Modeling the Growth of students’

covariational reasoning during an

introductory statistics course

 

Andrew S. Zieffler

University of Minnesota

zief0002@umn.edu

 

Joan B. Garfield

University of Minnesota

jbg@umn.edu

 

ABSTRACT

 

This study examined students’ development of reasoning about quantitative bivariate data during a one-semester university-level introductory statistics course. There were three research questions of interest: (1) What is the nature, or pattern of change in students’ development in reasoning throughout the course?; (2) Is the sequencing of quantitative bivariate data within the course associated with differences in the pattern of change in reasoning?; and (3) Are changes in reasoning about foundational concepts of distribution associated with differences in the pattern of change? Covariational and distributional reasoning were measured four times during the course, across four cohorts of students. A linear mixed-effects model was used to analyze the data, revealing some interesting trends and relationships regarding the development of covariational reasoning.

 

Keywords: Statistics education research; Growth modeling; Topic sequencing

 

 

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Statistics Education Research Journal, 8(1), 7-31, http://www.stat.auckland.ac.nz/serj

Ó International Association for Statistical Education (IASE/ISI), May, 2009

 

 

 

REFERENCES

 

Adi, H., Karplus, R., Lawson, A., & Pulos, S. (1978). Intellectual development beyond elementary school VI: Correlational reasoning. School Science & Mathematics, 78(8), 675-683.

American Statistical Association. (2005a). GAISE College Report.

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

American Statistical Association. (2005b). GAISE Endorsement.

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

Baterno, C., Estepa, A., Godino, J. D., & Green, D. R. (1996). Intuitive strategies and preconceptions about association in contingency tables. Journal for Research in Mathematics Education, 27(2), 151-169.

Batanero, C., Estepa, A., & Godino, J. D. (1997). Evolution of students’ undertanding of statistical association in a computer based teaching environment. In J. B. Garfield & G. Burrill (Eds.), Research on the role of technology in teaching and learning statistics: Proceedings of the 1996 IASE Round Table Conference (pp. 191-205). Voorburg, The Netherlands: International Statistical Institute.

Batanero, C., Godino, J. D., & Estepa, A. (1998). Building the meaning of statistical association through data analysis activities. In A. Olivier, & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 221-236). Stellenbosh, South Africa: University of Stellenbosh.

Bates, D., & Sarkar, D. (2005). The lme4 package. R package version 0.9975-13.

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.

Ben-Zvi, D., & Garfield, J. (2004). (Eds.). The challenge of developing statistical literacy, reasoning and thinking. Dordrecht, The Netherlands: Kluwer Academic Publishing.

Beyth-Marom, R. (1982). Perception of correlation reexamined. Memory & Cognition, 10, 511-519.

Boyle, M. H., & Willms, J. D. (2001). Multilevel modeling of hierarchical data in developmental studies. Journal of Child Psychology and Psychiatry, 42(1), 141-162.

Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events. Journal for Research in Mathematics Education, 33(5), 352-378.

Chance, B., & Rossman, A. (2001). Sequencing topics in introductory statistics: A debate on what to teach when. American Statistician, 55(2), 140-144.

Cobb, G. (1992). Teaching statistics. In L. A. Steen (Ed.), Heeding the call for change: Suggestions for curricular action, MAA Notes No. 22, 3-43.

Cobb, P. (1998). Theorizing about mathematical conversations and learning from practice. For the Learning of Mathematics, 18(1), 46-48.

Cobb, P., Gravemeijer, K. P. E., Bowers, J., & Doorman, M. (1997). Statistical Minitools [applets and applications]. Nashville, TN and Utrecht, The Netherlands: Vanderbilt University, TN & Freudenthal Institute, Utrecht University.

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

College Board (2003). Advanced Placement Statistics course guide. New York: Author.

Collins, L., Schafer, J., & Kam, C. (2001). A comparison of inclusive and restrictive strategies in modern missing data procedures. Psychological Methods, 6, 330-351.

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-333.

Cronbach, L. J., & Furby, L. (1970). How should we measure change – or should we? Psychological Bulletin, 74, 68-80.

Davis, F. B. (1964). Measurement of change. In F. B. Davis (Ed.), Educational measurements and their interpretation (pp. 234-252). Belmont, CA: Wadsworth.

Garfield, J., delMas, R., & Chance, B. (n.d.). Assessment Resource Tools for Improving Statistical Thinking. Retrieved April 8, 2006.

[Online: https://app.gen.umn.edu/artist/index.html]

Gravemeijer, K. P. E. (2000, April). A rationale for an instructional sequence for analyzing one- and two-dimensional data sets. Paper presented at the annual meeting of the American Educational Research Association, Montreal, Canada.

Hamilton, D. L., & Gifford, R. K. (1976). Illusory correlation in interpersonal perception: A cognitive basis for stereotypic judgments. Journal of Experimental Social Psychology, 12, 392-407.

Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. London: Routledge and Kegan Paul.

International Association for Statistical Education. (2005). SRTL-4 Report.

[Online: http://srtl.stat.auckland.ac.nz/]

Jennings, D., Amabile, T., & Ross, L. (1982). Informal covariation assessment: Data-based versus theory-based judgments. In D. Kahneman, P. Slovic, & A. Tversky (Eds.), Judgment under uncertainty: Heuristics and biases (pp. 211-230). Cambridge, England: Cambridge University Press.

Kanari, Z., & Millar, R. (2004). Reasoning from data: How students collect and interpret data in science investigations. Journal of Research in Science Teaching, 41(7), 748-769.

Kao, S. F., & Wasserman, E. A. (1993). Assessment of an information integration account of contingency judgment with examination of subjective cell importance and method of information presentation. Journal of Experimental Psychology: Learning, Memory, and Cognition, 19(6), 1363-1386.

Konold, C. (1999). Issues in assessing conceptual understanding in probability and statistics. Journal of Statistics Education, 3(1).

[Online: http://www.amstat.org/publications/jse/v3n1/konold.html]

Konold, C. (2002). Teaching concepts rather than conventions. New England Journal of Mathematics, 34(2), 69-81.

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.

Koslowski, B. (1996). Theory and evidence: The development of scientific reasoning (Learning, development & conceptual change). Cambridge, MA: MIT Press.

Kuhn, D., Amsel, E., & O’Loughlin, M. (1988). The development of scientific thinking skills. Orlando, FL: Academic Press.

McArdle, J. J., & Epstein, D. (1987). Latent growth curves within developmental structural equation models. Child Development, 58, 110-133.

McGahan, J. R., McDougal, B., Williamson, J. D., & Pryor, P. L. (2000). The equivalence of contingency structure for intuitive covariation judgments about height, weight, and body fat. Journal of Psychology, 134, 325-335.

McKenzie, C. R. M., & Mikkelsen, L. A. (2007). A Bayesian view of covariation assessment. Cognitive Psychology, 54, 33-61.

Min, R., Vos, H., Kommers, P., & van Dijkum, C. (2000). A concept model for learning: An attempt to define a proper relations scheme between instruction and learning and to establish the dynamics of learning in relation to motivation, intelligence and study-ability (‘studeerbaarheid’). Journal of Interactive Learning Research, 11(3/4), 485-506.

Monk, S., & Nemirovsky, R. (1994). The case of Dan: Student construction of a functional situation through visual attributes. In E. Dubinsky, J. Kaput, & A. Schoenfeld (Eds.), Research in collegiate mathematics education: Volume 1 (pp. 139-168). Providence, RI: American Mathematics Society.

Moritz, J. B. (2004). Reasoning about covariation. In D. Ben-Zvi, & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 227-256). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Murre, J. M. J., & Chessa, A. G. (2006). A model of learning and forgetting II: The learning curve. Unpublished manuscript.

Nemirovsky, R. (1996). A functional approach to algebra: Two issues that emerge. In N. Dedrarg, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 295-313). Boston: Kluwer Academic Publishers.

Pearl, R. (1925). The biology of population growth. New York: Knopf.

Pinheiro, J., & Bates, D. (2000). Mixed-effects models in S and S-PLUS. New York: Springer Verlag.

Pinheiro, J., Bates, D., DebRoy, S., & Sarkar, D. (2005). nlme: Linear and nonlinear mixed effects models. R package version 3.1-66.

R Development Core Team. (2008). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.

[Online: http://www.R-project.org]

Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods. Thousand Oaks, CA: Sage Publications, Inc.

Ross, J. A., & Cousins, J. B. (1993). Patterns of student growth in reasoning about correlational problems. Journal of Educational Psychology, 85(1), 49-65.

Sánchez, F. T. (1999). Significado de la correlación y regression para los estudiantes universitarios [Meanings of correlation and regression for undergraduates]. Unpublished doctoral dissertation, University of Granada, Spain.

Schauble, L. (1996). The development of scientific reasoning in knowledge-rich contexts. Developmental Psychology, 32(1), 102-119.

Siegler, R. S. (2000). The rebirth of children’s learning. Child Development, 71(1), 26-35.

Singer, J. D., & Willett, J. B. (2003). Applied longitudinal data analysis. New York: Oxford University Press.

Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26, 229-274.

Thurston, L. L. (1919). The learning curve equation. Psychological Monographs, 26(3), 1-51.

Truran, J. M. (1997). Understanding of association and regression by first year economics students from two different countries as revealed in responses to the same examination questions. In J. Garfield, & J. M. Truran (Eds.), Research papers on stochastics educations from 1997 (pp. 205-212). Minneapolis, MN: University of Minnesota.

United States Department of Education. (2005). Raising achievement: A new path for No Child Left Behind. News Release.

[Online: http://www.ed.gov/policy/elsec/guid/raising/new-path-long.html]

Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). According to the Book. Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Dordrecht: Kluwer Academic Publishers.

Verbeke, G, & Molenberghs, G. (2000). Linear mixed models for longitudinal data. New York: Springer Verlag.

Wavering, M. J. (1989). Logical reasoning necessary to make line graphs. Journal of Research in Science Teaching, 26(5), 373-379.

Willett, J. (1989a). Questions and answers in the measurement of change. Review of Research in Education, 15, 345-422.

Willett, J. B. (1989b). Some results on reliability for the longitudinal measurement of change: Implications for the design of studies of individual growth. Educational and Psychological Measurement, 49, 587–602.

Willet, J. B., Singer, J. D., & Martin, N. C. (1998). The design and analysis of longitudinal studies of development and psychopathology in context: Statistical models and methodological recommendations. Development and Psychopathology, 10, 395-426.

Wixted, J. T. (2004). The psychology and neuroscience of forgetting. Annual Review of Psychology, 55, 235-269.

Wozniak, P. A. (1990). Optimization of learning. Unpublished master’s thesis, Poznan University of Technology. Poznan, Poland.

Zieffler, A. (2006). A longitudinal investigation of the development of college students’ reasoning about bivariate data during an introductory statistics course. Unpublished doctoral dissertation, University of Minnesota.

Zimmerman, C. (2005). The development of scientific reasoning: What psychologists contribute to an understanding of elementary science learning. Paper commissioned by the National Academies of Science (National Research Council’s Board of Science Education, Consensus Study on Learning Science, Kindergarten through Eighth Grade).

[Online: www7.nationalacademies.org/bose/Corrine_Zimmerman_Final_Paper.pdf]

 

andrew s. zieffler

Educational Psychology

206 Burton Hall

178 Pillsbury Dr. SE

Minneapolis, MN 55455