Students' problem-solving performance in geometric extremum tasks through an integrated vector and analytical approach
DOI:
https://doi.org/10.29408/jel.v12i3.34344Keywords:
analytical tools, geometric extremum, problem-solving, vectorAbstract
This study addresses the inherent limitations of traditional Euclidean approaches to geometric extremum and distance problems, which frequently result in unsystematic and difficult-to-generalize solutions. To overcome these challenges, a structured three-step mathematical modeling framework is introduced: (1) vectorization of the geometric configuration; (2) formulation of an analytical objective function; and (3) systematic optimization via derivatives or gradients. Methodologically, the research employs a constructive modeling strategy that translates complex spatial relationships into vector-based algebraic expressions, rigorously tested across eight purposively selected examples ranging from elementary plane geometry to advanced spatial dynamics. The results indicate that this framework effectively transforms intricate geometric constraints into tractable, variable-dependent functions, allowing derivative-based optimization to yield precise extremum values while bypassing ad hoc geometric constructions. Ultimately, this research contributes to the literature by bridging synthetic geometry and mathematical analysis, offering a robust, generalizable tool that clarifies the underlying mathematical structures and fosters the development of abstract thinking and generalization capabilities for future pedagogical applications.
References
Alghadari, F., Turmudi, T., & Herman, T. (2018). The application of vector concepts on two skew lines. Journal of Physics: Conference Series, 948(1), Article 012030. https://doi.org/10.1088/1742-6596/948/1/012030
Aliu, F., Jusufi Zenku, T., Iseni, E., & Rexhepi, S. (2025). The advantage of using GeoGebra in the understanding of vectors and comparison with the classical method. International Electronic Journal of Mathematics Education, 20(2), Article em0824. https://doi.org/10.29333/iejme/16007
Anton, H., Bivens, I., & Davis, S. (2012). Calculus (10th ed.). John Wiley & Sons.
Axler, S. (2015). Linear algebra done right (3rd ed.). Springer.
Blum, W., & Leiss, D. (2011). How do students and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Modelling and applications in mathematics education (pp. 222–231). Springer. https://doi.org/10.1007/978-1-4419-0561-1_21
Bộ Giáo dục và Đào tạo. (2018). Chương trình giáo dục phổ thông môn Toán [General Education Curriculum for Mathematics]. Nhà xuất bản Giáo dục Việt Nam.
Grigorieva, E. (2025). Methods of solving stereometry problems. In Methods of solving solid geometry problems (pp. 1–152). Springer. https://doi.org/10.1007/978-3-031-86969-3_1
Hanna, G., & de Villiers, M. (Eds.). (2012). Proof and proving in mathematics education: The 19th ICMI study. Springer. https://doi.org/10.1007/978-94-007-2129-6
Leong, Y. H., Cheng, L. P., & Sze, W. Y. (2010). Teaching and learning of vectors: A review of literature. The Mathematics Educator, 12(1), 33–52.
Marsden, J. E., & Tromba, A. J. (2012). Vector calculus (6th ed.). W. H. Freeman and Company.
Saraçoğlu, H., & Kol, Ö. (2018). Evaluation of teacher candidates' knowledge about vectors. Asia-Pacific Forum on Science Learning and Teaching, 19(2), Article 14.
Shimizu, Y. (2022). Learning engagement as a moderator between self-efficacy, math anxiety, problem-solving strategy, and vector problem-solving performance. Psych, 4(4), 816–832. https://doi.org/10.3390/psych4040060
Stewart, J. (2016). Calculus: Early transcendentals (8th ed.). Cengage Learning.
Tall, D. (2013). How humans learn to think mathematically: Exploring the three worlds of mathematics. Cambridge University Press.
Watson, A., & Mason, J. (2015). Mathematics as a constructive activity: Learners generating examples. Routledge.
Zhao, Y. (2024). From tiles to worksheet: Exploring concreteness fading in learning vector addition. Education Sciences, 14(7), Article 730. https://doi.org/10.3390/educsci14070730
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Da Tien Nguyen, Anh Ngoc Nguyen, Kieu Anh Thi Phan
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Authors who publish with the Jurnal Elemen agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0).
- Authors are able to enter into separate, additional contractual arrangements for the distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
Jurnal Elemen is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License
