGoogle Scholar
  13. Cobb, P., McClain, K., & Gravemeijer, K. P. E. (2003). Learning about statistical covariation. Cognition and Instruction, 21(1), 1–78.CrossRefGoogle Scholar
  14. Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. In P. Cobb (Ed.), Learning mathematics (pp. 31–60). Dordrecht, The Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar
  15. Creswell, J. (2002). Educational research: Planning, conducting, and evaluating quantitative and qualitative research. Upper Saddle River, NJ: Prentice Hall.Google Scholar
  16. Dvir, M., & Ben-Zvi, D. (2018). The role of model comparison in young learners’ reasoning with statistical models and modeling. ZDM—International Journal on Mathematics Education. Scholar
  17. Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht, The Netherlands: Kluwer.Google Scholar
  18. Friel, S. (2007). The research frontier: Where technology interacts with the teaching and learning of data analysis and statistics. In G. W. Blume & M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics (Vol. 2, pp. 279–331). Greenwich, CT: Information Age.Google Scholar
  19. Garfield, J., & Ben-Zvi, D. (2005). A framework for teaching and assessing reasoning about variability. Statistics Education Research Journal, 4(1), 92–99.Google Scholar
  20. Garfield, J., & Ben-Zvi, D. (2008). Developing students’ statistical reasoning: Connecting research and teaching practice. Berlin: Springer.Google Scholar
  21. Garfield, J., delMas, R. C., & Chance, B. (2007). Using students’ informal notions of variability to develop an understanding of formal measures of variability. In M. C. Lovett & P. Shah (Eds.), Thinking with data (pp. 87–116). New York: Lawrence Erlbaum.Google Scholar
  22. Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177.CrossRefGoogle Scholar
  23. Hancock, C., Kaput, J. J., & Goldsmith, L. T. (1992). Authentic enquiry with data: Critical barriers to classroom implementation. Educational Psychologist, 27(3), 337–364.CrossRefGoogle Scholar
  24. Konold, C. (2002). Teaching concepts rather than conventions. New England Journal of Mathematics, 34(2), 69–81.Google Scholar
  25. Konold, C., Higgins, T., Russell, S. J., & Khalil, K. (2015). Data seen through different lenses. Educational Studies in Mathematics, 88(3), 305–325.CrossRefGoogle Scholar
  26. Konold, C., & Miller, C. (2011). TinkerPlots (Version 2.0) [Computer software]. Key Curriculum Press. Online:
  27. 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.CrossRefGoogle Scholar
  28. Lehrer, R., & English, L. (2017). Introducing children to modeling variability. In D. Ben-Zvi, J. Garfield, & K. Makar (Eds.), International handbook of research in statistics education. Springer international handbooks of Education (pp. 229–260). Cham: Springer.Google Scholar
  29. Lehrer, R., & Schauble, L. (2004). Modelling natural variation through distribution. American Educational Research Journal, 41(3), 635–679.CrossRefGoogle Scholar
  30. Lehrer, R., & Schauble, L. (2012). Seeding evolutionary thinking by engaging children in modeling its foundations. Science Education, 96(4), 701–724.CrossRefGoogle Scholar
  31. Lesh, R., Carmona, G., & Post, T. (2002). Models and modelling. In D. Mewborn, P. Sztajn, D. White, H. Wiegel, R. Bryant, et al. (Eds.), Proceedings of the 24th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 89–98). Columbus, OH: ERIC Clearinghouse.Google Scholar
  32. Makar, K., Bakker, A., & Ben-Zvi, D. (2011). The reasoning behind informal statistical inference. Mathematical Thinking and Learning, 13(1), 152–173.CrossRefGoogle Scholar
  33. Makar, K., & Rubin, A. (2009). A framework for thinking about informal statistical inference. Statistics Education Research Journal, 8(1), 82–105.Google Scholar
  34. Makar, K., & Rubin, A. (2017). Research on inference. In D. Ben-Zvi., K. Makar, & J. Garfield (Eds.), International handbook of research in statistics education. Springer international handbooks of education (pp. 261–294). Cham: Springer.Google Scholar
  35. Moore, D. S. (2004). The basic practice of statistics (3rd ed.). New York: W.H. Freeman.Google Scholar
  36. Moritz, J. B. (2004). Reasoning about covariation. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 227–256). Dordrecht, The Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar
  37. Pfannkuch, M., & Wild, C. (2004). Towards an understanding of statistical thinking. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 17–46). Dordrecht, Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar
  38. Reading, C., & Shaughnessy, C. (2004). Reasoning about variation. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 201–226). Dordrecht, The Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar
  39. Rubin, A., Hammerman, J. K. L., & Konold, C. (2006). Exploring informal inference with interactive visualization software. In Proceedings of the Seventh International Conference on Teaching Statistics [CD-ROM], Salvador, Brazil. International Association for Statistical Education.Google Scholar
  40. Schauble, L. (1996). The development of scientific reasoning in knowledge-rich contexts. Developmental Psychology, 32(1), 102–119.CrossRefGoogle Scholar
  41. Schoenfeld, A. H. (2007). Method. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 69–107). Charlotte, NC: Information Age Publishing.Google Scholar
  42. Shaughnessy, J. M. (2007). Research on statistics learning and reasoning. In F. K. Lester (Ed.), The second handbook of research on mathematics (pp. 957–1010). Charlotte: Information Age Publishing Inc.Google Scholar
  43. Siegler, R. S. (2006). Microgenetic analyses of learning. In D. Kuhn & R. S. Siegler (Eds.), Handbook of child psychology: Cognition, perception, and language (6th ed., Vol. 2, pp. 464–510). Hoboken, NJ: Wiley.Google Scholar
  44. Watkins, A. E., Scheaffer, R. L., & Cobb, G. W. (2004). Statistics in action: Understanding a world of data. Emeryville, CA: Key Curriculum Press.Google Scholar
  45. Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry (with discussion). International Statistical Review, 67, 223–265.CrossRefGoogle Scholar
  46. Zieffler, S. A., & Garfield, J. (2009). Modelling the growth of students’ covariational reasoning during an introductory statistical course. Statistics Education Research Journal, 8(1), 7–31.Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of EducationThe University of HaifaHaifaIsrael

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