Advertisement

ZDM

, Volume 44, Issue 7, pp 899–911 | Cite as

A conceptual pathway to confidence intervals

Original Article

Abstract

Finding ways for the majority of students to better understand conventional normal theory-based statistical inference seems to be an intractable problem area for researchers. In this paper we propose a conceptual pathway for developing confidence interval ideas for the one-sample situation only from an intuitive sense to bootstrapping for students from about age 14 to first-year university. We make the case that conceptual development should start early; that probability and statistical instruction should change so that both orientate students towards interconnected stochastic conceptions; and that the use of visual imagery has the potential to stimulate students towards such a perspective. We analyse our conceptual pathway based on a theoretical framework for a stochastic conception of statistical inference based on imagery and some research evidence. Our analysis suggests that the pathway has the potential for students to become conversant with the concepts underpinning inference, to view statistics probabilistically and to integrate concepts into a coherent comprehension of inference.

Keywords

Stochastic conception Probability–statistics instruction Secondary-university students Dynamic visual imagery Bootstrap 

Notes

Acknowledgments

This research is partly funded by a grant from the Teaching and Learning Research Initiative (http://www.tlri.org.nz).

References

  1. Arnold, P., Pfannkuch, M., Wild, C., Regan, M., & Budgett, S. (2011). Enhancing students’ inferential reasoning: From handson to “movies”. Journal of Statistics Education, 19(2), 1–32. http://www.amstat.org/publications/jse/v19n2/pfannkuch.pdf.Google Scholar
  2. Bakker, A. (2004). Reasoning about shape as a pattern in variability. Statistics Education Research Journal, 3(2), 64–83. http://www.stat.auckland.ac.nz/serj.
  3. Belia, S., Fidler, F., Williams, J., & Cumming, G. (2005). Researchers misunderstand confidence intervals and error bars. Psychological Methods, 10(4), 389–396.CrossRefGoogle Scholar
  4. Beyth-Marom, R., Fidler, F., & Cumming, G. (2008). Statistical cognition: Towards evidence-based practice in statistics and statistics education. Statistics Education Research Journal, 7(2), 20–39. http://www.stat.auckland.ac.nz/serj.
  5. 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, The Netherlands: International Statistical Institute. http://www.stat.auckland.ac.nz/~iase/publications.php (online).
  6. Bland, M. (2011). Reporting clinical trials with confidence. Paper presented at the Open University Statistics Conference, May 18, 2011. http://www-users.york.ac.uk/~mb55/talks/bland_ou.pdf.
  7. 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. 295–324). Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  8. Chernick, M. (2008). Bootstrap methods—A guide for practitioners and researchers (2nd ed.). New York: Wiley.Google Scholar
  9. Cumming, G. (2006). Understanding replication: Confidence intervals, p values, and what’s likely to happen next time. In A. Rossman & B. Chance (Eds.), Proceedings of the seventh International Conference on Teaching Statistics. Voorburg, The Netherlands: International Statistical Institute. http://www.stat.auckland.ac.nz/~iase/publications.php (online).
  10. Cumming, G. (2007). Inference by eye: Pictures of confidence intervals and thinking about levels of confidence. Teaching Statistics, 29(3), 89–93.CrossRefGoogle Scholar
  11. Cumming, G., Williams, J., & Fidler, F. (2004). Replication and researchers’ understanding of confidence intervals and standard error bars. Understanding Statistics, 3(4), 299–311.CrossRefGoogle Scholar
  12. DelMas, R., Garfield, J., & Chance, B. (1999). A model of classroom research in action: Developing simulation activities to improve students’ statistical reasoning. Journal of Statistics Education, 7(3). http://www.amstat.org/publications/jse/v7n3.
  13. Efron, B. (2000). The bootstrap and modern statistics. Journal of the American Statistics Association, 95(452), 1293–1296.CrossRefGoogle Scholar
  14. Engel, J. (2010). On teaching bootstrap confidence intervals. In C. Reading (Ed.), Proceedings of the eighth International Conference on Teaching Statistics. Voorburg, The Netherlands: International Statistical Institute. http://www.stat.auckland.ac.nz/~iase/publications.php (online).
  15. Fidler, F. (2006). Should psychology abandon p values and teach CIs instead? Evidence-based reforms in statistics education. In A. Rossman & B. Chance (Eds.), Proceedings of the seventh International Conference on Teaching Statistics. Voorburg, The Netherlands: International Statistical Institute. http://www.stat.auckland.ac.nz/~iase/publications.php (online).
  16. Garfield, J., & Ben-Zvi, D. (2008). Developing students’ statistical reasoning: connecting research and teaching practice. New York: Springer.Google Scholar
  17. Garfield, J., delMas, R., & Chance, B. (1999).Tools for teaching and assessing statistical inference. http://www.tc.umn.edu/~delma001/stat_tools/.
  18. Hegarty, M. (2004). Dynamic visualizations and learning: Getting to the difficult questions. Learning and Instruction, 14, 343–351.CrossRefGoogle Scholar
  19. Hesterberg, T. (2006). Bootstrapping students’ understanding of statistical concepts. In G. Burrill (Ed.), Thinking and reasoning with data and chance. Sixty-eighth National Council of Teachers of Mathematics Yearbook (pp. 391–416). Reston, VA: NCTM.Google Scholar
  20. Hesterberg, T., Moore, D., Monaghan, S., Clipson, A., & Epstein, R. (2009). Bootstrap methods and permutation tests. In D. Moore, G. McCabe & B. Craig (Eds.), Introduction to the practice of statistics (6th ed., pp. 16-1–16-60). New York, NY: Freeman.Google Scholar
  21. Jones, P., Lipson, K., & Phillips, B. (1994). A role for computer intensive methods in introducing statistical inference. In L. Brunelli & G. Cicchitelli (Eds.), Proceedings of the First Scientific Meeting of the International Association for Statistical Education (pp. 199–211). Perugia, Italy: University of Perugia.Google Scholar
  22. Kahneman, D., Slovic, P., & Tversky, A. (Eds.). (1982). Judgment under uncertainty: Heuristics and biases. New York: Press Syndicate of the University of Cambridge.Google Scholar
  23. Konold, C., & Kazak, S. (2008). Reconnecting data and chance. Technology Innovations in Statistics Education, 2(1). http://repositories.cdlib.org/uclastat/cts/tise/vol2/iss1/art1/.
  24. Konold, C., Madden, S., Pollatsek, A., Pfannkuch, M., Wild, C., Ziedins, I., et al. (2011). Conceptual challenges in coordinating theoretical and data-centered estimates of probability. Mathematical Thinking and Learning, 13(1 & 2), 68–86.CrossRefGoogle Scholar
  25. Liu, Y., & Thompson, P. (2007). Teachers’ understandings of probability. Cognition and Instruction, 25(2), 113–160.CrossRefGoogle Scholar
  26. Liu, Y., & Thompson, P. (2009). Mathematics teachers’ understandings of proto-hypothesis testing. Pedagogies, 4(2), 126–138.Google Scholar
  27. Makar, K., & Confrey, J. (2005). “Variation-Talk”: Articulating meaning in statistics. Statistics Education Research Journal, 4(1), 27–54. http://www.stat.auckland.ac.nz/serj.
  28. Makar, K., & Rubin, A. (2009).A framework to support research on informal inferential reasoning. Statistics Education Research Journal, 8(1), 82–105. http://www.stat.auckland.ac.nz/serj.
  29. Meletiou-Mavrotheris, M., Lee, C., & Fouladi, R. (2007). Introductory statistics, college student attitudes and knowledge—a qualitative analysis of the impact of technology-based instruction. International Journal of Mathematical Education in Science and Technology, 38(1), 65–83.CrossRefGoogle Scholar
  30. Nickerson, R. (2004). Cognition and chance: The psychology of probabilistic reasoning. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  31. Perkins, D., & Unger, C. (1994). A new look in representations for mathematics and science learning. Instructional Science, 22, 1–37.CrossRefGoogle Scholar
  32. Pfannkuch, M. (2008). Building sampling concepts for statistical inference: A case study. In 11th International Congress of Mathematics Education Proceedings, Monterrey, Mexico. http://tsg.icme11.org/tsg/show/15.
  33. Pfannkuch, M. (2011). The role of context in developing informal statistical inferential reasoning: A classroom study. Mathematical Thinking and Learning, 13(1 & 2), 27–46.CrossRefGoogle Scholar
  34. Pfannkuch, M., Regan, M., Wild, C.J., & Horton, N. (2010). Telling data stories: essential dialogues for comparative reasoning. Journal of Statistics Education, 18(1). http://www.amstat.org/publications/jse/v18n1/pfannkuch.pdf.
  35. Pratt, D., & Ainley, J. (2008). Introducing the special issue on informal inferential reasoning. Statistics Education Research Journal, 7(2), 3–4. http://www.stat.auckland.ac.nz/serj.
  36. Rossman, A., & Chance, B. (2004). Anticipating and addressing student misconceptions. Paper presented at the ARTIST Conference on assessment in Statistics, Lawrence University, 1–4 August, 2004. http://www.rossmanchance.com/artist/proceedings/rossman.pdf.
  37. Schwartz, D., & Goldman, S. (1996). Why people are not like marbles in an urn: An effect of context on statistical reasoning. Applied Cognitive Psychology, 10, S99–S112.CrossRefGoogle Scholar
  38. Shaughnessy, M. (2006). Research on students’ understanding of some big concepts in statistics. In G. Burrill (Ed.), Thinking and reasoning with data and chance. Sixty-eighth National Council of Teachers of Mathematics Yearbook (pp. 77–98). Reston, VA: NCTM.Google Scholar
  39. Shaughnessy, M. (2007). Research on statistics learning and reasoning. In F. Lester (Ed.), Second handbook of research on the teaching and learning of mathematics (Vol. 2, pp. 957–1009). Charlotte, NC: Information Age Publishers.Google Scholar
  40. Sotos, A., Vanhoof, S., Noortgate, W., & Onghena, P. (2007). Students’ misconceptions of statistical inference: A review of the empirical evidence from research on statistics education. Educational Research Review, 2, 98–113.CrossRefGoogle Scholar
  41. Thompson, P., Liu, Y., & Saldanha, L. (2007). Intricacies of statistical inference and teachers’ understandings of them. In M. Lovett & P. Shaw (Eds.), Thinking with data (pp. 207–231). Mawah, NJ: Erlbaum.Google Scholar
  42. Wild, C. J. (2006).The concept of distribution. Statistics Education Research Journal, 5(2), 10–26. http://www.stat.auckland.ac.nz/serj.Google Scholar
  43. Wild, C. J., Pfannkuch, M., Regan, M., & Horton, N. (2011). Towards more accessible conceptions of statistical inference. Journal of the Royal Statistical Society: Series A (Statistics in Society), 174(2), 247–295.Google Scholar

Copyright information

© FIZ Karlsruhe 2012

Authors and Affiliations

  • Maxine Pfannkuch
    • 1
  • Chris J. Wild
    • 1
  • Ross Parsonage
    • 1
  1. 1.The University of AucklandAucklandNew Zealand

Personalised recommendations