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Supporting the Development of Conceptions of Statistics by Engaging Students in Measuring and Modeling Variability

  • Richard Lehrer
  • Min-joung Kim
  • Leona Schauble
Article

Abstract

New capabilities in TinkerPlots 2.0 supported the conceptual development of fifth- and sixth-grade students as they pursued several weeks of instruction that emphasized data modeling. The instruction highlighted links between data analysis, chance, and modeling in the context of describing and explaining the distributions of measures that result from repeatedly measuring multiple objects (i.e., the height of the school’s flagpole, a teacher’s head circumference, the arm-span of a peer). We describe the variety of data representations, statistics, and models that students invented and how these inscriptions were grounded both in their personal experience as measurers and in the affordances of TinkerPlots, which assisted them in quantifying what they could readily display with the computer tool. By inventing statistics, students explored the relation between qualities of distribution and methods for expressing these qualities as a quantity. Attention to different aspects of distribution resulted in the invention of different statistics. Variable invention invited attention to the qualities of “good” measures (statistics), thus meshing conceptual and procedural knowledge. Students used chance simulations, built into TinkerPlots, to generate models that explained variability in a sample of measurements as a composition of true value and chance error. Error was, in turn, decomposed into a variety of sources and associated magnitudes—a form of analysis of variance for children. The dynamic notations of TinkerPlots altered the conceptual landscape of modeling, placing simulation and world on more equal footing, as first suggested by Kaput (Journal of Mathematical Behavior, 17(2), 265–281, 1998).

Keywords

Statistics education Modeling Learning 

Notes

Acknowledgements

Support for this work was provided by the National Science Foundation, REC 0337675. The views expressed do not necessarily represent those of the Foundation.

References

  1. delMas, R. C., & Liu, Y. (2007). Students’ conceptual understanding of the standard deviation. In M. C. Lovett, &P. Shah (Eds.), Thinking with Data (pp. 87–116). New York: Lawrence Erlbaum Associates (Taylor & Francis).Google Scholar
  2. diSessa, A. (2004). Metarepresentation: Native competence and targets for instruction. Cognition and Instruction, 22(3), 293–331.CrossRefGoogle Scholar
  3. 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. 117–147). New York: Lawrence Erlbaum Associates (Taylor & Francis).Google Scholar
  4. Giere, R. N. (1992). Cognitive models of science. Minneapolis, MN: University of Minnesota Press.Google Scholar
  5. Goodman, N. (1976). Languages of art. Indianapolis: Hackett Publishing.Google Scholar
  6. Greeno, J. G., & Hall, R. (1997). Practicing representation: Learning with and about representational forms. Phi Delta Kappan, 78(5), 361–367.Google Scholar
  7. Hall, R., Wright, K., & Wieckert, K. (2007). Interactive and historical processes of distributing statistical concepts through work organization. Mind, Culture, and Activity, 14(1–2), 103–127.Google Scholar
  8. Kaput, J. J. (1998). Representations, inscriptions, descriptions and learning: A kaleidoscope of windows. Journal of Mathematical Behavior, 17(2), 265–281.CrossRefGoogle Scholar
  9. Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), Research on mathematics teaching and learning (pp. 515–556). New York: Macmillian Publishing.Google Scholar
  10. Konold, C., & Miller, C.D. (2005). TinkerPlots: Dynamic data exploration. [Computer software] Emeryville, CA: Key Curriculum Press.Google Scholar
  11. 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
  12. Lehrer, R., & Romberg, T. (1996). Exploring children’s data modeling. Cognition and Instruction, 14(1), 69–108.CrossRefGoogle Scholar
  13. Lehrer R., & Schauble L. (Eds.) (2002). Investigating real data in the classroom: Expanding children’s understanding of math and science. New York: Teachers College Press.Google Scholar
  14. Lehrer, R., & Schauble, L. (2004). Modeling natural variation through distribution. American Educational Research Journal, 41(3), 635–679.CrossRefGoogle Scholar
  15. Lehrer, R., & Schauble, L. (2006). Cultivating model-based reasoning in science education. In R. Keith Sawyer (Ed.), Cambridge handbook of the learning sciences (pp. 371–387). Cambridge: Cambridge University Press.Google Scholar
  16. Lehrer, R., & Schauble, L. (2007). Contrasting emerging conceptions of distribution in contexts of error, natural variation. In M. Lovett & P. Shah (Eds.), Carnegie symposium on cognition: Thinking with data (pp. 149–176). New York: Lawrence Erlbaum Associates.Google Scholar
  17. Lehrer, R., Konold, C., & Kim, M. (2006). Constructing data, modeling chance in the middle school. Paper presented at the annual meeting of the American Educational Research Association, San Francisco, CA.Google Scholar
  18. Lehrer, R., Carpenter, S., Schauble, L., & Putz, A. (2000). Designing classrooms that support inquiry. In J. Minstrell, & E. H. van Zee (Eds.), Inquiring into inquiry: Learning and teaching in science (pp. 80–99). Washington, DC: American Association for the Advancement of Science.Google Scholar
  19. Petrosino, A., Lehrer, R., & Schauble, L. (2003). Structuring error and experimental variation as distribution in the fourth grade. Mathematical Thinking and Learing, 5(2&3), 131–156. CrossRefGoogle Scholar
  20. Porter, T. M. (1986). The rise of statistical thinking 1820–1900. Princeton, NJ: Princeton University Press.Google Scholar
  21. Stigler, S. M. (1986). The history of statistics. Cambridge, MA: Harvard University Press.Google Scholar
  22. Wertsch, J. V. (1998). Mind as action. New York: Oxford University Press. Google Scholar
  23. Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry (with discussion). International Statistical Review, 67(3), 223–265.CrossRefGoogle Scholar
  24. Wilensky, U. (1996). Modeling rugby: Kick first, generalize later? International Journal of Computers for Mathematical Learning, 1, 125–131.Google Scholar
  25. Wilensky, U., & Resnick, M. (1999). Thinking in levels: A dynamic systems approach to making sense of the world. Journal of Science Education and Technology., 8(1), 3–19.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Vanderbilt UniversityNashvilleUS

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