Didactic proposal for teaching the differential in real functions of one variable through dynamic visualizations
DOI:
https://doi.org/10.61174/recacym.v21i1.231Keywords:
teacing of Calculus, real function, differential, dynamic geometry softwareAbstract
This article presents a didactic proposal for introducing the concept of differential in real functions of one variable, focusing on its geometric meaning and its visualization through dynamic objects designed in GeoGebra software. Starting from the study of linear functions as an ideal model, the idea of local approximation is gradually introduced, along with the exploration of approximation error and the relative error quotient in the case of nonlinear functions, with the purpose of formalizing the differentiability condition for this type of functions. Through the integration of diverse forms of representation, this proposal seeks to foster a solid understanding of the concept of differential as a linear model of a local nature, emphasizing both its visualization and the analytical foundations that support it.
References
Apostol, T. M. (1972). Calculus, Volume I: One-variable calculus, with an introduction to linear algebra (2nd ed.). John Wiley y Sons
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241
Borba, M. C., y Villarreal, M. E. (2005). Humans-with-Media and the Reorganization of Mathematical Thinking: Information and Communication Technologies, Modeling, Visualization and Experimentation. Springer.
Borji, V., Martínez-Planell, R. y Trigueros, M. (2023). University students’ understanding of directional derivative: An APOS analysis. International Journal of Research in Undergraduate Mathematics Education
Courant, R. y Fritz, J. (1974). Introducción al cálculo y al análisis matemático. Vol. I (2.ª ed., trad. E. García de la Vega). Limusa
Cullen, C. J., Hertel, J. T. y Nickels, M. (2020). The roles of Technology in Mathematics Education. The Educational Forum, 84(2), 166–178
https://recacym.org/index.php/recacym/submission/wizard/2?submissionId=231#step-2
Dikovic, L. (2009). Applications of GeoGebra into teaching some topics of mathematics at the college level. Obnovljeni Zivot, 64(1), 73–78.
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131. https://doi.org/10.1007/s10649-006-0400-z
Finesilver, C. (2022). Beyond categories: Dynamic qualitative analysis of visuospatial representation in arithmetic. Educational Studies in Mathematics, 110(2), 271–290.
Hähkiöniemi, M. (2006). The role of representations in learning the derivative (No. 104). University of Jyväskylä.
Hohenwarter, M., y Preiner, J. (2007). Dynamic mathematics with GeoGebra. Journal of Online Mathematics and Its Applications, 7, Article ID 1448.
Larson, R., Edwards, B., Young, H., y Freedman, R. (2011). Cálculo. De 1 y de 2 variables. McGraw Hill.
Martínez-Planell, R., Gaisman, M. T., y McGee, D. (2015). Student understanding of directional derivatives of functions of two variables. Proceedings of the . . . PME Conference.
Martínez-Planell, R., Trigueros, M., y McGee, D. (2015). Student understanding of directional derivatives of functions of two variables. En T. G. Bartell, K. N. Bieda, R. T. Putnam, K. Bradfield y H. Domínguez (Eds.), Proceedings of the 37th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, (pp. 355–362). Michigan State University.
Piskunov, N. (1984). Differential and integral calculus (Vol. 1). MIR Publishers
Rojas, L. C., Mejía, H. R., y Esteban, P. V. (2019). Conceptualización de la derivada direccional a partir de la pendiente de una recta en el espacio. El cálculo y su enseñanza, 12, 13-26.
Rojas, L. C. (2019). Enseñanza y aprendizaje de la derivada direccional a través de la interacción con objetos dinámicos [Tesis doctoral]. Centro de Investigaciones y Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV).
Schoenherr, J., y Schukajlow, S. (2023). Characterizing external visualization in mathematics education research: A scoping review. ZDM – Mathematics Education, 56(1), 73–85.
Spivak, M. (2006). Calculus (4th ed.). Cambridge University Press.
Stewart, J. (2016). Cálculo de una variable (7.ª ed.). Cengage Learning.
Tall, D. (2004). Building theories: The three worlds of mathematics. For the Learning of Mathematics, 24(1), 29–32.
Tall, D. (2013). How humans learn to think mathematically: Exploring the three worlds of mathematics. Cambridge University Press.
Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. H. Schoenfeld y J. Kaput (Eds.), Research in Collegiate Mathematics Education I (pp. 21–44). AMS/MAA.
Trigueros, M., Badillo, E., Sánchez-Matamoros, G., y Hernández-Rebollar, L. A. (2024). Contributions to the characterization of the Schema using APOS theory: Graphing with derivative. ZDM, 56(6), 1093–1108.
Trouche, L., Rocha, K., Gueudet, G., y Pepin, B. (2020). Transition to digital resources as a critical process in teachers’ trajectories: the case of Anna’s documentation work. ZDM, 52(7), 1243–1257.
Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the derivative. CBMS Issues in Mathematics Education, 8, 103–127.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Rojas, L. C., Villamizar. F., Y.
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
