# Pre-service teachers’ flexibility with referent units in solving a fraction division problem

## Abstract

This study investigated 111 pre-service teachers’ (PSTs’) flexibility with referent units in solving a fraction division problem using a length model. Participants’ written solutions to a measurement fraction division problem were analyzed in terms of strategies and types of errors, using an inductive content analysis approach. Findings suggest that most PSTs could calculate fraction division and make equivalent fractions procedurally but did not have the quantitative meanings of measurement division with fraction quantities or of making equivalent fractions. Implications are discussed for the improvement of PSTs’ specialized knowledge for teaching fraction division.

### Keywords

Fraction division Referent units Visual representations Pedagogical content knowledge Quantitative approach## Notes

### Acknowledgements

An earlier version of this paper was presented at the NIMS (National Institute for Mathematical Sciences) and KSME (The Korean Society of Mathematical Education) International Workshop in 2017.

### References

- Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division.
*Journal for Research in Mathematics Education, 21*(2), 132–144.Google Scholar - Borko, H., Eisenhart, M., Brown, C., Underhill, R., Jones, D., & Agard, P. (1992). Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily?
*Journal for Research in Mathematics Education*,*23*(3), 194–222.CrossRefGoogle Scholar - Carraher, D. W., Schliemann, A. D., & Schwartz, J. L. (2008). Early algebra is not the same as algebra early. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.),
*Algebra in the early grades*(pp. 235–272). New York: Lawrence Erlbaum.Google Scholar - Common Core State Standards Initiative [CCSSI]. (2010).
*Common core state standards for mathematics*. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Retrieved February 21, 2017, http://corestandards.org/assets/CCSSI_Math%20Standards.pdf - Creswell, J. (2014).
*Research design: Qualitative, quantitative, and mixed methods approaches*(4th ed.). Thousand Oaks: SAGE.Google Scholar - Grbich, C. (2007).
*Qualitative data analysis: An introduction*. London: SAGE Publications.Google Scholar - Hackenberg, A. J. (2013). The fractional knowledge and algebraic reasoning of students with the first multiplicative concept.
*Journal of Mathematical Behavior, 32*(3), 538–563.Google Scholar - Hackenberg, A. J., & Lee, M. Y. (2015). Relationships between students’ fractional knowledge and equation writing.
*Journal for Research in Mathematics Education*,*46*(2), 196–243.CrossRefGoogle Scholar - Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students.
*Journal for Research in Mathematics Education*,*39*(4), 372–400.Google Scholar - Isiksal, M., & Cakiroglu, E. (2011). The nature of prospective mathematics teachers’ pedagogical content knowledge: The case of multiplication of fractions.
*Journal of Mathematics Teacher Education*,*14*(3), 213–230.CrossRefGoogle Scholar - Izsák, A. (2008). Mathematical knowledge for teaching fraction multiplication.
*Cognition and Instruction*,*26*(1), 95–143.CrossRefGoogle Scholar - Jacobs, V. R., & Empson, S. B. (2016). Responding to children’s mathematical thinking in the moment: An emerging framework of teaching moves.
*ZDM Mathematics Education, 48*(1), 185-197. doi: 10.1007/s11858-015-0717-0 - Jacobson, E., & Izsák, A. (2015). Knowledge and motivation as mediators in mathematics teaching practice: The case of drawn models for fraction arithmetic.
*Journal of Mathematics Teacher Education, 18*(5), 467–488.Google Scholar - Jansen, A., & Hohensee, C. (2016). Examining and elaborating upon the nature of elementary prospective teachers’ conceptions of partitive division with fractions.
*Journal of Mathematics Teacher Education*,*19*(6), 503–522.CrossRefGoogle Scholar - Lamon, S. (1994). Ratio and proportion: Cognitive foundations in unitizing and norming. In G. Harel & J. Confrey (Eds.),
*The development of multiplicative reasoning in the learning of mathematics*(pp. 89–120). Albany: State University Press of New York.Google Scholar - Lee, S. J., Brown, R. E., & Orrill, C. H. (2011). Mathematics teachers’ reasoning about fractions and decimals using drawn representations.
*Mathematical Thinking and Learning*,*13*(3), 198–220.CrossRefGoogle Scholar - Lee, M. Y., & Hackenberg, A. J. (2014). Relationships between fractional knowledge and algebraic reasoning: The case of Willa.
*International Journal of Science and Mathematics Education, 12*(4), 975–1000. doi: 10.1007/s10763-013-9442-8 - Lo, J. J., & Luo, F. (2012). Prospective elementary teachers’ knowledge of fraction division.
*Journal of Mathematics Teacher Education*,*15*(6), 481–500.CrossRefGoogle Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States*. Mahwah: Lawrence Erlbaum.Google Scholar - National Council of Teachers of Mathematics [NCTM]. (2000).
*Principles and standards for school mathematics*. Reston: Author.Google Scholar - National Council of Teachers of Mathematics [NCTM]. (2014).
*Principles to actions: Ensuring mathematical success for all*. Reston: Author.Google Scholar - National Mathematics Advisory Panel [NMAP]. (2008).
*The Final Report of the National Mathematics Advisory Panel*. Retrieved January 17, 2017, from http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf - 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 - 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 - Rosli, R., Han, S., Capraro, R., & Capraro, M. (2013). Exploring preservice teachers’ computational and representational knowledge of content and teaching fractions.
*Journal of Korean Society of Mathematics Education, 17*(4), 221–241.Google Scholar - Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching.
*Educational Researcher*,*15*(2), 4–14.CrossRefGoogle Scholar - Smith, J. P., & Thompson, P. W. (2008). 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: Lawrence Erlbaum.Google Scholar - Son, J. W., & Crespo, S. (2009). Prospective teachers’ reasoning and response to a student’s non-traditional strategy when dividing fractions.
*Journal of Mathematics Teacher Education*,*12*(4), 235–261.CrossRefGoogle Scholar - Sowder, J., Philipp, R., Armstrong, B., & Schappelle, B. (1998).
*Middle-grade teachers’ mathematical knowledge and its relationship to instruction: A research monograph*. Albany: State University of New York Press.Google Scholar - Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge.
*Journal of Mathematical Behavior, 20*(3), 267–307.Google Scholar - Steffe, L. P., & Olive, J. (2010).
*Children’s fractional knowledge*. New York: Springer.CrossRefGoogle Scholar - Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures.
*Educational Studies in Mathematics, 25*(3), 165–208.Google Scholar - Thompson, P. W. (1995). Notation, convention, and quantity in elementary mathematics. In J. T. Sowder & B. P. Schappelle (Eds.),
*Providing a foundation for teaching mathematics in the middle grades*(pp. 199–219). Albany: State University of New York Press.Google Scholar - Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions.
*Journal for Research of Mathematics Education*,*31*(1), 5–25.CrossRefGoogle Scholar