Advertisement

Promoting sophisticated fraction constructs through instructional changes in a mathematics course for PreK-8 prospective teachers

  • Alexis L. Stevens
  • Jesse L. M. Wilkins
  • LouAnn H. Lovin
  • John Siegfried
  • Anderson Norton
  • Richard Busi
Article
  • 58 Downloads

Abstract

In a previous study, we validated the fractions schemes and operations trajectory (Norton and Wilkins 2012; Steffe in J Math Behav 20(3): 267–307, 2002; Steffe and Olive Children’s fractional knowledge. Springer, New York, 2010; Wilkins and Norton 2011) with PreK-8 prospective teachers and explored PreK-8 prospective teachers’ understanding of fractions based on fraction schemes and operations (Lovin et al. 2016). In that study, we found that most of the prospective teachers had constructed the lower-level fraction schemes and operations, but less than half had constructed the more sophisticated ones. In the current study, we explored ways of evoking the more sophisticated fraction schemes and operations, in particular, by implementing simple instructional changes to a mathematics course for PreK-8 prospective teachers. One change focused on fraction language, specifically encouraging the use of iterative fraction language (e.g., 3/5 is three equal-size pieces, each of which is 1/5 of the whole). Another change focused on intentional number choice, specifically emphasizing improper fractions. After focused fraction instruction that included these instructional changes, we found significant increases in PreK-8 prospective teachers’ interiorization of coordinating three levels of units as well as in PreK-8 prospective teachers’ construction of the reversible partitive fraction scheme and the iterative fraction scheme. We conclude with implications for practice and research in mathematics courses intended for PreK-8 prospective teachers.

Keywords

Fractions Mathematical content knowledge Prospective elementary school teachers Prospective middle school teachers Teacher education Instructional changes 

References

  1. Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90, 449–466.CrossRefGoogle Scholar
  2. Ball, D. L. (1993). Halves, pieces, and twoths: Constructing and using representational contexts in teaching fractions. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 157–195). Mahwah, NJ: Lawrence Erlbaum Associates Inc.Google Scholar
  3. Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D., & Agard, P. C. (1992). Learning to teach hard mathematics: Do novice teachers give up too easily? Journal for Research in Mathematics Education, 23(3), 194–222.CrossRefGoogle Scholar
  4. Boyce, S. & Moss, D. (2017). Role of representation in prospective elementary teachers’ fraction schemes. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th annual meeting of the North American chapter of the international group for the psychology of mathematics education (pp. 829–836). Indianapolis, IN: Hoosier Association of Mathematics Teacher Educators.Google Scholar
  5. Busi, R., Lovin, L. A., Norton, A., Siegfried, J. M., Stevens, A. L., & Wilkins, J. L. M. (2015). An investigation of PreK–8 preservice teachers’ construction of fraction schemes and operations. Proceedings of the 37rd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 616–623). Michigan State University: East Lansing, MI.Google Scholar
  6. Campbell, D. T., & Stanley, J. C. (1963). Experimental and quasi-experimental designs for research on teaching. In N. L. Gage (Ed.), Handbook of research on teaching (pp. 171–246). Chicago: Rand McNally.Google Scholar
  7. Chinnappan, M. (2000). Preservice teachers’ understanding and representation of fractions in a JavaBars environment. Mathematics Education Research Journal, 12(3), 234–253.CrossRefGoogle Scholar
  8. Clements, M. A., & Del Campo, G. (1990). How natural is fraction knowledge? In L. P. Steffe & T. Wood (Eds.), Transforming children’s mathematics education: International perspectives (pp. 181–188). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  9. Conference Board of the Mathematical Sciences, (2012). The mathematical education of teachers II (Issues in Mathematics Education, Volume 17). Providence, RI: American Mathematical Society.Google Scholar
  10. Council of Chief State School Officers (CCSSO). (2010). Common core state standards. Retrieved from http://corestandards.org.
  11. Domoney, B. (2002). Student teachers’ understanding of rational number: Part-whole and numerical constructs. Research in Mathematics Education, 4(1), 53–67.CrossRefGoogle Scholar
  12. Hackenberg, A. J. (2005). Construction of algebraic reasoning and mathematical caring relations (Unpublished doctoral dissertation). Athens: University of Georgia.Google Scholar
  13. Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. The Journal of Mathematical Behavior, 26(1), 27–47.CrossRefGoogle Scholar
  14. Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383–432.CrossRefGoogle Scholar
  15. Hackenberg, A. J., & Tillema, E. S. (2009). Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. The Journal of Mathematical Behavior, 28, 1–18.CrossRefGoogle Scholar
  16. Ho, S. Y. & Lai, M. Y. (2012). Pre–service teachers’ specialized content knowledge on multiplication of fractions. In T. Y. Tso (Ed.), Proceedings of the 36th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 291–298). Taipei: PME.Google Scholar
  17. Howson, G. (1995). Mathematics textbooks: A comparative study of grade 8 textbooks. Vancouver: TIMSS, Pacific Educational Press.Google Scholar
  18. Izsák, A., Orrill, C. H., Cohen, A. S., & Brown, R. E. (2010). Measuring middle grades teachers’ understanding of rational numbers with the Mixture Rasch model. The Elementary School Journal, 110(3), 279–300.CrossRefGoogle Scholar
  19. Lamon, S. (2005). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (2nd ed.). Mahwah, NJ: Erlbaum.Google Scholar
  20. Landis, J. R., & Koch, G. G. (1977). The measurement of observer agreement for categorical data. Biometrics, 33(1), 159–174.CrossRefGoogle Scholar
  21. Lo, J., & Grant, T. (2012). Prospective elementary teachers’ conceptions of fractional units. In T. Y. Tso (Ed.), Proceedings of the 36th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 169–176). Taipei: PME.Google Scholar
  22. Lovin, L. H., Stevens, A. L., Siegfried, J., Wilkins, J. L. M., & Norton, A. (2016). Pre-K–8 prospective teachers’ understanding of fractions: An extension of fractions schemes and operations research. Journal of Mathematics Teacher Education.  https://doi.org/10.1007/s10857-016-9357-8.Google Scholar
  23. Luo, F., Lo, J., & Leu, Y. (2011). Fundamental fraction knowledge of preservice elementary teachers: A cross-national study in the United States and Taiwan. School Science and Mathematics, 111, 164–177.CrossRefGoogle Scholar
  24. Mack, N. K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267–296.CrossRefGoogle Scholar
  25. McCloskey, A., & Norton, A. (2009). Modeling students’ mathematics using steffe’s advanced fractions schemes. Mathematics Teaching in the Middle School, 15(1), 44–56.Google Scholar
  26. Menon, R. (2009). Preservice teachers’ subject matter knowledge of mathematics. International Journal for Mathematics Teaching and Learning, 12, 89–109.Google Scholar
  27. Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45(4), 1080–1110.CrossRefGoogle Scholar
  28. Norton, A., & Boyce, S. (2013). A cognitive core for common state standards. Journal of Mathematical Behavior, 32(2), 266–279.CrossRefGoogle Scholar
  29. Norton, A., & Wilkins, J. L. M. (2009). A quantitative analysis of children’s splitting operations and fraction schemes. Journal of Mathematical Behavior, 28(2/3), 150–161.CrossRefGoogle Scholar
  30. Norton, A., & Wilkins, J. L. M. (2012). The splitting group. Journal for Research in Mathematics Education, 43(5), 557–583.CrossRefGoogle Scholar
  31. Norton, A., & Wilkins, J. L. M. (2013). Supporting students’ constructions of the splitting operation. Cognition and Instruction, 31(1), 2–28.CrossRefGoogle Scholar
  32. Norton, A., Wilkins, J. L. M., & Xu, C. Z. (in press). A progression of fractions schemes that crosses Chinese and US classrooms.Google Scholar
  33. Norton, A., Wilkins, J. L. M., Evans, M. A., Deater-Deckard, K., Balci, O., & Chang, M. (2014). Technology helps students transcend part-whole concepts. Mathematics Teaching in the Middle School, 19(6), 352–359.CrossRefGoogle Scholar
  34. Olanoff, D., Lo, J., & Tobias, J. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast, 11(2), 267–310.Google Scholar
  35. Olive, J., & Steffe, L. P. (2002). The construction of an iterative fraction scheme: The case of Joe. Journal of Mathematical Behavior, 20, 413–437.CrossRefGoogle Scholar
  36. Pitkethly, A., & Hunting, R. (1996). A review of recent research in the area of initial fraction concepts. Educational Studies in Mathematics, 31(1), 5–38.CrossRefGoogle Scholar
  37. Rathouz, M. (2010). Ambiguity in the units and the referents; Two cases in rational number operations. For the Learning of Mathematics, 30(1), 43–51.Google Scholar
  38. Rizvi, N. F., & Lawson, M. J. (2007). Prospective teachers’ knowledge: Concept of division. International Education Journal, 8(2), 377–392.Google Scholar
  39. Rosli, R., Gonzalez, E. G., & Capraro, M. M. (2011). A case study of three preservice teachers on the units and unitizing of fractions. In L. R. Wiest & T. Lamberg (Eds.), Proceedings of the 33rd annual meeting of the North American chapter of the international group for the psychology of mathematics education (pp. 1682–1689). Reno, NV: University of Nevada.Google Scholar
  40. Siegel, S. C., & Castellan, J. (1988). Nonparametric statistics for the behavioural sciences. New York: McGraw-Hill.Google Scholar
  41. Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & 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 of Education Sciences, U.S. Department of Education. Retrieved from whatworks.ed.gov/publications/practiceguides.Google Scholar
  42. Sloane, F. C. (2008). Randomized trials in mathematics education: Recalibrating the proposed high watermark. Educational Researcher, 37(9), 624–630.CrossRefGoogle Scholar
  43. Sloane, F. C., & Wilkins, J. L. M. (2017). Aligning statistical modeling with theories of learning in mathematics education research. Compendium for Research in Mathematics Education, 183–207.Google Scholar
  44. Sowder, J., Sowder, L., & Nickerson, S. (2014). Reconceptualizing mathematics (2nd ed.). W. H: Freeman.Google Scholar
  45. Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–40). Albany: State University of New York Press.Google Scholar
  46. Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20(3), 267–307.CrossRefGoogle Scholar
  47. Steffe, L. P., & Olive, J. (2002). Design and use of computer tools for interactive mathematical activity (TIMA). Journal of Educational Computing Research, 27, 55–76.CrossRefGoogle Scholar
  48. Steffe, L. P., & Olive, J. (Eds.). (2010). Children’s fractional knowledge. New York: Springer.Google Scholar
  49. Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5–25.CrossRefGoogle Scholar
  50. Tirosh, D., & Graeber, A. O. (1990). Evoking cognitive conflict to explore preservice teachers’ thinking about division. Journal for Research in Mathematics Education, 21(2), 98–108.CrossRefGoogle Scholar
  51. Tobias, J. M. (2009). Prospective elementary teachers’ development of rational number understanding through the social perspective and the relationship among social and individual environments. Unpublished Dissertation, University of Central Florida, Orlando, FL.Google Scholar
  52. Tobias, J. M. (2013). Prospective elementary teachers’ development of fraction language for defining the whole. Journal of Mathematics Teacher Education, 16(2), 85–103.CrossRefGoogle Scholar
  53. 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
  54. Utley, J., & Reeder, S. (2012). Prospective elementary teachers’ development of fraction number sense. Investigations in Mathematics Learning, 5(2), 1–13.CrossRefGoogle Scholar
  55. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London: Routledge Falmer.CrossRefGoogle Scholar
  56. Watanabe, T. (2006). The teaching and learning of fractions: A Japanese perspective. Teaching Children Mathematics, 12(7), 368–374.Google Scholar
  57. Watanabe, T. (2007). Initial treatment of fractions in Japanese textbooks. Focus on Learning Problems in Mathematics, 29(2), 41–60.Google Scholar
  58. Weller, K., Arnon, I., & Dubinsky, E. (2009). Preservice teachers’ understanding of the relation between a fraction or integer and its decimal expansion. Canadian Journal of Science, 9(1), 5–28.Google Scholar
  59. Whitacre, I. & Nickerson, S. D. (2011). Prospective elementary teachers develop improved number sense in reasoning about fraction magnitude. In L. R. Wiest & T. Lamberg (Eds.), Proceedings of the 33rd annual meeting of the North American Chapter of the international group for the psychology of mathematics education (pp. 559–567). Reno, NV: University of Nevada.Google Scholar
  60. Wilkins, J. L. M., & Norton, A. (2011). The splitting loope. Journal for Research in Mathematics Education, 42(4), 386–416.CrossRefGoogle Scholar
  61. Wilkins, J. L. M., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. International Journal of STEM Education.  https://doi.org/10.1186/s40594-018-0119-2.Google Scholar
  62. Yang, D. C., Reys, R. E., & Wu, L. L. (2010). Comparing the development of fractions in the fifth- and sixth-graders’ textbooks of Singapore, Taiwan, and the USA. School Science and Mathematics, 110(3), 118–127.CrossRefGoogle Scholar
  63. Zhou, Z., Peverly, S. T., & Xin, T. (2006). Knowing and teaching fractions: A cross-cultural study of American and Chinese mathematics teachers. Contemporary Educational Psychology, 31, 438–457.CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsJames Madison UniversityHarrisonburgUSA
  2. 2.School of EducationVirginia TechBlacksburgUSA
  3. 3.Department of MathematicsVirginia TechBlacksburgUSA

Personalised recommendations