Invariantes operatórios relacionados a generalização: uma investigação com estudantes do 9º ano a partir de situações que envolvem função a fim

Abstract

The basic ideas of the concept of function - variable, correspondence, dependence, regularity and generalization - are considered essential for the understanding of this concept, and must be present from the first years of Elementary School. However, the formalization of the concept of function must occur in the last year of that level of education. Some researches have shown that it is the idea of generalization that causes bigger difficulty for students, making it impossible to understand the concept of function in its essence. In this way, in order to investigate the knowledge related to generalization mobilized by students of basic education, this research proposes to look for answers to the following question: which theorems in action related to generalization can be mobilized by 9th grade students, from situations involving a linear function? For that, we elaborated a didactic sequence, along the lines of Didactic Engineering, which was applied with a class of 32 students from the 9th grade of Elementary School in a public school in the countyside of Paraná. The activities were designed with attention to the diversity of situations and concepts present in the Conceptual Field of the related function. The organization of activities and the implementation of the didactic sequence by the researcher were supported by the Theory of Didactic Situations. The data produced in this research come from the protocols of the students and from the audio recordings produced during the development of the activities, and were analyzed in the light of the Conceptual Fields theory, with attention to the implicit knowledge possible to be manifested in the students' resolutions. The results of this investigation point out that the organized and implemented didactic sequence allowed to unravel, from the students' resolution strategies, twelve theorems in action implicit in the responses of the participants of this research, seven of which related to true knowledge and five to mistaken knowledge. Among these identified erroneous knowledge, three are related specifically to generalization, the other false theorems in action refer to particular cases of situations, a necessary step to reach the generalization process. From the resolution of the activities it was also possible to identify learning and advances in students' knowledge related to generalization for the students collaborating in this research, especially with regard to the algebraic expressions developed by the students and improved during the didactic sequence.