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Educational Studies in Mathematics

, Volume 88, Issue 3, pp 305–325 | Cite as

Data seen through different lenses

  • Clifford KonoldEmail author
  • Traci Higgins
  • Susan Jo Russell
  • Khalimahtul Khalil
Article

Abstract

Statistical reasoning focuses on properties that belong not to individual data values but to the entire aggregate. We analyze students’ statements from three different sources to explore possible building blocks of the idea of data as aggregate and speculate on how young students go about putting these ideas together. We identify four general perspectives that students use in working with data, which in addition to an aggregate perspective include regarding data as pointers, as case values, and as classifiers. Some students seem inclined to view data from only one of these three alternative perspectives, which then influences the types of questions they ask, the data representations they generate or prefer, the interpretations they give to notions such as the average, and the conclusions they draw from the data.

Keywords

Interpreting graphs Interpreting distributions Conceptions of averages Statistical reasoning Informal statistical inference 

References

  1. Bakker, A. (2004). Reasoning about shape as a pattern in variability. Statistics Education Research Journal, 3(2), 64–83.Google Scholar
  2. Bakker, A. (2007). Diagrammatic reasoning and hypostatic abstraction in statistics education. Semiotica, 164(1/4), 9–29.CrossRefGoogle Scholar
  3. Bakker, A., & Gravemeijer, K. P. E. (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). Dordrecht, The Netherlands: Kluwer Academic Press.CrossRefGoogle Scholar
  4. Ben-Zvi, D. (2004). Reasoning about data analysis. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 121–145). Dordrecht, The Netherlands: Kluwer Academic Press.CrossRefGoogle Scholar
  5. Ben-Zvi, D., & Amir, Y. (2005). How do primary school students begin to reason about distributions? In K. Makar (Ed.), Reasoning about distribution. Proceedings of the Fourth International Research Forum on Statistical Reasoning, Thinking, and Literacy, University of Auckland, New Zealand, 2-7 July, 2005. Brisbane: The University of Queensland. (CD-ROM).Google Scholar
  6. Ben-Zvi, D., & Arcavi, A. (2001). Junior high school students’ construction of global views of data and data representations. Educational Studies in Mathematics, 45, 35–65.CrossRefGoogle Scholar
  7. Bertin, J. (1983). Semiology of graphics: Diagrams networks maps (W. J. Berg, Trans.). Madison: University of Wisconsin Press. Original work published 1967.Google Scholar
  8. Biehler, R. (1994). Probabilistic thinking, statistical reasoning, and the search for causes—Do we need a probabilistic revolution after we have taught data analysis? In J. Garfield (Ed.), Research papers from ICOTS 4 (pp. 20–37). Minneapolis, MN: University of Minnesota.Google Scholar
  9. Bowker, G. C., & Star, S. L. (1999). Sorting things out: Classification and its consequences. Cambridge, MA: MIT Press.Google Scholar
  10. Bright, G. W., & Friel, S. N. (1998). Helpingstudents interpret data. In S.P. Lajoie (Ed.), Reflections on statistics:Learning, teaching, and assessment in grades K-12 (pp. 63–88). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  11. Cobb, P. (1999). Individual and collective mathematical development: The case of statistical data analysis. Mathematical Thinking and Learning, 1(1), 5–43.CrossRefGoogle Scholar
  12. Cobb, P., McClain, K., & Gravemeijer, K. (2003). Learning about statistical covariation. Cognition and Instruction, 21(1), 1–78.CrossRefGoogle Scholar
  13. Curcio, F. R. (1987). Comprehension of mathematical relationships expressed in graphs. Journal for Research in Mathematics Education, 18(5), 382–393.CrossRefGoogle Scholar
  14. English, L. D. (2012). Data modelling with first-grade students. Educational Studies in Mathematics, 81(1), 15–30.CrossRefGoogle Scholar
  15. Fisher, R. A. (1990). Statistical methods experimental design and scientific inference. Oxford: Oxford University Press. First published in 1925.Google Scholar
  16. Friel, S. N., Curcio, F. R., & Bright, G. W. (2001). Making sense of graphs: Critical factors influencing comprehension and instructional implications. Journal for Research in Mathematics Education, 32(2), 124–158.CrossRefGoogle Scholar
  17. Gal, I., Rothschild, K., & Wagner, D. A. (1990). Statistical concepts and statistical reasoning in school children: Convergence or divergence? Boston: Paper presented at the annual meeting of the American Educational Research Association, Boston.Google Scholar
  18. Gould, S. J. (1996). Full house. New York: Harmony Books.CrossRefGoogle Scholar
  19. Hammerman, J., & Rubin, A. (2004). Strategies for managing statistical complexity with new software tools. Statistics Education Research Journal, 3(2), 17–41.Google Scholar
  20. 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
  21. Jones, G. A., Thornton, C. A., Langrall, C. W., Mooney, E. S., Perry, B., & Putt, I. J. (1999). A framework for assessing students’ statistical thinking. San Francisco: Paper presented at the annual meeting of the Research Presession of the National Council of Teachers of Mathematics, San Francisco.Google Scholar
  22. Konold, C., & Higgins, T. (2002). Highlights of related research. In S. J. Russell, D. Schifter, & V. Bastable (Eds.), Developing mathematical ideas: Working with data (pp. 165–201). Parsippany, NJ: Dale Seymour Publications.Google Scholar
  23. Konold, C., & Higgins, T. L. (2003). Reasoning about data. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 193–215). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  24. Konold, C., & Khalil, K. (2003). If u can graff these numbers — 2, 15, 6 — your stat literit. Paper presented at the annual meeting of the American Educational Research Association, Chicago.Google Scholar
  25. Konold, C., & Lehrer, R. (2008). Technology and mathematics education: An essay in honor of Jim Kaput. In L. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 49–72). New York: Routledge.Google Scholar
  26. 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
  27. Konold, C., Pollatsek, A., Well, A., & Gagnon, A. (1997). Students analyzing data: Research of critical barriers. In J. B. Garfield & G. Burrill (Eds.), Research on the role of technology in teaching and learning statistics: 1996 Proceedings of the 1996 IASE Round Table Conference (pp. 151–167). Voorburg, The Netherlands: International Statistical Institute. Retrieved from http://www.dartmouth.edu/~chance/teaching_aids/IASE/IASE.book.pdf.
  28. Konold, C., Robinson, A., Khalil, K., Pollatsek, A., Well, A., Wing, R., et al. (2002). Students’ use of modal clumps to summarize data. In B. Phillips (Ed.), Proceedings of the Sixth International Conference on Teaching of Statistics (CD-ROM). Voorburg, The Netherlands: International Statistical Institute.Google Scholar
  29. Lehrer, R., & Schauble, L. (2004). Modeling natural variation through distribution. American Educational Research Journal, 41(3), 635–679.CrossRefGoogle Scholar
  30. Lehrer, R., Schauble, L., Carpenter, S., & Penner, D. (2000). The inter-related development of inscriptions and conceptual understanding. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools and instructional design (pp. 325–360). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  31. Makar, K., & Confrey, J. (2004). Secondary teachers’ reasoning about comparing two groups. In D. Ben-Zv i & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 353–373). Dordrecht, the Netherlands: Kluwer Academic Publisher.Google Scholar
  32. Mokros, J., & Russell, S. (1995). Children’s concepts of average and representativeness. Journal for Research in Mathematics Education, 26(1), 20–39.CrossRefGoogle Scholar
  33. Moore, D. S. (1990). Uncertainty. In L. A. Steen (Ed.), On the shoulders of giants (pp. 95–137). Washington, DC: National Academy Press.Google Scholar
  34. Prodromou, T., & Pratt, D. (2006). The role of causality in the co-ordination of two perspectives on distribution within a virtual simulation. Statistics Education Research Journal, 5(2), 69–88.Google Scholar
  35. Reading, C., & Canada, D. (2011). Teachers’ knowledge of distribution. In C. Batanero, G. Burrill, & C. Reading (Eds.), Teaching statistics in school mathematics—Challenges for teaching and teacher education: A joint ICMI/IASE study (pp. 223–234). New York: Springer.CrossRefGoogle Scholar
  36. Reading, C., & Reid, J. (2006). An emerging hierarchy of reasoning about distribution: From a variation perspective. Statistics Education Research Journal, 5(2), 46–68.Google Scholar
  37. Roth, W.-M. (2005). Mathematical inscriptions and the reflexive elaboration of understanding: An ethnography of graphing and numeracy in a fish hatchery. Mathematical Thinking and Learning, 7(2), 75–110.CrossRefGoogle Scholar
  38. Russell, S. J., Schifter, D., & Bastable, V. (2002). Developing mathematical ideas: Working with data casebook. Glenview, IL: Pearson Education, Inc.Google Scholar
  39. Shaughnessy, J. M., Watson, J., Moritz, J., & Reading, C. (1999). School mathematics students’ acknowledgment of statistical variation. Paper presented at the 77th annual meeting of the National Council of Teachers of Mathematics, San Francisco.Google Scholar
  40. Stigler, S. M. (1999). Statistics on the table: The history of statistical concepts and methods. Cambridge, MA: Harvard University Press.Google Scholar
  41. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press.Google Scholar
  42. Wainer, H. (2001). Order in the court. Chance, 14(1), 43–46.Google Scholar
  43. Watson, J. M. (2009). The influence of variation and expectation on the developing awareness of distribution. Statistics Education Research Journal, 8(1), 32–61.Google Scholar
  44. Watson, J. M., & Moritz, J. B. (1999). The beginning of statistical inference: Comparing two sets of data. Educational Studies in Mathematics, 37(2), 145–168.CrossRefGoogle Scholar
  45. Watson, J. M., & Moritz, J. B. (2000). The longitudinal development of understanding of average. Mathematical Thinking and Learning, 2(1), 9–48.Google Scholar
  46. Wild, C. (2006). The concept of distribution. Statistics Education Research Journal, 5(2), 10–25.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Clifford Konold
    • 1
    Email author
  • Traci Higgins
    • 2
  • Susan Jo Russell
    • 2
  • Khalimahtul Khalil
    • 1
  1. 1.Scientific Reasoning Research InstituteUniversity of Massachusetts AmherstAmherstUSA
  2. 2.TERCCambridgeUSA

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