THE INFLUENCE OF VARIATION AND EXPECTATION ON THE DEVELOPING AWARENESS OF DISTRIBUTION
DOI:
https://doi.org/10.52041/serj.v8i1.456Abstract
This study considers the evolving influence of variation and expectation on the development of school students’ appreciation of distribution as displayed in their construction of graphical representations of data sets. Three interview protocols are employed, presenting different contexts within which 109 students, ranging in age from 6 to 15 years, could display and interpret their understanding. Responses are analyzed within a hierarchical cognitive framework. It is hypothesized from the analysis that, contrary to the order in which expectation and variation are introduced in the school curriculum, the natural tendency for students is to acknowledge variation first and then expectation.
First published May 2009 at Statistics Education Research Journal Archives
References
Akker, 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: Kluwer Academic Publishers.
Ben-Zvi, D., & Amir, Y. (2005). How do primary students begin to reason about distributions? In K. Makar (Ed.), Reasoning about Distribution: A collection of studies. Proceedings of the Fourth International Research Forum on Statistical Reasoning, Thinking and Literacy (SRTL-4). [CDROM, with video segments]. Brisbane, Australia: University of Queensland.
Biggs, J. B. (1992). Modes of learning, forms of knowing, and ways of schooling. In A. Demetriou, M. Shayer, & A. Efklides (Eds.), Neo-Piagetian theories of cognitive development: Implications and applications for education (pp. 31-51). London: Routledge.
Biggs, J. B., & Collis, K. F. (1982). Evaluating the quality of learning: The SOLO taxonomy. New York: Academic Press.
Burns, R. B. (2000). Introduction to research methods (4th ed.). London: Sage.
Chick, H. (2004). Simple strategies for dealing with data. Australian Mathematics Teacher, 60(3), 20-24.
Chick, H. L., & Watson, J. M. (2001). Data representation and interpretation by primary school students working in groups. Mathematics Education Research Journal, 13, 91-111.
Curcio, F. R. (1987). Comprehension of mathematical relationships expressed in graphs. Journal for Research in Mathematics Education, 18, 382-393.
Curcio, F. R., & Artzt, A. F. (1996). Assessing students’ ability to analyze data: Reaching beyond computation. Mathematics Teacher, 89(8), 668-673.
delMas, R. C., & Liu, Y. (2003). Exploring students’ understanding of statistical variation. In C. Lee (Ed.), Reasoning about variability: Proceedings of the Third International Research Forum on Statistical Reasoning, Thinking, and Literacy [CDROM]. Mt. Pleasant, MI: Central Michigan University.
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.
Friel, S. N., O’Connor, W., & Mamer, J. D. (2006). More than “Meanmedianmode” and a bar graph: What’s needed to have a statistical conversation. In G. F. Burrill (Ed.), Thinking and reasoning with data and chance (pp. 117-137). Reston, VA: National Council of Teachers of Mathematics.
Green, D. (1993). Data analysis: What research do we need? In L. Pereira-Mendoza (Ed.), Introducing data analysis in the schools: Who should teach it? (pp. 219-239). Voorburg, The Netherlands: International Statistical Institute.
Kelly, B. A., & Watson, J. M. (2002). Variation in a chance sampling setting: The lollies task. In B. Barton, K. C. Irwin, M. Pfannkuch, & M. O. J. Thomas (Eds.), Mathematics education in the South Pacific: Proceedings of the twenty-sixth annual conference of the Mathematics Education Research Group of Australasia (Vol. 2, pp. 366-373). Sydney, NSW: MERGA.
Kerslake, D. (1981). Graphs. In K. M. Hart (Ed.), Children’s understanding of mathematics: 11-26 (pp. 120-136). London: John Murray.
Kirkpatrick, E. M. (Ed.). (1983). Chambers 20th century dictionary. Edinburgh: W. & R. Chambers.
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.
Konold, C., Higgins, T. L., Russell, S. J., & Khalil, K. (2003). Data seen through different lenses. Amherst, MA: University of Massachusetts. [Online: www.umass.edu/srri/serg/papers.html]
Langrall, C., Nisbet, S., & Mooney, E. (2006). The interplay between students’ statistical knowledge and context knowledge in analyzing data. In A. Rossman & B. Chance (Eds.), Working Cooperatively in Statistics Education. Proceedings of the Seventh International Conference on Teaching Statistics, Salvador, Brazil. [CDROM]. Voorburg, The Netherlands: International Statistical Institute. [Online: http://www.stat.auckland.ac.nz/~iase/publications/17/2A3_LANG.pdf]
Lehrer, R., & Romberg, T. (1996). Exploring children’s data modeling. Cognition and Instruction, 14(1), 69-108.
Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs and graphing: Tasks, learning and teaching. Review of Educational Research, 60(1), 1-64.
Mevarech, Z. R., & Kramarsky, B. (1997). From verbal descriptions to graphic representations: Stability and change in students’ alternative conceptions. Educational Studies in Mathematics, 32, 229-263.
Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). Thousand Oaks, CA: Sage.
Mooney, E., Langrall, C., & Nisbet, S. (2006). Developing a model to describe the use of context knowledge in data explorations. In A. Rossman & B. Chance (Eds.), Working Cooperatively in Statistics Education. Proceedings of the Seventh International Conference on Teaching Statistics, Salvador, Brazil. [CDROM]. Voorburg, The Netherlands: International Statistical Institute. [Online: http://www.stat.auckland.ac.nz/~iase/publications/17/2A4_MOON.pdf]
Moore, D. S. (1990). Uncertainty. In L. S. Steen (Ed.), On the shoulders of giants: New approaches to numeracy (pp. 95-137). Washington, DC: National Academy Press.
Moore, D. S. (1997). New pedagogy and new content: The case of statistics. International Statistical Review, 65(2), 123-165.
Moore, D. S., & McCabe, G. P. (1993). Introduction to the practice of statistics (2nd ed.). New York: W. H. Freeman.
Moritz, J. B. (2000). Graphical representations of statistical associations by upper primary students. In J. Bana & A. Chapman (Eds.), Mathematics education beyond 2000: Proceedings of the twenty-third annual conference of the Mathematics Education Research Group of Australasia (Vol. 2, pp. 440-447). Sydney: MERGA.
Moritz, J. B. (2002). Study times and test scores: What students’ graphs show. Australian Primary Mathematics Classroom, 7(1), 24-31.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Pegg, J. E. (2002a). Assessment in mathematics: A developmental approach. In J. M.
Royer (Ed.), Mathematical cognition (pp. 227-259). Greenwich, CT: Information Age Publishing.
Pegg, J. E. (2002b). Fundamental cycles of cognitive growth. In A. Cockburn & E. Nardi (Eds.), Proceedings of the 26th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 41-48). Norwich, UK: University of East Anglia.
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.
Pfannkuch, M., Rubick, A., & Yoon, C. (2002). Statistical thinking and transnumeration. In B. Barton, K. C. Irwin, M. Pfannkuch, & M. O. J. Thomas (Eds.), Mathematics education in the South Pacific: Proceedings of the twenty-fifth annual conference of the Mathematics Education Research Group of Australasia (Vol. 2, pp. 567-574). Sydney: MERGA.
Reading, C., & Shaughnessy, M. (2000). Student perceptions of variation in a sampling situation. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 89-96). Hiroshima, Japan: Hiroshima University.
Reading, C., & Shaughnessy, 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: Kluwer Academic Publishers.
Roth, W-H., & Bowen, G. M. (2003). When are graphs worth ten thousand words? An expert -expert study. Cognition and Instruction, 21(4), 429-473.
Russell, S. J. (2006). What does it mean that “5 has a lot”? From the world to data and back. In G. F. Burrill (Ed.), Thinking and reasoning with data and chance (pp. 17-29). Reston, VA: National Council of Teachers of Mathematics.
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 twentieth annual conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 6-22). Waikato, NZ: MERGA.
Shaughnessy, J. M. (2006). Research on students’ understanding of some big concepts in statistics. In G. F. Burrill (Ed.), Thinking and reasoning with data and chance (pp. 77-98). Reston, VA: National Council of Teachers of Mathematics.
Shaughnessy, J. M. (2007). Research on statistics learning and reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 957-1009). Charlotte, NC: Information Age Publishing.
Swan, M. (1988). Learning the language of functions and graphs. In J. Pegg (Ed.), Mathematics Interfaces: Proceedings of the twelfth biennial conference of The Australian Association of Mathematics Teachers (pp. 76-80). Newcastle, NSW: The New England Mathematical Association.
Watson, J. M. (2001). Longitudinal development of inferential reasoning by school students. Educational Studies in Mathematics, 47, 337-372.
Watson, J. M. (2002). Inferential reasoning and the influence of cognitive conflict. Educational Studies in Mathematics, 51, 225-256.
Watson, J. M. (2005). Variation and expectation as foundations for the chance and data curriculum. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, & A. Roche (Eds.), Building connections: Theory, research and practice (Proceedings of the 28th annual conference of the Mathematics Education Research Group of Australasia, Melbourne, pp. 35-42). Sydney: MERGA.
Watson, J. M. (2006). Statistical literacy at school: Growth and goals. Mahwah, NJ: Lawrence Erlbaum.
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, 83-130.
Watson, J. M., & Kelly, B. A. (2002a). Can grade 3 students learn about variation? In B. Phillips (Ed.), Proceedings of the Sixth International Conference on Teaching Statistics: Developing a statistically literate society, Cape Town, South Africa. [CDROM]. 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). Emerging concepts in chance and data. Australian Journal of Early Childhood, 27(4), 24-28.
Watson, J. M., & Kelly, B. A. (2004a). Statistical variation in a chance setting: A twoyear study.Educational Studies in Mathematics, 57, 121-144.
Watson, J. M., & Kelly, B. A. (2004b). Expectation versus variation: Students’ decision
making in a chance environment. Canadian Journal of Science, Mathematics and
Technology Education, 4, 371-396.
Watson, J. M., & Kelly, B. A. (2005). The winds are variable: Student intuitions about variation. School Science and Mathematics, 105, 252-269.
Watson, J. M., & Kelly, B. A. (2006). Expectation versus variation: Students’ decision
making in a sampling environment. Canadian Journal of Science, Mathematics and Technology Education, 6, 145-166.
Watson, J. M., & Moritz, J. B. (2000). Developing concepts of sampling. Journal for Research in Mathematics Education, 31, 44-70.
Watson, J. M., & Moritz, J. B. (2001). Development of reasoning associated with pictographs: representing, interpreting, and predicting. Educational Studies in Mathematics, 48, 47-81.
