Uma abordagem alternativa de ensino de cálculo utilizando infinitésimos

Abstract

The teaching of Differential and Integral Calculas in Brazilian Universities, as in several other countries, has been revealed to be inadequate both by the high failure rate in these courses and the low leveis of funcional knowledge acquired by the students. This implies that there is a need for more effective teaching strategies. Accordingly, we elaborated, tested and evaluated an altemative approach to the teaching of Differential Calculus for Engineering Students. We also analyzed the effects of this approach on the students’ construction of their mental schema. The approach included, among other things, working on activities in small groups and using infinitesimais to construct the concept of infinitesimal approximation, continuity and the derivate. We elaborated a teaching module, complemented by containing problems of interest to the student. The problems were elaborated with regard to the prior knowledge which we identified as being necessary to the construction and application of the concepts to be constructed. We also took into account abilities and attitudes that we wished to foster in the student; the later were taken from the Curriculum Standards for Engineering Courses. We undertook an investigation of the history of Differential and Integral Calculus in order to delineate the evolution of its main concepts. Three classroom interventions were made. In the classroom interventions we applied our altemative approach using a learning theory based on Construtivism and an ontology of mathematical objects consistent with it. We analyzed the resuits obtained relating to the a list of activities viability, meanning and functionality to the concepts to be constructed. Both the students’ performance and the time necessary for the realization of the program were taken into consideration. Based on our description of the historical development of the Calculus, we classifies the various mental schema which the students formed in their construction of infinitesimal approximation of a point when using either the altemative infinitesimal approach or the traditional approach using infinite sequences. We concluded by pointing out the necessity for more studies on this topic as well as the necessity for extending this approach to the concepts involved in Integral Calculus.