This work was motivated by the poor performance of first year undergraduates in college. Therefore, our interest was to investigate how students learn mathematics, how they construct knowledge, and which is the level of their constructed learning of a mathematical content that is especially difficult for them, i.e. the concept of derivative and its applications. This work is part of a broader research that considered a multiple case study. Here we present the results obtained from two of them. To investigate constructed learning we used Dubinsky's APOS cognitive model. We created an interactive teaching environment, supported by classroom lessons and class modules of practical work based on Computer Workshops. Some results will be mentioned briefly. It was found that if the concept of "Derivative at a point" is not understood at the level of action, the students have difficulty in extending it and moving to a higher level of understanding and interpreting the concept geometrically. They also have difficulties understanding when they have to discriminate between true and false propositions set out on properties of the derivative related to monotony and concavity of a function. Conclusions were drawn on learning achieved and on the teaching strategies being implemented.
APOS Cognitive Model; Genetic Decomposition; Derivative; Case Studies
