Flexibility in partitioning strategies of fourth graders

Yeliz Yazgan


This study combines the concepts of flexibility and partitioning, and aims to probe fourth grade students’ flexibility in partitioning strategies. Seven students participated in this descriptive case study. Students were given three partitioning tasks. Forty-eight answers produced by students were evaluated and classified based on the strategies defined in the taxonomy developed by Charles and Nason (2000). Results showed that students could easily change their strategies both within and across tasks.  Namely, they displayed both inter- and intra-task strategy flexibility to a large extent even though they did not have any intervention on partitioning. Another point that findings have implicated was that the fourth graders’ flexibility in partitioning strategies may be utilized to introduce concepts of equivalent fractions and mixed numbers. Results are discussed in terms of their implications related to mathematics education, and some recommendations aimed at learning environments and future studies are presented.


Flexibility; Fractions; Partitioning; Partitioning strategies;

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Charles, K., & Nason, R. (2000). Young children’s partitioning strategies. Educational Studies in Mathematics, 43(2), 191-221. https://doi.org/10.2307/3483090

Cohen, L., Manion, L., & Morrison, K. (2007). Research methods in education. Routledge.

Confrey, J., Maloney, A. P., Nguyen, K. H., & Rupp, A. A. (2014). Equipartitioning, a foundation for rational number reasoning. In A. P. Maloney, J. Confrey, & K. H. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (pp. 61-96). Information Age.

Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five approaches. Sage Publications.

Cutting, C. (2019). Re-thinking fraction instruction in primary school: The case for an alternative approach in the early years. In G. Hine, S. Blackley, & A. Cooke (Eds.), Proceedings of the 42nd Annual Conference of the Mathematics Education Research Group of Australasia (pp. 204–211). Perth: MERGA.

Elia, I., van den Heuvel-Panhuizen, M., & Kolovou, A. (2009). Exploring strategy use and strategy flexibility in nonroutine problem solving by primary school high achievers in mathematics, ZDM, 41(5), 605-618. https://doi.org/10.1007/s11858-009-0184-6

Empson, S. B. (1999). Equal sharing and shared meaning: The development of fraction concepts in a first-grade classroom. Cognition and Instruction, 17(3), 283-342. https://doi.org/10.1207/S1532690XCI1703_3

Empson, S. B., Junk, D., Dominguez, H. and 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), 1-28. https://doi.org/10.1007/s10649-005-9000-6

Jausovec, N. (1991). Flexible strategy use: A characteristic of gifted problem solving. Creativity Research Journal, 4(4), 349-366. https://doi.org/10.1080/10400419109534411

Kalyuga, S., Renkl, A., & Paas, F. (2010). Facilitating flexible problem solving: A cognitive load perspective. Educational Psychology Review, 22(2), 175-186. https://doi.org/10.1007/s10648-010-9132-9

Lamon, S. J. (1996). The development of unitizing: Its role in children's partitioning strategies. Journal for Research in Mathematics Education, 27 (2), 170-193. https://doi.org/10.2307/749599

Lamon, S. J. (1999). Teaching fractions and ratios for understanding: Essential content knowledge and instruction strategies for teachers. Lawrence Erlb.

Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman and B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–145). Sense Publishers.

Leikin, R., Berman, A., & Koichu, B. (2009). Creativity in mathematics and the education of gifted students. Sense Publishers.

Levav-Waynberg, A., & Leikin, R. (2012). Using multiple solution tasks for the evaluation of students’ problem-solving performance in geometry. Canadian Journal of Science, Mathematics and Technology Education, 12(4), 311-333. https://doi.org/10.1080/14926156.2012.732191

Liu, R. D., Wang, J., Star, J. R., Zhen, R., Jiang, R. H., & Fu, X. C. (2018). Turning potential flexibility into flexible performance: Moderating effect of self-efficacy and use of flexible cognition. Frontiers in Psychology, 9, 1–10. https://doi.org/10.3389/fpsyg.2018.00646

Low, C. S., & Chew, C. M. (2019). Strategy flexibility in mathematics education. In C. S. Lim, C. M. Chew, & B. Sriraman (Eds.), Mathematics Education from an Asian Perspective (pp. 85-105). USM Press.

Merriam, S. B. (2009). Qualitative research: A guide to design and implementation (3rd ed.). Jossey-Bass.

Nguyen, H. A., Guo, Y., Stamper, J., & McLaren, B. M. (2020). Improving students’ problem-solving flexibility in non-routine mathematics. In I. Bittencourt, M. Cukurova, K. Muldner, R. Luckin, & E. Millán (Eds), Artificial Intelligence in Education (pp. 409-413). Springer.

Norton, A. H., & McCloskey, A. V. (2008). Modeling students' mathematics using Steffe's fraction schemes. Teaching Children Mathematics, 15(1), 48-54.

Norton, A., & Wilkins, J. L. (2010). Students’ partitive reasoning. The Journal of Mathematical Behavior, 29(4), 181-194. https://doi.org/10.1016/j.jmathb.2010.10.001

Petit, M. M., Laird, R. E., Marsden, E. L., & Ebby, C. B. (2015). A focus on fractions: Bringing research to the classroom. Routledge.

Pitkethly, A., & Hunting, R. P. (1996). A review of recent research in the area of initial fraction concepts. Educational Studies in Mathematics, 30(1), 5–38. https://doi.org/10.1007/bf00163751

Pothier, Y., & Sawada, D. (1990). Partitioning: An approach to fractions. The Arithmetic Teacher, 38(4), 12-16.

Pothier, Y., & Sawada, D. (1983) Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Education, 14 (5), 307-317. https://doi.org/10.2307/748675

Selter, C. (2001). Addition and subtraction of three-digit numbers: German elementary children's success, methods and strategies. Educational Studies in Mathematics, 47(2), 145-173. https://doi.org/10.1023/a:1014521221809

Selter, C. (2009). Creativity, flexibility, adaptivity, and strategy use in mathematics. ZDM, 41(5), 619-625. https://doi.org/10.1007/s11858-009-0203-7

Siemon, D. (2003). Partitioning: the missing link in building fraction knowledge and confidence. Australian Mathematics Teacher, 59(3), 22-24.

Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. Springer.

Star, J. (2018). Flexibility in mathematical problem solving: The state of the field. In Hsieh, F. J. (Ed.), Proceedings of the 8th ICMI-East Asia Regional Conference on Mathematics Education (Vol. 1, pp.15-25). EARCOME.

Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18, 565–579. https://doi.org/10.1016/j.learninstruc.2007.09.018

Star, J. R., Rittle-Johnson, B., Lynch, K., & Perova, N. (2009). The role of prior knowledge in the development of strategy flexibility: The case of computational estimation. ZDM, 41(5), 569-579. https://doi.org/10.1007/s11858-009-0181-9

Star, J. R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31, 280–300. https://doi.org/ 10.1016/j.cedpsych.2005.08.001

Streefland, L., (1991). Fractions in Realistic Mathematics Education. Kluwer Academic, Dordrecht.

Toluk, Z. (1999). Children's conceptualizations of the quotient subconstruct of rational numbers [Doctoral dissertation, Arizona State University]. ProQuest Dissertations & Theses Global. https://www.proquest.com/pqdtglobal/docview/304494304/742314C2FBBB4464PQ/1?accountid=17219

Torbeyns, J., De Smedt, B., Ghesquière, P., & Verschaffel, L. (2009). Jump or compensate? Strategy flexibility in the number domain up to 100. ZDM, 41(5), 581-590. https://doi.org/10.1007/s11858-009-0187-3

Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualising, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335–359. https://doi.org/ 10.1007/bf03174765

Wang, J., Liu, R. D., Star, J., Liu, Y., & Zhen, R. (2019). The moderating effect of regulatory focus in the relationship between potential flexibility and practical flexibility. Contemporary Educational Psychology, 56, 218-227. https://doi.org/10.1016/j.cedpsych.2019.01.013

Xu, L., Liu, R., Star, J.R., Wang, J., Liu, Y., & Zhen, R. (2017). Measures of potential flexibility and practical flexibility in equation solving. Frontiers in Psychology, 8, 1368. https://doi.org/10.3389/fpsyg.2017.01368

Yazgan, Y (2010), Partitioning strategies of fourth and fifth grade students. E-Journal of New World Sciences Academy, 5 (4), 1628-1642.

Yin, R. (2003). Applications of case study research (2nd ed.). Sage

DOI: https://doi.org/10.29103/mjml.v4i2.4451

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