The role of the relationships between the solution function and its variation in the solution scheme of systems of differential equations
DOI:
https://doi.org/10.23925/1983-3156.2023v25i2p439-458Keywords:
Systems of differential equations, Calculus, APOE theory, Dynamical systems, Parametric functions, SchemeAbstract
This article contributes to the knowledge about the learning of systems of differential equations from the point of view of dynamical systems. It analyzes the evolution of the Schema of dynamic systems of two variables in university students, after finishing a course of dynamic systems designed with the Action Process Object Object Schema (APOE) theory as a support for the design of activities used throughout the course. In particular, this study focuses on how students give meaning to the strategies used to represent and interpret systems of differential equations and the relationships they establish between the structures that make up the Systems of Equations Schema and in particular the relationships between the function and its derivative through the different representations used to study them. This work also contributes to enrich the notion of Schema and interaction between Schemas of the APOE theory, as well as to the analysis of the relationships between the different concepts involved in and between the different representations of the solutions that play a role in the context of systems of differential equations and, importantly, in the understanding of parametric functions.
Metrics
References
Arnon, I, Cottril, J, Dubinsky, E. Roa Fuentes, S , Trigueros, M, Weller, K (2014). APOS Theory: Framework for research and curriculum development in Mathematics Education, Springer.
Arslan, S. (2010). Do students really understand what an ordinary differential equation is? International Journal of Mathematical Education in Science and Technology, 41(7), 873-888. doi.org/ 10.1080/0020739X.2010.486448
Baker, B., L. Cooley and M. Trigueros. (2000). The Schema Triad: A Calculus Example. Journal for Research in Mathematics Education. Vol. 31, No. 5, 557-578.
Blanchard, P., Devaney, R. L. & Hall, G. R.& Hall G. R. (2011). Differential Equations, 4th Edition. Brooks and Cole.
Blumenfeld, H. L. (2006). Student’s reinvention of straight-line solutions to systems of linear ordinary differential equations. San Diego State University. Retrieved from http://faculty.sdmiramar.edu/faculty/sdccd/hblumenf/thesis.pdf
Chaachoua, H. & Saglam, A. (2006). Modelling by differential equations. Teaching mathematics and its applications, 25, 15–22. Oxford University Press.
Dana-Picard T, Kidron I (2008) Exploring the phase space of a system of differential equations: different mathematical registers. Int J Sci Math Educ 6(4):695–717
Fuentealba C., Trigueros, M., Sánchez-Matamoros, G. & Badillo, E. (2022). Los mecanismos de asimilación y acomodación en la tematización de un Esquema de derivada. en Gloria Sánchez Matamoros y María Trigueros (Eds.). El aprendizaje y la enseñanza de las matemáticas en la universidad.. Avances De Investigación En Educación Matemática, (21), 23–44. https://doi.org/10.35763/aiem21.4241
Kwon, O.N. (2020). Differential Equations Teaching and Learning. In: Lerman, S. (eds) Encyclopedia of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-15789-0_100023
Lopes, A. (2021). Modelagem Matemática e Equações Diferenciais: um mapeamento das pesquisas em Educação Matemática. REnCiMa: Revista de Ensino de Ciências e Matemática, 12(4), 16–31. https://doi.org/10.26843/rencima.v12n4a16
Martin Bracke, J. & Lantau, M. (2017). Mathematical modelling of dynamical systems and implementation at school. CERME10, Dublin, Ireland. hal-01933489
Martínez Planell, R. & Trigueros, M. (2019) “Using cycles of research in APOS: The case of functions of two variables”. The Journal of Mathematical Behavior, Vol. 55, 100687. ISNN: 0732-3123.
Martínez-Planell, R., & Trigueros, M. (2013). “Graphs of functions of two variables: results from the design of instruction”. International Journal of Mathematical Education in Science and Technology, vol. 44, No. 5, pp. 663-672. http://dx.doi.org/10.1080/0020739X.2013.780214
Perez Campos, A. & Da Silva Reis, F. (2022). Contributions of Mathematical Modelling for Learning Differential Equations in the Remote Teaching Context. Acta Sci. (Canoas), 24(3), 184-215. ISSN: 2178-7727
Piaget J. & García, R. (1982). Psicogénesis e Historia de la Ciencia, SigloXXI ed.
Rasmussen, C. L. (2001). New directions in differential equations: A framework for interpreting students’ understandings and difficulties. The Journal of Mathematical Behavior, 20(1), 55-87. https://doi.org/10.1016/S0732-3123(01)00062-1
Rowland, D. R. (2006). Student difficulties with units in differential equations in modelling contexts. International Journal of Mathematical Education in Science and Technology, 37(5), 553-558. https://doi.org/10.1080/00207390600597690
Trigueros, M., & Martínez Planell, R. (2010). “Geometrical representations in the learning of two-variable functions” in Educational Studies in Mathematics, Vol. 73, Issue 1, pp. 3-19. Published on line: 24 June 2009. http://www.springerlink.com/openuri.asp?genre=article&id=doi:10.1007/s10649-009-9201-5.
Trigueros M. (2021). Un acercamiento a la Física a través de un modelo matemático de variación. Revista UNO 93, 38-49 ISSN:1133-9853
Trigueros, M. (2014). Vínculo entre la modelación y el uso de representaciones en la comprensión de los conceptos de ecuación diferencial de primer orden y de solución. Educación Matemática, 25 años (número especial), 207–226.
Trigueros, M. (2004). Understanding the meaning and representation of straight line solutions of systems of differential equations. In D. McDougall & J. Ross (Eds.), Proceedings of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 127-134). Ontario, Canada.
Trigueros, M. (2000). Students' conceptions of solution curves and equilibrium in systems of differential equations. In Fernandez, M. L. (Ed.), Proceedings of the 22nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 93-97). Columbus, OH: ERIC.
Vajravelu, K. (2018). Innovative strategies for learning and teaching of large differential equations classes. International Electronic Journal of Mathematics Education, 13(2), https://doi.org/10.12973/iejme/2699
Zandieh, M. & McDonald, M. (1999). Student understanding of equilibrium solution in differential equations. In F. Hitt & M. Santos (Eds.), Proceedings of the 21st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 253-258). Columbus, OH: ERIC.
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Autores que publicam nesta revista concordam com os seguintes termos:- Autores mantém os direitos autorais e concedem à revista o direito de primeira publicação, com o trabalho simultaneamente licenciado sob a Licença Creative Commons Attribution que permite o compartilhamento do trabalho com reconhecimento da autoria e publicação inicial nesta revista.
- Autores têm autorização para assumir contratos adicionais separadamente, para distribuição não-exclusiva da versão do trabalho publicada nesta revista (ex.: publicar em repositório institucional ou como capítulo de livro), com reconhecimento de autoria e publicação inicial nesta revista.
- Autores têm permissão e são estimulados a publicar e distribuir seu trabalho online (ex.: em repositórios institucionais ou na sua página pessoal) a qualquer ponto antes ou durante o processo editorial, já que isso pode gerar alterações produtivas, bem como aumentar o impacto e a citação do trabalho publicado (Veja O Efeito do Acesso Livre).
