• 2019 Accesses

Part of the Springer International Handbooks of Education book series (SIHE)

## Abstract

Many decisions in politics, economics, and society are based on data and statistics. In order to participate as a responsible citizen, it is essential to have a solid grounding in reasoning about data. Reasoning about data is a fundamental human activity; its components can be found in nearly every profession and in most school curricula in the world. This chapter reviews past and recent research on reasoning about data across all ages of learners from primary school to adults. Specifically in this chapter, the term reasoning about data is defined, the implementation of reasoning about data in the curricula of different countries is investigated, and research studies of learner reasoning about distribution, variation, comparing groups, and association, which are fundamental concepts when reasoning about data, are reviewed. The research review presented includes references to existing frameworks and taxonomies that can assess learner reasoning in regard to these concepts and discusses the influence of digital tools to enhance learner statistical reasoning. Finally, some insights for future directions in research about reasoning about data are provided.

### Keywords

• Statistical reasoning
• SOLO
• Data
• Distribution
• Variation
• Comparing groups
• Association

This is a preview of subscription content, access via your institution.

## Notes

1. 1.

There is also a GAISE report for the statistical education of college students.

2. 2.

Mean absolute deviation, or MAD, is the (sum of the distances of all values from the mean )÷ (number of values). In analysis and measure theory, it is the L1 norm.

3. 3.

For a more comprehensive categorization of statistical questions, see Biehler (2001, p. 98) and also Arnold (2013).

4. 4.

Summary statistics refer to measures of central tendency in this case.

5. 5.

The development and the implementation of these minitools were the starting point to develop the educational software TinkerPlots , which includes the features of the minitools.

## References

• Arnold, P. (2013). Statistical investigative questions: An enquiry into posing and answering investigative questions from existing data (Doctoral thesis). Retrieved from https://researchspace.auckland.ac.nz/handle/2292/21305.

• Australian Curriculum Assessment and Reporting Authority. (n.d.). Australian curriculum. ACARA. Retrieved from http://www.australiancurriculum.edu.au

• Bakker, A., Biehler, R., & Konold, C. (2005). Should young students learn about box plots? In G. Burrill & M. Camden (Eds.), Curricular development in statistics education: International Association for Statistical Education (IASE) roundtable (pp. 163–173). Voorburg, The Netherlands: International Statistical Institute.

• Bakker, A., & Gravemeijer, K. P. E. (2004). Learning to reason about distribution. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 147–168). Dordrecht, The Netherlands: Kluwer.

• Batanero, C., Estepa, A., & Godino, J. (1997). Evolution of students’ understanding of statistical association in a computer-based teaching environment. In J. Garfield & G. Burrill (Eds.), Research on the role of technology in teaching and learning statistics (pp. 198–212). Voorburg: International Statistical Institute.

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

• Batanero, C., Godino, J., & Estepa, A. (1998). Building the meaning of statistical association through data analysis activities, Proceedings of PME-22, Stellenbosch, South Africa. Alwyn: Olivier.

• Ben-Zvi, D. (2004). Reasoning about variability in comparing distributions. Statistics Education Research Journal, 3(2), 42–63.

• Ben-Zvi, D., Bakker, A., & Makar, K. (2015). Learning to reason from samples. Educational Studies in Mathematics, 88, 291–303.

• Biehler, R. (1997). Students’ difficulties in practicing computer supported data analysis—Some hypothetical generalizations from results of two exploratory studies. In J. Garfield & G. Burrill (Eds.), Research on the role of technology in teaching and learning statistics (pp. 169–190). Voorburg: ISI.

• Biehler, R. (2001). Statistische Kompetenz von Schülerinnen und Schülern—Konzepte und Ergebnisse empirischer Studien am Beispiel des Vergleichens empirischer Verteilungen [Statistical competence of students—Concepts and results of empirical studies on the example of comparing groups]. In M. Borovcnik, J. Engel, & D. Wickmann (Eds.), Anregungen zum Stochastikunterricht (pp. 97–114). Franzbecker: Hildesheim.

• Biehler, R. (2004, July). Variation, co-variation, and statistical group comparison: Some results from epistemological and empirical research on technology supported statistics education. Paper presented at the Tenth International Congress on Mathematics Education, Copenhagen.

• Biehler, R. (2005, February). Strength and weaknesses in students’ project work in exploratory data analysis. In M. Bosch (Eds.), Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education, Sant Feliu de Guíxols, Spain (pp. 580–590). Retrieved from http://ermeweb.free.fr/CERME4/CERME4_WG5.pdf.

• Biehler, R. (2007a). Denken in Verteilungen—Vergleichen von Verteilungen [Thinking in distributions—Comparing distributions]. Der Mathematikunterricht, 53(3), 3–11.

• Biehler, R. (2007b). Students’ strategies of comparing distributions in an exploratory data analysis context. CD-ROM Proceedings of 56th Session of the International Statistical Institute. Retrieved from http://www.stat.auckland.ac.nz/~iase/publications/isi56/IPM37_Biehler.pdf

• Biehler, R., Ben-Zvi, D., Bakker, A., & Makar, K. (2013). Technology for enhancing statistical reasoning at the school level. In M. A. Clements, A. J. Bishop, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Third international handbook of mathematics education (Vol. 27, pp. 643–689). New York: Springer.

• Biggs, J. B., & Collis, K. F. (1982). Evaluating the quality of learning: The SOLO taxonomy. New York: Academic Press.

• Biggs, J. B., & Collis, K. (1991). Multimodal learning and the quality of intelligent behavior. In H. Rowe (Ed.), Intelligence, reconceptualization and measurement (pp. 57–76). NJ: Laurence Erlbaum Associates.

• Burrill, G., & Biehler, R. (2011). Fundamental statistical ideas in the school curriculum and in training teachers. In C. Batanero, G. Burrill, & C. Reading (Eds.), Teaching statistics in school mathematics—Challenges for teaching and teacher education—A joint ICMI/IASE study: The 18th ICMI study (pp. 57–69). Dordrecht: Springer.

• Canada, D. (2006). Elementary pre-service teachers’ conceptions of variation in a probability context. Statistics Education Research Journal, 5(1), 36–64.

• Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378.

• Casey, S. A., & Nagle, C. (2016). Students’ use of slope conceptualizations when reasoning about the line of best fit. Educational Studies in Mathematics, 92(2), 163–177.

• Ciancetta, M. (2007). Statistics students reasoning when comparing distributions of data (Unpublished doctoral dissertation). Portland State University, Portland, OR.

• Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.

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

• Contreras, J. M., Batanero, C., Diaz, C., & Fernandes, J. A. (2011). Prospective teachers’ common and specialized knowledge in a probability task. In M. Pytlak, T. Rowland, & E. Swoboda (Eds.), Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (pp. 766–775). Rzeszów: University of Rzeszów, Poland.

• Corbin, J., & Strauss, A. (1994). Grounded theory methodology: An overview. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 273–285). Thousand Oaks, CA: Sage.

• Engel, J., & Sedlmeier, P. (2011). Correlation and regression in the training of teachers. In C. Batanero, G. Burrill, & C. Reading (Eds.), Teaching statistics in school mathematics—Challenges for teaching and teacher education—A joint ICMI/IASE study: The 18th ICMI study (pp. 247–258). Dordrecht, The Netherlands: Springer.

• Estrada Roca, A., & Batanero, C. D. (2006). Computing probabilities from two way tables: An exploratory study with future teachers. In A. Rossman & B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics: Working cooperatively in statistics education, Salvador, Brazil [CD-ROM]. Voorburg, The Netherlands: International Association for Statistical Education and the International Statistical Institute.

• Finzer, W. (2007). Fathom dynamic data software [computer software]. Emeryville, CA: Key Curriculum Press. [Software is available from www.concord.org]

• Fitzallen, N. (2012). Reasoning about covariation with TinkerPlots (Unpublished Ph.D. Thesis). University of Tasmania, Australia.

• Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M., et al. (2007). Guidelines for assessment and instruction in statistics education (GAISE) report: A pre-K–12 curriculum framework. Alexandria, VA: American Statistical Association.

• Friel, S., Curcio, F. R., & Bright, G. W. (2001). Making sense of graphs: Critical factors influencing comprehension and instructional implications. Journal for Research in Mathematics Education, 32(2), 124–158.

• Friel, S. N., O’Connor, W., & Mamer, J. D. (2006). More than “meanmedianmode” and a bar graph: What’s needed to have a statistical conversation. In G. Burrill & P. C. Elliott (Eds.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 117–137). Reston, VA: National Council of Teachers of Mathematics.

• Frischemeier, D. (2014, July). Comparing groups by using TinkerPlots as part of a data analysis task—Tertiary students’ strategies and difficulties. In K. Makar, B. de Sousa, & R. Gould (Eds.), Sustainability in statistics education. Proceedings of the Ninth International Conference on Teaching Statistics (ICOTS9), Flagstaff, AZ. Voorburg, The Netherlands: International Statistical Institute.

• Frischemeier, D. (2017). Statistisch denken und forschen lernen mit der Software TinkerPlots. Wiesbaden: Springer Spektrum.

• Frischemeier, D., & Biehler, R. (2016). Preservice teachers’ statistical reasoning when comparing groups facilitated by software. In K. Krainer & N. Vondrova (Eds.), Proceedings of the 9th Congress of the European Society for Research in Mathematics Education (pp. 643–650). Faculty of Education and ERME: Charles University in Prague.

• Gal, I., Rothschild, K., & Wagner, D. A. (1989). Which group is better? The development of statistical reasoning in elementary school children. Paper presented at the meeting of the Society for Research in Child Development, Kansas City.

• Garfield, J., & Ben-Zvi, D. (2005). A framework for teaching and assessing reasoning about variability. Statistics Education Research Journal, 4(1), 92–99.

• Garfield, J., & Ben-Zvi, D. (2008). Developing students’ statistical reasoning: Connecting research and practice. The Netherlands: Springer.

• Gigerenzer, G. (1996). The psychology of good judgment frequency formats and simple algorithms. Medical Decision Making, 16(3), 273–280.

• Gigerenzer, G. (2002). Reckoning with risk: Learning to live with uncertainty. London, UK: Penguin.

• Graham, A. (1987). Statistical investigations in the secondary school. Cambridge, UK: Cambridge University Press.

• Hasemann, K., & Mirwald, E. (2012). Daten, Häufigkeit und Wahrscheinlichkeit. In G. Walther, M. van den Heuvel-Panhuizen, D. Granzer, & O. Köller (Eds.), Bildungsstandards für die Grundschule: Mathematik konkret (pp. 141–161). Berlin: Cornelsen Scriptor.

• Hogan, T. P., Zaboski, B. A., & Perry, T. R. (2015). College students’ interpretation of research reports on group differences: the tall-tale effect. Statistics Education Research Journal, 14(1), 90–111.

• Inhelder, B., & Piaget, J. (1955). De la logique de l’enfant á la logique de l’adolescent [The growth of logical thinking from childhood to adolescence]. Paris: Presses Universitaires de France.

• Jones, G. A., Langrall, C. W., Mooney, E. S., & Thornton, C. A. (2004). Models of development in statistical reasoning. In J. Garfield & D. Ben-Zvi (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 201–226). Dordrecht, The Netherlands: Kluwer.

• Kader, G. D., & Perry, M. (1994). Learning statistics. Mathematics Teaching in the Middle School, 1(2), 130–136.

• KMK. (2004). Bildungsstandards im Fach Mathematik für den mittleren Schulabschluss. München: Wolters Kluwer.

• Konold, C. (2002). Alternatives to scatterplots. Proceedings of the Sixth International Conference on Teaching Statistics, Cape Town, South Africa.

• Konold, C., & Higgins, T. L. (2003). Reasoning about data. A research companion to principles and standards for school mathematics (pp. 193–215). Reston, VA: National Council of Teachers of Mathematics.

• Konold, C., Higgins, T., Russell, S. J., & Khalil, K. (2015). Data seen through different lenses. Educational Studies in Mathematics, 88(3), 305–325.

• Konold, C., & Miller, C. (2011). TinkerPlots TM Version 2 [computer software]. Emeryville, CA: Key Curriculum Press. [Software available from www.tinkerplots.com]

• Konold, C., & Pollatsek, A. (2002). Data analysis as a search for signals in noisy processes. Journal for Research in Mathematics Education, 33(4), 259–289.

• Konold, C., Pollatsek, A., Well, A., & Gagnon, A. (1997). Students analyzing data: Research of critical barriers. In J. B. Garfield & G. Burrill (Eds.), Research on the role of technology in teaching and learning statistics: 1996 Proceedings of the 1996 IASE Round Table Conference (pp. 169–190). Voorburg, The Netherlands: International Statistical Institute.

• Konold, C., Robinson, A., Khalil, K., Pollatsek, A., Well, A., Wing, R., et al. (2002). Students’ use of modal clumps to summarize data. Paper presented at the Sixth International Conference on Teaching Statistics. South Africa: Cape Town.

• Lane, A. (2015). Simulations of the distribution of the mean do not necessarily mislead and can facilitate learning. Journal of Statistics Education, 23(2).

• Langrall, C., Nisbet, S., Mooney, E., & Jansem, S. (2011). The role of context expertise when comparing groups. Mathematical Thinking and Learning, 13(1–2), 47–67.

• Lehrer, R., Kim, M., & Schauble, L. (2007). Supporting the development of conceptions of statistics by engaging students in measuring and modeling variability. International Journal of Computers for Mathematical Learning, 12(3), 195–216.

• Lehrer, R., & Schauble, L. (2004). Modeling natural variation through distribution. American Educational Research Journal, 41(3), 635–679.

• Lem, S., Kempen, G., Ceulemans, E., Onghena, P., Verschaffel, L., & Van Dooren, W. (2014, July). Teaching box plots: An intervention using refutational text and multiple external representations. In K. Makar, B. de Sousa, & R. Gould (Eds.), Sustainability in statistics education. Proceedings of the Ninth International Conference on Teaching Statistics (ICOTS9), Flagstaff, AZ. Voorburg, The Netherlands: International Statistical Institute.

• Lem, S., Onghena, P., Verschaffel, L., & Van Dooren, W. (2013). External representations for data distributions: In search of cognitive fit. Statistics Education Research Journal, 12(1), 4–19.

• Madden, S. R. (2008). High school mathematics teachers’ evolving understanding of comparing distributions (Unpublished Dissertation). Western Michigan University.

• Makar, K. (2004). Developing statistical inquiry: Prospective secondary math and science teachers’ investigations of equity and fairness through analysis of accountability data (Ph.D. Thesis). University of Texas at Austin.

• Makar, K., Bakker, A., & Ben-Zvi, D. (2011). The reasoning behind informal inferential inference. Mathematical Thinking and Learning, 13(1–2), 152–173.

• Makar, K., & Ben-Zvi, D. (2011). The role of context in developing reasoning about informal statistical inference. Mathematical Thinking and Learning, 13(1–2), 1–4.

• Makar, K., & Confrey, J. (2002). Comparing two distributions: Investigating secondary teachers’ statistical thinking. Paper presented at the Sixth International Conference on Teaching Statistics, Cape Town, South Africa.

• Makar, K., & Confrey, J. (2004). Secondary teachers’ statistical reasoning in comparing two groups. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 353–373). Dordrecht, The Netherlands: Kluwer Academic Publishers.

• Makar, K., & Confrey, J. (2005). “Variation-talk”: Articulating meaning in statistics. Statistics Education Research Journal, 4(1), 27–54.

• Makar, K., & Confrey, J. (2014). Wondering, wandering or unwavering? Learners’ statistical investigations with Fathom. In T. Wassong, D. Frischemeier, P. R. Fischer, R. Hochmuth, & P. Bender (Eds.), Mit Werkzeugen Mathematik und Stochastik lernen—Using tools for learning mathematics and statistics. Springer Spektrum: Wiesbaden.

• Mayring, P. (2015). Qualitative content analysis: Theoretical background and procedures. In A. Bikaner-Ahsbahs, C. Knipping, & N. C. Presmeg (Eds.), Approaches to qualitative research in mathematics education (pp. 365–380). Dordrecht: Springer.

• McKenzie, C. R., & Mikkelsen, L. A. (2007). A Bayesian view of covariation assessment. Cognitive Psychology, 54(1), 33–61.

• Meletiou-Mavrotheris, M., & Lee, C. (2002). Teaching students the stochastic nature of statistical concepts in an introductory statistics course. Statistics Education Research Journal, 1(2), 22–37.

• Ministry of Education. (2007). The New Zealand curriculum. Ministry of Education, New Zealand Government. Retrieved from http://nzcurriculum.tki.org.nz/The-New-Zealand-Curriculum

• Mokros, J., & Russell, S. J. (1995). Children’s concepts of average and representativeness. Journal for Research in Mathematics Education, 26(1), 20–39.

• Mooney, E., Duni, D., VanMeenen, E., & Langrall, C. (2014). Preservice teachers’ awareness of variability. In K. Makar, B. de Sousa, & R. Gould (Eds.), Sustainability in statistics education. Proceedings of the Ninth International Conference on Teaching Statistics (ICOTS-9), Flagstaff, AZ. Voorburg, The Netherlands: International Statistical Institute.

• Moritz, J. (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.

• National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

• Noll, J. (2011). Graduate teaching assistants’ statistical content knowledge of sampling. Statistics Education Research Journal, 10(2), 48–74.

• Noll, J., & Shaughnessy, M. (2012). Aspects of students’ reasoning about variation in empirical sampling distributions. Journal for Research in Mathematics Education, 43(5), 509–556.

• Noss, R., Hoyles, C., & Pozzi, S. (2002). Abstraction in expertise: A study of nurses’ conceptions of concentration. Journal for Research in Mathematics Education, 33(3), 204–229.

• Noss, R., Pozzi, S., & Hoyles, C. (1999). Touching epistemologies: Meanings of average and variation in nursing practice. Educational Studies in Mathematics, 40(1), 25–51.

• Obersteiner, A., Bernhard, M., & Reiss, K. (2015). Primary school children’s strategies in solving contingency table problems: the role of intuition and inhibition. ZDM, 47(5), 825–836.

• Pérez Echevarría, M. P. (1990). Psicología del razonamiento probabilístico [The psychology of probabilistic reasoning]. Madrid: Ediciones de la Universitad Autónoma Madrid.

• Peters, S. (2011). Robust understanding of statistical variation. Statistics Education Research Journal, 10(1), 52–88.

• Peters, S. (2014). Developing understanding of statistical variation: Secondary statistics teachers’ perceptions and recollections of learning factors. Journal of Mathematics Teacher Education, 17(6), 539–582.

• Pfannkuch, M. (2005). Thinking tools and variation. Statistics Education Research Journal, 4(1), 83–91.

• Pfannkuch, M. (2007). Year 11 Students’ informal inferential reasoning: A case study about the interpretation of box plots. International Electronic Journal of Mathematics Education, 2(3), 149–167.

• Pfannkuch, M., & Budgett, S. (2017). Reasoning from an eikosogram: An exploratory study. International Journal of Research in Undergraduate Mathematics Education, 3(2), 283–310. https://doi.org/10.1007/s40753-016-0043-0

• Pfannkuch, M., Budgett, S., & Parsonage, R. (2004). Comparison of data plots: Building a pedagogical framework. Paper presented at the Tenth Meeting of the International Congress on Mathematics Education, Copenhagen, Denmark.

• Pfannkuch, M., & Reading, C. (2006). Reasoning about distribution: A complex process. Statistics Education Research Journal, 5(2), 4–9.

• Pfannkuch, M., & Wild, C. (2004). Towards an understanding of statistical thinking. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, thinking and reasoning (pp. 17–46). Dordrecht, The Netherlands: Kluwer Academic Press.

• Reaburn, R. (2012). Strategies used by students to compare two data sets. In J. Dindyal, L. P. Cheng, & S. F. Ng (Eds.), Mathematics education: Expanding horizons (Proceedings of the 35th annual conference of the Mathematics Education Research Group of Australasia). MERGA: Singapore.

• Reading, C. (2004). Student description of variation while working with weather data. Statistics Education Research Journal, 3(2), 84–105.

• Reading, C., & Reid, J. (2006). An emerging hierarchy of reasoning about distribution: From a variation perspective. Statistics Education Research Journal, 5(2), 46–68.

• Reading, C., & Reid, J. (2007). Reasoning about variation: Student voice. International Electronic Journal of Mathematics Education, 2(3), 111–127.

• Reading, C., & Reid, J. (2010). Reasoning about variation: Rethinking theoretical frameworks to inform practice. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. Proceedings of the Eighth International Conference on Teaching Statistics (ICOTS-8), Ljubljana, Slovenia. Voorburg, The Netherlands: International Statistics Institute.

• Reading, C., & Shaughnessy, J. M. (2004). Reasoning about variation. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 201–226). Dordrecht, The Netherlands: Kluwer.

• Reid, J., & Reading, C. (2006). A hierarchy of tertiary students’ consideration of variation. In A. Rossman & B. Chance (Eds.), Working cooperatively in statistics education: Proceedings of the Seventh International Conference on Teaching Statistics (ICOTS-7), Salvador, Brazil. International Statistics Institute: Voorburg, The Netherlands.

• Reid, J., & Reading, C. (2008). Measuring the development of students’ consideration of variation. Statistics Education Research Journal, 7(1), 40–59.

• Reid, J., & Reading, C. (2010). Developing a framework for reasoning about explained and unexplained variation. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. Proceedings of the Eighth International Conference on Teaching Statistics (ICOTS-8), Ljubljana, Slovenia. Voorburg, The Netherlands: International Statistics Institute.

• Rossman, A., & Chance, B. (2001). Workshop statistics: Discovery with data (2nd ed.). Emeryville, CA: Key College Publishing.

• Rossman, A. J., & Chance, B. L. (2014) Using simulation-based inference for learning introductory statistics. Wiley Interdisciplinary Reviews: Computational Statistics 6 (4):211–221. Doi:10.1002/wics.1302

• Rubin, A., Bruce, B., & Tenney, Y. (1991). Learning about sampling: Trouble at the core of statistics. In D. Vere-Jones (Ed.), Proceedings of the Third International Conference on Teaching Statistics (Vol. Vol. 1, pp. 314–319). Voorburg, The Netherlands: International Statistical Institute.

• Saldanha, L., & Thompson, P. (2003). Conceptions of sample and their relationship to statistical inference. Educational Studies in Mathematics, 51(3), 257–270.

• Sanchez, E., Borim da Silva, C., & Coutinho, C. (2011). Teachers’ understanding of variation. In C. Batanero, G. Burrill, & C. Reading (Eds.), Teaching statistics in school mathematics—Challenges for teaching and teacher education (pp. 211–221). Dordrecht, The Netherlands: Springer.

• Schnell, S., & Büscher, C. (2015). Individual concepts of students comparing distributions. In K. Krainer & N. Vondrova (Eds.), Proceedings of the 9th Congress of the European Society for Research in Mathematics Education (pp. 754–760). Prague: Charles University in Prague and the European Society for Research in Mathematics Education (ERME).

• Shaughnessy, J. M. (2007). Research on statistics learning and reasoning. In F. Lester & NCTM (Eds.), Second handbook of research on mathematics teaching and learning (pp. 957–1009). Information Age Publications: Charlotte, NC.

• Shaughnessy, J. M., Chance, B., & Kranendonk, H. (2009). Focus in high school mathematics. Reasoning and sense making: Statistics and probability. National Council of Teachers of Mathematics. Reston: VA.

• Shaughnessy, J. M., Ciancetta, M., & Canada, D. (2004). Types of student reasoning on sampling tasks. In M. Johnsen Høines & A. Berit Fuglestad (Eds.), Proceedings of the 28th meeting of the International Group for Psychology and Mathematics Education (Vol. 4, pp. 177–184). Bergen, Norway: Bergen University College Press.

• Slauson, L. V. (2008). Students’ conceptual understanding of variability (Unpublished PhD Dissertation). The Ohio State University.

• Taylor, L., & Doehler, K. (2015). Reinforcing sampling distributions through a randomization-based activity for introduction ANOVA. Journal of Statistics Education, 23(3).

• Vergnaud, G. (1996). The theory of conceptual fields. In L. P. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 219–239). Hillsdale, NJ: Erlbaum.

• Wassner, C. (2004). Förderung Bayesianischen Denkens: kognitionspsychologische Grundlagen und didaktische Analysen; mit Arbeitsmaterialien und didaktischen Kommentaren zum Thema “Authentisches Bewerten und Urteilen unter Unsicherheit” für den Stochastikunterricht der Sekundarstufe I. Hildesheim [u.a.]: Franzbecker.

• Watkins, A., Bargagliotti, A., & Franklin, C. (2014). Simulation of the sampling distribution of the mean can mislead. Journal of Statistics Education, 22(3).

• Watson, J. M. (2009). The influence of variation and expectation on the developing awareness of distribution. Statistics Education Research Journal, 8(1), 32–61.

• Watson, J., & Callingham, R. (2014). Two-way tables: Issues at the heart of statistics and probability for students and teachers. Mathematical Thinking and Learning, 16(4), 254–284.

• Watson, J., & Callingham, R. (2015). Lung disease, indigestion, and two-way tables. Investigations in Mathematics Learning, 8(2), 1–16.

• Watson, J., Callingham, R., & Donne, J. (2008). Proportional reasoning: Student knowledge and teachers’ pedagogical content knowledge. In M. Goos, R. Brown, & K. Makar (Eds.), Navigating currents and charting directions. Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 563–571). Brisbane, Australia: Mathematics Education Research Group of Australasia.

• Watson, J. M., Callingham, R. A., & Kelly, B. A. (2007). Students’ appreciation of expectation and variation as a foundation for statistical understanding. Mathematical Thinking and Learning, 9(3), 83–130.

• Watson, J., Fitzallen, N., Wilson, K., & Creed, J. (2008). The representational value of HATS. Mathematics Teaching in Middle School, 14(1), 4–10.

• Watson, J. M., & Kelly, B. A. (2004a). Expectation versus variation: Students’ decision making in a chance environment. Canadian Journal of Science, Mathematics and Technology Education, 4(3), 371–396.

• Watson, J. M., & Kelly, B. A. (2004b). Statistical variation in a chance setting: A two-year study. Educational Studies in Mathematics, 57(1), 121–144.

• Watson, J. M., & Kelly, B. A. (2006). Expectation versus variation: Students’ decision making in a sampling environment. Canadian Journal of Science, Mathematics and Technology Education, 6(2), 145–166.

• Watson, J. M., Kelly, B. A., Callingham, R. A., & Shaughnessy, J. M. (2003). The measurement of school students’ understanding of statistical variation. International Journal of Mathematical Education in Science and Technology, 34(1), 1–29.

• Watson, J. M., & Moritz, J. B. (1999). The beginnings of statistical inference: Comparing two data sets. Educational Studies in Mathematics, 37, 145–168.

• Watson, J. M., & Moritz, J. B. (2000). The longitudinal development of understanding of average. Mathematical Thinking and Learning, 2(1), 11–50.

• Watson, J. M., & Shaughnessy, J. M. (2004). Proportional reasoning: Lessons from research in data and chance. Mathematics Teaching in the Middle School, 10(1), 104–109.

• Wild, C. J. (2006). The concept of distribution. Statistics Education Research Journal, 5(2), 10–26.

• Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(1), 223–265.

• Zieffler, A. S., & Garfield, J. B. (2009). Modeling the growth of students’ covariational reasoning during an introductory statistics course. Statistics Education Research Journal, 8(1), 7–31.

• Zieffler, A., Harring, J., & Long, J. D. (2011). Comparing groups: Randomization and bootstrap methods using R. Hoboken, NJ: Wiley.

## Author information

Authors

### Corresponding author

Correspondence to Rolf Biehler .

## Rights and permissions

Reprints and Permissions

© 2018 Springer International Publishing AG

### Cite this chapter

Biehler, R., Frischemeier, D., Reading, C., Shaughnessy, J.M. (2018). Reasoning About Data. In: Ben-Zvi, D., Makar, K., Garfield, J. (eds) International Handbook of Research in Statistics Education. Springer International Handbooks of Education. Springer, Cham. https://doi.org/10.1007/978-3-319-66195-7_5