Abstract
The aim of this paper is on the one hand to discuss from an APOS (Action–Process–Object–Schema) theory perspective the mental constructions involved in the learning of linear algebra, through examples concerning the linear transformation concept and related notions. On the other hand, methodological issues related to the design of research instruments and implementation of didactic interviews are discussed, supported by empirical data. Detailed analysis of transcripts from an interview with a student focuses on strategies used in interviewing as well as the mental stages involved in the construction of some linear algebra concepts. Due to the strategies employed, during the interview it is possible to witness the transition between different conceptions. A discussion of the relationships and interactions between different mental structures and mechanisms that play a role in the development of knowledge is provided, including theoretical considerations on the matter. Recommendations about pedagogical strategies are included.
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References
Alcock, L., & Simpson, A. (2009). Ideas from mathematics education: An introduction for mathematicians. UK: The Higher Education Academy. Maths Stats and OR Network.
Alves Dias, M., & Artigue, M. (1995). Articulation problems between different systems of symbolic representations in linear algebra. In L. Meira & D. Carraher (Eds.) Proceedings of the 19th PME Conference, vol. 2 (pp. 34–41). Brazil: Atual Editora.
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.
Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M. & Weller, K. (2014). APOS theory—A framework for research and curriculum development in mathematics education. Cham: Springer.
Dogan, H. (2018). Mental schemes of: linear algebra visual constructs. In S. Stewart, C. Andrews-Larson, A. Berman & M. Zandieh (Eds.), Challenges and strategies in teaching linear algebra (pp. 219–239). Cham: Springer.
Dorier, J.-L. (1995). Meta level in the teaching of unifying and generalizing concepts in mathematics. Educational Studies in Mathematics, 29(2), 175–197.
Dorier, J.-L., Robert, A., Robinet, J., & Rogalsiu, M. (2000). The obstacle of formalism in linear algebra. In Dorier (Ed.), On the teaching of linear algebra (pp. 85–124). Dordrecht: Springer.
Dorier, J.-L., & Sierpinska, A. (2001). Research into the teaching and learning of linear algebra. In D. Holton & M. Artigue (Eds.), The teaching and learning of mathematics at university level: An ICMI study (pp. 255–273). Dordrecht: Springer.
Dreyfus, T., Hillel, J., & Sierpinska, A. (1998). Cabri-based linear algebra: transformations. Proceedings of CERME-1 (First Conference on European Research in Mathematics Education), Osnabrück. http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/papers/g2-dreyfus-et-al.pdf. Accessed 29 Mar 2018.
Dubinsky, E. (1997). Some thoughts on a first course in linear algebra at the college level. In D. Carlson, C. Johnson, D. Lay, D. Porter, A. Watkins & W. Watkins (Eds.), Resources for teaching linear algebra (MAA Notes) (vol. 42, pp. 85–106). Washington, DC: Mathematical Association of America.
Harel, G., & Kaput, J. (1991). The role of conceptual entities and their symbols in building advanced mathematical concepts. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–79). Dordrecht: Kluwer.
Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In Dorier (Ed.), On the Teaching of Linear Algebra (pp. 191–207). Dordrecht: Springer.
Kú, D., Trigueros, M., & Oktaç, A. (2008). Comprensión del concepto de base de un espacio vectorial desde el punto de vista de la teoría APOE. Educación Matemática, 20(2), 65–89.
Molina, G., & Oktaç, A. (2007). Concepciones de la Transformación Lineal en Contexto Geométrico. Revista Latinoamericana de Investigación en Matemática Educativa, 10(2), 241–273.
Oktaç, A. (2018). Understanding and visualizing linear transformations. In G. Kaiser et al. (Eds.), Invited Lectures from the 13th International Congress on Mathematical Education (pp. 436–481). Cham: SpringerOpen.
Parraguez, M., & Oktaç, A. (2010). Construction of the vector space concept from the viewpoint of APOS theory. Linear Algebra and Its Applications, 432(8), 2112–2124.
Possani, E., Trigueros, M., Preciado, J. G., & Lozano, M. D. (2010). Use of models in the teaching of linear algebra. Linear Algebra and Its Applications, 432(8), 2125–2140.
Ramírez Sandoval, O., & Oktaç, A. (2012). Modelos intuitivos sobre el concepto de transformación lineal. Actes du Colloque Hommage à Michèle Artigue. Université Paris Diderot—Paris 7, Paris, France. https://www.uqat.ca/telechargements/info_entites/Didactiques%20des%20math%C3%A9matiques.%20Approches%20et%20enjeux_Atelier7.pdf. Accessed 4 May 2018.
Ramírez-Sandoval, O., Romero-Félix, C., & Oktaç, A. (2014). Coordinación de registros de representación semiótica en el uso de transformaciones lineales en el plano. Annales de Didactique et de Sciences Cognitives, 19, 225–250.
Roa-Fuentes, S., & Oktaç, A. (2010). Construcción de una descomposición genética: Análisis teórico del concepto transformación lineal. Revista Latinoamericana de Investigación en Matemática Educativa, 13(1), 89–112.
Robinet, J. (1986). Esquisse d’une genèse des concepts d’algèbre linéaire. Cahier de Didactique des Mathématiques, 29, IREM de Paris VII.
Romero Félix, C. F., & Oktaç, A. (2015). Coordinación de registros y construcciones mentales en un ambiente dinámico para el aprendizaje de transformaciones lineales. In I. M. Gómez-Chacón et al. (eds.) Actas Cuarto Simposio Internacional ETM (pp. 387–400), Madrid, Spain.
Salgado, H., & Trigueros, M. (2015). Teaching eigenvalues and eigenvectors using models and APOS Theory. Journal of Mathematical Behavior, 39, 100–120.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.
Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification—The case of function. In E. Dubinsky & G. Harel (Eds.), The Concept of Function: Aspects of Epistemology and Pedagogy (pp. 59–84). USA: Mathematical Association of America.
Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J.-L. Dorier (Ed.), On the Teaching of Linear Algebra (pp. 209–246). Dordrecht: Kluwer.
Sierpinska, A., Dreyfus, T., & Hillel, J. (1999). Evaluation of a teaching design in linear algebra: the case of linear transformations. Recherches en Didactique des Mathématiques, 19(1), 7–40.
Stewart, S., & Thomas, M. O. J. (2010). Student learning of basis, span and linear independence in linear algebra. International Journal of Mathematical Education in Science and Technology, 41, 173–188.
Thomas, M. O. J., & Stewart, S. (2011). Eigenvalues and eigenvectors: embodied, symbolic and formal thinking. Mathematics Education Research Journal, 23(3), 275–296.
Trigueros, M., & Oktaç, A. (2005). La théorie APOS et l’enseignement de l’algèbre linéaire. Annales de Didactique et de Sciences Cognitives, 10, 157–176.
Trigueros, M., & Possani, E. (2013). Using an economics model for teaching linear algebra. Linear Algebra and Its Applications, 438(4), 1779–1792.
Zazkis, R., Dubinsky, E., & Dautermann, J. (1996). Coordinating visual and analytic strategies: a study of students’ understanding of the group D4. Journal for Research in Mathematics Education, 27(4), 435–457.
Acknowledgements
I wish to thank the colleagues of the seminar that I am directing, for the insights they always bring during discussion sessions: Gisela Camacho, Osiel Ramírez, Avenilde Romo, José Rosales, Mario Sánchez, María Trigueros, Rita Vázquez, Diana Villabona.
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Oktaç, A. Mental constructions in linear algebra. ZDM Mathematics Education 51, 1043–1054 (2019). https://doi.org/10.1007/s11858-019-01037-9
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DOI: https://doi.org/10.1007/s11858-019-01037-9