APOS Theory and the Teaching of Linear Algebra: Engineering Students’ Understanding of Systems of Linear Equations and Their Solution Sets
DOI:
https://doi.org/10.15649/2346075X.5861Keywords:
APOS Theory, Linear Algebra, Engineering Education, Systems of Linear Equations, Solution Set, Teaching StrategiesAbstract
Introduction. The teaching and learning of linear algebra remain a persistent challenge in engineering education, particularly regarding students’ understanding of systems of linear equations and their associated solution sets. From the perspective of APOS Theory, these concepts are understood as mental constructions that develop through a progression from Actions to Processes and, in some cases, toward Objects, involving the articulation of algebraic, matrix, and geometric representations. Objectives. This study examines the mental constructions that engineering students develop when solving systems of linear equations within a teaching strategy informed by a genetic decomposition grounded in APOS Theory. Materials and methods. A descriptive qualitative study was conducted with 26 engineering students enrolled in a Linear Algebra course. Instruction was designed according to the ACE teaching cycle and supported by tasks intended to foster transitions among mental structures. Data sources included students’ written productions and their articulated reasoning, which were analyzed using APOS Theory as an analytical framework. Results. The analysis reveals a progression in students’ understanding of systems of linear equations and their solution sets, moving from predominantly procedural and algorithmic approaches toward reasoning that coordinates algebraic, matrix, and geometric representations. Conclusions. The findings suggest that a teaching strategy grounded in APOS Theory and structured through the ACE cycle supports students’ development from operational treatments of systems of linear equations toward more structural forms of understanding. These results contribute to ongoing discussions on the design of instructional approaches that align with the conceptual demands of linear algebra in engineering education.
References
1. Stewart S, Axler S, Beezer R, Boman E, Catral M, Harel G, et al. The Linear Algebra Curriculum Study Group (LACSG 2.0) Recommendations. Notices Amer Math Soc. 2022;69(05):1. doi: https://doi.org/10.1090/noti2479.
2. Osorio Pérez JP. Resolución de problemas de sistemas de ecuaciones lineales 2x2 a partir de la comprensión matemática y la teoría APOE [Internet]. 2021 Jun 3 [citado 2025 oct 29]. Disponible en:https://repositorio.unal.edu.co/handle/unal/79609.
3. Rodríguez Jara MA, Mena Lorca A, Mena Lorca J, Vásquez Saldias P, Del Valle Leo ME. Construcción cognitiva del conjunto solución de un sistema de ecuaciones lineales con dos incógnitas. Ensciencias. 2019;37(1):71-92. doi: https://doi.org/10.5565/rev/ensciencias.2194.
4. Oliveros Tasco LF. Un modelo cognitivo de construcción de los sistemas de ecuaciones lineales [Internet]. 2019 [citado 2025 oct 30]. Disponible en: https://noesis.uis.edu.co/handle/20.500.14071/13871.
5. Costa VA, Trípoli MM. Encuentro de docentes de Ciencias Básicas [Internet]. Facultad de Ingeniería (UNLP); 2022 [citado 2025 oct 29]. Disponible en: http://sedici.unlp.edu.ar/handle/10915/133644.
6. Kú D, Trigueros M, Oktaç A. Comprensión del concepto de base de un espacio vectorial desde el punto de vista de la teoría APOE [Internet]. Educación Matemática. 2008;20(2):65-89. Disponible en: https://www.redalyc.org/articulo.oa?id=40512062004.
7. Parraguez González M, Bozt Ortiz J. Conexiones entre los conceptos de dependencia e independencia lineal de vectores y el de solución de sistemas de ecuaciones lineales en R² y R³ desde el punto de vista de los modos de pensamiento. Revista Electrónica de Investigación en Educación en Ciencias. 2012;7(1):49-72. doi: https://doi.org/10.54343/reiec.v7i1.82.
8. Trigueros M, Sánchez-Matamoros G. El aprendizaje y la enseñanza de las matemáticas en la Universidad. Avances de Investigación en Educación Matemática. 2022;(21):1. https://doi.org/10.35763/aiem21.4445.
9. Dubinsky E. Reflective Abstraction in Advanced Mathematical Thinking. En: Tall D, editor. Advanced Mathematical Thinking [Internet]. Dordrecht: Springer Netherlands; 1991 [citado 2025 jun 1]. p. 95-126. Disponible en: https://doi.org/10.1007/0-306-47203-1_7.
10. Oktaç A, Trigueros M. ¿Cómo se aprenden los conceptos de álgebra lineal? Revista Latinoamericana de Investigación en Matemática Educativa. 2010;13(4-II):373-385. Disponible en: https://www.redalyc.org/articulo.oa?id=33558827009.
11. Trigueros M, Oktaç A. Task design in APOS Theory. AIEM. 2019;15:43-55. doi: https://doi.org/10.35763/aiem.v0i15.256.
12. Possani E, Trigueros M, Preciado JG, Lozano MD. Use of models in the teaching of linear algebra. Linear Algebra Appl. 2010;432(8):2125-2140. doi: https://doi.org/10.1016/j.laa.2009.05.004.
13. Trigueros M, Possani E. Using an economics model for teaching linear algebra. Linear Algebra Appl. 2013;438(4):1779-1792. doi: https://doi.org/10.1016/j.laa.2011.04.009.
14. Parraguez M, Oktaç A. Desarrollo de un esquema del concepto espacio vectorial [Internet]. Paradígma. 2012;33(1):103-134. Disponible en: https://ve.scielo.org/scielo.php?pid=S1011-22512012000100007&script=sci_arttext.
15. Andía S, Repetto L. El lenguaje relacionado con el aprendizaje del concepto de base de un espacio vectorial utilizando textos: una mirada desde la teoría APOE. Serie Selección de Textos. 2013;3:235-250. doi: https://doi.org/10.22370/sst.2013.3.5223.
16. Betancur Sánchez A, Roa-Fuentes S, Parraguez González M. Construcciones mentales asociadas a los eigenvalores y eigenvectores: refinación de un modelo cognitivo. Avances de Investigación en Educación Matemática. 2022;(22):23-46. doi: https://doi.org/10.35763/aiem22.4005.
17. Parraguez González M, Roa-Fuentes S, Jiménez Alarcón R, Betancur Sánchez A. Mental structures and mechanisms that from a geometric perspective model and articulate the learning of eigenvalue and eigenvector in ℝ2. Revista Latinoamericana de Investigación en Matemática Educativa. 2022;25(1):63-92. doi: https://doi.org/10.12802/relime.22.2513.
18. Roa-Fuentes S, Oktaç A. Construcción de una descomposición genética: análisis teórico del concepto transformación lineal [Internet]. Revista Latinoamericana de Investigación en Matemática Educativa. 2010;13(1):89-112. Disponible en: https://www.scielo.org.mx/pdf/relime/v13n1/v13n1a5.pdf.
19. Franco Carzolio GD, Olave Baggi MI, Rosas Mendoza AM. La descomposición genética como marco de referencia para el análisis a priori de actividades. Educación Matemática. 2021;33(3):39-66. doi: https://doi.org/10.24844/em3303.02.
20. Oktaç A, Trigueros M, Romo A. APOS Theory: connecting research and teaching. For the Learning of Mathematics. 2019;39(1):33-37. Disponible en: https://eric.ed.gov/?id=EJ1211397.
21. Dubinsky E. Reflective abstraction in advanced mathematical thinking. En: Tall D, editor. Advanced Mathematical Thinking. Mathematics Education Library. Vol. 11. Dordrecht: Springer; 2002. doi:https://doi.org/10.1007/0-306-47203-1_7.
22. Arnon I, Cottrill J, Dubinsky E, Oktaç A, Roa Fuentes S, Trigueros M, et al. APOS Theory: A Framework for Research and Curriculum Development in Mathematics Education. New York, NY: Springer New York; 2014. doi: https://doi.org/10.1007/978-1-4614-7966-6.
23. Hernández Sampieri R, Fernández Collado C, Baptista Lucio P. Metodología de la investigación [Internet]. McGraw Hill España; 2014 [citado 2025 nov 7]. Disponible en: https://dialnet.unirioja.es/servlet/libro?codigo=775008.
24. Grossman SSI, Godoy JJF. Álgebra Lineal. 7a ed. México D.F.: McGraw-Hill Education; 2012.
25. Poole D. Álgebra Lineal. Una Introducción Moderna. 3a ed. México: CENGAGE Learning; 2011.
26. Anaya Puebla FJ. Ambiente virtual de aprendizaje para la construcción del concepto Sistema de Ecuaciones Lineales fundamentado en la teoría APOE [Internet]. 2020 Feb [citado 2025 jul 16]. Disponible en: https://hdl.handle.net/20.500.12371/11526.
27. Ochoviet Filgueiras TC. Sobre el concepto de solución de un sistema de ecuaciones lineales con dos incógnitas [Internet]. Montevideo, Uruguay: Instituto Politécnico Nacional, Centro de Investigación en Ciencia Aplicada y Tecnología Avanzada, Unidad Legaria; 2009 [citado 2025 oct 26]. Disponible en: https://repositoriodigital.ipn.mx/bitstream/123456789/11405/1/ochoviet_2009.pdf.
28. Rozas-Torres E, Cárcamo A, Fortuny JM. Concepciones previas sobre sistemas de ecuaciones lineales: un estudio exploratorio con estudiantes universitarios. Bolema. 2024;38:e230153. doi: https://doi.org/10.1590/1980-4415v38a230153.
29. Fox A, Quickfall A, Reiss M, Brown N, Chong SW, Courtney M, et al. Ethical guidelines for educational research. 5th ed. London: British Educational Research Association; 2024. Disponible en: https://www.bera.ac.uk/publication/ethical-guidelines-for-educational-research-fifth-edition-2024.
30. National Commission for the Protection of Human Subjects of Biomedical and Behavioral Research. The Belmont Report: ethical principles and guidelines for the protection of human subjects of research [Internet]. Washington, DC: Department of Health, Education, and Welfare; 1979 Apr 18 [citado 2025 oct 26]. Disponible en: https://www.hhs.gov/ohrp/regulations-and-policy/belmont-report/read-the-belmont-report/index.html.
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