Analyzing Students’ Cognitive Processes on Trigonometric Functions beyond 90 Degrees through the DAPIC Framework
DOI:
https://doi.org/10.32938/jpm.v7i1.9683Keywords:
Adaptive Strategies, Cognitive Process, Mathematical Representation, Trigonometric Functions, Qualitative Case StudyAbstract
Understanding trigonometric functions at angles beyond 90 degrees presents unique cognitive challenges for students, requiring the integration of conceptual, procedural, and representational knowledge. However, research exploring how students cognitively process such problems, especially within a structured framework, remains limited. This study aims to analyze students’ cognitive processes in solving trigonometric problems involving beyond-90-degree angles through the DAPIC framework (Define, Assess, Plan, Implement, Communicate), offering a novel application of DAPIC to this underexplored context. A qualitative case study approach was employed, involving six 11th-grade students from a public high school with varying cognitive levels. Data were collected through diagnostic tasks, think-aloud protocols, and semi-structured interviews, and were analyzed by mapping students’ thinking patterns, defined as the recurring sequences of cognitive moves and representation use observed across DAPIC stages. The results reveal that high-performing students demonstrated flexible shifts between symbolic, graphical, and unit circle representations and were capable of self-regulating their errors reflectively. In contrast, students with moderate and low performance encountered difficulties in identifying the quadrant of angles and understanding the periodic nature of trigonometric functions, particularly during the Assess and Plan stages. Adaptive strategies, such as visual re-checking or intuitive quadrant reasoning, emerged spontaneously but were not always effective. These findings suggest that DAPIC serves as a systematic tool for capturing the dynamics of students’ thinking processes and offers valuable insights for developing deeper instructional strategies in trigonometry, especially for topics involving beyond-90-degree angles.
References
Adler, I., Warren, S., Norris, C., & Soloway, E. (2025). Leveraging opportunities for self-regulated learning in smart learning environments. Smart Learning Environments 2025 12:1, 12(1), 1–34. https://doi.org/10.1186/S40561-024-00359-W
Ainsworth, S. (1999). The functions of multiple representations. Computers & Education, 33(2–3), 131–152. https://doi.org/10.1016/S0360-1315(99)00029-9
Ainsworth, S. E., & Scheiter, K. (2021). Learning by Drawing Visual Representations: Potential, Purposes, and Practical Implications. Current Directions in Psychological Science, 30(1), 61–67. https://doi.org/10.1177/0963721420979582
Almutairi, T. S., & Shraid, N. S. (2021). Teacher evaluation by different internal evaluators: Head of departments, teachers themselves, peers and students. International Journal of Evaluation and Research in Education (IJERE), 10(2), 588–596. https://doi.org/10.11591/IJERE.V10I2.20838
Bekene Bedada, T., & Machaba, F. (2022). The effect of GeoGebra on STEM students learning trigonometric functions. Cogent Education, 9(1). https://doi.org/10.1080/2331186X.2022.2034240
Brown, A., & Danaher, P. A. (2019). CHE Principles: facilitating authentic and dialogical semi-structured interviews in educational research. International Journal of Research & Method in Education, 42(1), 76–90. https://doi.org/10.1080/1743727X.2017.1379987
Carron, T., Domeisen Benedetti, F., Fringer, A., Fierz, K., & Peytremann-Bridevaux, I. (2023). Integrated care models in Swiss primary care: An embedded multiple case study. Journal of Evaluation in Clinical Practice, 29(6), 1025–1038. https://doi.org/10.1111/JEP.13891
Cash, P., Isaksson, O., Maier, A., & Summers, J. (2022). Sampling in design research: Eight key considerations. Design Studies, 78, 101077. https://doi.org/10.1016/J.DESTUD.2021.101077
Cirneanu, A. L., & Moldoveanu, C. E. (2024). Use of Digital Technology in Integrated Mathematics Education. Applied System Innovation 2024, Vol. 7, Page 66, 7(4), 66. https://doi.org/10.3390/ASI7040066
Cohen, L., Manion, L., & Morrison, K. (2017). Research Methods in Education. Research Methods in Education. https://doi.org/10.4324/9781315456539
Craig, S. L., McInroy, L. B., Goulden, A., & Eaton, A. D. (2021). Engaging the Senses in Qualitative Research via Multimodal Coding: Triangulating Transcript, Audio, and Video Data in a Study With Sexual and Gender Minority Youth. International Journal of Qualitative Methods, 20. https://doi.org/10.1177/16094069211013659
Dehaene, S., Al Roumi, F., Lakretz, Y., Planton, S., & Sablé-Meyer, M. (2022). Symbols and mental programs: a hypothesis about human singularity. Trends in Cognitive Sciences, 26(9), 751–766. https://doi.org/10.1016/J.TICS.2022.06.010
Gillen, A. L., Grohs, J. R., Matusovich, H. M., & Kirk, G. R. (2021). A multiple case study of an interorganizational collaboration: Exploring the first year of an industry partnership focused on middle school engineering education. Journal of Engineering Education, 110(3), 545–571. https://doi.org/10.1002/JEE.20403
Giyanti, G., & Oktaviyanthi, R. (2024). Graphing Quadratics Worksheet Performance in Optimizing Mathematical Visual Thinking: A Single Subject Research. Mosharafa: Jurnal Pendidikan Matematika, 13(2), 371–386. https://doi.org/10.31980/MOSHARAFA.V13I2.1470
Haigh, J. (2019). Mathematics in everyday life: Second edition. Mathematics in Everyday Life: Second Edition, 1–173. https://doi.org/10.1007/978-3-030-33087-3/COVER
Haleva, L., Hershkovitz, A., & Tabach, M. (2021). Students’ Activity in an Online Learning Environment for Mathematics: The Role of Thinking Levels. Journal of Educational Computing Research, 59(4), 686–712. https://doi.org/10.1177/0735633120972057
Hegedus, S. J., & Otálora, Y. (2023). Mathematical strategies and emergence of socially mediated metacognition within a multi-touch Dynamic Geometry Environment. Educational Studies in Mathematics, 112(2), 289–307. https://doi.org/10.1007/S10649-022-10170-4
Hidayat, W., Rohaeti, E. E., Hamidah, I., & Putri, R. I. I. (2023). How can android-based trigonometry learning improve the math learning process? Frontiers in Education, 7, 1101161. https://doi.org/10.3389/FEDUC.2022.1101161
Hunziker, S., & Blankenagel, M. (2024). Multiple Case Research Design. Research Design in Business and Management, 171–186. https://doi.org/10.1007/978-3-658-42739-9_9
İncelenmesi, O., Topal, K., & Yenmez, A. A. (2024). Students’ Mathematical Literacy in the Process of Teaching Problem Solving Strategies. Anadolu University Journal of Education Faculty, 8(3), 1040–1066. https://doi.org/10.34056/AUJEF.1240354
Isaev, R. A., & Podvesovskii, A. G. (2022). Cognitive Clarity of Graph Models: an Approach to Understanding the Idea and a Way to Identify Influencing Factors Based on Visual Analysis. Scientific Visualization, 14(4), 38–51. https://doi.org/10.26583/SV.14.4.04
Kholid, M. N., Sa’Dijah, C., Hidayanto, E., & Permadi, H. (2022). Students’ reflective thinking pattern changes and characteristics of problem solving. Reflective Practice, 23(3), 319–341. https://doi.org/10.1080/14623943.2021.2025353
Li, S., Huang, X., Wang, T., Zheng, J., & Lajoie, S. P. (2024). Using text mining and machine learning to predict reasoning activities from think-aloud transcripts in computer assisted learning. Journal of Computing in Higher Education, 37(1), 477–496. https://doi.org/10.1007/S12528-024-09404-6
Liu, X., Li, A., Sun, J., & Lu, Z. (2023). Trigonometric projection statistics histograms for 3D local feature representation and shape description. Pattern Recognition, 143, 109727. https://doi.org/10.1016/J.PATCOG.2023.109727
Meier, S. L., Hovde, R. L., & Meier, R. L. (1996). Problem Solving: Teachers’ Perceptions, Content Area Models, and Interdisciplinary Connections. School Science and Mathematics, 96(5), 230–237. https://doi.org/10.1111/J.1949-8594.1996.TB10234.X
Modabbernia, N., Yan, X., & Zazkis, R. (2023). When algebra is not enough: a dialogue on the composition of even and odd functions. Educational Studies in Mathematics, 112(3), 397–414. https://doi.org/10.1007/S10649-022-10189-7
Moore, S. L., Howard, C. D., Boling, E., Leary, H., & Hodges, C. B. (2024). Research methods for design knowledge: clarifying definitions, characteristics, and areas of confusion. Educational Technology Research and Development, 72(5), 2679–2703. https://doi.org/10.1007/S11423-023-10271-8
Nardi, E. (2014). Reflections on Visualization in Mathematics and in Mathematics Education. 193–220. https://doi.org/10.1007/978-94-007-7473-5_12
Ndari, W., Suyatno, Sukirman, & Mahmudah, F. N. (2023). Implementation of the Merdeka Curriculum and Its Challenges. European Journal of Education and Pedagogy, 4(3), 111–116. https://doi.org/10.24018/EJEDU.2023.4.3.648
Newton, G., & Martin, E. (2013). Blooming, SOLO Taxonomy, and Phenomenography as Assessment Strategies in Undergraduate Science Education. Journal of College Science Teaching, 43(2), 78–90. https://doi.org/10.2505/4/JCST13_043_02_78
Ngu, B. H., & Phan, H. P. (2023). Differential instructional effectiveness: overcoming the challenge of learning to solve trigonometry problems that involved algebraic transformation skills. European Journal of Psychology of Education, 38(4), 1505–1525. https://doi.org/10.1007/S10212-022-00670-5
Nielsen, M. E., & Bostic, J. D. (2018). Connecting and Using Multiple Representations. Mathematics Teaching in the Middle School, 23(7), 386–393. https://doi.org/10.5951/MATHTEACMIDDSCHO.23.7.0386
Nordlander, M. C. (2022). Lifting the understanding of trigonometric limits from procedural towards conceptual. International Journal of Mathematical Education in Science and Technology, 53(11), 2973–2986. https://doi.org/10.1080/0020739X.2021.1927226
Obeng, B. A., Banson, G. M., Owusu, E., & Owusu, R. (2024). Analysis of senior high school students’ errors in solving trigonometry. Cogent Education, 11(1). https://doi.org/10.1080/2331186X.2024.2385119
Parameswaran, U. D., Ozawa-Kirk, J. L., & Latendresse, G. (2020). To live (code) or to not: A new method for coding in qualitative research. Qualitative Social Work, 19(4), 630–644. https://doi.org/10.1177/1473325019840394
Planas-Lladó, A., Feliu, L., Arbat, G., Pujol, J., Suñol, J. J., Castro, F., & Martí, C. (2021). An analysis of teamwork based on self and peer evaluation in higher education. Assessment and Evaluation in Higher Education, 46(2), 191–207. https://doi.org/10.1080/02602938.2020.1763254
Popkewitz, T. S. (2022). Comparative Reasoning, Fabrication, and International Education Assessments: Desires about Nations, Society, and Populations. International Journal of Educational Research, 112, 101940. https://doi.org/10.1016/J.IJER.2022.101940
Pratt, S. M., & Hodges, T. S. (2023). The Think-Aloud Observation Protocol: Developing a Literacy Instruction Tool for Teacher Reflection and Growth. Reading Psychology, 44(1), 1–31. https://doi.org/10.1080/02702711.2022.2126572
Pulferer, H. S., Guan, C., & Müller-Putz, G. R. (2024). Investigating multilevel cognitive processing within error-free and error-prone feedback conditions in executed and observed car driving. Frontiers in Human Neuroscience , 18, 1383956. https://doi.org/10.3389/FNHUM.2024.1383956
Ramírez-Uclés, R., & Ruiz-Hidalgo, J. F. (2022). Reasoning, Representing, and Generalizing in Geometric Proof Problems among 8th Grade Talented Students. Mathematics 2022, Vol. 10, Page 789, 10(5), 789. https://doi.org/10.3390/MATH10050789
Resnick, I., & Lowrie, T. (2023). Spatial Reasoning Supports Preschool Numeracy: Findings From a Large-Scale Nationally Representative Randomized Control Trial. Journal for Research in Mathematics Education, 54(5), 295–316. https://doi.org/10.5951/JRESEMATHEDUC-2022-0051
Robinson, R. S. (2023). Purposive Sampling. Encyclopedia of Quality of Life and Well-Being Research, 5645–5647. https://doi.org/10.1007/978-3-031-17299-1_2337
Santos-Trigo, M. (2020). Problem-Solving in Mathematics Education. Encyclopedia of Mathematics Education, 686–693. https://doi.org/10.1007/978-3-030-15789-0_129
Schraw, G., & Moshman, D. (1995). Metacognitive theories. Educational Psychology Review, 7(4), 351–371. https://doi.org/10.1007/BF02212307
Sekgoma, A., & Salani, E. (2023). Analyzing Common Trigonometric Errors Among First-Year Primary School Student Teachers at The University of Botswana. European Journal of Education Studies, 10(12). https://doi.org/10.46827/EJES.V10I12.5128
Sholahudin, U., & Oktaviyanthi, R. (2025). Impact of Phet-Based Vs. Paper-Based Trigonometric Worksheets on Students’ Adaptive Thinking Skills. International Journal of Pedagogy and Teacher Education, 9(1), 98–118. https://doi.org/10.20961/IJPTE.V9I1.98470
Sweller, J. (2011). Cognitive Load Theory. Psychology of Learning and Motivation - Advances in Research and Theory, 55, 37–76. https://doi.org/10.1016/B978-0-12-387691-1.00002-8
Sweller, J. (2020). Cognitive load theory and educational technology. Educational Technology Research and Development, 68(1), 1–16. https://link.springer.com/article/10.1007/s11423-019-09701-3
Sweller, J. (2022). The role of evolutionary psychology in our understanding of human cognition: Consequences for cognitive load theory and instructional procedures. Educational Psychology Review, 34(4), 2229–2241. https://doi.org/10.1007/s10648-021-09647-0
Tallman, M. A. (2021). Investigating the transformation of a secondary teacher’s knowledge of trigonometric functions. The Journal of Mathematical Behavior, 62, 100869. https://doi.org/10.1016/J.JMATHB.2021.100869
von Thienen, J. P. A., Weinstein, T. J., & Meinel, C. (2023). Creative metacognition in design thinking: exploring theories, educational practices, and their implications for measurement. Frontiers in Psychology, 14, 1157001. https://doi.org/10.3389/FPSYG.2023.1157001/BIBTEX
Wallsten, T. S. (1980). Cognitive Processes in Choice and Decision Behavior. Cognitive Processes in Choice and Decision Behavior, 1–285. https://doi.org/10.4324/9781003469544
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