Retractive Thinking Characteristics of Mathematics Education Students in Solving Differential Equation Problems: Evidence from a Qualitative Case Study
DOI:
https://doi.org/10.61255/jupiter.v4i3.1266Keywords:
Differential Equations, Mathematics Education Students, Metacognition, Problem Solving, Retractive ThinkingAbstract
Purpose: This study aims to identify and describe the characteristics of students’ retractive thinking in solving differential equation problems. The research focuses on how students recognize errors, evaluate solution procedures, and revise their mathematical reasoning during the problem-solving process. Methods: This qualitative study employed a case study design involving eight mathematics education students who had completed a Differential Equations course. Participants were selected through purposive sampling. Data were collected using think-aloud protocols, semi-structured interviews, and documentation of students’ written work. Data were analyzed through data condensation, data display, and conclusion drawing, supported by triangulation and member checking to ensure credibility. Findings: The findings reveal three main characteristics of retractive thinking. First, students demonstrate error recognition, shown by their ability to identify inconsistencies in their solution steps during or after problem solving. Second, students exhibit procedural re-examination, where they return to earlier steps to verify formulas and computational processes. Third, students apply solution revision strategies, ranging from correcting algebraic errors to restructuring entire solution procedures. However, retractive thinking is not consistently supported by conceptual understanding, as many revisions remain procedural rather than conceptual. This indicates a gap between procedural fluency and conceptual awareness in differential equation problem solving. Research Implications: The findings suggest the importance of integrating metacognitive strategies in teaching differential equations, particularly activities that promote self-monitoring, error analysis, and reflective revision. These strategies can strengthen students’ retractive thinking abilities. However, this study is limited by its small sample size and case study design, which restrict generalization. The findings are therefore context-bound and intended to provide analytical insight rather than statistical generalization. Originality: This study contributes to the literature by specifically identifying and describing retractive thinking characteristics in the context of differential equation problem solving, offering a process-oriented perspective beyond procedural error analysis
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References
Alam, A., & Mohanty, A. (2024). Unveiling the complexities of ‘Abstract Algebra’ in University Mathematics Education (UME): Fostering ‘Conceptualization and Understanding’ through advanced pedagogical approaches. Cogent Education, 11(1), 2355400. https://doi.org/10.1080/2331186X.2024.2355400
Bascandziev, I. (2024). Thought Experiments as an Error Detection and Correction Tool. Cognitive Science, 48(1), e13401. https://doi.org/10.1111/cogs.13401
Becker, L. C. (2026). The Wonderful World of Differential Equations. In L. C. Becker (Ed.), Ordinary Differential Equations: Concepts, Methods, and Models: A Course after Calculus 2 (pp. 1–67). Springer Nature Switzerland. https://doi.org/10.1007/978-3-032-15150-6_1
Bednar, P. M., & Welch, C. (2020). Socio-Technical Perspectives on Smart Working: Creating Meaningful and Sustainable Systems. Information Systems Frontiers, 22(2), 281–298. https://doi.org/10.1007/s10796-019-09921-1
Bhattacharya, K. (2017). Fundamentals of Qualitative Research: A Practical Guide. Routledge. https://doi.org/10.4324/9781315231747
Bleecker, D. (2018). Basic Partial Differential Equations (1st ed). CRC Press. https://doi.org/10.1201/9781351070089-1
Bressoud, D., Ghedamsi, I., Martinez-Luaces, V., & Törner, G. (2016). Teaching and Learning of Calculus. In D. Bressoud, I. Ghedamsi, V. Martinez-Luaces, & G. Törner (Eds.), Teaching and Learning of Calculus (pp. 1–37). Springer International Publishing. https://doi.org/10.1007/978-3-319-32975-8_1
Brigo, D., & Pistone, G. (2017). Dimensionality Reduction for Measure Valued Evolution Equations in Statistical Manifolds. In F. Nielsen, F. Critchley, & C. T. J. Dodson (Eds.), Computational Information Geometry: For Image and Signal Processing (pp. 217–265). Springer International Publishing. https://doi.org/10.1007/978-3-319-47058-0_10
Bryce, T. G. K., & Blown, E. J. (2024). Ausubel’s meaningful learning re-visited. Current Psychology, 43(5), 4579–4598. https://doi.org/10.1007/s12144-023-04440-4
Çini, A., Järvelä, S., Dindar, M., & Malmberg, J. (2023). How multiple levels of metacognitive awareness operate in collaborative problem solving. Metacognition and Learning, 18(3), 891–922. https://doi.org/10.1007/s11409-023-09358-7
Damayanti, N., Hartono, Subali, B., Nugroho, S. E., & Sureeporn, K. (2020). Items analysis of physics assessment based on cognitive level of high order thinking skills in bloom taxonomy. Journal of Physics: Conference Series, 1521(2), 022022. https://doi.org/10.1088/1742-6596/1521/2/022022
Demirel, M., Derman, I., & Karagedik, E. (2015). A Study on the Relationship between Reflective Thinking Skills towards Problem Solving and Attitudes towards Mathematics. Procedia - Social and Behavioral Sciences, 7th World Conference on Educational Sciences, 197, 2086–2096. https://doi.org/10.1016/j.sbspro.2015.07.326
Elahi, M., Afolaranmi, S. O., Martinez Lastra, J. L., & Perez Garcia, J. A. (2023). A comprehensive literature review of the applications of AI techniques through the lifecycle of industrial equipment. Discover Artificial Intelligence, 3(1), 43. https://doi.org/10.1007/s44163-023-00089-x
Engelbrecht, J., Llinares, S., & Borba, M. C. (2020). Transformation of the mathematics classroom with the internet. ZDM, 52(5), 825–841. https://doi.org/10.1007/s11858-020-01176-4
Fiorella, L., & Zhang, Q. (2018). Drawing Boundary Conditions for Learning by Drawing. Educational Psychology Review, 30(3), 1115–1137. https://doi.org/10.1007/s10648-018-9444-8
Ghanizadeh, A. (2017). The interplay between reflective thinking, critical thinking, self-monitoring, and academic achievement in higher education. Higher Education, 74(1), 101–114. https://doi.org/10.1007/s10734-016-0031-y
Goos, M., & Kaya, S. (2020). Understanding and promoting students’ mathematical thinking: A review of research published in ESM. Educational Studies in Mathematics, 103(1), 7–25. https://doi.org/10.1007/s10649-019-09921-7
Izzati, L. R., & Mahmudi, A. (2018). The influence of metacognition in mathematical problem solving. Journal of Physics: Conference Series, 1097(1), 012107. https://doi.org/10.1088/1742-6596/1097/1/012107
Jablonka, E. (2020). Critical Thinking in Mathematics Education. In Encyclopedia of Mathematics Education (pp. 159–163). Springer, Cham. https://doi.org/10.1007/978-3-030-15789-0_35
Jin, T. (2023). Online interactive face-to-face learning in mathematics in engineering education. European Journal of Engineering Education, 48(2), 300–320. https://doi.org/10.1080/03043797.2022.2117023
Kloeden, P. E. (2025). Numerical Methods for Stochastic Differential Equations. In International Encyclopedia of Statistical Science (pp. 1794–1796). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-69359-9_437
Lee, H.-Y., Wu, T.-T., Lin, C.-J., Wang, W.-S., & Huang, Y.-M. (2023). Integrating Computational Thinking Into Scaffolding Learning: An Innovative Approach to Enhance Science, Technology, Engineering, and Mathematics Hands-On Learning. Journal of Educational Computing Research. (pub.1165817518). https://doi.org/10.1177/07356331231211916
List, A., & Sun, Y. (2023). To Clarity and Beyond: Situating Higher-Order, Critical, and Critical-Analytic Thinking in the Literature on Learning from Multiple Texts. Educational Psychology Review, 35(2), 40. https://doi.org/10.1007/s10648-023-09756-y
Mathaba, P. N., Bayaga, A., Tîrnovan, D., & Bossé, M. J. (2024). Error Analysis in Algebra Learning: Exploring Misconceptions and Cognitive Levels. Journal on Mathematics Education, 15(2), 575–592. https://doi.org/10.22342/jme.v15i2.pp575-592
Melker, S., Gabrils, E., Villavicencio, V., Faraon, M., & Rönkkö, K. (2025). Artificial intelligence for design education: A conceptual approach to enhance students’ divergent and convergent thinking in ideation processes. International Journal of Technology and Design Education, 35(5), 1871–1899. https://doi.org/10.1007/s10798-025-09964-3
Nedaei, M., Radmehr, F., & Drake, M. (2022). Exploring undergraduate engineering students’ mathematical problem-posing: The case of integral-area relationships in integral calculus. Mathematical Thinking and Learning, 24(2), 149–175. https://doi.org/10.1080/10986065.2020.1858516
Österman, T., & Bråting, K. (2019). Dewey and mathematical practice: Revisiting the distinction between procedural and conceptual knowledge. Journal of Curriculum Studies, 51(4), 457–470. https://doi.org/10.1080/00220272.2019.1594388
Özdemir, A. Z., & Toker, Z. (2025). Analysis of distractors in mathematics questions and their potential to lead misconceptions. Thinking Skills and Creativity, 56, 101730. https://doi.org/10.1016/j.tsc.2024.101730
Rach, S., & Ufer, S. (2020). Which Prior Mathematical Knowledge Is Necessary for Study Success in the University Study Entrance Phase? Results on a New Model of Knowledge Levels Based on a Reanalysis of Data from Existing Studies. International Journal of Research in Undergraduate Mathematics Education, 6(3), 375–403. https://doi.org/10.1007/s40753-020-00112-x
Radmehr, F., & Drake, M. (2019). Revised Bloom’s taxonomy and major theories and frameworks that influence the teaching, learning, and assessment of mathematics: A comparison. International Journal of Mathematical Education in Science and Technology, 50(6), 895–920. https://doi.org/10.1080/0020739X.2018.1549336
Scherer, P., Beswick, K., DeBlois, L., Healy, L., & Opitz, E. M. (2016). Assistance of students with mathematical learning difficulties: How can research support practice? ZDM, 48(5), 633–649. https://doi.org/10.1007/s11858-016-0800-1
Schneider, M. (2020). Epistemological Obstacles in Mathematics Education. In Encyclopedia of Mathematics Education (pp. 276–279). Springer, Cham. https://doi.org/10.1007/978-3-030-15789-0_57
Shidaganti, G., Karthik, K. N., Anvith, & Kantikar, N. A. (2023). Integration of RPA and AI in Industry 4.0. In S. Bhattacharyya, J. S. Banerjee, & D. De (Eds.), Confluence of Artificial Intelligence and Robotic Process Automation (pp. 267–288). Springer Nature. https://doi.org/10.1007/978-981-19-8296-5_11
Siregar, B. H., Dewi, I., & Andriani, A. (2018). Error analysis of mathematics students who are taught by using the book of mathematics learning strategy in solving pedagogical problems based on Polya’s four-step approach. Journal of Physics: Conference Series, 970(1), 012004. https://doi.org/10.1088/1742-6596/970/1/012004
Sweller, J. (2016). Working Memory, Long-term Memory, and Instructional Design. Journal of Applied Research in Memory and Cognition, Working Memory in the Wild: Applied Research in Working Memory, 5(4), 360–367. https://doi.org/10.1016/j.jarmac.2015.12.002
Thakral, P. P., Barberio, N. M., Devitt, A. L., & Schacter, D. L. (2023). Constructive episodic retrieval processes underlying memory distortion contribute to creative thinking and everyday problem solving. Memory & Cognition, 51(5), 1125–1144. https://doi.org/10.3758/s13421-022-01377-0
Tisngati, U., & Genarsih, T. (2021). Reflective thinking process of students in completing mathematical problems based on mathematical reasoning ability. Journal of Physics: Conference Series, 1776(1), 012035. https://doi.org/10.1088/1742-6596/1776/1/012035
Vasileiou, K., Barnett, J., Thorpe, S., & Young, T. (2018). Characterising and justifying sample size sufficiency in interview-based studies: Systematic analysis of qualitative health research over a 15-year period. BMC Medical Research Methodology, 18(1), 148. https://doi.org/10.1186/s12874-018-0594-7
Verschaffel, L., Schukajlow, S., Star, J., & Van Dooren, W. (2020). Word problems in mathematics education: A survey. ZDM, 52(1), 1–16. https://doi.org/10.1007/s11858-020-01130-4
Voss, C., Johnson, M., & Godsell, J. (2016). Case research. In Research Methods for Operations Management (2nd ed.). Routledge.
Walther, J., Sochacka, N. W., Benson, L. C., Bumbaco, A. E., Kellam, N., Pawley, A. L., & Phillips, C. M. L. (2017). Qualitative Research Quality: A Collaborative Inquiry Across Multiple Methodological Perspectives. Journal of Engineering Education, 106(3), 398–430. https://doi.org/10.1002/jee.20170
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