Compreensões sobre derivada e integral com o uso de um cas on line: um estudo com alunos do terceiro ano do ensino médio

Abstract

The present study aimed to investigate understandings of the concepts of Derivative and Integral constructed by a group of third-year students of a Brazilian private high school in digital environment with the CAS online WolframAlpha and applets generated by the Geogebra. Although the literature on teaching and learning Calculus is quite extensive, on the other hand, particularly in Brazil, it is almost always restricted to Higher Education settings. The study proposed a new approach to the concepts of Derivative and Integral with focus on conceptual understanding as well as weakening procedural algebraic manipulations typical in Calculus classes in undergraduate level. For this reason, we explored the concept of derivative as instantaneous rate of change and the integral as accumulation function and net area under a curve. Participants worked in pairs with the CAS online WolframAlpha as a tool for all algebraic and algorithmic manipulations deemed as necessary during the proposed activities. Methodologically, the study was conducted as a teaching experiment with the theoretical framework based on the ideas of Concept Image and Concept Definition. Data have suggested that the participants were able to deal the CAS WolframAlpha to some extent with ease, partially due to its quite flexible syntax. They also had an initial difficulty in terms of comparative thinking, which led to an initial difficulty with the idea of instantaneous rate of change. The participants had a good understanding of the integral as the net area under a curve, but we detected some difficulties with the accumulation function. Despite these initial problems, data suggest that the rate of change had a role as a cognitive root for the concept of derivative. Finally, the research suggests that both concepts of derivative and integral, as well as their connection via Fundamental Theorem of Calculus, can be explored conceptually in a significant way for high school students in similar settings under the proposed approach.