Developing students’ reasoning about samples and sampling variability as a path to expert statistical thinking
- 1.5k Downloads
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.
KeywordsStatistical thinking Expert and novice thinking Sampling variability Informal statistical inference Informal inferential reasoning
- 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.
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- Biggs, J. B., & Collis, K. F. (1982). Evaluating the quality of learning: The SOLO taxonomy. New York: Academic.Google Scholar
- Blessing, S., & Anderson, J. R. (1996). How people learn to skip steps. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, 576–598.Google Scholar
- Bransford, J., Brown, A., & Cocking, R. (2000). How people learn: Brain, mind, experience, and school (expanded edition). Washington, DC: National Academy Press.Google Scholar
- 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.
- 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.Google Scholar
- 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 TinkerPlotsTM 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.Google Scholar
- Finzer, W. (2006). Fathom TM dynamic data software. Emeryville, CA: Key Curriculum Press.Google Scholar
- Garfield, J., & Ben-Zvi, D. (2008). Developing students’ statistical reasoning: Connecting research and teaching practice. New York: Springer.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- Konold, C. (2002). Teaching concepts rather than conventions. New England Mathematics Journal, 34(2), 69–81.Google Scholar
- Konold, C. (2011). TinkerPlots TM : Dynamic data exploration. Emeryville, CA: Key Curriculum Press.Google Scholar
- Kuhn, D. (Ed.). (1990). Introduction to developmental perspectives on teaching and learning thinking skills. Basel: Karger.Google Scholar
- Makar, K., Bakker, A., & Ben-Zvi, D. (2011). The reasoning behind informal statistical inference. Mathematical Thinking and Learning, 13(1–2), 152–173.Google Scholar
- 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.Google Scholar
- 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.
- Novick, L. R. (1988). Analogical transfer, problem similarity, and expertise. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14(3), 510–520.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- Siegler, R. S., & Alibali, M. W. (2004). Children’s thinking (4th ed.). Upper Saddle River, NJ: Pearson.Google Scholar
- 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.Google Scholar
- Voss, J. F., Perkins, D. N., & Segal, J. W. (Eds.). (1991). Informal reasoning and education. Hillsdale, NJ: Erlbaum.Google Scholar
- 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.