Exploring cognitive pathways of mathematics education students in solving contextual modular arithmetic problems
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Abstract
Understanding how university students grasp modular arithmetic in real-world contexts remains underexplored, particularly regarding their use of cognitive pathways in solving contextualized problems. This study aims to examine the cognitive pathways utilized by mathematics education students when solving context-based modular arithmetic problems. 32 undergraduate students in the early stage of a mathematics education program who had successfully completed course in Number Theory were asked to solve three real-world-based problems involving work cycle patterns, container multiples, and congruence systems. The tasks were designed to represent fundamental real-world structures of modular arithmetic and to explore how contextual representations influence students’ cognitive pathways. The analysis was conducted using a descriptive qualitative approach through Cognitive Task Analysis (CTA) to identify five key cognitive pathways: quantitative reasoning, linguistic processing, working memory, pattern recognition, and cognitive flexibility. The result show that the first three pathways were consistently used across all task, indicating stable patterns in numerical reasoning, language processing, and information management. In contrast, pattern recognition and flexibility varied depending on how the problems were presented. Contextual narratives encouraged more reflective and adaptive thinking, whereas symbolic forms tended to limit exploration and reduce conceptual engagement. These results highlight the importance of problem design in supporting diverse cognitive approaches. The study's practical implications include the development of assessments and learning strategies that foster flexible and meaningful thinking, especially when understanding abstract mathematical concepts. Future research is encouraged to explore the integration of digital technologies and multiple representations to enhance students' cognitive flexibility.
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Novianti, M., Suratman, D., & Munthe, C. O. S. (2026). Exploring cognitive pathways of mathematics education students in solving contextual modular arithmetic problems. Kalamatika: Jurnal Pendidikan Matematika, 11(1), 178-196. https://doi.org/10.22236/KALAMATIKA.vol11no1.2026pp178-196
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Tofel-Grehl, C., & Feldon, D. F. (2013). Cognitive task analysis–based training: A meta-analysis of studies. Journal of Cognitive Engineering and Decision Making, 7(3), 293–304. https://doi.org/10.1177/1555343412474821
Verschaffel, L., Schukajlow, S., Star, J. R., & Van Dooren, W. (2020).
Word problems in mathematics education: A survey. ZDM – Mathematics Education, 52, 1–16. https://doi.org/10.1007/s11858-020-01130-4
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A. (2024). Task characteristics associated with mathematical word problem solving performance among elementary school-aged children: A systematic review and metaanalysis. Educational Psychology Review, 36, Article 117. https://doi.org/10.1007/s10648-024-09954-2
Barumbun, M., & Kharisma, D. (2022). Procedural knowledge or conceptual knowledge? Developing the so-called proceptual knowledge in mathematics learning. Beta: Jurnal Tadris Matematika, 15(2), 167–180. https://doi.org/10.20414/betajtm.v15i2.472
Borji, V., Radmehr, F., & Font, V. (2019). The impact of procedural and conceptual teaching on students’ mathematical performance over time. International Journal of Mathematical Education in Science and Technology, 52(3), 404–426. https://doi.org/10.1080/0020739X.2019.1688404
Brown, O., Power, N., & Gore, J. (2022). Cognitive task analysis: Eliciting expert cognition in context. Cognition, Technology & Work, 24, 689–703. https://doi.org/10.1177/10944281241271216
Chavarría-Arroyo, G., & Albanese, V. (2022). Contextualized mathematical problems: Perspective of teachers about problem posing. Education and Information Technologies, 27(5), 6543–6565. https://doi.org/10.3390/educsci13010006
Crandall, B. W., & Hoffman, R. R. (2013). Cognitive task analysis. In J. D. Lee & A. Kirlik (Eds.), The Oxford handbook of cognitive engineering (pp. 229–239). Oxford University Press. https://doi.org/10.1093/oxfordhb/9780199757183.013.0014
Creswell, J. W., & Creswell, J. D. (2023). Research design: Qualitative, quantitative, and mixed methods approaches (6th ed.). SAGE Publications.
Getenet, S. T. (2022). Understanding how pre-service teachers design numeracy-rich activities in non-mathematic curriculum areas. Journal of Research in Innovative Teaching and Learning, 16(2), 241–259. 10.1108/JRIT-05-2024-0124
Gilmore, C., Keeble, S., Richardson, S., & Cragg, L. (2015). The role of cognitive inhibition in different components of arithmetic. ZDM Mathematics Education, 47(5), 771–782. https://doi.org/10.1007/s11858-014-0659-y
Gros, H., Thibaut, J., Raynal, L., & Sander, E. (2024). Revealing mental representations of arithmetic word problems through false memories: New insights into semantic congruence. Journal of experimental psychology. Learning, memory, and cognition. https://doi.org/10.1037/xlm0001373.
Jaffe, J. B., & Bolger, D. J.(2023). Cognitive processes, linguistic factors, and arithmetic word problem success: A review of behavioral studies. Cognitive Processing, 24(4), 645–663. https://doi.org/10.1007/s10648-023-09821-6
Kohen, Z., & Nitzan-Tamar, O. (2022). Contextual mathematical modelling: Problem-solving characterization and feasibility. Educational Studies in Mathematics, 110(2), 165–185. https://doi.org/10.3390/educsci12070454
LeFevre, J.-A., Skwarchuk, S.-L., Smith-Chant, B. L., Fast, L., Kamawar, D., & Bisanz, J. (2010). Pathways to mathematics: Longitudinal predictors of performance. Child Development, 81(6), 1753–1767. https://doi.org/10.1111/j.1467-8624.2010.01508.x
Medrano, J., & Miller‐Cotto, D. (2025). Understanding working memory as a facilitator of math problem‐solving: Offloading as a potential strategy. British Journal of Educational Psychology, 95(2), 467–484. https://doi.org/10.1111/bjep.12767
Muhtasyam, A., Syamsuri, & Santosa, C. A. H. F. (2024).
Meta-analysis of the effect of learning models on mathematical problem-solving skills. World Journal of Advanced Research and Reviews.
https://doi.org/10.30574/wjarr.2024.22.3.1941
Raghubar, K. P., Barnes, M. A., & Hecht, S. A. (2010).
Working memory and mathematics: A review of developmental, individual difference, and cognitive approaches. Learning and Individual Differences, 20(2), 110–122. https://doi.org/10.1016/j.lindif.2009.10.005
Rittle-Johnson, B. (2017). Developing mathematics knowledge. Child Development Perspectives, 11(3), 183–188. https://doi.org/10.1111/cdep.12229
Rosen, K. H. (2011). Elementary number theory and its applications (6th ed.). Pearson Education.
Scheibling-Sève, C., Sander, E., & Pasquinelli, E. (2017).
Developing cognitive flexibility in solving arithmetic word problems. In Proceedings of the 39th Annual Meeting of the Cognitive Science Society.
https://escholarship.org/uc/item/0dv9z04h
Schüler-Meyer, A. (2019). How do students revisit school mathematics in modular arithmetic? Conditions and affordances of the transition to tertiary mathematics with a focus on learning processes. International Journal of Research in Undergraduate Mathematics Education, 5(2), 163–182. https://doi.org/10.1007/s40753-019-00088-3
Shen, C., Chen, Q., Zhang, N., Diao, F., Liu, P., & Zhou, X. (2024). The development of situational mathematical ability lags behind the development of symbolic mathematical ability. European Journal of Psychology of Education, 40. https://doi.org/10.1007/s10212-024-00924-4.
Sowinski, C., LeFevre, J.-A., Skwarchuk, S.-L., Kamawar, D., & Bisanz, J. (2015). Refining the quantitative pathway of the Pathways to Mathematics model. Journal of Experimental Child Psychology, 131, 73–93. https://doi.org/10.1016/j.jecp.2014.11.004
Stylianou, D. A. (2011). An examination of middle school students’ representation practices in mathematical problem solving through the lens of expert work: Towards an organizing scheme. Educational Studies in Mathematics, 76(3), 265–280. https://doi.org/10.1007/s10649-010-9273-2
Subekti, F. E., Sukestiyarno, Y. L., Wardono, & Rosyida, I. (2022). Mathematics Pre-Service Teachers' Numerical Thinking Profiles. European Journal of Educational Research, 11(2), 1075–1087. https://doi.org/10.12973/eu-jer.11.2.1075
Träff, U., Olsson, L., Östergren, R., Skagerlund, K., & Ljungberg, T. (2019). Cognitive mechanisms underlying third graders’ arithmetic skills: Expanding the Pathways to Mathematics model. Journal of Experimental Child Psychology, 183, 40–60. https://doi.org/10.1016/j.jecp.2017.11.010
Tofel-Grehl, C., & Feldon, D. F. (2013). Cognitive task analysis–based training: A meta-analysis of studies. Journal of Cognitive Engineering and Decision Making, 7(3), 293–304. https://doi.org/10.1177/1555343412474821
Verschaffel, L., Schukajlow, S., Star, J. R., & Van Dooren, W. (2020).
Word problems in mathematics education: A survey. ZDM – Mathematics Education, 52, 1–16. https://doi.org/10.1007/s11858-020-01130-4
Vessonen, T., Dahlberg, M., Hellstrand, H., Widlund, A., Korhonen, J., Aunio, P., & Laine,
A. (2024). Task characteristics associated with mathematical word problem solving performance among elementary school-aged children: A systematic review and metaanalysis. Educational Psychology Review, 36, Article 117. https://doi.org/10.1007/s10648-024-09954-2