Journal of Mathematics Teacher Education

, Volume 19, Issue 1, pp 57–77 | Cite as

Investigating the improvement of prospective elementary teachers’ number sense in reasoning about fraction magnitude

  • Ian WhitacreEmail author
  • Susan D. Nickerson


We report on interview results from a classroom teaching experiment in a Number and Operations course for prospective elementary teachers. Improving the number sense of this population is an important goal for mathematics teacher education, and researchers have found this goal to be difficult to accomplish. In earlier work, we devised a local instruction theory for the development of number sense, which focused on whole-number mental computation. In this study, the local instruction theory was applied to the rational-number domain, with the help of a framework for reasoning about fraction magnitude, and it guided instruction in the content course. We interviewed seven participants pre- and post-instruction, and we found that their reasoning on fraction comparison tasks improved. The participants made more correct comparisons, reasoned more flexibly, and came to favor less conventional and more sophisticated strategies. These improvements in number sense parallel those that we found previously in mental computation. In addition to the overall results, we highlight two cases of improvement that illustrate ways in which prospective elementary teachers’ reasoning about fraction magnitude can change.


Prospective elementary teachers Fractions Number sense Local instruction theory 


  1. Australian Education Council. (1991). A national statement on mathematics for Australian schools: A joint project of the states, territories and the commonwealth of Australia/initiated by the Australian Education Council. Curriculum Corporation for the Australian Education Council.Google Scholar
  2. Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90, 449–466.CrossRefGoogle Scholar
  3. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389–407.CrossRefGoogle Scholar
  4. Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323–341.CrossRefGoogle Scholar
  5. Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1987). Written and oral mathematics. Journal for Research in Mathematics Education, 18, 83–97.CrossRefGoogle Scholar
  6. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, R. (2003). Design experiments in educational research. Educational Researcher, 32, 9–13.CrossRefGoogle Scholar
  7. Conference Board of Mathematical Sciences (CBMS). (2012). The mathematical education of teachers II. Washington, DC: Mathematical Association of America.CrossRefGoogle Scholar
  8. DfEE. (1999). The national curriculum handbook for primary teachers in England. London: Department for Education and Employment.Google Scholar
  9. Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1, 155–177.CrossRefGoogle Scholar
  10. Gravemeijer, K. (2004). Local instruction theories as a means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6, 105–128.CrossRefGoogle Scholar
  11. Greeno, J. (1991). Number sense as situated knowing in a conceptual domain. Journal for Research in Mathematics Education, 22, 170–218.CrossRefGoogle Scholar
  12. Heirdsfield, A. M., & Cooper, T. J. (2002). Flexibility and inflexibility in accurate mental addition and subtraction: Two case studies. Journal of Mathematical Behavior, 21, 57–74.CrossRefGoogle Scholar
  13. Heirdsfield, A. M., & Cooper, T. J. (2004). Factors affecting the process of proficient mental addition and subtraction: Case studies of flexible and inflexible computers. Journal of Mathematical Behavior, 23, 443–463.CrossRefGoogle Scholar
  14. Hope, J. A., & Sherrill, J. M. (1987). Characteristics of unskilled and skilled mental calculators. Journal for Research in Mathematics Education, 18, 98–111.CrossRefGoogle Scholar
  15. Hsu, C. Y., Yang, D. C., & Li, F. M. (2001). The design of the fifth and sixth grade number sense rating scale. Chinese Journal of Science Education (TW), 9, 351–374.Google Scholar
  16. Japanese Ministry of Education. (1989). Curriculum of mathematics for the elementary school. Tokyo: Printing Bureau.Google Scholar
  17. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. New Jersey: Erlbaum.Google Scholar
  18. Markovits, Z., & Sowder, J. (1994). Developing number sense: An intervention study in grade 7. Journal for Research in Mathematics Education, 25, 4–29.CrossRefGoogle Scholar
  19. McIntosh, A., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12(3), 2–8., 44.Google Scholar
  20. National Center for Education Statistics. (2014). National Assessment of Educational Progress (NAEP) Mathematics. Retrieved from
  21. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  22. National Governor’s Association and the Council of Chief State School Officers. (2010). Common core state standards for mathematics (CCSSM). Washington, DC: National Governor’s Association and the Council of Chief State School Officers.Google Scholar
  23. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: National Academies Press.Google Scholar
  24. National Research Council. (2001). Adding it up: Helping children learn mathematics. In J. Kilpatrick, J. Swafford, & B. Findel (Eds.), Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.Google Scholar
  25. Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45, 1080–1110.CrossRefGoogle Scholar
  26. Nickerson, S. D., & Whitacre, I. M. (2010). A local instruction theory for the development of number sense. Mathematical Thinking and Learning, 12, 227–252.CrossRefGoogle Scholar
  27. Parker, M., & Leinhardt, G. (1995). Percent: A privileged proportion. Review of Educational Research, 65, 421–481.CrossRefGoogle Scholar
  28. Philipp, R. A. (2008). Motivating prospective elementary school teachers to learn mathematics by focusing upon children’s mathematical thinking. Issues in Teacher Education, 17(2), 7–26.Google Scholar
  29. Reys, R. E., Reys, B. J., McIntosh, A., Emanuelsson, G., Johansson, B., & Yang, D. C. (1999). Assessing number sense of students in Australia, Sweden, Taiwan and the United States. School Science and Mathematics, 99(2), 61–70.CrossRefGoogle Scholar
  30. Reys, R. E., Reys, B. J., Nohda, N., & Emori, H. (1995). Mental computation performance and strategy use of Japanese students in Grades 2, 4, 6, and 8. Journal for Research in Mathematics Education, 26, 304–326.CrossRefGoogle Scholar
  31. Reys, R., Rybolt, J., Bestgen, B., & Wyatt, J. (1982). Processes used by good computational estimators. Journal for Research in Mathematics Education, 13, 183–201.CrossRefGoogle Scholar
  32. Reys, R. E., & Yang, D. C. (1998). Relationship between computational performance and number sense among sixth- and eighth-grade students in Taiwan. Journal for Research in Mathematics Education, 29, 225–237.CrossRefGoogle Scholar
  33. Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., … Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE #2010-4039). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute for Education Sciences, U.S. Department of Education. Retrieved from
  34. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114–145.CrossRefGoogle Scholar
  35. Smith, J. P, I. I. I. (1995). Competent reasoning with rational numbers. Cognition and Instruction, 13, 3–50.CrossRefGoogle Scholar
  36. Smith, J., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95–132). New York: Erlbaum.Google Scholar
  37. Sowder, J. (1992). Estimation and number sense. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 371–389). New York: Macmillan.Google Scholar
  38. Tsao, Y.-L. (2005). The number sense of preservice elementary teachers. College Student Journal, 39, 647–679.Google Scholar
  39. Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning. Journal for Research in Mathematics Education, 30, 390–416.CrossRefGoogle Scholar
  40. Whitacre, I. M. (2007). Preservice teachers’ number sensible mental computation strategies. In Proceedings of the Tenth Special Interest Group of the Mathematical Association of America on research in undergraduate mathematics education. San Diego, CA. Retrieved from
  41. Whitacre, I. M., & Nickerson, S. D. (2006). Pedagogy that makes (number) sense: A classroom teaching experiment around mental math. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the twenty-eighth annual meeting of the North American chapter of the International Group for the psychology of mathematics education (pp. 736–743). Mérida, México: Universidad Pedagógica Nacional.Google Scholar
  42. Yang, D. C. (2003). Teaching and learning number sense: An intervention study of fifth grade students in Taiwan. International Journal of Science and Mathematics Education, 1, 115–134.Google Scholar
  43. Yang, D. C. (2007). Investigating the strategies used by preservice teachers in Taiwan when responding to number sense questions. School Science and Mathematics, 107, 293–301.CrossRefGoogle Scholar
  44. Yang, D. C., Reys, R. E., & Reys, B. J. (2009). Number sense strategies used by pre-service teachers in Taiwan. International Journal of Science and Mathematics Education, 7, 383–403.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Florida State UniversityTallahasseeUSA
  2. 2.San Diego State University, CRMSESan DiegoUSA

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