Motivation Cards to Support Students’ Understanding on Fraction Division

Kamirsyah Wahyu, Siti Maghfirotun Amin, Agung Lukito


This design research aims to develop a learning activity which supports the fifth-grade students to understand measurement fraction division problems (A whole number divided by a fraction that result in a whole number answer) conceptually. Furthermore, how students solve the fraction division problem using models is also analyzed.  Data for the retrospective analysis is collected through two teaching experiments in the form of students’ work, field notes, and some part of classroom discussions. The important findings in this research are: 1) the developed learning activity namely Motivation Cards support students understand that  3 divided by one-half means how many one-half are in 3 through models. However, when the divisor is not a unit fraction they could not directly relate the unshaded part in area model for example. 2) area model is proper model to be firstly introduced when the students work on fraction division. 3) understanding this kind of fraction division help students understand other measurement fraction division where both divisor and dividend are fractions. 4) the learning activity supports the development of character values for students.    


Understanding; Measurement Fraction Division; Multiple Models; Design Research

Full Text:



Aksu, M. (1997). Student performance in dealing with fractions. The Journal of Educational Research, 90(6), 375-380.

Bakker, A. (2004). Design research in statistics education: On symbolizing and computer tools. Retrieved from

Bulgar, S. (2003). Children’s sense-making of division of fractions. The Journal of Mathematical Behavior, 22(3), 319-334.

Bulgar, S. (2009). A longitudinal study of students’ representations for division of fractions. The Montana Mathematics Enthusiast, 6(1&2), 165-200.

Cengiz, N., & Rathouz, M. (2011). Take a bite out of fraction division. Mathematics Teaching in the Middle School, 17(3), 146-153.

Cramer, K., Monson, D., Whitney, S., Leavitt, S., & Wyberg, T. (2010). Dividing fractions and problem solving: Reflect and discuss. Mathematics Teaching in the Middle School, 15(6), 338-346.

Cramer, K., Wyberg, T., & Leavitt, S. (2008). The role of representations in fraction addition and subtraction. Mathematics Teaching in the Middle School, 13(8), 490-496.

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in mathematics, 61(1), 103-131.

Ervin, H.K. (2015). The impact of instruction through models on preservice teachers' understanding of fraction multiplication and division. Retrieved from

Feil, Y.C. (2010). Can you teach in a normal way? Examining Chinese and US curricula's approach to teaching fraction division. University of Illinois at Urbana-Champaign.

Flores, A., & Priewe, M.D. (2014). Orange you glad I did say “Fraction Division”? Mathematics Teaching in the Middle School, 19(5), 288-293.

Flores, A., Turner, E.E., & Bachman, R.C. (2005). Posing problems to develop conceptual understanding: Two teachers make sense of division of fractions. Teaching Children Mathematics, 12(3), 117.

Gravemeijer, K.P.E., & Cobb, P. (2006). Design research from a learning design perspective. In J. van Den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.), Educational design research. New York: Routledge.

Gregg, J., & Gregg, D.U. (2007). Measurement and fair-sharing models for dividing fractions. Mathematics Teaching in the Middle School, 12(9), 490-496.

Hu, H.W., & Hsiao, W.Y. (2013). Developing pre-service teachers’ understanding in division of fractions by using TPACK. Paper presented at the Society for Information Technology & Teacher Education International Conference.

Jansen, A., & Hohensee, C. (2015). Examining and elaborating upon the nature of elementary prospective teachers’ conceptions of partitive division with fractions. Journal of Mathematics Teacher Education, 1-20.

Kribs-Zaleta, C. (2006). Invented strategies for division of fractions. Paper presented at the The 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.

Kribs-Zaleta, C. (2008). Oranges, posters, ribbons, and lemonade: Concrete computational strategies for dividing fractions. Mathematics Teaching in the Middle School, 13(8), 453-457.

Li, Y. (2008). What do students need to learn about division of fractions? Mathematics Teaching in the Middle School, 13(9), 546-552.

Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. New York: Routledge.

NCTM. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

NCTM. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.

Nillas, L. (2003). Division of fractions: Preservice teachers’ understanding and use of problem solving strategies. The Mathematics Educator, 7(2), 96-113.

Petit, M.M., Laird, R.E., & Marsden, E.L. (2010). A Focus on fractions: Bringing research to the classroom. New York: Routledge.

Plomp, T. (2010). Educational design research: An introduction. In T. Plomp & N. Nieeven (Eds.), An introduction to educational design research. Netherlands: SLO.

Schwartz, J.E. (2008). Elementary mathematics pedagogical content knowledge: Powerful ideas for teachers. New Jersey: Prentice Hall.

Sharon, V.V., & Swarthout, M.B. (2015). How many in one? Mathematics Teaching in the Middle School, 20(5), 308-312.

Sharp, J., & Adams, B. (2002). Children's constructions of knowledge for fraction division after solving realistic problems. The Journal of Educational Research, 95(6), 333-347.

Sharp, J., & Welder, R.M. (2014). Reveal limitations through fraction division problem posing. Mathematics Teaching in the Middle School, 19(9), 540-547.

Siebert, D. (2002). Connecting informal thinking and algorithms: The case of division of fractions. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions (pp. 247-256). Reston, VA.: National Council of Teachers of Mathematics, Inc.

Slattery, J., & Fitzmaurice, O. (2014). Ours is not to reason why, just invert and multiply: an insight into Irish prospective secondary teachers' conceptual understanding of the division of fractions. Irish Educational Studies, 33(4), 467-488.

Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research (Vol. 8). Springer Science & Business Media.

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.

van Eerde, D. (2013). Design research: Looking into the heart of mathematics education. Paper presented at the The 1st South-East Asian Design Research conference, Palembang.

Wahyu, K. (2015). Changing mathematics classroom setting: Looking into students’ response and performance in learning. Paper presented at the International Conference on Mathematics, Science and Education Mataram-Indonesia.

Warrington, M.A. (1997). How children think about division with fractions. Mathematics Teaching in the Middle School, 2(6), 390-394.

Zembat, I.O. (2004). Conceptual development of prospective elementary teachers: The case of division of fractions. The Pennsylvania State University. Retrieved from


Article Metrics

Abstract view : 220 times
PDF - 107 times


  • There are currently no refbacks.

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

International Journal on Emerging Mathematics Education
Kampus 2 Universitas Ahmad Dahlan
Jalan Pramuka No. 42, Pandeyan, Umbulharjo, Yogyakarta - 55161
Telp. (0274) 563515, ext. 4902; Fax. (0274) 564604, Hp. +6285774991187

p-ISSN: 2549-4996 | e-ISSN: 2548-5806

This work is licensed under a Creative Commons Attribution 4.0 International License

View IJEME's Stats