Email this article Exploring Polya's Problem-Solving Stages, Cognitive Components, and Role Change Dynamics in Mathematical Problem-Solving Discussions
This study aims to examine the effectiveness of integrating Polya's problem-solving stages with commognitive discourse practices in collaborative learning, and to describe the associations among Polya's stages, commognitive components, and the dynamics of role change in mathematical problem-solving discussions, as well as their contributions to the development of mathematical thinking quality. A quantitative approach with a quasi-experimental Non-Equivalent Control Group Design was employed. Participants were pre-service teacher students at STKIP Taman Siswa Bima, assigned to an experimental class (n = 30) and a control class (n = 30). Data were collected using a mathematics problem-solving test based on Polya's four stages and structured observation instruments that measured cognitive components and changes in discussion roles. Data analysis included Rasch-model-based instrument quality testing, normality and homogeneity prerequisite tests, a paired-samples t-test for each class separately to assess within-group pre-post differences (df = 29 per class), an independent-samples t-test on N-Gain scores to compare treatment effectiveness between classes (df = 58), and an N-Gain analysis to assess treatment effectiveness. Instruments demonstrated strong validity (PTMEA Corr. = 0.40–0.65) and high reliability (person reliability = 0.87–0.88). The experimental class showed significantly greater improvement in mathematical problem-solving ability than the control class (posttest means: 49.5 vs. 43.7). The paired t-test confirmed a significant pre–post improvement in the experimental class [t(29) = 2.435, p = 0.021] but not in the control class [t(29) = 1.295, p = 0.206]. Observational data indicated that activation of commognitive components: consistent use of mathematical terms, visual mediators, and justification narratives, and the emergence of dynamic role changes during discussion were associated with improvements in problem-solving quality. Mathematical problem solving develops more effectively when Polya's heuristic stages are integrated with cognitive-discursive practices that foster epistemic role changes in collaborative discussions. This integration offers an innovative framework for designing meaningful, discussion-based mathematics learning.
Keywords: mathematical problem solving, Polya’s stages, commognitive, collaborative discussion, role change.
Barnett, J. H. (2022). Primary source projects as textbook replacements: a commognitive analysis. ZDM - International Journal on Mathematics Education, 54(7), 1569–1582. https://doi.org/10.1007/s11858-022-01401-2
Ben-Dor, N., & Heyd-Metzuyanim, E. (2021). Standing on each other’s shoulders: A case of coalescence between geometric discourses in peer interaction. Journal of Mathematical Behavior, 64. https://doi.org/10.1016/j.jmathb.2021.100900
Ben-Dor, N., & Heyd-Metzuyanim, E. (2025). Shifts in meta-level learning agreement during mathematics peer learning: integrating positioning theory and commognitive framework. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-025-10433-w
Berman, A., Mahagna, A., Ram, I., & Wolf, A. (2024). Problem-solving before instruction: A case study of a matrix theory course. PRIMUS. https://doi.org/10.1080/10511970.2024.2352370
Clark, K. M., Can, C., Barnett, J. H., Watford, M., & Rubis, O. M. (2022). Tales of research initiatives on university-level mathematics and primary historical sources. ZDM - International Journal on Mathematics Education, 54(7), 1507–1520. https://doi.org/10.1007/s11858-022-01382-2
Edwards, A. (2025). Trusted together: A commognitive perspective on a primary source project in multivariable calculus. Mathematics Enthusiast, 22(1–2), 123–148. https://doi.org/10.54870/1551-3440.1655
Fabiani Marcatto, F. S. (2025). The three decades journey of problem solving in Brazil. ZDM - International Journal on Mathematics Education, 57(7), 1371–1381. https://doi.org/10.1007/s11858-025-01750-8
Fatmanissa, N., Siswono, T., Lukito, A., & Rahaju, E. B. (2022). Collaborative problem solving in mathematics: A systematic literature review. Pedagogika, 148(4), 45–65. https://doi.org/10.15823/p.2022.148.3
Felmer, P. (2023). Collaborative problem-solving in mathematics. Current Opinion in Behavioral Sciences, 52. https://doi.org/10.1016/j.cobeha.2023.101296
Gavilán Izquierdo, J. M., Gallego-Sánchez, I., González, A., & Puertas, M. L. (2022). A new tool for the teaching of graph theory: identification of commognitive conflicts. Mathematics Teaching-Research Journal, 14(2), 186–212.
Guo, F. Y., Liu, S. Y., Zheng, T. C., & Cao, T. W. (2025). Session-based recommendation with quaternion-enhanced attention calculation. Intelligent Data Analysis, 29(4), 850–865. https://doi.org/10.1177/1088467X241301378
Gustafsson, P. (2024). Productive mathematical whole-class discussions: A mixed-method approach exploring the potential of multiple-choice tasks supported by a classroom response system. International Journal of Science and Mathematics Education, 22(4), 861–884. https://doi.org/10.1007/s10763-023-10402-w
Gustiningsi, T., Putri, R. I., Zulkardi, Z., & Hapizah, H. (2024). Supporting students’ mathematical literacy skill using digital tools. In N. Happy, D. Wulandari, Muhtarom, Sutrisno, M. S. Zuhri, F. Nursyahidah, & D. Purwosetiyono (Eds.), AIP Conference Proceedings (Vol. 3046, Issue 1). American Institute of Physics Inc. https://doi.org/10.1063/5.0194695
Iilonga, H. K., & Ogbonnaya, U. I. (2023). Grade 10 namibian learners' problem-solving skill in algebraic word problems. Infinity Journal, 12(2), 275–290. https://doi.org/10.22460/infinity.v12i2.p275-290
Johnson, H. L., & Ohtani, M. (2025). Advances in task design in mathematics education. ZDM - International Journal on Mathematics Education, 57(4), 651–664. https://doi.org/10.1007/s11858-025-01690-3
Lathifaturrahmah, L., Nusantara, T., Subanji, S., & Muksar, M. (2024). Analysis of mathematics students’ problem-solving skills in making prediction mathematical representations. In L. Anwar, D. Rahmadani, D. E. Cahyani, T. Listiawan, I. Rofiki, P. Darmawan, & A. D. Pahrany (Eds.), AIP Conference Proceedings (Vol. 3049, Issue 1). American Institute of Physics Inc. https://doi.org/10.1063/5.0195480
Lefrida, R., Siswono, T. Y. E., & Lukito, A. (2023). Development of derivative understanding task instruments to explore student commognition. Mathematics Education Journal, 17(3), 343–360. https://doi.org/10.22342/jpm.17.3.20826.343-360
Liu, Q., Wu, J., & Cao, Y. (2025). Deep learning-based low-complexity hybrid precoding for massive MIMO systems. IEEE Communications Letters, 29(10), 2238–2242. https://doi.org/10.1109/LCOMM.2025.3591774
Liu, Y., & Cao, Y. (2025). Teacher intervention during collaborative problem solving in mathematics classrooms in Mainland China. Behavioral Sciences, 15(3). https://doi.org/10.3390/bs15030377
Lu, J., Wu, S., Wang, Y., & Zhang, Y. (2023). Visualizing the commognitive processes of collaborative problem solving in mathematics classrooms. Asia-Pacific Education Researcher, 32(5), 615–628. https://doi.org/10.1007/s40299-022-00681-2
Martín Molina, V., González-Regaña, A. J., Toscano, R., & Gavilán Izquierdo, J. M. (2020). Differences between how undergraduate students define geometric solids and what their lecturers expect from them through the lens of the theory of commognition. Eurasia Journal of Mathematics, Science and Technology Education, 16(12), 1–10. https://doi.org/10.29333/ejmste/9159
Moustapha-Corrêa, B., Bernardes, A., Giraldo, V., Biza, I., & Nardi, E. (2021). Problematizing mathematics and its pedagogy through teacher engagement with history-focused and classroom situation-specific tasks. Journal of Mathematical Behavior, 61. https://doi.org/10.1016/j.jmathb.2021.100840
Muslim, M., Nusantara, T., Sudirman, S., & Irawati, S. (2024). The causes of changes in student positioning in group discussions using Polya’s problem-solving and commognitive approaches. Eurasia Journal of Mathematics, Science and Technology Education, 20(9), em2506. https://doi.org/10.29333/ejmste/15148
Nachlieli, T., & Elbaum-Cohen, A. (2021). Teaching practices aimed at promoting meta-level learning: The case of complex numbers. Journal of Mathematical Behavior, 62. https://doi.org/10.1016/j.jmathb.2021.100872
Oktaviani, N. K. C., Prastika, A. P. A., Fajaroh, F., & Suharti, S. (2020). Incorporation of Polya’s problem solving into process guide inquiry in learning buffer solution. In H. Habiddin, S. Majid, I. Suhadi, N. Farida, & I. W. Dasna (Eds.), AIP Conference Proceedings (Vol. 2215). American Institute of Physics Inc. subs@aip.org. https://doi.org/10.1063/5.0000692
Sihlangu, S., Maphutha, K., & Mokwana, L. (2025). A commognitive analysis of learners’ mathematical thinking on mathematics vocabulary used during classroom discourse. Journal on Mathematics Education, 16(4), 1389–1406. https://doi.org/10.22342/jme.v16i4.pp1389-1406
Smith, J. (2024). Supporting metacognitive talk during collaborative problem solving: A case study in Scottish primary school mathematics. Education 3-13, 52(8), 1578–1593. https://doi.org/10.1080/03004279.2023.2187670
Sumalan, A.-L., & Moldoveanu, C.-E. (2024). Use of digital technology in integrated mathematics education. Applied System Innovation, 7(4). https://doi.org/10.3390/asi7040066
Suparatulatorn, R., Jun-On, N., Hong, Y.-Y., Intaros, P., & Suwannaut, S. (2023). Exploring problem-solving through the intervention of technology and realistic mathematics education in the calculus content course. Journal on Mathematics Education, 14(1), 103–128. https://doi.org/10.22342/JME.V14I1.PP103-128
Sutama, S., Fuadi, D., Narimo, S., Hafida, S. H. N., Novitasari, M., Anif, S., Prayitno, H. J., Sunanih, S., & Adnan, M. (2022). Collaborative mathematics learning management: Critical thinking skills in problem solving. International Journal of Evaluation and Research in Education, 11(3), 1015–1027. https://doi.org/10.11591/ijere.v11i3.22193
Tran, D., & Díez-Palomar, J. (2021). Measuring agency in mathematics collaborative problem solving. In M. Inprasitha, N. Changsri, & N. Boonsena (Eds.), Proceedings of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 153–160). Psychology of Mathematics Education (PME).
Tseng, T.-N., Lee, T.-M., & Lin, J.-Y. (2025). Enhancing inquiry-based learning in human factors engineering with generative AI: A case study in industrial design education. In B. K. Smith & M. Borge (Eds.), Lecture Notes in Computer Science: Vol. 15808 LNCS (pp. 402–414). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-031-93746-0_28
Tyskerud, A., Mosvold, R., & Bjuland, R. (2023). Commognitive conflicts in mathematics teachers’ pedagogical discourse in lesson study. Mathematics Teacher Education and Development, 25(1), 81–92.
Weingarden, M., & Heyd-Metzuyanim, E. (2023). What can the realization tree assessment tool reveal about explorative classroom discussions? Journal for Research in Mathematics Education, 54(2), 97–117. https://doi.org/10.5951/jresematheduc-2020-0084
Weingarden, M., & Heyd-Metzuyanim, E. (2025). Fostering pre-service teachers’ attention to mathematical objects: The realization tree mediator as a teaching representation. Journal of Mathematics Teacher Education, 28(4), 851–877. https://doi.org/10.1007/s10857-024-09622-w
Wiyah, R. B., & Nurjanah, N. (2021). Error analysis in solving the linear equation system of three variables using Polya’s problem-solving steps. In R. Oktavia, E. Yusibani, Mailizar, Suhartono, Rahmi, Elizar, Irwandi, E. Etkina, G. Planinsic, N. Mansour, N. Idris, C. O’Donnell, K. Kagawa, R. Sheffield, V. M. Mistades, & O. Kaosaiyaporn (Eds.), Journal of Physics: Conference Series (Vol. 1882, Issue 1). IOP Publishing Ltd. https://doi.org/10.1088/1742-6596/1882/1/012084
Zhang, L., & Ma, Y. (2023). A study of the impact of project-based learning on student learning effects: a meta-analysis study. Frontiers in Psychology, 14. https://doi.org/10.3389/fpsyg.2023.1202728
Zhang, S., Cao, Y., Chan, M. C. E., & Wan, M. E. V. (2022). A comparison of meaning negotiation during collaborative problem solving in mathematics between students in China and Australia. ZDM - International Journal on Mathematics Education, 54(2), 287–302. https://doi.org/10.1007/s11858-022-01335-9
Zhang, S., Esther Chan, M. C., & Cao, Y. (2021). Examining student participation in collaborative mathematics problem solving through the lens of commognitive framework.. In M. Inprasitha, N. Changsri, & N. Boonsena (Eds.), Proceedings of the International Group for the Psychology of Mathematics Education (Vol. 1, p. 197). Psychology of Mathematics Education (PME).
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