The Gauss-Jordan Method as a Representation Framework for HOTS in Rotational Dynamics
Developing higher-order thinking skills (HOTS) in physics requires learning experiences that encourage students to construct mathematical representations rather than rely solely on procedural problem solving. Although the Gauss–Jordan method has frequently been introduced as a computational technique for solving systems of linear equations, its potential as a mathematical representation framework has received little attention. This study investigated whether integrating the Gauss–Jordan elimination method into rotational dynamics instruction could support the development of students' higher-order thinking skills through mathematical representation. A quasi-experimental study involving 68 eleventh-grade students at MAN 1 Jember, Indonesia, used a nonequivalent control-class design. Students in the experimental class learned rotational dynamics through a representation-oriented Gauss–Jordan approach, whereas those in the control class received conventional instruction based on substitution and elimination methods. Students' mathematical thinking was assessed using five essay questions covering reasoning, generalizing, critical thinking, problem solving, and communicating. The results showed greater improvement in the experimental class (N-gain = 0.63) than in the control class (N-gain = 0.26). The difference between classes was statistically significant (Mann–Whitney, p < 0.001) with a large effect size (r = 0.78). The strongest gains were observed in generalizing, reasoning, and critical thinking. Analysis of students' written responses further indicated that learning through the Gauss–Jordan method helped students translate physical situations into mathematical models and recognize explicit relationships among variables. These findings suggest that the educational value of the Gauss–Jordan method extends beyond solving systems of linear equations, highlighting its role as a mathematical representation framework that supports higher-order thinking in physics learning.
Keywords: mathematical representation, Gauss–Jordan Method, mathematical thinking, Higher-order thinking skills, Rotational dynamics.
Ahlamy, S. M., Susilawati, S., Padilah, H. N., & Izzatulhaq, A. (2022). Analisis pembelajaran fisika materi dinamika rotasi: studi literatur publikasi ilmiah [analysis of physics learning of rotational dynamics material: literature study of scientific publications]. Mitra Pilar: Jurnal Pendidikan, Inovasi, Dan Terapan Teknologi, 1(2), 213–230. https://doi.org/10.58797/pilar.0102.11
Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183–198. https://doi.org/10.1016/j.learninstruc.2006.03.001
Álvarez-Tinajero, N., Basantes-Andrade, A., Ayala-Vásquez, O., Pereira-González, L.-M., & Arciniegas-Romero, G. (2026). Mathematical competencies and critical thinking in secondary education: A PRISMA-Based Systematic Review (2019–2025). F1000Research, 14, 1407. https://doi.org/10.12688/f1000research.173462.2
Castillo, D., Carrión, J., Chamba, C., Jiménez-Gaona, Y., Rodríguez-Álvarez, M. J., & Lakshminarayanan, V. (2025). Didactic strategies for conceptual understanding and motivation in university mathematics: a systematic review. Frontiers in Education, 10, 1–18. https://doi.org/10.3389/feduc.2025.1536470
Creswell, J. W. (2014). Research design: qualitative, quantitative, and mixed methods approaches (Fourth). SAGE Publications.
Creswell, J., & Creswell, D. (2023). Research Design: Qualitative, Quantitative, and Mixed Methods Approaches. In Sage: Sixth Edition (Sixth Edition, Number 1). https://medium.com/@arifwicaksanaa/pengertian-use-case-a7e576e1b6bf
Devlin, K. (2021). Teaching mathematics as a way of thinking – not calculating. Eesti Haridusteaduste Ajakiri. Estonian Journal of Education, 9(1), 33–59. https://doi.org/10.12697/eha.2021.9.1.02b
Dick, A. S., Comer, J. S., Bayat, M., Curtis, M., Hayes, T., Pruden, S. M., Hawes, S., Gonzalez, R., Laird, A. R., & Graziano, P. (2025). Meaningful associations redux: quantifying and interpreting effect size in the context of the adolescent brain and cognitive development study. Developmental Cognitive Neuroscience, 101630. https://doi.org/10.1016/j.dcn.2025.101630
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1), 103–131. https://doi.org/10.1007/s10649-006-0400-z
English, L. D. (2023). Ways of thinking in STEM-based problem solving. ZDM - Mathematics Education, 55(7), 1219–1230. https://doi.org/10.1007/s11858-023-01474-7
Fabrigar, L. R., Vaughan-Johnston, T. I., & Wegener, D. T. (2024). Quasi-experimental designs. In HT Reis, T. West, & CM Judd (Eds.), Handbook of research methods in social and personality psychology (p. 696). Cambridge University Press.
Fitriyah, I. M., Arrifadah, Y., & Lailiyah, S. (2021). Semiotics of mathematics problem-solving in Mason’s generalization. Jurnal Riset Pendidikan Matematika, 8(1), 1–21. https://doi.org/10.21831/jrpm.v8i1.39621
Fraenkel, J. R., Wallen, N. E., & Hyun, H. H. (2012). How to design and evaluate research in education (Eighth Edi, Number 0). McGraw-Hill.
Ghimire, A., & Singh, C. (2025). Perspectives of women and men students and faculty on conceptual and quantitative problem-solving in physics from introductory to graduate levels. Education Sciences, 15(12), 1–36. https://doi.org/10.3390/educsci15121602
González‐pérez, L. I., & Ramírez‐montoya, M. S. (2022). Components of education 4.0 in 21st century skills frameworks: systematic review. Sustainability (Switzerland), 14(3), 1–31. https://doi.org/10.3390/su14031493
Hake, R. R. (1998). Interactive-engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses. American Journal of Physics, 66(1), 64–74. https://doi.org/10.1119/1.18809
Harahap, S. D., Ahmad, M., Sabri, & Lubis, I. S. (2023). Implementation of the connected mathematics project learning model on students’ mathematical problem solving abilities sinar. Jurnal Pendidikan MIPA, 24(3), 556–570. http://dx.doi.org/10.23960/jpmipa/v24i3.pp556-570
Hartmann, L. M., Krawitz, J., & Schukajlow, S. (2021). Create your own problem! When given descriptions of real-world situations, do students pose and solve modelling problems? ZDM - Mathematics Education, 53(4), 919–935. https://doi.org/10.1007/s11858-021-01224-7
Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics. (pp. 65–97). Macmillan Publishing Co, Inc.
Leng, M. (2021). Models, structures, and the explanatory role of mathematics in empirical science. Synthese, 199(3–4), 10415–10440. https://doi.org/10.1007/s11229-021-03253-x
Lu, K., Yang, H. H., Shi, Y., & Wang, X. (2021). Examining the key influencing factors on college students’ higher-order thinking skills in the smart classroom environment. International Journal of Educational Technology in Higher Education, 18(1), 1–13. https://doi.org/10.1186/s41239-020-00238-7
Ndraha, I. S., Mendrofa, R. N., & Lase, R. (2022). Analisis hubungan minat belajar dengan hasil belajar matematika [analysis of the relationship between learning interest and mathematics learning outcomes]. Educativo: Jurnal Pendidikan, 1(2), 672–681. https://doi.org/10.56248/educativo.v1i2.92
Neumann, I., Sorge, S., Hoth, J., Lindmeier, A., Neumann, K., & Heinze, A. (2021). Synergy effects in learning? The influence of mathematics as a second subject on teacher students’ physics content knowledge. Studies in Higher Education, 46(10), 2035–2046. https://doi.org/10.1080/03075079.2021.1953335
OECD. (2023). PISA 2022 Results. OECD Publishing, I, 10. https://www.oecd.org/publication/pisa-2022-results/country-notes/malaysia-1dbe2061/
Palmgren, E., & Rasa, T. (2024). Modelling roles of mathematics in physics: perspectives for physics education. Science and Education, 33(2), 365–382. https://doi.org/10.1007/s11191-022-00393-5
Peña-Ayala, A. (2021). A learning design cooperative framework to instill 21st century education. Telematics and Informatics, 62, 101632. https://doi.org/10.1016/j.tele.2021.101632
Pols, C. F. J., & Dekkers, P. J. J. M. (2024). Redesigning a first year physics lab course on the basis of the procedural and conceptual knowledge in science model. Physical Review Physics Education Research, 20(1), 10117. https://doi.org/10.1103/PhysRevPhysEducRes.20.010117
Polya, G. (2014). How to Solve It: A New Aspect of Mathematical Method (p. 288). Princeton University Press.
Sugiyono. (2013). Metode penelitian kuantitatif, kualitatif dan R&D. In Alfabeta (p. 330). ALFABETA.
Rasyid, F., Jumadi, J., & Hawur, P. K. (2025). How to improve multiple representation skills in physics learning: a systematic literature review. Journal of Science Learning, 8(1), 25–40. https://doi.org/10.17509/jsl.v8i1.76514
Redish, E. F. (2003). Teaching Physics with the Physics Suite. In American Journal of Physics (p. 216). Wiley. https://doi.org/10.1119/1.1691552
Rexigel, E., Kuhn, J., Becker, S., & Malone, S. (2024). The more the better? a systematic review and meta-analysis of the benefits of more than two external representations in STEM Education. In Educational Psychology Review (Vol. 124, Number 36). Springer US. https://doi.org/10.1007/s10648-024-09958-y
Schoenfeld, A. H. (1985). Mathematical Problem Solving (p. 403). Academic Press Inc. https://vjs.ac.vn/index.php/jse/article/view/13829
Serpe, A. (2023). Applied mathematical modelling in the physics problem-solving classroom. International Conference on Numerical Computations: Theory and Algorithms, 193–206. https://doi.org/10.1007/978-3-031-81247-7_15
Shadish, W. R., Cook, T, D., & Camp, D. T. (2002). Experimental and quasi-experimental designs for generalized causal inference. In Experimental and quasi-experimental design for causal inference (p. 609). Houghton mifflin company. https://moodle2.units.it/pluginfile.php/132646/mod_resource/content/1/Estratto_ShadishCookCampbellExperimental2002.pdf
Shidqiya, A. I., & Sukestiyarno, S. (2022). Analysis of students’ mathematical thinking ability in terms of self efficacy. Unnes Journal of Mathematics Education, 11(3), 272–281. https://doi.org/10.15294/ujme.v11i3.58772
Sirianansopa, K. (2024). Evaluating students’ learning achievements using the formative assessment technique: a retrospective study. BMC Medical Education, 24(1), 1–7. https://doi.org/10.1186/s12909-024-06347-5
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20–26. https://doi.org/10.5951/MTMS.12.2.0088
Supriadi, B., Arsita, M., Ni'mah, S., Wardhany, M. K. K., Afidah, Z., & Kinanti, A. Z. L. (2025). The effectiveness of Cramer's rule on the system of linear equations (LES) of 2-loop electrical circuits in improving mathematical thinking ability and learning outcomes. Unnes Science Education Journal, 14(1), 153–163. https://doi.org/10.15294/usej.v13i1.20924
Supriadi, B., Maryani, M., Putri, F. A., Zahro, R. F., Rohman, M. F., Sari, P. A. E., & Nafisah, N. (2026). The effect of algorithmic gauss–jordan method on mathematical reasoning and learning outcomes in dynamic electricity. Jurnal Ilmiah Ilmu Terapan Universitas Jambi, 10(1), 181–200. https://doi.org/10.22437/jiituj.v10i1.52626
Syaiful, S., Mukminin, A., & Puspayanti, P. (2025). Using APOS theory in learning basic mathematics to promote students’ mathematical critical thinking skills. Discover Education, 4(1), 1–19. https://doi.org/10.1007/s44217-025-00863-2
Szabo, Z. K., Körtesi, P., Guncaga, J., Szabo, D., & Neag, R. (2020). Examples of problem-solving strategies in mathematics education supporting the sustainability of 21st-century skills. Sustainability (Switzerland), 12(23), 1–28. https://doi.org/10.3390/su122310113
Tasarib, A., Rosli, R., & Rambely, A. S. (2025). Impacts and challenges of mathematical modelling activities on students’ learning development: A systematic literature review. Eurasia Journal of Mathematics, Science and Technology Education, 21(5). https://doi.org/10.29333/ejmste/16398
Tong, D. H., Uyen, B. P., & Quoc, N. V. A. (2021). The improvement of 10th students’ mathematical communication skills through learning ellipse topics. Heliyon, 7, 08282. https://doi.org/10.1016/j.heliyon.2021.e08282
Tong, T., Pi, F., Zheng, S., Zhong, Y., Lin, X., & Wei, Y. (2025). Exploring the effect of mathematics skills on student performance in physics problem-solving: a structural equation modeling analysis. Research in Science Education, 55(3), 489–509. https://doi.org/10.1007/s11165-024-10201-5
Tupulu, N., Jamiah, Y., Rustam, R., & Fitriawan, D. (2023). Pengembangan kemampuan berpikir matematis untuk penguatan disposisi matematis melalui kolaborasi antara siswa dan guru [developing mathematical thinking skills to strengthen mathematical dispositions through collaboration between students and teachers]. Media Pendidikan Matematika, 11(1), 131. https://doi.org/10.33394/mpm.v11i1.7853
Wang, Q., & Abdullah, A. H. (2024). Enhancing students’ critical thinking through mathematics in higher education: a systemic review. SAGE Open, 14(3), 1–15. https://doi.org/10.1177/21582440241275651
Wasis, Widodo, W., Sunarti, T., Setyarsih, W., Jauhariyah, M. N. R., & Zainuddin, A. (2023). The relationship between multiple representational skills and understanding of physics concepts in the pre-service science teacher. Journal of Physics: Conference Series, 2623(1). https://doi.org/10.1088/1742-6596/2623/1/012031
Wei, X., Saab, N., & Admiraal, W. (2023). Do learners share the same perceived learning outcomes in MOOCs? Identifying the role of motivation, perceived learning support, learning engagement, and self-regulated learning strategies. Internet and Higher Education, 56(August 2022), 100880. https://doi.org/10.1016/j.iheduc.2022.100880
Woitkowski, D. (2020). Tracing physics content knowledge gains using content complexity levels. International Journal of Science Education, 42(10), 1585–1608.
Wu, H., Saleh, A. R., & Molnár, G. (2022). Inductive and combinatorial reasoning in international educational context: assessment, measurement invariance, and latent mean differences. Asia Pacific Education Review, 23(2), 297–310. https://doi.org/10.1007/s12564-022-09750-z
Xu, X., Shi, Z., Bos, N. A., & Wu, H. (2023). Student engagement and learning outcomes: an empirical study applying a four-dimensional framework. Medical Education Online, 28(1), 1–13. https://doi.org/10.1080/10872981.2023.2268347
Zhao, F. F., & Schuchardt, A. (2021). Development of the Sci-math Sensemaking Framework: categorizing sensemaking of mathematical equations in science. International Journal of STEM Education, 8(10), 1–18. https://doi.org/10.1186/s40594-020-00264-x
Refbacks
- There are currently no refbacks.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.