Skip to main content

Statistics Learning Trajectories

  • Chapter
  • First Online:

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

Abstract

Statistics curricula and pedagogy are changing rapidly in response to a growing body of research findings involving students’ reasoning processes, technology capability, attention to underpinning conceptual infrastructure, and new ways of statistical practice. Because many of the statistical ideas being considered are currently not in the curriculum, many researchers in statistics education have investigated students’ reasoning processes through the use of learning trajectories in conjunction with design-based research methods. In this chapter, we outline the characteristics of learning trajectories and exemplify how learning trajectories have been used in three case studies in statistics education. Commonalities and differences across the learning trajectories are discussed as well as recommendations for future research.

This is a preview of subscription content, log in via an institution.

Notes

  1. 1.

    The popliteal length is a measurement taken on the back of the leg from behind the knee to the floor when a student is seated.

References

  • Ainley, J., Pratt, D., & Hansen, A. (2006). Connecting engagement and focus in pedagogic task design. British Educational Research Journal, 32(1), 23–38.

    Article  Google Scholar 

  • 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

  • Arnold, P., Pfannkuch, M., Wild, C. J., Regan, M., & Budgett, S. (2011). Enhancing students’ inferential reasoning: From hands-on to “movies”. Journal of Statistics Education, 19(2), 1–32.

    Article  Google Scholar 

  • Bakker, A. (2004). Design research in statistics education: On symbolizing and computer tools. Utrecht, The Netherlands: Freudenthal Institute.

    Google Scholar 

  • Bakker, A., & Gravemeijer, K. (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.

    Chapter  Google Scholar 

  • Bakker, A., & van Eerde, D. (2015). An introduction to design-based research with an example from statistics education. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education (pp. 429–466). Dordrecht, The Netherlands: Springer. https://doi.org/10.1007/978-94-017-9181-6_16

    Google Scholar 

  • Baroody, A., Cibulskis, M., Lai, M., & Li, X. (2004). Comments on the use of learning trajectories in curriculum development and research. Mathematical Thinking and Learning, 6(2), 227–260. https://doi.org/10.1207/s153277833mtl0602_8

    Article  Google Scholar 

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

    Google Scholar 

  • Ben-Zvi, D., & Garfield, J. (2004). Research on statistical literacy, reasoning, and thinking: Issues, challenges, and implications. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking (pp. 397–409). Dordrecht, The Netherlands: Kluwer Academic Publishers.

    Chapter  Google Scholar 

  • Bodemer, D., Ploetzner, R., Feuerlein, I., & Spada, H. (2004). The active integration of information during learning with dynamic and interactive visualisations. Learning and Instruction, 14(3), 325–341. https://doi.org/10.1016/j.learninstruc.2004.06.006

    Article  Google Scholar 

  • Catalysts for Change. (2012). Statistical thinking: A simulation approach to modeling uncertainty. Minnesota, MN: Catalyst Press.

    Google Scholar 

  • CensusAtSchool New Zealand. (2003). Retrieved from www.censusatschool.org.nz

  • Clements, D., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81–89.

    Article  Google Scholar 

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

    Google Scholar 

  • Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. (College- and Career-Readiness Standards and K12 standards in English Language Arts and Math). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Retrieved from http://www.corestandards.org

  • Confrey, J. (1991). Learning to listen: A student's understanding of powers of ten. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 111–138). Dordrecht, The Netherlands: Kluwer Academic Press.

    Chapter  Google Scholar 

  • Confrey, J. (2002). Sixth grade pre-algebra curriculum at St. Francis School, Austin, TX. Unpublished document.

    Google Scholar 

  • Confrey, J. (2006). The evolution of design studies as methodology. In K. Sawyer (Ed.), Cambridge handbook of the learning sciences (pp. 135–151). Cambridge, UK: Cambridge University Press.

    Google Scholar 

  • Confrey, J. (2015). Some possible implications of data-intensive research in education—The value of learning maps and evidence-centered design of assessment to educational data mining. In C. Dede (Ed.), Data-intensive research in education: Current work and next steps. Report on two National Science Foundation-sponsored Computing Research Association workshops (pp. 79–87). Washington, DC: Computing Research Association.

    Google Scholar 

  • Confrey, J., & Lachance, A. (2000). Transformative teaching experiments through conjecture-driven research design. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 231–265). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Design-Based Implementation Research. (2016). Retrieved from http://learndbir.org/

  • Dierdorp, A., Bakker, A., Eijkelhof, H., & van Maanen, J. (2011). Authentic practices as contexts for learning to draw inferences beyond correlated data. Mathematical Thinking and Learning, 13(1&2), 132–151. https://doi.org/10.1080/10986065.2011.538294

    Article  Google Scholar 

  • Finzer, B., Konold, C. & Erickson, T. (2012). Data games. Retrieved from http://play.ccssgames.com

  • 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. Alexandria, VA: American Statistical Association.

    Google Scholar 

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

    Google Scholar 

  • Garfield, J., & Ben-Zvi, D. (2007). How students learn statistics revisited: A current review of research on teaching and learning statistics. International Statistical Review, 75(3), 372–396.

    Article  Google Scholar 

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

    Google Scholar 

  • Garfield, J., delMas, R., & Zieffler, A. (2012). Developing statistical modelers and thinkers in an introductory, tertiary-level statistics course. ZDM Mathematics Education, 44(7), 883–898.

    Article  Google Scholar 

  • Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105–128.

    Article  Google Scholar 

  • Gravemeijer, K., & Cobb, P. (2006). Design research from a learning design perspective. In J. van den Akker (Ed.), Educational design research (pp. 17–51). London: Routledge.

    Google Scholar 

  • Konold, C., & Miller, C. (2005). TinkerPlots dynamic data exploration: Statistics software for middle school curricula. Emeryville, CA: Key Curriculum Press.

    Google Scholar 

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

    Article  Google Scholar 

  • Lane-Getaz, S. J. (2006). What is statistical thinking and how is it developed? In G. Burrill (Ed.), Thinking and reasoning with data and chance: Sixty-eighth NCTM yearbook (pp. 272–289). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Lee, H. S., Angotti, R. L., & Tarr, J. E. (2010). Making comparisons between observed data and expected outcomes: Students’ informal hypothesis testing with probability simulation tools. Statistics Education Research Journal, 9(1), 68–96.

    Google Scholar 

  • Lee, H. S., Doerr, H. M., Tran, D., & Lovett, J. N. (2016). The role of probability in developing learners’ models of simulation approaches to inference. Statistics Education Research Journal, 15(2), 216–238.

    Google Scholar 

  • Lee, H. S., Starling, T. T., & Gonzalez, M. D. (2014). Connecting research to practice: Using data to motivate empirical sampling distributions. Mathematics Teacher, 107(6), 465–469.

    Article  Google Scholar 

  • Lehrer, R. (2016). Data modeling. Retrieved from http://modelingdata.org/

  • Lehrer, R., & Kim, M. J. (2009). Structuring variability by negotiating its measure. Mathematics Education Research Journal, 21(2), 116–133.

    Article  Google Scholar 

  • Lehrer, R., Kim, M. J., Ayers, E., & Wilson, M. (2014). Toward establishing a learning progression to support the development of statistical reasoning. In A. Maloney, J. Confrey, & K. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (pp. 32–59). Charlotte, NC: Information Age Publishers.

    Google Scholar 

  • Lehrer, R., Kim, M. J., & Jones, R. S. (2011). Developing conceptions of statistics by designing measures of distribution. ZDM Mathematics Education, 43(5), 723–736.

    Article  Google Scholar 

  • Lehrer, R., & Schauble, L. (Eds.) (2002). Investigating real data in the classroom: Expanding children’s understanding of math and science. New York, NY: Teachers College Press.

    Google Scholar 

  • Lesh, R. A., & Doerr, H. M. (Eds.). (2003). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Lock, R., Lock, P., Morgan, K., Lock, E., & Lock, D. (2013). StatKey. Retrieved from http://lock5stat.com/statkey

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

    Article  Google Scholar 

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

    Google Scholar 

  • Makar, K., & Rubin, A. (2009). A framework to support research on informal inferential reasoning. Statistics Education Research Journal, 8(1), 82–105.

    Google Scholar 

  • Ministry of Education. (2007). The New Zealand curriculum. Wellington, New Zealand: Learning Media.

    Google Scholar 

  • Petrosino, A. J., Lehrer, R., & Schauble, L. (2003). Structuring error and experimental variation as distribution in the fourth grade. Mathematical Thinking and Learning, 5(2&3), 131–156.

    Article  Google Scholar 

  • 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.

    Google Scholar 

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

    Chapter  Google Scholar 

  • Prediger, S., Gravemeijer, K., & Confrey, J. (2015). Design research with a focus on learning processes – An overview on achievements and challenges. ZDM Mathematics Education, 47(6), 877–891.

    Article  Google Scholar 

  • Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability for mathematical meanings. Educational Studies in Mathematics, 70(2), 111–126.

    Article  Google Scholar 

  • Rubin, A., Bruce, B., & Tenney, Y. (1990). Learning about sampling: Trouble at the core of statistics. In D. Vere-Jones (Ed.), Proceedings of the 3rd International Conference on Teaching Statistics, Dunedin, New Zealand (Vol. 2, pp. 314–319). Voorburg, The Netherlands: International Statistical Institute. Retrieved from http://www.stat.auckland.ac.nz/~iase/publications/18/BOOK1/A9-4.pdf

    Google Scholar 

  • Saldanha, L., & Liu, Y. (2014). Challenges of developing coherent probabilistic reasoning: Rethinking randomness and probability from a stochastic perspective. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic thinking: Presenting plural perspectives (pp. 367–396). Dordrecht, The Netherlands: Springer.

    Google Scholar 

  • Saldanha, L., & Thompson, P. (2002). Conceptions of sample and their relationship to statistical inference. Educational Studies in Mathematics, 51(3), 257–270.

    Article  Google Scholar 

  • Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction. Cognition and Instruction, 22(2), 129–184.

    Article  Google Scholar 

  • Sfard, A. (2005). What could be more practical than good research? On mutual relations between research and practice of mathematics education. Educational Studies in Mathematics Education, 58(3), 393–413.

    Article  Google Scholar 

  • Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145.

    Article  Google Scholar 

  • Watson, A., & Ohtani, M. (Eds.). (2015). Task design in mathematics education. New York, NY: Springer.

    Google Scholar 

  • Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223–248.

    Article  Google Scholar 

  • Wild, C. J., Pfannkuch, M., Regan, M., & Horton, N. (2011). Towards more accessible conceptions of statistical inference (with discussion). Journal of the Royal Statistical Society: Series A (Statistics in Society), 174(2), 247–295.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Maxine Pfannkuch .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Arnold, P., Confrey, J., Jones, R.S., Lee, H.S., Pfannkuch, M. (2018). Statistics Learning Trajectories. 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_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-66195-7_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66193-3

  • Online ISBN: 978-3-319-66195-7

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics