Invariantes operatórios associados ao conceito de função mobilizados por alunos do 5º ano do ensino fundamental

Abstract

This research, based on the Conceptual Fields Theory, idealized by Gérard Vergnaud, considers that the construction of a concept by the subject may be observed through operative invariants mobilized by him/her in different situations. Among these situations, mixed problems are mentioned, which involve additive and multiplicative relationships in the same problem. By this perspective, this research was carried out with the aim at analyzing operative invariants associated with the conceptual field of functions mobilized by the Elementary School 5th grade students to solve mixed problems type simple proportion and measurement transformation. To develop the investigation, four mixed problems were proposed to be individually solved by 12 5th grade students from different public schools in the State of Paraná countryside, through the Google Meet app. Mixed problems elaborated belong to the subclasses different from the simple proportion and measurement transformation classes. To perform the analysis, solutions written by the students were considered, and dialogues among students and the researcher as well, enabling analyses of schemas and operative invariants which were modeled as theorems in action and made explicit as concepts in action. Analyses of students’ schemes enabled indicate mobilization of fifteen (15) true theorems in action and two (2) false theorems in action; associated to them, sixteen (16) concepts in action belonging to the conceptual field of functions were identified. Most theorems in action mobilized by students are composed by linear function isomorphic properties. It was verified that even theorem in action might be manifested in different schemes from the same class of problems, or different ones. Among the concepts in action, mobilization of the basic ideas of function by students is highlighted: variable, correspondence, dependence, regularity, and the idea of proportionality. The mobilization of basic function ideas varies according to the problem subclass and with the scheme used by the student to solve the problem. Results still point that 5th grade students bring their operational invariant schemes implicitly, associated with the concept of function. Consequently, the importance of teacher being awareness on the implicit invariant operatory associated to the concept of function is highlighted because they are possible potentialize the function thoughts of the students. This allows us to indicate that it is necessary to consider the development of the concept of function along the schooling process.