Preservice Teachers’ Understanding of Fraction Multiplication, Representational Knowledge, and Computational Skills

Ji-Won Son, Ji-Eun Lee


Despite the importance of teacher fractional knowledge, there are several questions unanswered. The purpose of this study was to characterize profiles of preservice teachers’ (PSTs) mathematical competence on the topic of fraction multiplication by examining PSTs’ understanding of multiplication of fractions in three different contexts. Analyses of 60 PSTs’ written responses revealed that there are distinct gradations of competency, ranging from the PSTs who were unable to solve a given problem in any context to those capable of flexibly portraying understanding of fraction multiplication in three contexts. Most PSTs who recognized the word problem as a multiplication of fractions were able to explain their thinking using graphical representations. However, we also observed various types of errors PSTs made in representing the word problem in graphical representations and translating it to a correct multiplication expression. These findings offer descriptors of how the PSTs understand fraction multiplication in different contexts and provide information for the design of intervention in teacher education. One objective would be to support recognition of the connectivity of fraction multiplication in different contexts.


Teacher knowledge; Fraction multiplication; Preservice teachers

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Aksu, M. (1997). Student performance in dealing with fractions. The Journal of Educational Research, 90, 375-380.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389-407.

Bezuk, N. S., & Armstrong, B. E. (1995). Understanding of fraction multiplication and division of fractions. Mathematics Teacher, 86(1), 120-131.

Cramer, K., & Henry, A. (2002). Using manipulative models to build number sense for addition of fractions. In B. Litwiller & G. Bright (Eds.), National Council of Teachers of Mathematics 2002 yearbook: Making sense of fractions, ratios, and proportions (pp. 41-48). Reston, VA: National

Council of Teachers of Mathematics.

Cramer, K. A., Post, T. R., & delMas, R. C. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33(2), 111–144.

Cramer, K., & Wyberg, T. (2009). Efficacy of different concrete models for teaching the part-whole construct for fractions. Math. Think. Learn, 11(4), 226–257.

Creswell, J. W. (1994). Research design: qualitative and quantitative approaches. Thousand Oaks: Sage Publications.

Depaepe, F., Torbeyns, J., Vermeersch, N., Janssens, D., Janssen, R., Verschaffel, L., & Van Dooren, W. (2015). Teachers’ content and pedagogical content knowledge on rational numbers: A comparison of prospective elementary and lower secondary school teachers. Teaching and Teacher Education, 47, 82-92.

Empson, S. B. (2002). Organising diversity in early fraction thinking. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios and proportions. 2002 Yearbook of the National Council of Teachers of Mathematics (pp. 29– 40). Reston, VA: National Council of Teachers of Mathematics.

Empson, S. B., Junk, D., Dominguez, H., & Turner, E. (2006). Fractions as the coordination of multiplicatively related quantities: A cross-sectional study of children’s thinking. Educational Studies in Mathematics, 63, 1–28.

Graeber, A. O., & Tanenhaus, E. (1993). Multiplication and division: From whole number to rational numbers. In T. Owens (Ed.), Research ideas for the classroom middle grade mathematics (pp. 99-117). New York: Macmillan.

Hackenberg, A. J., & Tillema, E. S. (2009). Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. Journal of Mathematical Behavior, 28, 1-18.

Hiebert, J., & Behr, M. (1988). Number concepts and operations in the middle grades (Vol. 2). Reston, VA: National Council of Teachers of Mathematics.

Hill, H. C., Rowan, B., & Ball, D. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42, 371-406.

Izsák, A. (2003). “We want a statement that is always true”: Criteria for good algebraic representations and the development of modelling knowledge. Journal for Research in Mathematics Education, 34(3), 191-227.

Izsák, A. (2008). Mathematical knowledge for teaching fraction multiplication. Cognition and Instruction, 26, 95–143.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

Krauss, S., Baumert, J. & Blum, W. (2008). Secondary mathematics teachers' pedagogical content knowledge and content knowledge: Validation of the COACTIV constructs. The International Journal on Mathematics Education, 40(5), 873-892.

Lamon, S. J. (2001). Presenting and representing from fractions to rational numbers. In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics (pp. 146–165). Reston, VA: National Council of Teachers of Mathematics.

Lannin, J., Chval, K., & Jones, D. (2013). Putting essential understanding of multiplication and division into practice 3-5. Reston, VA: National Council of Teachers of Mathematics.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum.

Mack, N. K. (1998). Building a foundation for understanding the multiplication of fractions. Teaching Children Mathematics, 5(1), 34-38.

Mack, N. K. (2000). Long-term effects of building on informal knowledge in a complex content domain: The case of multiplication of fractions. Journal of Mathematical Behavior, 19(3), 307–332.

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), 267–295.

McClain, K. (2003). Supporting pre-service teachers’ understanding of place value and multidigit arithmetic. Mathematical Thinking and Learning, 5(4), 281-306.

Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: Understanding the real numbers. In M.

Limon, & L. Mason (Eds.), Reconsidering conceptual change. Issues in theory and practice (pp. 233-258). Dordrecht, The Netherlands: Kluwer Academic.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author.

National Research Council. (2003). Improving undergraduate instruction in science, technology, engineering, and mathematics: Report of a workshop. Washington, DC: National Academies Press.

Newton, K. J. (2008). An extensive analysis of pre-service elementary teachers' knowledge of fractions. American Educational Research Journal, 45, 1080-1110.

Olive, J. (1999). From fractions to rational numbers of arithmetic: A reorganisation hypothesis. Mathematical Thinking and Learning, 1(4), 279–314.

Olive, J., & Steffe, L. P. (2002). The construction of an iterative fractional scheme: The case of Joe. Journal of Mathematical Behavior, 20, 413–437.

Senk, S. L., Tatto, M. T., Reckase, M., Rowley, G., Peck, R., & Bankov, K. (2012). Knowledge of future primary teachers for teaching mathematics: An international comparative study. ZDM Mathematics Education, 44, 307-324.

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

Son, J. (2012). Fraction multiplication from a Korean perspective. Mathematics Teaching in the Middle School, 17(7), 388-393.

Son, J. (2013). How pre-service teachers interpret and respond to student errors: Ratio and proportion in similar rectangles. Educational Studies in Mathematics, 84(1), 49-70.

Son, J. (2016a). Pre-service teachers' response and feedback type to correct and incorrect student-invented strategies for subtracting whole numbers. The Journal of Mathematical Behavior, 42, 49-68.

Son, J. (2016b). Moving beyond a traditional algorithm in whole number subtraction: Pre-service teachers' responses to a student's invented strategy. Educational Studies in Mathematics, 93, 105-129.

Son, J., & Sinclaire, N. (2010). How pre-service teachers interpret and respond to student geometric errors. School Science and Mathematics, 110(1), 31-46.

Son, J., & Crespo, S. (2009). Prospective teachers’ reasoning about students’ non-traditional strategies when dividing fractions. Journal of Mathematics Teacher Education, 12(4), 236-261.

Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20, 267–307.

Steffe, L. P. (2003). Fractional commensurate, composition, and adding schemes: Learning trajectories of Jason and Laura: Grade 5. Journal of Mathematical Behavior, 22, 237–295.

Steffe, L. P. (2004). On the construction of learning trajectories of children: The case of commensurate fractions. Mathematical Thinking and Learning, 6(2), 129–162.

Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), Research companion to the principles and standards for school mathematics (pp. 95–113). Reston, VA: National Council of Teachers of Mathematics.

Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31, 5-25.

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7th ed.). Boston, MA: Pearson Allyn & Bacon.

Whitin, D. J., & Whitin, P. (2012). Making sense of fractions and percentages. Teaching Children Mathematics, 18, 490-496.

Wu, Z. (2001). Multiplying fractions. Teaching Children Mathematics, 8(3), 174.

Zazkis, R., & Liljedahl, P. (2004). Understanding primes: The role of representation. Journal for Research in Mathematics Education, 35(3), 164-186.


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