THE CONCEPT OF DISTRIBUTION

“Variation-talk”: articulating meaning in statistics

Katie Makar

University of Queensland, Australia

k.makar@uq.edu.au

Jere Confrey

Washington University in St. Louis, U.S.A.

jconfrey@wustl.edu

SUMMARY

Little is known about the way that teachers articulate notions of variation in their own words. The study reported here was conducted with 17 prospective secondary math and science teachers enrolled in a preservice teacher education course which engaged them in statistical inquiry of testing data. This qualitative study examines how these preservice teachers articulated notions of variation as they compared two distributions. Although the teachers made use of standard statistical language, they also expressed rich views of variation through nonstandard terminology. This paper details the statistical language used by the prospective teachers, categorizing both standard and nonstandard expressions. Their nonstandard language revealed strong relationships between expressions of variation and expressions of distribution. Implications and the benefits of nonstandard language in statistics are outlined.

Keywords: Statistical reasoning; Statistics education; Reasoning about variation and distribution; Mathematics education; Teacher education; Nonstandard language

__________________________

Statistics Education Research Journal, 4(1), 27-54, http://www.stat.auckland.ac.nz/serj

Ó International Association for Statistical Education (IASE/ISI), May, 2005

References

Bakker, A. (2004). Design research in statistics education: On symbolizing and computer tools. Published Doctoral dissertation. Utrecht, The Netherlands: Freudenthal Institute. [Online: www.stat.auckland.ac.nz/~iase/publications/dissertations/dissertations.php]

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 Publishers.

Biehler, R. (1997). Student’s difficulties in practicing computer-supported data analysis: Some hypothetical generalizations from results of two exploratory studies. In J. Garfield & G. Burrill (Eds.), Research on the role of technology in teaching and learning statistics (pp. 169-190). Voorburg, The Netherlands: International Statistics Institute.

Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex, and setting. Buckingham: Open University Press.

Canada, D. (2004). Preservice teachers’ understanding of variation. Unpublished doctoral dissertation, Portland State University, Portland, Oregon (USA). [Online: www.stat.auckland.ac.nz/~iase/publications/dissertations/dissertations.php]

Cobb, P. (1999). Individual and collective mathematical development: The case of statistical data analysis. Mathematical Thinking and Learning, 1(1), 5-43.

Confrey, J. (1998). Voice and perspective: Hearing epistemological innovation in students’ words. In M. Larochelle, N. Bednarz, & J. Garrison (Eds.), Constructivism and education (pp. 104-120). New York, N.Y.: Cambridge University Press.

Confrey, J., & Makar, K. (2002). Developing secondary teachers’ statistical inquiry through immersion in high-stakes accountability data. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the Twenty-fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA), Athens, GA, (Vol. 3, pp. 1267-1279). Columbus, OH: ERIC Clearinghouse, Mathematics and Environment Education.

Confrey, J., & Makar, K. (2005). Critiquing and improving the use of data from high-stakes tests with the aid of dynamic statistics software. In C. Dede, J. Honan, & L. Peters (Eds.), Scaling up for success: Lessons from technology-based educational improvement (pp. 198-226). Cambridge, MA: Harvard College of Education.

Confrey, J., Makar, K., & Kazak, S. (2004). Undertaking data analysis of student outcomes as professional development for teachers. International Reviews on Mathematical Education (Zentralblatt für Didaktik der Mathematik), 36(1), 32-40.

Cramer, D. (1997). Basic statistics for social research. London: Routledge.

Finzer, W. (2001). Fathom! (Version 1.16) [Computer Software]. Emeryville, CA: Key Curriculum Press.

Fischbein, E. (1987). Intuition in mathematics and science: An educational approach. Dordrecht, The Netherlands: D. Reidel Publishing Company.

Flyvbjerg, B. (2001). Making social science matter: Why social inquiry fails and how it can succeed again. Cambridge, UK: Cambridge University Press.

Hammerman, J., & Rubin, A. (2004). Strategies for managing statistical complexity with new software tools. Statistics Education Research Journal, 3(2), 17-41. [Online: www.stat.auckland.ac.nz/serj]

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

Hawkins, A. (Ed.). (1990). Teaching teachers to teach statistics. Voorburg, The Netherlands: International Statistics Institute.

Konold, C. (2002). “Hat Plots? ” Unpublished manuscript, University of Massachusetts, Amherst.

Konold, C., & Higgins, T. (2002). Highlights of related research. In S. J. Russell, D. Schifter, & V. Bastable, Developing mathematical ideas: Working with data, (pp. 165-201). Parsippany, NJ: Dale Seymour Publications.

Konold, C., Higgins, T., Russell, S. J., & Khalil, K. (submitted). Data seen through different lenses. Journal for Research in Mathematics Education.

Konold, C., & Miller, C. (2004). Tinkerplots (Version 1.0) [Computer Software]. Emeryville, CA: Key Curriculum Press.

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.

Konold, C., Pollatsek, A., Well, A., & Gagnon, A. (1997). Students analyzing data: research of critical barriers. In J. Garfield & G. Burrill (Eds.), Research on the role of technology in teaching and learning statistics (pp. 151-167). Voorburg, The Netherlands: International Statistics Institute.

Konold, C., Robinson, A., Khalil, K., Pollatsek, A., Well, A., Wing, R., & Mayr, S. (2002). Students’ use of modal clumps to summarize data. In B. Phillips (Ed.), Proceedings of the Sixth International Conference on Teaching Statistics, Cape Town, South Africa [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.

Kosslyn, S. (1994). Elements of graph design. New York: W. H. Freeman and Company.

Lemke, J. (1990). Talking science: Language, learning, and values. Norwood, NJ: Ablex Publishing.

Makar, K. (2004). Developing statistical inquiry: Prospective secondary mathematics and science teachers’ investigations of equity and fairness through analysis of accountability data. Unpublished Doctoral dissertation, University of Texas at Austin (USA). [Online: www.stat.auckland.ac.nz/~iase/publications/dissertations/dissertations.php]

Makar, K., & Confrey, J. (submitted). Modeling fairness in student achievement in mathematics using statistical software by preservice secondary teachers. ICMI Study Volume 14: Applications and modeling in mathematics education.

Makar, K., & Confrey, J. (2004). Secondary teachers’ reasoning about comparing two groups. In D. Ben-Zvi & J. Garfield (Eds.), The challenges of developing statistical literacy, reasoning, and thinking (pp. 353-373). Dordrecht, the Netherlands: Kluwer Academic Publishers.

Marshall, J., Makar, K., & Kazak, S. (2002). Young urban students’ conceptions of data uses, representation, and analysis. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the Twenty-fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA), Athens, GA, (Vol. 3, pp. 1293-1304). Columbus, OH: ERIC Clearinghouse, Mathematics and Environment Education.

McClain, K. (2002). Supporting teachers’ understanding of statistical data analysis: Learning trajectories as tools for change. In B. Phillips (Ed.), Proceedings of the Sixth International Conference on Teaching Statistics, Cape Town, South Africa [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.

Moore, D. (1990). Uncertainty. In L. A. Steen (Ed.), On the shoulders of giants (pp. 95-137). Washington, D.C.: Academy Press.

Moore, D., & McCabe, G. (1993). Introduction to the practice of statistics (2nd ed.). New York: Freeman.

National Research Council (1999). High stakes: Testing for tracking, promotion, and graduation. Heubert, J. P., & Hauser, R. M. (Eds.), Washington, D.C.: National Academy Press.

National Research Council (2000). Inquiry and the National Science Education Standards: A guide for teaching and learning. Washington, D.C.: National Academy Press.

National Research Council (2001). Knowing what students know. Washington, D.C.: National Academy Press.

Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings: Learning cultures and computers. Dordrecht, The Netherlands: Kluwer Academic Publishers.

Pfannkuch, M., & Begg, A. (2004). The school statistics curriculum: Statistics and probability education literature review. Mathematics Curriculum Reference Group Report to the Ministry of Education. Auckland, New Zealand.

Pfannkuch, M., Budgett, S., Parsonage, R., & Horring, J. (2004). Comparison of data plots: Building a pedagogical framework. Paper presented at the Tenth International Congress on Mathematics Education (ICME-10), Copenhagen, Denmark, 4-11 July.

Reading, C. (2004). Student description of variation while working with weather data. Statistics Education Research Journal, 3(2), 84-105. [Online: www.stat.auckland.ac.nz/serj]

Reading, C., & Shaughnessy, J. M. (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.

Russell, S. J., & Mokros, J. R. (1990). What’s typical: Teacher’s descriptions of data. Paper presented at the Annual Meeting of the American Educational Research Association, Boston, MA.

Schoenfeld, A. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In J. Voss, D. Perkins, & J. Segal (Eds.), Informal reasoning and education (pp. 311-343). Hillsdale, NJ: Lawrence Erlbaum Associates.

Shaughnessy, J. M. (1992). Research in probability and statistics: Reflections and directions. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 465-494). New York: MacMillan.

Shaughnessy, J. M. (2001). Draft of grant proposal sent to the National Science Foundation.

Shaughnessy, J. M., Watson, J., Moritz, J., & Reading, C. (1999, April). School mathematics students’ acknowledgement of statistical variation. In C. Maher (Chair), There’s More to Life than Centers. Presession Research Symposium, 77th Annual National Council of Teachers of Mathematics Conference, San Francisco, CA.

von Glasersfeld, E. (Ed.) (1991). Radical constructivism in mathematics education. Dordrecht, the Netherlands: Kluwer Academic Publisher.

Watkins, A. E., Scheaffer, R., & Cobb, G. (2003). Statistics in action: Understanding a world of data. Emeryville, CA: Key Curriculum Press.

Watson, J., & Moritz, J. B. (1999). The beginning of statistical inference: Comparing two data sets. Educational Studies in Mathematics, 37, 145-168.

Katie Makar

School of Education, Social Sciences Bldg

University of Queensland

Brisbane, Queensland 4072

Australia