Aspectos Puntuales de la derivada desde el análisis no estándar: Una experiencia de aula

Abstract

The main objective of this monograph is to identify which aspects of the concept of the point derivative are favored by teaching it from DGSs such as GeoGebra and the theory of non-standard analysis constructed by Abraham Robinson. For this purpose, a didactic sequence is constructed that revolves around the first world of mathematics of David Tall's didactic theory, where intuition and representations have an essential role in the development of ideas and concepts related, in this case, to infinitesimal quantities and the point derivative on the part of the student. The application of the didactic sequence was done with high school students, eleventh grade. From the analysis of class protocols, satisfactory results were obtained around the main objective, highlighting: 1) The infinitesimal quantities allowed conceiving that from two points that are infinitely close, the instantaneous rate of change in the vicinity of a point can be found. 2) Students conceived the point derivative as a numerical value, obtained when the increments, between the two points through which secant passes, become infinitesimal, reinforcing the interpretation of the derivative in terms of instantaneous variation. 3) The relationship between infinitesimal quantities and small changes in velocity was understood, connecting this idea with the analysis of the motion of an object, which allowed the students to associate minute changes in velocity with the instantaneous variation.