Building Concept Images of Fundamental Ideas in Statistics: The Role of Technology

  • Gail BurrillEmail author
Part of the ICME-13 Monographs book series (ICME13Mo)


Having a coherent mental structure for a concept is necessary for students to make sense of and use the concept in appropriate and meaningful ways. Dynamically linked documents based on TI© Nspire technology can provide students with opportunities to build such mental structures by taking meaningful statistical actions, identifying the consequences, and reflecting on those consequences, with appropriate instructional guidance. The collection of carefully sequenced documents is based on research about student misconceptions and challenges in learning statistics. Initial analysis of data from preservice elementary teachers in an introductory statistics course highlights their progress in using the documents to cope with variability in a variety of contextual situations.


Concept image Deviation Distribution Interactive dynamic visualization Mean Variability 


  1. Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245–274.CrossRefGoogle Scholar
  2. Bakker, A., Biehler, R., & Konold, C. (2005). Should young students learn about boxplots? In G. Burrill & M. Camden (Eds.), Curriculum development in statistics education: International association for statistics education 2004 roundtable (pp. 163–173). Voorburg, the Netherlands: International Statistics Institute.Google Scholar
  3. Bakker, A., & Gravemeijer, K. (2004). Learning to reason about distributions. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 147–168). Dordrecht, The Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar
  4. Bakker, A., & Van Eerde, H. (2014). An introduction to design-based research with an example from statistics education. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Doing qualitative research: Methodology and methods in mathematics education (pp. 429–466). New York: Springer.Google Scholar
  5. Batanero, C. (2015). Understanding randomness: Challenges for research and teaching. In K. Kriner (Ed.), Proceedings of the ninth congress of the European Society for Research in Mathematics Education (pp. 34–49).Google Scholar
  6. Baumgartner, L. M. (2001). An update on transformational learning. In S. B. Merriam (Ed.), New directions for adult and continuing education, no. 89 (pp. 15–24). San Francisco, CA: Jossey-Bass.CrossRefGoogle Scholar
  7. Ben-Zvi, D. (2000). Toward understanding the role of technological tools in statistical learning. Mathematical Thinking and Learning, 2(1–2), 127–155.CrossRefGoogle Scholar
  8. Ben-Zvi, D., & Arcavi, A. (2001). Junior high school students’ construction of global views of data and data representations. Educational Studies in Mathematics, 45(1–3), 35–65.CrossRefGoogle Scholar
  9. Ben-Zvi, D., & Friedlander, A. (1997). Statistical thinking in a technological environment. In J. B. Garfield & G. Burrill (Eds.), Research on the role of technology in teaching and learning statistics (pp. 45–55). Voorburg, The Netherlands: International Statistical Institute.Google Scholar
  10. Biehler, R., Ben-Zvi, D., Bakker, A., & Makar, 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). Springer.Google Scholar
  11. Building Concepts: Statistics and Probability. (2016). Texas Instruments Education Technology.
  12. Black, P., & Wiliam, D. (1998). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappan, 80(2), 139–144.Google Scholar
  13. Breen, C. (1997). Exploring imagery in P, M and E. In E. Pehkonen (Ed.), Proceedings of the 21st PME International Conference, 2, 97–104.Google Scholar
  14. Burrill, G. (2014). Tools for learning statistics: Fundamental ideas in statistics and the role of technology. In Mit Werkzeugen Mathematik und Stochastik lernen[Using Tools for Learning Mathematics and Statistics], (pp. 153–162). Springer Fachmedien Wiesbaden.Google Scholar
  15. 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 (2).
  16. Cobb, P., McClain, K., & Gravemeijer, K. (2003). Learning about statistical covariation. Cognition and Instruction, 21(1), 1–78.CrossRefGoogle Scholar
  17. Common Core State Standards. (2010). College and career standards for mathematics. Council of Chief State School Officers (CCSSO) and National Governor’s Association (NGA).Google Scholar
  18. Cranton, P. (2002). Teaching for transformation. In J. M. Ross-Gordon (Ed.), New directions for adult and continuing education, no. 93 (pp. 63–71). San Francisco, CA: Jossey-Bass.Google Scholar
  19. 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, [Online] 7(3). (
  20. delMas, R., & Liu, Y. (2005). Exploring students’ conceptions of the standard deviation. Statistics Education Research Journal, 4(1), 55–82.Google Scholar
  21. Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.), Proceedings of the 15th PME International Conference, 1, 33–48.Google Scholar
  22. 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.Google Scholar
  23. Fischbein, E., Nello, M. S., & Marino, M. S. (1991). Factors affecting probabilistic judgments in children and adolescents. Educational Studies in Mathematics, 22(6), 523–549.CrossRefGoogle Scholar
  24. Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic, intuitively based misconceptions. Journal for Research in Mathematics Education, 28(1), 96–105.CrossRefGoogle Scholar
  25. Friel, S. (1998). Teaching statistics: What’s average? In L. J. Morrow (Ed.), The teaching and learning of algorithms in school mathematics (pp. 208–217). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  26. 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
  27. Groth, R., & Bergner, J. (2006). Preservice elementary teachers’ conceptual and procedural knowledge of mean, median, and mode. Mathematical Thinking and Learning, 8(1), 37–63.CrossRefGoogle Scholar
  28. Gould, R. (2011). Statistics and the modern student. Department of statistics papers. Department of Statistics, University of California Los Angeles.Google Scholar
  29. Guin, D., & Trouche, L. (1999). The complex process of converting tools into mathematical instruments: The case of calculators. International Journal of Computers for Mathematical Learning, 3(3), 195–227.CrossRefGoogle Scholar
  30. Hancock, C., Kaput, J., & Goldsmith, L. (1992). Authentic inquiry with data: Critical barriers to classroom implementation. Educational Psychologist, 27(3), 337–364.CrossRefGoogle Scholar
  31. Hodgson, T. (1996). The effects of hands-on activities on students’ understanding of selected statistical concepts. In E. Jakbowski, D. Watkins & H. Biske (Eds.), Proceedings of the eighteenth annual meeting of the North American chapter of the international group for the psychology of mathematics education (pp. 241–246).Google Scholar
  32. Johnston-Wilder, P., & Pratt, D. (2007). Developing stochastic thinking. In R. Biehler, M. Meletiou, M. Ottaviani & D. Pratt (Eds.), A working group report of CERME 5 (pp. 742–751).Google Scholar
  33. Jones, G., Langrall, C., & Mooney, E. (2007). Research in probability: Responding to classroom realities. In F. K. Lester (Ed.), The second handbook of research on mathematics (pp. 909–956). Reston, VA: National Council of Teachers of Mathematics (NCTM).Google Scholar
  34. Kader, G., & Mamer, J. (2008). Contemporary curricular issues: Statistics in the middle school: Understanding center and spread. Mathematics Teaching in the Middle School, 14(1), 38–43.Google Scholar
  35. Kolb, D. (1984). Experiential learning: Experience as the source of learning and development. New Jersey: Prentice-Hall.Google Scholar
  36. Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6(1), 59–98.CrossRefGoogle Scholar
  37. Langer, E. J. (1975). The illusion of control. Journal of Personality and Social Psychology, 32(2), 311–328.CrossRefGoogle Scholar
  38. Learning Progressions for the Common Core Standards in Mathematics: 6–8 Progression probability and statistics (Draft). (2011). Common Core State Standards Writing Team.Google Scholar
  39. Mathematics Education of Teachers II. (2012). Conference Board of the Mathematical Sciences. Providence, RI and Washington, DC: American Mathematical Society and Mathematical Association of America.Google Scholar
  40. Mathews, D., & Clark, J. (2003). Successful students’ conceptions of mean, standard deviation and the central limit theorem. Unpublished paper.Google Scholar
  41. Mezirow, J. (1997). Transformative learning: Theory to practice. In P. Cranton (Ed.), New directions for adult and continuing education, no. 74. (pp. 5–12). San Francisco, CA: Jossey-Bass.CrossRefGoogle Scholar
  42. Mezirow, J. (2000). Learning to think like an adult: Core concepts of transformation theory. In J. Mezirow & Associates (Eds.), Learning as transformation: Critical perspectives on a theory in progress (pp. 3–34). San Francisco, CA: Jossey-Bass.Google Scholar
  43. Michael, J., & Modell, H. (2003). Active learning in secondary and college science classrooms: A working model of helping the learner to learn. Mahwah, NJ: Erlbaum.Google Scholar
  44. Mokros, J., & Russell, S. (1995). Children’s concepts of average and representativeness. Journal for Research in Mathematics Education, 26(1), 20–39.CrossRefGoogle Scholar
  45. National Research Council. (1999). In J. Bransford, A. Brown & R. Cocking (Eds.), How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.Google Scholar
  46. Oehrtman, M. (2008). Layers of abstraction: Theory and design for the instruction of limit concepts. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (pp. 1–21).Google Scholar
  47. Peirce, C. S. (1932). In C. Hartshorne & P. Weiss (Eds.), Collected papers of Charles Sanders Peirce 1931–1958. Cambridge, MA: Harvard University Press.Google Scholar
  48. Peirce, C. S. (1998). The essential Peirce: Selected philosophical writings, Vol. 2 (1893–1913). The Peirce Edition Project. Bloomington, Indiana: Indiana University Press.Google Scholar
  49. Piaget, J. (1970). Structuralism, (C. Maschler, Trans.). New York: Basic Books, Inc.Google Scholar
  50. Piaget, J. (1985). The equilibration of cognitive structures (T. Brown & K. J. Thampy, Trans.). Chicago: The University of Chicago Press.Google Scholar
  51. Posner, G., Strike, K., Hewson, P., & Gertzog, W. (1982). Accommodation of a scientific conception: Toward a theory of conceptual change. Science Education, 66(2), 211–227.CrossRefGoogle Scholar
  52. Presmeg, N. C. (1994). The role of visually mediated processes in classroom mathematics. Zentralblatt für Didaktik der Mathematik: International Reviews on Mathematics Education, 26(4), 114–117.Google Scholar
  53. Sacristan, A., Calder, N., Rojano, T., Santos-Trigo, M., Friedlander, A., & Meissner, H. (2010). The influence and shaping of digital technologies on the learning— and learning trajectories—of mathematical concepts. In C. Hoyles & J. Lagrange (Eds.), Mathematics education and technology—Rethinking the mathematics education and technology—Rethinking the terrain: The 17th ICMI Study (pp. 179–226). New York, NY: Springer.Google Scholar
  54. Shaughnessy, J., Watson, J., Moritz, J., & Reading, C. (1999). School mathematics students’ acknowledgement of statistical variation. In C. Maher (Chair), There’s more to life than centers. Presession Research Symposium, 77th Annual National Council of Teachers of Mathematics Conference, San Francisco, CA.Google Scholar
  55. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.CrossRefGoogle Scholar
  56. Taylor, E. W. (2007). An update of transformative learning theory: A critical review of the empirical research (1999–2005). International Journal of Lifelong Education, 26(2), 173–191.CrossRefGoogle Scholar
  57. Thompson, P. (2002). Didactic objects and didactic models in radical constructivism. In K. Gravemeijer, R. Lehrer, B. V. Oers & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 191–212). Dordrecht, The Netherlands: Kluwer Academic Publishers.CrossRefGoogle Scholar
  58. Watson, J., & Fitzallen, N. (2016). Statistical software and mathematics education: Affordances for learning. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (3rd ed., pp. 563–594). New York, NY: Routledge.Google Scholar
  59. Watson, J., & Moritz, J. (2000). Developing concepts of sampling. Journal for Research in Mathematics Education, 31(1), 44–70.CrossRefGoogle Scholar
  60. Wild, C. (2006). The concept of distribution. Statistics Education Research Journal, 5(2), 10–26.
  61. Zehavi, N., & Mann, G. (2003). Task design in a CAS environment: Introducing (in)equations. In J. Fey, A. Couco, C. Kieran, L. McCullin, & R. Zbiek (Eds.), Computer algebra systems in secondary school mathematics education (pp. 173–191). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  62. Zull, J. (2002). The art of changing the brain: Enriching the practice of teaching by exploring the biology of learning. Alexandria VA: Association for Supervision and Curriculum Development.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Program in Mathematics EducationMichigan State UniversityEast LansingUSA

Personalised recommendations