# Examining and elaborating upon the nature of elementary prospective teachers’ conceptions of partitive division with fractions

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## Abstract

The purpose of this study was to examine and elaborate upon elementary prospective teachers’ (PSTs) conceptions of partitive division with fractions. We examined the degree to which PSTs’ conceptions were connected (i.e., capable of translating between representations correctly; aware that partitive division generates a unit rate for its quotient) and flexible (i.e., capable of differentiating between opportunities to partition or iterate (or both) when solving a partitive division task; aware that partitioning or iterating (or both) could be associated with the operation of division, as appropriate). Seventeen PSTs participated in task-based interviews prior to instruction in a mathematics content course for teachers. These PSTs demonstrated disconnected conceptions of partitive division with fractions when they incorrectly translated between representations and either inconsistently or did not express awareness that the purpose of the task was to generate a unit rate. These PSTs demonstrated rigid conceptions of partitive division with fractions such that they did not express awareness that the process of iterating could be associated with the operation of division, even when they obtained a correct answer by iterating. Results extend prior research by looking beyond PSTs’ performance on tasks to elaborate upon PSTs’ conceptions of the operation of partitive division. This study contributes new insights into PSTs’ conceptions that can be used by mathematics teacher educators to inform the design of future instructional interventions.

## Keywords

Prospective elementary teachers Partitive division Division with fractions## Notes

### Acknowledgments

We would like to thank the mathematics education community at the University of Delaware for feedback at our Friday seminar presentation about this work during its early stages. The first author also appreciates the support of her women’s writing retreat group (AnnaMarie Conner, Amy Ellis, Beth Herbel-Eisenmann, Heather Johnson, and Eva Thanheiser) who talked with her about ideas for the initial draft of this paper.

## References

- Armstrong, B. E., & Bezuk, N. (1995). Multiplication and division of fractions: The search for meaning. In J. T. Sowder & B. P. Schappelle (Eds.),
*Providing a foundation for teaching mathematics in the middle grades*(pp. 85–119). Albany, NY: SUNY Press.Google Scholar - Ball, D. (1988).
*The subject matter preparation of prospective mathematics teachers: Challenging the myths*. East Lansing, MI: The National Center for Research on Teacher Education.Google Scholar - Ball, D. (1990). Prospective elementary and secondary teachers’ understanding of division.
*Journal for Research in Mathematics Education,**21*(2), 132–144.CrossRefGoogle Scholar - Berk, D., & Hiebert, J. (2009). Improving the mathematics preparation of elementary teachers, one lesson at a time.
*Teachers and Teaching—Theory and Practice,**15*(3), 337–356.CrossRefGoogle Scholar - Bernard, H. E. (1988). Structured and unstructured interviewing. In
*Research methods in cultural anthropology*(pp. 203–240). Newbury Park, CA: Sage.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*, 194–222.CrossRefGoogle Scholar - Cengiz, N., & Rathouz, M. (2011). Take a bite out of fraction division.
*Mathematics Teaching in the Middle School,**17*(3), 147–153.Google Scholar - Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly & R. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 547–589). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Empson, S. B., Junk, D., Dominguez, H., & Turner, E. (2005). Fractions as the coordination of multiplicatively related quantities: A cross-sectional study of children’s thinking.
*Educational Studies in Mathematics,**63*(1), 1–28.CrossRefGoogle Scholar - Ginsburg, H. P. (1997).
*Entering the child’s mind: The clinical interview in Psychological research and Practice*. Cambridge, MA: Cambridge University Press.CrossRefGoogle Scholar - Jansen, A., Bartell, T., & Berk, D. (2009). The role of learning goals in building a knowledge base for elementary mathematics teacher education.
*Elementary School Journal*,*109*(5), 525–536.Google Scholar - Lloyd, G. M., & Wilson, M. (1998). Supporting innovation: The impact of a teacher’s conceptions of functions on his implementation of a reform curriculum.
*Journal for Research in Mathematics Education,**29*(3), 248–274.Google Scholar - Lo, J.-J., & Luo, F. (2012). Prospective elementary teachers’ knowledge of fraction division.
*Journal of Mathematics Teacher Education,**15*(6), 481–500.CrossRefGoogle Scholar - Lo, J.-J., & Watanabe, T. (1997). Developing ratio and proportion schemes: A story of a fifth grader.
*Journal for Research in Mathematics Education,**28*(2), 216–236.CrossRefGoogle Scholar - Lobato, J., Hohensee, C., Rhodehamel, B., & Diamond, J. (2012). Using student reasoning to inform the development of conceptual learning goals: The case of quadratic functions.
*Mathematical Thinking and Learning,**14*(2), 85–119.CrossRefGoogle Scholar - Lubinski, C. A., Fox, T., & Thomason, R. (1998). Learning to make sense of division of fractions: One K-8 preservice teacher’s perspective.
*School Science and Mathematics,**98*(5), 247–251.CrossRefGoogle Scholar - Luo, F., Lo, J.-J., & Leu, Y.-C. (2011). Fundamental fraction knowledge of preservice elementary teachers: A cross-national study in the United States and Taiwan.
*School Science and Mathematics,**111*(4), 164–177.CrossRefGoogle Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics*. Mahwah, NJ: Lawrence Erlbaum and Associates.Google Scholar - Okazaki, M., & Koyama, M. (2005). Characteristics of 5th graders’ logical development through learning division with decimals.
*Educational Studies in Mathematics,**60*(2), 217–251.CrossRefGoogle Scholar - Orrill, C. H., & Brown, R. E. (2012). Making sense of double number lines in professional development: Exploring teachers’ understandings of proportional relationships.
*Journal of Mathematics Teacher Education,**15*, 381–403.CrossRefGoogle Scholar - Ott, J. M., Snook, D. L., & Gibson, D. L. (1991). Understanding partitive division of fractions.
*Arithmetic Teacher,**39*, 7–11.Google Scholar - Rizvi, N. F., & Lawson, M. J. (2007). Prospective teachers’ knowledge: Concept of division.
*International Education Journal,**8*(2), 377–392.Google Scholar - Roche, A., & Clarke, D. (2009). Making sense of partitive and quotitive division: A snapshot of teacher’s pedagogical content knowledge. In R. Hunter, B. Bicknell, & T. Burgess (Eds.),
*Crossing divides (Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia Incorporated*(pp. 467–474). Palmerston: MERGA.Google Scholar - Schoenfeld, A. H. (1992). On paradigms and methods: What do you do when the ones you know don’t do what you want them to? Issues in the analysis of data in the form of videotapes.
*The Journal of the Learning Sciences,**2*(2), 179–214.CrossRefGoogle Scholar - Siebert, D., & Gaskin, N. (2006). Creating, naming, and justifying fractions.
*Teaching Children Mathematics,**12*(8), 394–400.Google Scholar - Simon, M. A. (1993). Prospective elementary teachers’ knowledge of division.
*Journal for Research in Mathematics Education,**24*(3), 233–254.CrossRefGoogle Scholar - Sinicrope, R., Mick, H. W., & Kolb, J. R. (2002). Interpretations of fractions. In B. Litwiller & G. Bright (Eds.),
*Making sense of fractions, ratios, and proportions, NCTM 2002 yearbook*(pp. 153–161). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Smith, J. P., diSessa, A. A., & Roschelle, J. (1993–1994). Misconceptions reconceived: A constructivist analysis of knowledge in transition.
*Journal of the Learning Sciences*,*3*, 115–163.Google Scholar - Son, J., & 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 - Sun, X. (2011). “Variation problems” and their roles in the topic of fraction division in Chinese mathematics textbook examples.
*Educational Studies in Mathematics,**76*(1), 65–85.CrossRefGoogle Scholar - Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.),
*A research companion to principles and standards for school mathematics*(pp. 95–113). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions.
*Journal for Research in Mathematics Education,**22*, 125–147.Google Scholar - 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 - Tirosh, D., & Graeber, A. O. (1991). The effect of problem type and common misconceptions on preservice elementary teachers’ thinking about division.
*School Science and Mathematics,**91*(4), 157–163.CrossRefGoogle Scholar - Unlu, M., & Ertekin, E. (2012). Why do pre-service teachers pose multiplication problems instead of division problems in fractions?
*Procedia—Social and Behavioral Sciences,**46*, 490–494.CrossRefGoogle Scholar - Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.),
*Acquisition of mathematical concepts and processes*(pp. 127–174). New York: Academic press.Google Scholar - Zembat, I. O. (2004).
*Conceptual development of prospective elementary teachers: The case of division of fractions*. Ph.D. dissertation, The Pennsylvania State University. ProQuest Digital Dissertations Database. (Publication No. AAT 3148695).Google Scholar - Zembat, I. O. (2007). Working on the same problem—concepts; with the usual subjects—prospective elementary teachers.
*Elementary Education Online,**6*(2), 305–312.Google Scholar