Comparing Integer and Fraction Multiplication Problem-Solving Across Multiplicative Structures Among Rural Sixth-Grade Students in Taiwan

Yen-Ting Lai; Erh-Tsung Chin

doi:10.59525/inestera.1573

Authors

Yen-Ting Lai National Changhua University of Education, Taiwan.
Erh-Tsung Chin National Changhua University of Education, Taiwan

DOI:

https://doi.org/10.59525/inestera.1573

Keywords:

Multiplicative Reasoning; Fractions; Multiplicative Structures; Rural Students.

Abstract

Multiplication is fundamental in primary mathematics, yet students often struggle when extending their understanding from integers to fractions. This study investigated the multiplication problem-solving performance of sixth-grade students in rural primary schools in Taiwan, focusing on differences across four multiplicative structures: equal groups, multiplicative comparison, rectangular area, and Cartesian product. A researcher-developed paper-and-pencil test was administered to a total of 178 sixth-grade students (87 boys and 91 girls) from 15 rural schools, and the data were analysed quantitatively. The results show that students consistently performed better in integer multiplication than in fraction multiplication across all structures. Equal groups problems yielded the highest accuracy rates, whereas rectangular area and Cartesian product problems showed lower performance. In addition, the performance gap between integer and fraction contexts varied by structure, ranging from 10.8% in equal groups problems to 23.4% in rectangular area problems. These findings suggest that different multiplicative structures may impose varying cognitive demands on students, which may contribute to differences in fraction multiplication performance. By highlighting structure-based variations in performance, this study provides a more nuanced understanding of students’ multiplicative reasoning and raises questions about whether instruction that explicitly addresses structural differences could support student learning, particularly in rural educational contexts.

References

Bell, A., Greer, B., Grimison, L., & Mangan, C. (1989). Children’s performance on multiplicative word problems: Elements of a descriptive theory. Journal for Research in Mathematics Education, 20(5), 434–449. https://doi.org/10.5951/jresematheduc.20.5.0434

Björkhammer, C., Hedman, M., Krey, A.-K., Samuelsson, J., Träff, U., & Östergren, R. (2025). Effects of a fraction intervention for students with special educational needs in mathematics. Scandinavian Journal of Educational Research, 1–18. https://doi.org/10.1080/00313831.2025.2552232.

Braithwaite, D. W., & Siegler, R. S. (2021). Putting fractions together. Journal of Educational Psychology, 113(3), 556–571. https://doi.org/10.1037/edu0000477

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. https://doi.org/10.2307/749646

D’Erchie, M., Forsmann, C. L., Schneider, M., & Obersteiner, A. (2026). Early algebra knowledge predicts later fraction understanding. Journal of Experimental Child Psychology, 264, 106418. https://doi.org/10.1016/j.jecp.2025.106418

Ding, R., Tzur, R., Davis, A., Wei, B., Jin, X., & Jin, X. (2022). Distinguishing conceptions of multiplicative reasoning in Chinese elementary students: What sense correct and incorrect solutions might make to them? Asian Journal for Mathematics Education, 1(2), 242–262. https://doi.org/10.1177/27527263221110629

Empson, S. B., & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals: Innovations in Cognitively Guided Instruction. Heinemann.

Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Hamlett, C. L., Cirino, P. T., & Changas, P. (2013). Improving at risk learners’ understanding of fractions. Journal of Educational Psychology, 105(3), 683–700. https://doi.org/10.1037/a0032446

Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). Macmillan.

Jordan, N. C., Dyson, N., Guba, T.-P., Botello, M., Suchanec-Cooper, H., & May, H. (2024). Exploring the impact of a fraction sense intervention in authentic school environments: An initial investigation. Journal of Experimental Child Psychology, 244, 105954. https://doi.org/10.1016/j.jecp.2024.105954

Liu, Q., & Braithwaite, D. W. (2023). Affordances of fractions and decimals for arithmetic. Journal of Experimental Psychology: Learning, Memory, and Cognition, 49(9), 1459–1470. https://doi.org/10.1037/xlm0001161

Ministry of Education. (2008). Grade 1–9 curriculum guidelines: Mathematics learning area. Taiwan.

Ministry of Education. (2018). Curriculum guidelines of 12-year basic education: Mathematics domain. Taiwan.

Xu, C., Di Lonardo Burr, S., McNally-Edwards, N., McShane, C., Wan, L., & Wylie, J. (2024). A holistic investigation of fraction learning. Journal of Cognitive Psychology, 36(7), 815–833. https://doi.org/10.1080/20445911.2024.2388907

Zwanch, K., & Wilkins, J. L. M. (2021). Releasing the conceptual spring to construct multiplicative reasoning. Educational Studies in Mathematics, 106(1), 151–170. https://doi.org/10.1007/s10649-020-09999-4

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.

Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91–126). Academic Press.

Clarke, D. M., Roche, A., & Mitchell, A. (2008). Ten practical tips for making fractions come alive and make sense. Mathematics Teaching in the Middle School, 13(7), 372–380.

Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions. Journal of Mathematical Behavior, 26(1), 27–56. https://doi.org/10.1016/j.jmathb.2007.03.002

Hackenberg, A. J., & Lee, M. Y. (2015). Relationships between students’ fractional knowledge and algebraic reasoning. Journal of Mathematical Behavior, 40, 228–244.

Lamon, S. J. (2012). Teaching fractions and ratios for understanding (3rd ed.). Routledge.

Lamon, S. J. (2007). Rational numbers and proportional reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Information Age Publishing.

Lesh, R., Post, T. R., & Behr, M. J. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93–118). Lawrence Erlbaum.

Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21(1), 16–32. https://doi.org/10.2307/749455

Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52. https://doi.org/10.1207/s15326985ep4001_3

Norton, A., & Wilkins, J. L. M. (2009). A quantitative analysis of children's splitting operations and fraction schemes. Journal of Mathematical Behavior, 28(2–3), 150–161.

Post, T. R., Harel, G., Behr, M. J., & Lesh, R. (1991). Intermediate teachers’ knowledge of rational number concepts. In E. Fennema, T. Carpenter, & S. Lamon (Eds.), Integrating research on teaching and learning mathematics (pp. 177–198). SUNY Press.

Prediger, S., & Wessel, L. (2013). Fostering German-language learners’ conceptual understanding of fractions. Journal of Mathematics Education, 6(2), 1–22.

Resnick, L. B., Cauzinille-Marmèche, E., & Mathieu, J. (1987). Understanding algebra. In J. Sloboda & D. Rogers (Eds.), Cognitive processes in mathematics. Oxford University Press.

Siegler, R. S., Fazio, L. K., Bailey, D. H., & Zhou, X. (2013). Fractions: The new frontier for theories of numerical development. Trends in Cognitive Sciences, 17(1), 13–19. https://doi.org/10.1016/j.tics.2012.11.004

Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). 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

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145.

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

Streefland, L. (1991). Fractions in realistic mathematics education. Kluwer Academic Publishers.

Torbeyns, J., Schneider, M., Xin, Z., & Siegler, R. S. (2015). Bridging the gap: Fraction understanding is central to mathematics achievement. Learning and Instruction, 37, 1–11.

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2019). Elementary and middle school mathematics: Teaching developmentally (10th ed.). Pearson.

Vamvakoussi, X., Van Dooren, W., & Verschaffel, L. (2012). Naturally biased? In search for reaction time evidence for the natural number bias. Journal for Research in Mathematics Education, 43(4), 381–404.

Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 557–628). Information Age Publishing.

Wilkins, J. L. M., Norton, A., & Boyce, S. J. (2013). Validating a learning progression for fractions. Mathematics Education Research Journal, 25(4), 657–681.

Wu, H. (2011). Understanding numbers in elementary school mathematics. American Mathematical Society.