# Statistics students’ identification of inferential model elements within contexts of their own invention

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## Abstract

Statistical thinking partially depends upon an iterative process by which essential features of a problem setting are identified and mapped onto an abstract model or archetype, and then translated back into the context of the original problem setting (Wild and Pfannkuch, Int Stat Rev 67(3):223–248, 1999). Assessment in introductory statistics often relies on tasks that present students with data in context and expects them to choose and describe an appropriate model. This study explores post-secondary student responses to an alternative task that prompts students to clearly identify a sample, population, statistic, and parameter using a context of their own invention. The data include free-text narrative responses of a random sample of 500 students from a sample of more than 1600 introductory statistics students. Results suggest that students’ responses often portrayed sample and population accurately. Portrayals of statistic and parameter were less reliable and were associated with descriptions of a wide variety of other concepts. Responses frequently attributed a variable of some kind to the statistic, or a study design detail to the parameter. Implications for instruction and research are discussed, including a call for emphasis on a modeling paradigm in introductory statistics.

## Keywords

Statistics education Statistical modeling Statistical inference Assessment Parameter## Notes

### Acknowledgements

The authors wish to express their sincere gratitude to Joan Garfield for her thoughtful direction and influence on development of early research leading to this work. The authors are also grateful for the thoughtful feedback and constructive suggestions of colleagues during the SRTL-10 forum.

## References

- Allmond, S., & Makar, K. (2010). Developing primary students’ ability to pose questions in statistical investigations. In C. Reading (Ed.),
*Data and context in statistics education: Towards an evidence*-*based society. Proceedings of the 8th international conference on teaching statistics*. Voorburg: International Statistical Institute.Google Scholar - Beckman, M. D. (2015).
*Assessment of cognitive transfer outcomes for students of introductory statistics*(Doctoral dissertation, University of Minnesota—Twin Cities). Retrieved from http://iase-web.org/documents/dissertations/15.MatthewBeckman.Dissertation.pdf. Accessed 1 Dec 2016. - 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: Kluwer Academic Publishers.CrossRefGoogle Scholar - Bransford, J. D., Brown, A. L., & Cocking, R. R. (2000).
*How people learn: Brain, mind, experience, and school: Expanded edition*. Washington, DC: National Academies.Google Scholar - Chance, B. (2002). Components of statistical thinking and implications for instruction and assessment.
*Journal of Statistics Education, 10*(3). Retrieved from http://ww2.amstat.org/publications/jse/v10n3/chance.html. Accessed 1 Dec 2016. - 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.Google Scholar - Garfield, J., & Ben-Zvi, D. (2008).
*Developing students*’*statistical reasoning: Connecting research and teaching practice*. Berlin: Springer Science & Business Media.Google Scholar - Graham, A. (2006).
*Developing thinking in statistics*. London: Paul Chapman.Google Scholar - Haller, H., & Krauss, S. (2002). Misinterpretations of significance: A problem students share with their teachers?
*Methods of Psychological Research, 7*(1), 1–20.Google Scholar - Kaplan, J. J., Fisher, D. G., & Rogness, N. T. (2009). Lexical ambiguity in statistics: What do students know about the words association, average, confidence, random and spread?
*Journal of Statistics Education, 17*(3). Retrieved from http://www.amstat.org/publications/jse/v17n3/kaplan.html. Accessed 17 Nov 2017. - Kaplan, J. J., Fisher, D. G., & Rogness, N. T. (2010). Lexical ambiguity in statistics: How students use and define the words: Association, average, confidence, random and spread.
*Journal of Statistics Education, 18*(2). Retrieved from http://www.amstat.org/publications/jse/v18n2/kaplan.pdf. Accessed 17 Nov 2017. - Kaplan, J. J., & Rogness, N. (2018). Increasing statistical literacy by exploiting lexical ambiguity of technical terms.
*Numeracy, 18*(1), 1–14.Google Scholar - Lavigne, N. C., & Lajoie, S. P. (2007). Statistical reasoning of middle school children engaging in survey inquiry.
*Contemporary Educational Psychology, 23*(4), 630–666.CrossRefGoogle Scholar - Makar, K., & Rubin, A. (2009). A framework for thinking about informal statistical inference.
*Statistics Education Research Journal, 8*(1), 82–105.Google Scholar - McCullagh, P. (2002). What is a statistical model?
*The Annals of Statistics*,*30*(5), 1225–1267.CrossRefGoogle Scholar - Meletiou-Mavrotheris, M., & Paparistodemou, E. (2015). Developing students’ reasoning about samples and sampling in the context of informal inferences.
*Educational Studies in Mathematics, 88*(3), 385–404.CrossRefGoogle Scholar - Pfannkuch, M. (2006). Informal inferential reasoning. In A. Rossman & B. Chance (Eds.),
*Working cooperatively in statistics education. Proceedings of the 7th international conference on teaching statistics*. Voorburg: International Statistical Institute.Google Scholar - Pfannkuch, M., Ben-Zvi, D., & Budgett, S. (2018). Innovations in statistical modeling to connect data, chance, and context.
*ZDM Mathematics Education*. https://doi.org/10.1007/s11858-018-0989-2**(this issue)**.CrossRefGoogle Scholar - R Core Team. (2017).
*R: A language and environment for statistical computing*. Vienna: R Foundation for Statistical Computing. https://www.R-project.org/. Accessed 10 June 2017. - Reed, S. K., Dempster, A., & Ettinger, M. (1985). Usefulness of analogous solutions for solving algebra word problems.
*Journal of Experimental Psychology: Learning, Memory, and Cognition, 11*(1), 106–125.Google Scholar - Rossman, A. J., & Chance, B. L. (2001).
*Workshop statistics: Discovery with data*(2nd ed.). Emeryville: Key College Publishing.Google Scholar - Singley, M. K., & Anderson, J. R. (1989).
*The transfer of cognitive skill*. Cambridge: Harvard University Press.Google Scholar - Vallecillos, A. (1999). Some empirical evidence on learning difficulties about testing hypotheses.
*Bulletin of the International Statistical Institute: Bulletin of the 52nd Session of the International Statistical Institute, 58*, 201–204.Google Scholar - Watson, J. M., & Kelly, B. A. (2005). Cognition and instruction: Reasoning about bias in sampling.
*Mathematics Education Research Journal, 17*(1), 25–27.CrossRefGoogle Scholar - Watson, J. M., & Moritz, J. B. (2000). Development of understanding of sampling for statistical literacy.
*The Journal of Mathematical Behavior, 19*(1), 109–136.CrossRefGoogle Scholar - Well, A. D., Pollatsek, A., & Boyce, S. J. (1990). Understanding the effects of sample size on the variability of the mean.
*Organizational Behavior and Human Decision Processes, 47*, 289–312.CrossRefGoogle Scholar - Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry.
*International Statistical Review, 67*(3), 223–248.CrossRefGoogle Scholar - Williams, A. M. (1999). Novice students’ conceptual knowledge of statistical hypothesis testing. In J. M. Truran, & K. M. Truran (Eds.),
*Making the difference: Proceedings of the twenty*-*second annual conference of the mathematics education research group of Australasia*(pp. 554–560). Adelaide: MERGA.Google Scholar