## Abstract

This paper describes the importance of developing students’ reasoning about samples and sampling variability as a foundation for statistical thinking. Research on expert–novice thinking as well as statistical thinking is reviewed and compared. A case is made that statistical thinking is a type of expert thinking, and as such, research comparing novice and expert thinking can inform the research on developing statistical thinking in students. It is also posited that developing students’ informal inferential reasoning, akin to novice thinking, can help build the foundations of experts’ statistical thinking.

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## References

American Statistical Association (2005).

*Statistics guidelines for the assessment and instruction in education: College report*. Alexandria, VA: American Statistical Association. Retrieved from http://www.amstat.org/education/gaise/.Bakker, A. (2004). Reasoning about shape as a pattern in variability.

*Statistics Education Research Journal*,*3*(2), 64–83. Retrieved from http://iase-web.org/Publications.php?p=SERJ.Bakker, A., & Derry, J. (2011). Lessons from inferentialism for statistics education.

*Mathematical Thinking and Learning, 13*(1), 5–26.Ben-Zvi, D. (2006). Scaffolding students’ informal inference and argumentation. In A. Rossman & B. Chance (Eds.),

*Proceedings of the seventh international conference on teaching of statistics [CD-ROM], Salvador, Bahia, Brazil, July 2006*. Voorburg: International Statistical Institute.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., & Arcavi, A. (2004). Reasoning about variability in comparing distributions.

*Statistics Education Research Journal*,*3*(2), 42–63. Retrieved from http://iase-web.org/Publications.php?p=SERJ.Ben-Zvi, D., Gil, E., & Apel, N. (2007). What is hidden beyond the data? Helping young students to reason and argue about some wider universe. In D. Pratt & J. Ainley (Eds.),

*Reasoning about informal inferential statistical reasoning: A collection of current research studies: Proceedings of the fifth international research forum on statistical reasoning, thinking, and literacy (SRTL-5)*. UK: University of Warwick.Ben-Zvi, D., Aridor, K., Makar, K., & Bakker, A. (2012). Students’ emergent articulations of uncertainty while making informal statistical inferences.

*ZDM—The International Journal on Mathematics Education, 44*(7), 913–925.Biehler, R. (1995).

*Toward requirements for more adequate software tools that support both: Learning and doing statistics*. Bielefeld: University of Bielefeld. Revised version of paper presented at ICOTS 4. Occasional Paper 157.Biehler, R., Ben-Zvi, D., Bakker, A., & Maker, K. (2013). Technology for enhancing statistical reasoning at the school level. In M. A. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. Leung (Eds.),

*Third international handbook of mathematics education*(pp. 643–690). New York: Springer.Biggs, J. B., & Collis, K. F. (1982).

*Evaluating the quality of learning: The SOLO taxonomy*. New York: Academic.Blessing, S., & Anderson, J. R. (1996). How people learn to skip steps.

*Journal of Experimental Psychology: Learning, Memory, and Cognition, 22*, 576–598.Bransford, J., Brown, A., & Cocking, R. (2000).

*How people learn: Brain, mind, experience, and school (expanded edition)*. Washington, DC: National Academy Press.Chance, B. (2002). Components of statistical thinking and implications for instruction and assessment.

*Journal of Statistics Education*,*10*(3), 1–17. Retrieved from http://www.amstat.org/publications/jse/.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: Kluwer.Chance, B., Ben-Zvi, D., Garfield, J., & Medina, E. (2007). The role of technology in improving student learning of statistics.

*Technology Innovations in Statistics Education*,*1*(1), 1–26. Retrieved from http://www.escholarship.org/uc/item/8sd2t4rr.Chi, M., Feltovich, P., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices.

*Cognitive Science, 5*, 121–152.Cobb, G. W. (1992). Teaching statistics. In L. A. Steen (Ed.),

*Heeding the call for change: Suggestions for curricular action*(pp. 3–43). Washington, DC: Mathematical Association of America.Cobb, G. W. (2007). The introductory statistics course: A ptolemaic curriculum?

*Technology Innovations in Statistics Education*,*1*(1), 1–14. Retrieved from http://www.escholarship.org/uc/item/6hb3k0nz.delMas, R. (2002). Sampling SIM (Version 5.4) [Software]. Retrieved from http://www.tc.umn.edu/delma001/stat_tools/.

delMas, R., Garfield, J., & Chance, B. (2004, April).

*Using assessment to study the development of students’ reasoning about sampling distributions*. Paper presented at the Annual Meeting of the American Educational Research Association, San Diego, CA. Retrieved from http://www.gen.umn.edu/faculty_staff/delmas/AERA_2004_samp_dist.pdf.delMas, R., Garfield, J., & Zieffler, A. (2014). Using TinkerPlots

^{TM}to develop tertiary students’ statistical thinking in a modeling-based introductory statistics class. In T Wassong, D. Frischemeier, P. Fischer, R. Hochmuth & P. Bender (Eds.),*Mit werkzeugen mathematik und stochastik lernen*(Using tools for Learning mathematics and statistics) (pp. 405–420). Wiesbaden, Germany: Springer Spektrum.Finzer, W. (2006).

*Fathom*^{TM}*dynamic data software*. Emeryville, CA: Key Curriculum Press.Garfield, J., & Ben-Zvi, D. (2008).

*Developing students’ statistical reasoning: Connecting research and teaching practice*. New York: Springer.Garfield, J., delMas, R., & Zieffler, A. (2012). Developing statistical modelers and thinkers in an introductory, tertiary-level statistics course.

*ZDM—The International Journal on Mathematics Education, 44*(7), 883–898.Gould, R. (2004). Variability: One statistician’s view.

*Statistics Education Research Journal*,*3*(2), 7–16. Retrieved from http://iase-web.org/Publications.php?p=SERJ.Gravemeijer, K. P. E., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing, modeling, and instructional design. In P. Cobb, E. Yackel, & K. McClain (Eds.),

*Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design*(pp. 225–273). Mahwah, NJ: Lawrence Erlbaum Associates.Groth, R. E. (2005). An investigation of statistical thinking in two different contexts: Detecting a signal in a noisy process and determining a typical value.

*Journal of Mathematical Behavior, 24*, 109–124.Hershkowitz, R., Dreyfus, T., Schwarz, B., Ben-Zvi, D., Friedlander, A., Hadas, N., et al. (2002). Mathematics curriculum development for computerized environments: A designer–researcher–teacher–learner activity. In L. D. English (Ed.),

*Handbook of international research in mathematics education*(pp. 657–694). London: Lawrence Erlbaum Associates.Jones, G. A., Thornton, C. A., Langrall, C. W., Mooney, E. S., Perry, B., & Putt, I. J. (2000). A framework for characterizing children’s statistical thinking.

*Mathematical Thinking and Learning, 2*(4), 269–307.Konold, C. (2002). Teaching concepts rather than conventions.

*New England Mathematics Journal, 34*(2), 69–81.Konold, C. (2011).

*TinkerPlots*^{TM}*: Dynamic data exploration*. Emeryville, CA: Key Curriculum Press.Kozma, R. B., & Russell, J. (1997). Multimedia and understanding: Expert and novice responses to different representations of chemical phenomena.

*Journal of Research in Science Teaching, 34*, 949–968.Kuhn, D. (Ed.). (1990).

*Introduction to developmental perspectives on teaching and learning thinking skills*. Basel: Karger.Larkin, J. H. D., McDermott, D., Simon, D. P., & Simon, H. A. (1980). Models of competence in solving physics problems.

*Cognitive Science, 4*, 317–345.Lave, J. (1988).

*Cognition in practice: Mind, mathematics and culture in everyday life*. New York: Cambridge University Press.MacKay, J., & Elam, J. (1992). A comparative study of how experts and novices use a decision aid to solve problems in complex knowledge domains.

*Information Systems Research, 3*(2), 150–172.Makar, K., Bakker, A., & Ben-Zvi, D. (2011). The reasoning behind informal statistical inference.

*Mathematical Thinking and Learning, 13*(1–2), 152–173.Makar, K., & Ben-Zvi, D. (2011). The role of context and evidence in informal inferential reasoning.

*Mathematical Thinking and Learning, 13*(1–2), 1–4.Makar, K., & Confrey, J. (2002). Comparing two distributions: Investigating secondary teachers statistical thinking. In B. Phillips (Ed.),

*Proceedings of the sixth international conference on teaching statistics*(pp. 1–4). Cape Town: International Association for Statistics Education.Makar, K., & Rubin, A. (2009). A framework for thinking about informal statistical inference.

*Statistics Education Research Journal*,*8*(1), 82–105. Retrieved from http://iase-web.org/Publications.php?p=SERJ.Martin, J. (2013). Differences between experts’ and students’ conceptual images of the mathematical structure of Taylor series convergence.

*Educational Studies in Mathematics, 82*(2), 267–283.Melton, K. I. (2004). Statistical thinking activities: Some simple exercises with powerful lessons.

*Journal of Statistics Education*,*12*(2), 1–10. Retrieved from http://www.amstat.org/publications/jse/.Nesher, P., & Peled, I. (1986). Shifts in reasoning.

*Educational Studies in Mathematics, 17*(1), 67–79.Novick, L. R. (1988). Analogical transfer, problem similarity, and expertise.

*Journal of Experimental Psychology: Learning, Memory, and Cognition, 14*(3), 510–520.Pfannkuch, M. (2006). Informal inferential reasoning. In A. Rossman & B. Chance (Eds.),

*Proceedings of the seventh international conference on teaching of statistics [CD-ROM], Salvador, Bahia, Brazil, July 2006*. Voorburg: International Statistical Institute.Pfannkuch, M., & Ben-Zvi, D. (2011). Developing teachers’ statistical thinking. In C. Batanero, G. Burrill, & C. Reading (Eds.),

*Teaching statistics in school mathematics—challenges for teaching and teacher education: A joint ICMI/IASE study*(Vol. 14, pp. 323–346). Dordrecht, The Netherlands: Springer.Pfannkuch, M., & Horring, J. (2005). Developing statistical thinking in a secondary school: A collaborative curriculum development. In G. Burrill & M. Camden (Eds.),

*Curricular development in statistics education: International Association for Statistical Education (IASE) roundtable, Lund, Sweden, 28 June–3 July 2004*(pp. 163–173). Voorburg: International Statistical Institute.Pratt, D., & Ainley, J. (2008). Introducing the special issue on informal inference.

*Statistics Education Research Journal*,*7*(2), 3–4. Retrieved from http://iase-web.org/Publications.php?p=SERJ.Pratt, D., & Noss, R. (2010). Designing for mathematical abstraction.

*International Journal of Computers for Mathematical Learning, 15*(2), 81–97.Resnick, L. (1989). Treating mathematics as an ill-structured discipline. In R. Charles & E. Silver (Eds.),

*The teaching and assessing of mathematical problem solving*(pp. 32–60). Reston, VA: National Council of Teachers of Mathematics.Rossman, A. J. (2008). Reasoning about informal statistical inference: One statistician’s view.

*Statistics Education Research Journal*,*7*(2), 5–19. Retrieved from http://iase-web.org/Publications.php?p=SERJ.Rüede, C. (2013). How secondary level teachers and students impose personal structure on fractional expressions and equations—an expert–novice study.

*Educational Studies in Mathematics, 83*(3), 387–408.Saldanha, L., & Thompson, P. (2002). Conceptions of sample and their relationship to statistical inference.

*Educational Studies in Mathematics, 51*(3), 257–270.Sanchez, E., & Blancarte, A. (2010). Training in-service teachers to develop statistical thinking. In C. Reading (Ed.),

*Data and context in statistics education: Towards an evidence-based society. Proceedings of the eighth international conference on teaching statistics (icots8, July, 2010), Ljubljana, Slovenia*. Voorburg, the Netherlands: International Statistical Institute. Retrieved from http://www.stat.auckland.ac.nz/iase/publications.Schenk, K., Vitalari, N., & Davis, K. (1998). Differences between novice and expert systems analysts: What do we know and what do we do?

*Journal of Management Information Systems, 15*(1), 9–50.Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.),

*Handbook for research on mathematics teaching and learning*(pp. 334–370). New York: Macmillan.Schoenfeld, A. H. (1998). Making mathematics and making pasta: From cookbook procedures to really cooking. In J. G. Greeno & S. V. Goldman (Eds.),

*Thinking practices in mathematics and science learning*(pp. 1–30). Mahwah, NJ: Lawrence Erlbaum.Schwartz, D. L., Bransford, J. D., & Sears, D. (2005). Efficiency and innovation in transfer. In J. Mestre (Ed.),

*Transfer of learning from a modern multidisciplinary perspective*(pp. 1–51). Greenwich, CT: Information Age.Siegler, R. S., & Alibali, M. W. (2004).

*Children’s thinking*(4th ed.). Upper Saddle River, NJ: Pearson.Smith, J. P., III, diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition.

*The Journal of the Learning Sciences, 3*(2), 115–163.Tintle, N., VanderStoep, J., Holmes, V. L., Quisenberry, B., & Swanson, T. (2011). Development and assessment of a preliminary randomization-based introductory statistics curriculum.

*Journal of Statistics Education*,*19*(1). Retrieved from http://www.amstat.org/publications/jse/.Tukey, J. (1977).

*Exploratory data analysis*. Reading: Addison-Wesley.Tversky, A., & Kahneman, D. (1971). Belief in the law of small numbers.

*Psychological Bulletin, 76*, 105–110.Voss, J. F., Perkins, D. N., & Segal, J. W. (Eds.). (1991).

*Informal reasoning and education*. Hillsdale, NJ: Erlbaum.Watson, J. M. (2004). Developing reasoning about samples. In D. Ben-Zvi & J. Garfield (Eds.),

*The challenge of developing statistical literacy, reasoning and thinking*(pp. 277–294). Dordrecht: Kluwer Academic.Wild, C. (2006). The concept of distribution.

*Statistics Education Research Journal*,*5*(2), 10–26. Retrieved from http://iase-web.org/Publications.php?p=SERJ.Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry.

*International Statistical Review, 67*(3), 223–248.Wilensky, U. (1997). What is normal anyway? Therapy for epistemological anxiety.

*Educational Studies in Mathematics, 33*(2), 171–202.

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Garfield, J., Le, L., Zieffler, A. *et al.* Developing students’ reasoning about samples and sampling variability as a path to expert statistical thinking.
*Educ Stud Math* **88**, 327–342 (2015). https://doi.org/10.1007/s10649-014-9541-7

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DOI: https://doi.org/10.1007/s10649-014-9541-7