robust understanding of statistical variation

Susan A. Peters

University of Louisville

s.peters@louisville.edu

ABSTRACT

This paper presents a framework that captures the complexity of reasoning about variation in ways that are indicative of robust understanding and describes reasoning as a blend of design, data-centric, and modeling perspectives. Robust understanding is indicated by integrated reasoning about variation within each perspective and across perspectives for four elements: variational disposition, variability in data for contextual variables, variability in relationships among data and variables, and effects of sample size on variability. This holistic image of robust understanding of variation arises from existing expository and empirical literature, and additional empirical study.

Keywords: Statistics education research; Understanding variation; Framework for robust understanding of variation; SOLO Taxonomy

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Statistics Education Research Journal, 10(1), 52-88, http://www.stat.auckland.ac.nz/serj

Ó International Association for Statistical Education (IASE/ISI), May, 2011

REFERENCES

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.

Batanero, C. (2003). Thematic working group 5, Stochastic thinking theme: Curricular issues and teacher education. In M. A. Mariotti (Ed.), European research in mathematics education III: Proceedings of the Third Congress of the European Society for Research in Mathematics Education. Bellaria, Italia: University of Pisa.

[Online: www.dm.unipi.it/~didattica/CERME3/proceedings/Groups/TG5/TG5_introduction1_cerme3.pdf ]

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), 35-65.

Ben-Zvi, D., & Garfield, J. (2004). Statistical literacy, reasoning, and thinking: Goals, definitions, and challenges. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 3-15). Dordrecht, The Netherlands: Kluwer.

Ben-Zvi, D., Gil, E., & Apel, N. (2007, July). What is hidden beyond the data? Helping young students to reason and argue about some wider universe. Paper presented at Fifth International Research Forum on Statistical Reasoning, Thinking, and Literacy (SRTL-5), University of Warwick, UK.

Biggs, J. B., & Collis, K. F. (1982). Evaluating the quality of learning: The SOLO taxonomy (Structure of the Observed Learning Outcome). New York: Academic Press.

Biggs, J. B., & Collis, K. F. (1991). Multimodal learning and the quality of intelligent behavior. In H. A. H. Rowe (Ed.), Intelligence: Reconceptualization and measurement (pp. 57-76). Hillsdale, NJ: Erlbaum.

Bullard, F. (2006, October 11). Washington Post article on penmanship [online ap-stat Electronic Discussion Group of The College Board comment]. Retrieved from http://mathforum.org/kb/thread.jspa?forumID=67&threadID=1467874&messageID=5239082#5239082

Callingham, R. (1997). Teachers' multimodal functioning in relation to the concept of average. Mathematics Education Research Journal, 9(2), 205-224.

Chance, B. L. (2002). Components of statistical thinking and implications for instruction and assessment. Journal of Statistics Education, 10(3).

[Online: http://www.amstat.org/publications/jse/v10n3/chance.html ]

Chance, B., delMas, R., & Garfield, J. (2004). Reasoning about sampling distributions. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 295-323). Dordrecht, The Netherlands: Kluwer.

Clark, J. M., Kraut, G., Mathews, D., & Wimbish, J. (2007). The "fundamental theorem" of statistics: Classifying student understanding of basic statistical concepts. Unpublished manuscript.

[Online: http://www1.hollins.edu/faculty/clarkjm/stat2c.pdf ]

Cobb, G. W. (2005). The introductory statistics course: A saber tooth curriculum. Plenary talk presented at the United States Conference on Teaching Statistics, Columbus, OH.

[Online: http://www.causeweb.org/uscots/uscots05/plenary/ ]

Cobb, G. W., & Moore, D. S. (1997). Mathematics, statistics, and teaching. The American Mathematical Monthly, 104(9), 801-823.

Cobb, P., McClain, K., & Gravemeijer, K. (2003). Learning about statistical covariation. Cognition and Instruction, 21(1), 1-78.

Curcio, F. R. (1987). Comprehension of mathematical relationships expressed in graphs. Journal for Research in Mathematics Education, 18(5), 382-393.

delMas, R. C. (2004). A comparison of mathematical and statistical reasoning. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 79-95). Dordrecht, The Netherlands: Kluwer.

delMas, R., Garfield, J., Ooms, A., & Chance, B. (2007). Assessing students' conceptual understanding after a first course in statistics. Statistics Education Research Journal, 6(2), 28-58.

[Online: http://www.stat.auckland.ac.nz/~iase/serj/SERJ6%282%29_delMas.pdf ]

delMas, R., & Liu, Y. (2005). Exploring students' conceptions of the standard deviation. Statistics Education Research Journal, 4(1), 55-82.

[Online: www.stat.auckland.ac.nz/~iase/serj/SERJ4%281%29_delMas_Liu.pdf ]

Derry, S. J., Levin, J. R., Osana, H. P., Jones, M. S., & Peterson, M. (2000). Fostering students' statistical and scientific thinking: Lessons learned from an innovative college course. American Educational Research Journal, 37(3), 747-773.

Dierdorp, A., Bakker, A., Eijkelhof, H., & van Maanen, J. (2011). Authentic practices as contexts for learning to draw inferences beyond correlated data. Mathematical Thinking and Learning, 13(1-2), 132-151.

Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic, intuitively based misconceptions. Journal for Research in Mathematics Education, 28(1), 96-105.

Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M., & Scheafer, R. (2007). Guidelines for assessment and instruction in statistics education (GAISE) report: A pre-K-12 curriculum framework. Alexandria, VA: American Statistical Association.

[Online: http://www.amstat.org/education/gaise/ ]

Franklin, C., & Mewborn, D. S. (2006). The statistical education of grades pre-K-12 teachers: A shared responsibility. In G. F. Burrill (Ed.), Thinking and reasoning with data and chance: Sixty-eighth annual yearbook of the National Council of Teachers of Mathematics (pp. 335-344). Reston, VA: National Council of Teachers of Mathematics.

Friel, S. N., Curcio, R. F., & 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.

Garfield, J. B. (2002). The challenge of developing statistical reasoning. Journal of Statistics Education, 10(3).

[Online: http://www.amstat.org/publications/jse/v10n3/garfield.html ]

Garfield, J., & Ben-Zvi, D. (2005). A framework for teaching and assessing reasoning about variability. Statistics Education Research Journal, 4(1), 92-99.

[Online: www.stat.auckland.ac.nz/~iase/serj/SERJ4%281%29_Garfield_BenZvi.pdf ]

Garfield, J. B., & Ben-Zvi, D. (with Chance, B., Medina, E., Roseth, C., & Zieffler, A.). (2008). Developing students' statistical reasoning: Connecting research and teaching practice. New York: Springer.

Garfield, J., delMas, R., & 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. 117-148). Mahwah, NJ: Erlbaum.

Gil, E., & Ben-Zvi, D. (2011). Explanations and context in the emergence of students' informal inferential reasoning, Mathematical Thinking and Learning, 13(1&2), 87-108.

Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. New Brunswick, NJ: Aldine Transaction.

Groth, R. (2003). High school students' levels of thinking in regard to statistical study design. Mathematics Education Research Journal, 15(3), 252-269.

Groth, R. E., & Bergner, J. A. (2006). Preservice elementary teachers' conceptual and procedural knowledge of mean, median, and mode. Mathematical Thinking and Learning, 8(1), 37-63.

Hammerman, J. K., & 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/~iase/serj/SERJ3%282%29_Hammerman_Rubin.pdf ]

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

Holcomb, J., Chance, B., Rossman, A., Tietjen, E., & Cobb, G. (2010). Introducing concepts of statistical inference via randomization tests. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. Proceedings of the Eighth International Conference on Teaching Statistics (ICOTS-8), Ljubljana, Slovenia. Voorburg, The Netherlands: International Statistical Institute.

[Online: www.stat.auckland.ac.nz/~iase/publications/icots8/ICOTS8_8D1_HOLCOMB.pdf ]

Konold, C., Harradine, A., & Kazak, S. (2007). Understanding distributions by modeling them. International Journal of Computers for Mathematical Learning, 12(3), 217-230.

Langrall, C. W., & Mooney, E. S. (2002). The development of a framework characterizing middle school students' statistical thinking. In B. Philips (Ed.), Developing a statistically literate society. Proceedings of the Sixth International Conference on Teaching Statistics, Cape Town, South Africa. [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/1/6b3_lang.pdf ]

Langrall, C., Nisbet, S., Mooney, E., & Jansem, S. (2011). The role of context expertise when comparing data. Mathematical Thinking and Learning, 13(1&2), 47-67.

Lann, A., & Falk, R. (2003). What are the clues for intuitive assessment of variability? In C. Lee (Ed.), Reasoning about variability: A collection of research studies. Proceedings of the Third International Research Forum on Statistical Reasoning, Thinking, and Literacy (STRL-3). Lincoln, NE. Mount Pleasant: Central Michigan University.

MacCullough, D. L. (2007). A study of experts' understanding of arithmetic mean. (Unpublished doctoral dissertation). Pennsylvania State University, University Park, PA.

[Online: http://etda.libraries.psu.edu/theses/approved/WorldWideIndex/ETD-1789/ ]

Makar, K., & Confrey, J. (2002). Comparing two distributions: Investigating secondary teachers' statistical thinking. In B. Philips (Ed.), Developing a statistically literate society. Proceedings of the Sixth International Conference on Teaching Statistics, Cape Town, South Africa. [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/1/10_18_ma.pdf ]

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

Makar, K., & Confrey, J. (2005). "Variation-talk": Articulating meaning in statistics. Statistics Education Research Journal, 4(1), 27-54.

[Online: www.stat.auckland.ac.nz/~iase/serj/SERJ4%281%29_Makar_Confrey.pdf ]

Masnick, A. M., & Klahr, D. (2003). Error matters: An initial exploration of elementary school children's understanding of experimental error. Journal of Cognition and Development, 4(1), 67-98.

McClain, K., & Cobb, P. (2001). Supporting students' ability to reason about data. Educational Studies in Mathematics, 45(1-3), 103-129.

Meletiou, M., & Lee, C. (2002). Student understanding of histograms: A stumbling stone to the development of intuitions about variation. In B. Philips (Ed.), Developing a statistically literate society. Proceedings of the Sixth International Conference on Teaching Statistics, Cape Town, South Africa. [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/1/10_19_me.pdf ]

Meletiou-Mavrotheris, M. (2007). The formalist mathematical tradition as an obstacle to stochastical reasoning. In K. Francois & J. P. Van Bendegem (Eds.), Philosophical dimensions in mathematics education (pp. 131-155). New York: Springer.

Meletiou-Mavrotheris, M., & Lee, C. (2003). Studying the evolution of students' conceptions of variation using the transformative and conjecture-driven research design. In C. Lee (Ed.), Reasoning about variability: A collection of research studies. Proceedings of the Third International Research Forum on Statistical Reasoning, Thinking, and Literacy (SRTL-3). Lincoln, NE. Mount Pleasant: Central Michigan University.

Mokros, J., & Russell, S. J. (1995). Children's concepts of average and representativeness. Journal for Research in Mathematics Education, 26(1), 20-39.

Moore, D. S. (1990). Uncertainty. In L. A. Steen (Ed.), On the shoulders of giants: New approaches to numeracy (pp. 95-137). Washington, DC: National Academy Press.

Moore, D. S. (1998). Statistics among the liberal arts. Journal of the American Statistical Association, 93(444), 1253-1259.

Moritz, J. (2004). Reasoning about covariation. In D. Ben-Zvi & J. B. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 227-255). Dordrecht, The Netherlands: Kluwer Academic Publishers.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Pegg, J. (2003). Assessment in mathematics: A developmental approach. In J. M. Royer (Ed.), Mathematical cognition: A volume in current perspectives on cognition, learning, and instruction (pp. 227-259). Greenwich, CT: Information Age.

Pegg, J., & Tall, D. (2001, December). Fundamental cycles in learning algebra: An analysis. Paper presented at the 12^th ICMI study conference on the future of the teaching and learning of Algebra, Melbourne.

Pegg, J., & Tall, D. (2005). The fundamental cycle of concept construction underlying various theoretical frameworks. ZDM, 37(6), 468-475.

Peters, S. A. (2009). Developing an understanding of variation: AP Statistics teachers' perceptions and recollections of critical moments (Unpublished doctoral dissertation). Pennsylvania State University, University Park, PA.

[Online: http://etda.libraries.psu.edu/theses/approved/WorldWideIndex/ETD-4200/ ]

Petrosino, A. J., Lehrer, R., & Schauble, L. (2003). Structuring error and experimental variation as distribution in the fourth grade. Mathematical Thinking and Learning, 5(2&3), 131-156.

Porter, A. L. (2001). Improving statistical education through the experience of reflective practice (Unpublished doctoral dissertation). University of Wollongong, New South Wales, Australia.

[Online: www.stat.auckland.ac.nz/~iase/publications/dissertations/01.Porter.Dissertation.pdf ]

Pressler, M. W. (2006, October 11). The handwriting is on the wall: Researchers see a downside as keyboards replace pens in schools. The Washington Post, p. A01.

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.

[Online: http://www.stat.auckland.ac.nz/~iase/serj/SERJ5%282%29_Prod_Pratt.pdf ]

Reading, C. (2002). Profile for statistical understanding. In B. Philips (Ed.), Developing a statistically literate society. Proceedings of the Sixth International Conference on Teaching Statistics, Cape Town, South Africa. [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/1/1a4_read.pdf ]

Reading, C. (2004). Student description of variation while working with weather data. Statistics Education Research Journal, 3(2), 84-105.

[Online: http://www.stat.auckland.ac.nz/~iase/serj/SERJ3%282%29_Reading.pdf ]

Reading, C., & Reid, J. (2004). Consideration of variation: A model for curriculum development. Paper presented at the International Association for Statistical Education roundtable on curricular development in statistics education, Lund, Sweden.

[Online: www.stat.auckland.ac.nz/~iase/publications/rt04/2.3_Reading&Reid.pdf ]

Reading, C., & Reid, J. (2010). Reasoning about variation: Rethinking theoretical frameworks to inform practice. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. Proceedings of the Eighth International Conference on Teaching Statistics (ICOTS-8), Ljubljana, Slovenia. Voorburg, The Netherlands: International Statistical Institute.

[Online: www.stat.auckland.ac.nz/~iase/publications/icots8/ICOTS8_8E2_READING.pdf ]

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.

Reid, J., & Reading, C. (2005). Developing consideration of variation: Case studies from a tertiary introductory service statistics course. Paper presented at the 55th session of the International Statistical Institute, Sydney.

Reid, J., & Reading, C. (2006). A hierarchy of tertiary students' consideration of variation. In A. Rossman & B. Chance (Eds.), Working cooperatively in statistics education: Proceedings of the Seventh International Conference on Teaching Statistics (ICOTS-7), Salvador, Brazil. [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/17/C122.pdf ]

Reid, J., & Reading, C. (2008). Measuring the development of students' consideration of variation. Statistics Education Research Journal, 7(1), 40-59.

[Online: www.stat.auckland.ac.nz/~iase/serj/SERJ7%281%29_Reid_Reading.pdf ]

Reid, J., & Reading, C. (2010). Developing a framework for reasoning about explained and unexplained variation. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. Proceedings of the Eighth International Conference on Teaching Statistics (ICOTS-8), Ljubljana, Slovenia. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/icots8/ICOTS8_C169_REID.pdf ]

Rubin, A., Bruce, B., & Tenney, Y. (1990). Learning about sampling: Trouble at the core of statistics. In D. Vere-Jones (Ed.), Proceedings of the Third International Conference on Teaching Statistics (Vol. 1, pp. 314-319). Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/18/BOOK1/A9-4.pdf ]

Saldanha, L., & Thompson, P. (2002). Conceptions of sample and their relationship to statistical inference. Educational Studies in Mathematics, 51(3), 257-270.

Shaughnessy, J. M. (1992). Research on 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. (1997). Missed opportunities in research on the teaching and learning of data and chance. In F. Biddulph & K. Carr (Eds.), People in mathematics education. Proceedings of the 20^th annual conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 6-22). Waikato, New Zealand: Mathematics Education Research Group of Australasia.

Shaughnessy, J. M. (2007). Research on statistics learning and reasoning. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 957-1009). Greenwich, CT: Information Age.

Shaughnessy, J. M., Canada, D., & Ciancetta, M. (2003). Middle school students' thinking about variability in repeated trials: A cross-task comparison. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the joint meeting of the 27^th conference of the International Group for the Psychology of Mathematics Education and the 25^th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 159-165). Honolulu, HI: University of Hawaii.

Shaughnessy, J. M., & Ciancetta, M. (2002). Students' understanding of variability in a probability environment. In B. Philips (Ed.), Developing a statistically literate society. Proceedings of the Sixth International Conference on Teaching Statistics, Cape Town, South Africa. [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/1/6a6_shau.pdf ]

Shaughnessy, J. M., Ciancetta, M., Best, K., & Canada, D. (2004). Students' attention to variability when comparing distributions. Paper presented at the Research Presession of the 82nd annual meeting of the National Council of Teachers of Mathematics, Philadelphia, PA.

Shaughnessy, J. M., Ciancetta, M., & Canada, D. (2004). Types of student reasoning on sampling tasks. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 177-184). Bergen, Norway: Psychology of Mathematics Education.

Silva, C. B., & Coutinho, C. Q. S. (2006). The variation concept: A study with secondary school mathematics teachers. In A. Rossman & B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics (ICOTS-7), Salvador, Brazil. [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.

Silva, C. B., & Coutinho, C. Q. S. (2008). Reasoning about variation of a univariate distribution: A study with secondary math teachers. In C. Batanero, G. Burrill, C. Reading, & A. Rossman (Eds.), Joint ICMI/IASE study: Teaching statistics in school mathematics. Challenges for teaching and teacher education. Proceedings of the ICMI Study 18 and 2008 IASE Round Table Conference. Monterrey, Mexico: International Commission on Mathematical Instruction and the International Association for Statistical Education.

Snee, R. D. (1999). Discussion: Development and use of statistical thinking: A new era. International Statistical Review, 67(3), 255-258.

Sorto, M. A. (2004). Prospective middle school teachers' knowledge about data analysis and its applications to teaching (Unpublished doctoral dissertation). Michigan State University, East Lansing, MI.

[Online: www.stat.auckland.ac.nz/~iase/publications/dissertations/04.Sorto.Dissertation.pdf ]

Truran, J. (1995). Some undergraduates' understanding of the meaning of a correlation coefficient. In W. Atweh & S. Flavel (Eds.), Proceedings of the 18^th annual conference of the Mathematics Education Research Group of Australasia (pp. 524-529), Darwin, Australia: Mathematics Education Research Group of Australasia.

Watson, J. M. (2002). Lessons from variation research I: Student understanding. In M. Goos & T. Spencer (Eds.), Mathematics-making waves. Proceedings of the 19^th biennial conference of the Australian Association of Mathematics Teachers Inc., Brisbane (pp. 261-268). Adelaide, SA: Australian Association of Mathematics Teachers.

Watson, J. M., & Callingham, R. A. (2003). Statistical literacy: A complex hierarchical construct. Statistics Education Research Journal, 2(2), 3-46.

[Online: www.stat.auckland.ac.nz/~iase/serj/SERJ2%282%29_Watson_Callingham.pdf ]

Watson, J. M., Callingham, R. A., & Kelly, B. A. (2007). Students' appreciation of expectation and variation as a foundation for statistical understanding. Mathematical Thinking and Learning, 9(2), 83-130.

Watson, J. M., Collis, K. F., Callingham, R. A., & Moritz, J. B. (1995). A model for assessing higher order thinking in statistics. Educational Research and Evaluation: An International Journal on Theory and Practice, 1(3), 247-275.

Watson, J. M., & Kelly, B. A. (2002a). Can grade 3 students learn about variation? In B. Philips (Ed.), Developing a statistically literate society. Proceedings of the Sixth International Conference on Teaching Statistics, Cape Town, South Africa. [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/1/2a1_wats.pdf ]

Watson, J. M., & Kelly, B. A. (2002b). Grade 5 students' appreciation of variation. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 385-392). Norwich, UK: University of East Anglia.

Watson, J. M., & Kelly, B. A. (2004). Expectation versus variation: Students' decision making in a chance environment. Canadian Journal of Science, Mathematics, and Technology Education, 4(3), 371-396.

Watson, J. M., Kelly, B. A., Callingham, R. A., & Shaughnessy, J. M. (2003). The measurement of school students' understanding of statistical variation. International Journal of Mathematical Education in Science and Technology, 34(1), 1-29.

Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223-265.

Wilensky, U. (1995). Learning probability through building computational models. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 152-159). Recife, Brazil: Program Committee of the 19th PME Conference.

Wilensky, U. (1997). What is normal anyway? Therapy for epistemological anxiety. Educational Studies in Mathematics, 33(2), 171-202.

Zieffler, A. S., & Garfield, J. B. (2009). Modeling the growth of students' covariational reasoning during an introductory statistics course. Statistics Education Research Journal, 8(1), 7-31.

[Online: www.stat.auckland.ac.nz/~iase/serj/SERJ8%281%29_Zieffler_Garfield.pdf ]

SUSan A. Peters

University of Louisville

College of Education and Human Development

Louisville, KY 40292