Identification and characterization of the development sub-levels of the derivative schema

The results of the research related to the learning process of the derivative notion show that its understanding is very complex and that only a significant number of students acquire a partial comprehension of the concept. In this context, this research presents an exploratory/confirmatory analysis in which we identify and characterize, through mixed analysis methods, the different sub-levels of development of the derivative schema that university students previously instructed in Differential Calculus managed to reach. The results show how useful these methods are in this study, since they made it possible to identify the 9 sub-levels of development and to characterize 5 of them according to the dominant variables and their associated mathematical elements.