Aspectos históricos del teorema fundamental del cálculo y posibles mediaciones tecnológicas

Abstract

Este artículo presenta aspectos históricos del Teorema Fundamental del Cálculo (TFC) que podrían ser mediados por herramientas tecnológicas. Esta exploración pretende identificar en los argumentos de Newton y de Leibniz, un potencial heurístico para ser incorporados en ambientes de aprendizaje del cálculo en las universidades. Se presentan los resultados de una exploración de actualización de los argumentos de Newton y de Leibniz con la mediación del software matemático Geogebra. Los resultados de esta exploración articulan factores históricos y tecnológicos del TFC, que podrían usarse de manera curricular, por ejemplo, en el diseño de tareas para la formación universitaria.

Fundamental Theorem of Calculus: Cognitive Demands and Learning Limitations on Mathematical Tasks

“…The lack of use of these records, and the little transit between them, reveals possible implications of the deduction of characteristics of the integrals, for their analysis and treatment, or also, for a correct interpretation and application of the FTC. This also shows the difficulty previously pointed out by Muñoz-Villate (2021) in relation to the lack of previous knowledge about the FTC.…”

“…In relation to learning limitations, we must take into account that the university professor may not be aware of them, which may result in obstacles in the teaching and learning process of the various mathematical concepts. In the case of the FTC, we found that the limitations detected in the literature (Muñoz-Villate, 2021;Reyna-Segura, 2019) were part of those detected in this study for the proposed mathematical tasks. In particular, we noted that in the university reality, an algebraic and/or algorithmic treatment predominates in the teaching of the FTC, i.e., an imbalance between the conceptual and the algorithmic, which privileges the automation of operating techniques of the mathematical object (isolated/disintegrated) and not the formation of future generations of engineers capable of transcending the use and relationships between integrated mathematical objects, as pointed out by Muñoz (2000) and Zavala, Vera, and Ruiz (2017).…”

“…In relation to learning limitations in the teaching of the FTC, some studies point to its identification. Among the most common obstacles in the understanding of the FTC are the fuzzy memories about previous mathematical objects and the basis for the understanding of others typical of the theorem, such as: function, continuity, derivative, and integral, the reason for change and accumulation, and the lack of relationships between representations of the integral, by not identifying the link of the integral with the area that represents or not variability of the limits of the integral), or also dismissing the recognition of the importance of the FTC in engineering education programs (Muñoz-Villate, 2021). These ideas are confirmed by Reyna Segura, (2019) who points out that the learning difficulties of the FTC lie in the complexity of the notions of Calculus and the language used.…”

Relaciones entre historia de las matemáticas y formación de ingenieros

“…Don't forget that the algorithmic processes, of what we know today as integral calculus, were already given in Proposition 8. The detailed explanation of Problem 5 made by Newton in 1666 and the analysis of the first example that he offers, can be read in [38]. One of the direct conclusions of this manuscript regarding FTC is that:…”