A hypothetical learning trajectory for vector subspace: An example with engineering students

Andrea Cárcamo; Claudio Fuentealba

doi:10.29333/ejmste/18995

Full Text (PDF)

Andrea Cárcamo ¹ , Claudio Fuentealba ¹ ^*

More Detail

¹ Universidad Austral de Chile, Valdivia, CHILE^* Corresponding Author

Abstract

The concept of vector subspace presents significant challenges in linear algebra courses due to its abstract nature and the required mathematical formalism. This research characterizes how first-year engineering students construct the concept of a vector subspace through a hypothetical learning trajectory (HLT) grounded in the emergent models design heuristic and inclusive mathematics education. Two cycles of teaching experiments were implemented with 26 and 30 students, respectively, using design-based research. The results show conceptual progression through four milestones: identifying properties in solution sets of homogeneous systems of linear equations, recognizing these as vector subspaces, establishing verification procedures, and autonomously applying the test theorem in various contexts. The integrated inclusive strategies (activation of prior knowledge, collaborative learning, comparative tables, and differentiation) supported students with diverse mathematical backgrounds in moving from situational models to formal reasoning. This study provides empirical evidence on the design of inclusive HLTs for abstract algebraic concepts in university mathematics education.

Keywords

hypothetical learning trajectory
emergent models heuristic
inclusive mathematics education
subspace
homogeneous systems of linear equations

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, Volume 22, Issue 8, August 2026, Article No: em2883

https://doi.org/10.29333/ejmste/18995

Publication date: 22 Jul 2026

Article Views: 19

Article Downloads: 11

Open Access References How to cite this article

References

Abtahi, Y., & Planas, N. (2024). Mathematics teaching and teacher education against marginalisation, or towards equity, diversity and inclusion. ZDM Mathematics Education, 56, 307-318. https://doi.org/10.1007/s11858-024-01602-x
Alayont, F., & Schlicker, S. (2019). Linear algebra and applications: An inquiry-based approach. Grand Valley State University Libraries.
Alsina, Á., & Franco, J. (2020). Promoviendo la educación matemática inclusiva desde el enfoque de los itinerarios de enseñanza de las matemáticas: El caso de las fracciones [Promoting inclusive mathematics education from the perspective of mathematics teaching pathways: The case of fractions]. APEduC Revista-Investigação e Práticas em Educação em Ciências, Matemática e Tecnologia, 1(2), 13-29.
Alting, A., & Walser, A. (2006). The influence of mathematics preparation on the retention and academic achievement of underrepresented engineering students. In Proceedings of the 2006 American Society for Engineering Educations Annual Conference and Exposition (pp. 1-18). ASEE. https://doi.org/10.18260/1-2--665
Andrews-Larson, C., Wawro, M., & Zandieh, M. (2017). A hypothetical learning trajectory for conceptualizing matrices as linear transformations. International Journal of Mathematical Education in Science and Technology, 48(6), 809-829. https://doi.org/10.1080/0020739X.2016.1276225
Bakker, A., & van Eerde, D. (2015). An introduction to design-based research with an example from statistics education. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education: Examples of methodology and methods (pp. 429-466). Springer. https://doi.org/10.1007/978-94-017-9181-6_16
Boghossian, P. (2006). Behaviorism, constructivism, and Socratic pedagogy. Educational Philosophy and Theory, 38(6), 713-722. https://doi.org/10.1111/j.1469-5812.2006.00226.x
Britton, S., & Henderson, J. (2009). Linear algebra revisited: An attempt to understand students’ conceptual difficulties. International Journal of Mathematical Education in Science and Technology, 40(7), 963-974. https://doi.org/10.1080/00207390903206114
Caglayan, G. (2019). Is it a subspace or not? Making sense of subspaces of vector spaces in a technology-assisted learning environment. ZDM Mathematics Education, 51, 1215-1237. https://doi.org/10.1007/s11858-019-01101-4
Cárcamo, A., & Fuentealba, C. (2023). Un modelo para la construcción de trayectorias hipotéticas de aprendizaje preliminaries [A model for constructing preliminary hypothetical learning trajectories]. Bolema Boletim de Educação Matemática 37(76), 577-601. https://doi.org/10.1590/1980-4415v37n76a10
Cárcamo, A., Barquero, B., Badillo, E., & Fuentealba, C. (2023). Errores en subespacios vectoriales: Un estudio exploratorio con estudiantes de primer año de ingeniería [Errors in vector subspaces: An exploratory study with first-year engineering students]. Formación Universitaria, 16(2), 13-24. https://doi.org/10.4067/S0718-50062023000200013
Cárcamo, A., Fortuny, J. M., & Fuentealba, C. (2021). Las trayectorias hipotéticas de aprendizaje: Un ejemplo en un curso de álgebra lineal [Hypothetical learning trajectories: An example in a linear algebra course]. Enseñanza de las Ciencias, 39(1), 45-64. https://doi.org/10.5565/rev/ensciencias.2857
Chandler, F., & Taylor, D. (2008). Constructivist learning in undergraduate linear algebra. PRIMUS: Problems, Resources and Issues in Mathematics Undergraduate Studies, 18(3), 299-303. https://doi.org/10.1080/10511970701335171
Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (pp. 68-95). Lawrence Erlbaum Associates.
Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13. https://doi.org/10.3102/0013189X032001009
Dierdorp, A., Bakker, A., Eijkelhof, H., & van Maanen, J. (2011). Authentic practices as contexts for learning to draw inferences beyond correlated data. Mathematical Thinking and Learning, 13(1-2), 132-151. https://doi.org/10.1080/10986065.2011.538294
Dorier, J. L. (1995). A general outline of the genesis of vector space theory. Historia Mathematica, 22(3), 227-261. https://doi.org/10.1006/hmat.1995.1024
Espinoza, C. (2023). Percepción de los estudiantes de pedagogía en matemática sobre la educación inclusiva [Perceptions of mathematics pedagogy students regarding inclusive education]. In T. Loureiro (Ed.), Educação: Saberes em movimento, saberes que movimentam V (pp. 77-87). Artemis. https://doi.org/10.37572/EdArt_2807238807
Florian, L., & Black-Hawkins, K. (2011). Exploring inclusive pedagogy. British Educational Research Journal, 37(5), 813-828. https://doi.org/10.1080/01411926.2010.501096
Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155-177. https://doi.org/10.1207/s15327833mtl0102_4
Gravemeijer, K. (2004). Learning trajectories and local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105-128.
Gravemeijer, K. (2020). A socio-constructivist elaboration of realistic mathematics education. In M. V. den Heuvel-Panhuizen (Ed.), National reflections on the Netherlands didactics of mathematics: Teaching and learning in the context of realistic mathematics education (pp. 217-233). Springer.
Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1-3), 111-129. https://doi.org/10.1023/A:1003749919816
Gravemeijer, K., & van Eerde, D. (2009). Design research as a means for building a knowledge base for teachers and teaching in mathematics education. The Elementary School Journal, 109(5), 510-524. https://doi.org/10.1086/596999
Grossman, S., & Flores, I. (2019). Álgebra lineal [Linear algebra]. McGraw-Hill.
Jackson-Summers, A. G., Mrakovcich, K. L., Gray, J. P., Fleischmann, C. M., Emami, T., & Page, E. J. (2024). A systematic review of inclusive pedagogical research using the CIRTL inclusive pedagogy framework: Multi-disciplinary and STEM perspectives, current trends and a research agenda. Discover Education, 3, Article 30. https://doi.org/10.1007/s44217-024-00093-y
Kosunen, S. (2018). Access to higher education in Finland: Emerging processes of hidden privatization. Nordic Journal of Studies in Educational Policy, 4(2), 67-77. https://doi.org/10.1080/20020317.2018.1487756
Larson, R. (2014). Fundamentos de algebra lineal [Fundamentals of linear algebra]. Cengage Learning.
Laubenstein, D., Guthöhrlein, K., Lindmeier, C., Scheer, D., & Sponholz, D. (2019). The ‘learning office’ as an approach for inclusive education in mathematics: Opportunities and challenges. In D. Kollosche, R. Marcone, M. Knigge, M. G. Penteado, & O. Skovsmose (Eds.), Inclusive mathematics education: State-of-the-art research from Brazil and Germany (pp. 107-121). Springer. https://doi.org/10.1007/978-3-030-11518-0_9
Lay, D. (2012). Álgebra lineal elemental y sus aplicaciones [Elementary linear algebra and its applications]. Pearson.
Lockhart, M. E., Hakim, N., Chilukuri, V., Champagne, J., Rambo-Hernandez, K. E., & Hensel, R. A. (2023, June). Work in progress: Uncovering links between mathematical preparation and engineering persistence [Paper presentation]. The 2023 ASEE Annual Conference & Exposition.
Marchesi, Á., Durán, D., & Giné, C. (2009). Guía para la reflexión y valoración de prácticas inclusivas [Guide for reflection and evaluation of inclusive practices]. OEI - UNESCO - MAPFRE.
Mutambara, L. H. N., & Bansilal, S. (2018). Dealing with the abstraction of vector space concepts. In S. Stewart, C. Andrews-Larson, A. Berman, & M. Zandieh (Eds.), Challenges and strategies in teaching linear algebra (pp. 147-173). Springer. https://doi.org/10.1007/978-3-319-66811-6_7
Mutambara, L. H. N., & Bansilal, S. (2019). An exploratory study on the understanding of the vector subspace concept. African Journal of Research in Mathematics, Science and Technology Education, 23(1), 14-26. https://doi.org/10.1080/18117295.2018.1564496
Poole, D. (2011). Álgebra lineal. Una introducción moderna [Linear algebra: A modern introduction]. Cengage Learning.
Roos, H. (2019). Inclusion in mathematics education: An ideology, a way of teaching, or both? Educational Studies in Mathematics, 100(1), 25-41. https://doi.org/10.1007/s10649-018-9854-z
Scherer, P., Beswick, K., DeBlois, L., Healy, L., & Moser Opotiz, E. (2016). Assistance of students with mathematical learning difficulties, how can research support practice? ZDM Mathematics Education, 48, 633-649. https://doi.org/10.1007/s11858-016-0800-1
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145. https://doi.org/10.5951/jresematheduc.26.2.0114
Slavin, R. E. (1995). Cooperative learning: Theory, research, and practice (2nd ed.). Allyn & Bacon.
Stewart, S., & Thomas, M. O. (2010). Student learning of basis, span and linear independence in linear algebra. International Journal of Mathematical Education in Science and Technology, 41(2), 173-188. https://doi.org/10.1080/00207390903399620
Strang, G. ( 2007). Algebra lineal y sus aplicaciones [Linear algebra and its applications]. EParaninfo.
Tomlinson, C. A. (2014). The differentiated classroom: Responding to the needs of all learners (2nd ed.). ASCD.
UNESCO-IESALC. (2020). COVID-19 y educación superior: De los efectos inmediatos al día después. análisis de impactos, respuesta y recomendaciones [COVID-19 and higher education: From immediate effects to the day after. Impact analysis, response, and recommendations]. UNESCO. https://unesdoc.unesco.org/ark:/48223/pf0000375125
Wawro, M., Sweeney, G. F., & Rabin, J. M. (2011). Subspace in linear algebra: Investigating students’ concept images and interactions with the formal definition. Educational Studies in Mathematics, 78, 1-19. https://doi.org/10.1007/s10649-011-9307-4
Zandieh, M., & Rasmussen, C. (2010). Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning. The Journal of Mathematical Behavior, 29(2), 57-75. https://doi.org/10.1016/j.jmathb.2010.01.001

How to cite this article

APA

Cárcamo, A., & Fuentealba, C. (2026). A hypothetical learning trajectory for vector subspace: An example with engineering students. Eurasia Journal of Mathematics, Science and Technology Education, 22(8), em2883. https://doi.org/10.29333/ejmste/18995

Vancouver

Cárcamo A, Fuentealba C. A hypothetical learning trajectory for vector subspace: An example with engineering students. EURASIA J Math Sci Tech Ed. 2026;22(8):em2883. https://doi.org/10.29333/ejmste/18995

AMA

Cárcamo A, Fuentealba C. A hypothetical learning trajectory for vector subspace: An example with engineering students. EURASIA J Math Sci Tech Ed. 2026;22(8), em2883. https://doi.org/10.29333/ejmste/18995

Chicago

Cárcamo, Andrea, and Claudio Fuentealba. "A hypothetical learning trajectory for vector subspace: An example with engineering students". Eurasia Journal of Mathematics, Science and Technology Education 2026 22 no. 8 (2026): em2883. https://doi.org/10.29333/ejmste/18995

Harvard

Cárcamo, A., and Fuentealba, C. (2026). A hypothetical learning trajectory for vector subspace: An example with engineering students. Eurasia Journal of Mathematics, Science and Technology Education, 22(8), em2883. https://doi.org/10.29333/ejmste/18995

MLA

Cárcamo, Andrea et al. "A hypothetical learning trajectory for vector subspace: An example with engineering students". Eurasia Journal of Mathematics, Science and Technology Education, vol. 22, no. 8, 2026, em2883. https://doi.org/10.29333/ejmste/18995