**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

**__________________________**

*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 22*^{nd}*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