Analyzing students’ fraction strategies: a case study of high-achieving middle school learners

Abstract

The aim of this study is to examine the cognitive strategies that academically successful middle school students employ when solving problems involving fractions, and to explore how these strategies influence their mathematical thinking and problem-solving processes. Adopting a qualitative case study design, the research involved students from 5th to 8th grades who demonstrated high achievement in mathematics. Data were gathered through a semi-structured interview form that focused on key components associated with reasoning about fractions—such as estimation, operational understanding, reference benchmarks, magnitude comparison, and equivalence. Content analysis and thematic coding techniques were used to analyze the data. The findings suggest that students' strategy use is closely related to their conceptual and procedural understanding of fractions. Students who employed intuitive, flexible strategies demonstrated deeper mathematical reasoning, while those who relied on rule-based methods produced more procedural and limited responses. These cognitive patterns were directly reflected in the nature and effectiveness of their mathematical thinking. The study also suggests that reasoning about fractions develops through instruction and can be observed at varying levels—basic, intermediate, and advanced. Accordingly, number sense should be approached as a foundational competence in mathematics education. Instructional practices should integrate visual models, estimation activities, conceptual tools, and strategy-based tasks to better support students’ mathematical development.

Keywords

• fractionalnumbersense
• mathematiceducation
• qualitativeresearch
• middileschoolstudents
• problemsolvingstrategies

References

1. Akgün, L., & Yıldız, C. (2016). Kesir kavramlarının öğretiminde öğretmen rehberliğinde yapılan iş birlikli öğrenmenin başarıya ve kalıcılığa etkisi, (The effect of teacher-guided cooperative learning on success and retention in teaching fraction concepts). Eğitim ve Bilim, 41(186), 1–21. https://doi.org/10.15390/EB.2016.4526
2. Ahmed, W., & Williams, M. (1997). Case study method in educational research: From theory to practice. Contemporary Education, 68(1), 27–30.
3. Bailey, D. H., Hoard, M. K., Nugent, L., & Geary, D. C. (2012). Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology, 113(3), 447–455. https://doi.org/10.1016/j.jecp.2012.06.004
4. Bailey, D. H., Hoard, M. K., Nugent, L., & Geary, D. C. (2014). Competence with fractions predicts gains in mathematics achievement. Journal of Experimental Child Psychology, 118, 110–118. https://doi.org/10.1016/j.jecp.2013.09.009
5. Bailey, D. H., Siegler, R. S., & Geary, D. C. (2015). Developmental change in the influence of domain-general abilities and domain-specific knowledge on mathematics achievement. Journal of Educational Psychology, 107(3), 880–893. https://doi.org/10.1037/edu0000024
6. Behr, M. J., Harel, G., Post, T. R., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–333). Macmillan Publishing.
7. Cetin, H., & Yapıcı, M. (2023). 8. sınıf öğrencilerinin sayı duyusu performanslarının ve stratejilerinin incelenmesi(An Investigation of 8th Grade Students’ Number Sense Performance and Strategies). Trakya Eğitim Dergisi, 13(3), 1799–1817. https://doi.org/10.24315/tred.1251704
8. Clarke, D. M., & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72, 127–138. https://doi.org/10.1007/s10649-009-9198-9

9. Cramer, K. A., Behr, M. J., Post, T. R., & Lesh, R. (1997). Rational number project: Fraction lessons for the middle grades. Research in Mathematics Education.
10. Creswell, J. W. (2013). Qualitative inquiry and research design: Choosing among five approaches (3rd ed.). Sage Publications.
11. DePascale, J., & Ramani, G. B. (2024). Exploring the role of mathematics games in developing number sense in elementary students: A systematic review. Journal of Early Mathematics Education, 15(1), 50–75.
12. Durmuş, S., & Karaca, E. (2013). İlköğretim matematik öğretim programındaki kesirler konusunun öğretiminde karşılaşılan güçlükler (Challenges encountered in teaching the fractions topic in the primary school mathematics curriculum). Milli Eğitim Dergisi, 197, 106–123.
13. Fennell, F., Kobett, B., & Wray, J. (2014). Fractions are numbers too! Mathematics Teaching in the Middle School, 19(8), 486–493. https://www.jstor.org/stable/10.5951/mathteacmiddscho.19.8.0486
14. Jordan, N. C., Kaplan, D., Ramineni, C., & Locuniak, M. N. (2010). Early math matters: Kindergarten number competence and later mathematics outcomes. Developmental Psychology, 46(3), 850–867. https://doi.org/10.1037/a0019672
15. Kartal, A., & Pirasa, N. (2022). Examination of fractional number sense in eighth graders with high academic performance. Education Quarterly Reviews, 5(3), 297–308. https://doi.org/10.31014/aior.1993.05.03.546
16. Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Information Age Publishing.
17. Lin, C., & Powell, S. R. (2023). Supporting conceptual understanding in fractions through dynamic visualizations. Mathematics Teacher Education and Development, 25(1), 98–122.
18. McNamara, J., & Shaughnessy, M. M. (2015). Beyond pizzas & pies: 10 essential strategies for supporting fraction sense, grades 3–5. Math Solutions.
19. Ng, S. F. (2010). Teaching fractions through realistic problems in Chinese classrooms: A pilot study. International Journal of Science and Mathematics Education, 8(5), 911–935. https://doi.org/10.1007/s10763-009-9183-2
20. Reeder, S. L., & Utley, J. (2007). Assessing preservice teachers’ development of fraction number sense. Issues in the Undergraduate Mathematics Preparation of School Teachers, 1.
21. Rochaendi, R., & Ariyani, F. (2024). Enhancing students’ conceptual understanding of fractions through visual- based models in middle school mathematics. International Journal of Educational Research, 125, Article 102076. https://doi.org/10.1016/j.ijer.2023.102076
22. Siegler, R. S., Thompson, C. A., & Schneider, M. (2012). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296. https://doi.org/10.1016/j.cogpsych.2011.03.001
23. Sowder, J. T. (1992). Making sense of numbers in school mathematics. In G. Leinhardt, R. Putnam, & R. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 1–51). Lawrence Erlbaum.
24. Suh, J. M., Moyer, P. S., & Heo, H. J. (2005). Examining technology uses in the classroom: Developing fraction sense using virtual manipulatives. Journal of Computers in Mathematics and Science Teaching, 24(3), 303–324.
25. Utley, J., & Reeder, S. (2012). Prospective elementary teachers’ development of fraction number sense. Investigations in Mathematics Learning, 5(1), 1–24.
26. Yıldız, C., & Eryılmaz, A. (2018). Ortaokul matematik öğretiminde kesir sayılarının öğretimi ve öğrenci başarılarına etkisi (The effect of teaching fractional numbers in middle school mathematics on student achievement). Kuram ve Uygulamada Eğitim Bilimleri, 18(3), 625–648. https://doi.org/10.12738/estp.2018.3.0231
27. Yılmaz, E., & Yeşildere-İmre, S. (2018). Öğrencilerin sayı duyusu gelişiminde öğretim sürecinin etkisi. Elementary Education Online, 17(4), 2234–2250. https://doi.org/10.17051/ilkonline.2019.507441