Uma investigação sobre a introdução do estudo de derivadas associado ao conceito de velocidade instantânea para o ensino médio

Abstract

This paper has as its goal to investigate how we could introduce the differential equations to high school students and how these students could understand the mathematical models involved. We elaborated an investigation about the possibilities of studying the Variation Rates associated to the comprehension of the concept of Instantaneous velocity with high school students who took part in one the editions of the Canoa Project, which took place at the UFMT campus in Cuiabá-MT, between the 12th and the 25th of November, 2006. This is a 40 hours long course and it was divided into 10 (ten) meeting sessions; the intention of the Canoa Project is to create the opportunity of resources and methodologies experimentations and also the reformulation of the high school curriculum concerning the Education of Sciences. The teaching strategies were based on the intuitive notion of limit, consisting didactical material that presented the derivatives as being infinitesimal quotients. We presented texts with related information and graphics which expressed the movement of vehicles in day-by-day situations and by creating tangent straight lines we determine the instantaneous speeds. Such proposal was based on free software which we have used to compose the visual structures, focusing on the Dynamic Geometry by CABRI GEOMETRIE II, paralleling to the construction and manipulation of Cartesian graphics through WINPLOT and the composition of conceptual maps using the CMAP TOOLS program to provide the didactical organization and learning verification. We have used as theoretical references the concept of Qualitative Mathematics and the Significant Learning Theory started by David Ausubel. The focus of the learning analyzes of the apprentices is placed on the construction of conceptual maps which were built by ten students. We have used those instruments in order to analyze the relations established by the students when we opposed them with the concept of Differentiation of functions with a real variant.