As formas operatória e predicativa do conhecimento manifestadas por alunos do 5º ano mediante problemas de estrutura multiplicativa: uma investigação das ideias base de função

Abstract

The study of the concept of Function is formalized in the ninth year of Elementary School and consolidated in High School, but basic ideas associated with this concept, such as correspondence, dependence, regularity, variable and generalization can be worked on from the Early Years, especially through problems of the Multiplicative Conceptual Field. Considering the Theory of Conceptual Fields (TCC), which assumes that student`s learning occurs through the simultaneous manifestation of the operational form of knowledge, knowing how to do, and the predicative form of knowledge, knowing how to enunciate about objects, their relationships and properties, and the Theory of Didactic Situations (TSD), which, among other characteristics, provides support for the action of the researchers in a classroom situation with regard to the organization of the milieu, in this research it was investigated what contributions does the implementation, in the light of TSD, of a sequence of multiplicative structure problems provide for the understanding of basic function ideas by fifth-year students? The qualitative investigation was carried out with a group of thirty fifth-year students in the second semester of 2019 in Foz do Iguaçu/Paraná. The research instrument consisted of a sequence of twenty-two multiplicative structure problems, elaborated from the perspective of TCC and for its implementation students were organized into groups, following TSD assumptions for the structuring of the milieu. Data production resulted from the audio recording of the dialogic interactions during the resolutions and corrections of problems, from the written production protocols both results of the implementation, during the class period, and from the researcher's logbook. Data were analyzed with support from the TCC, considering the search for indications of the manifestation of the operational and predicative form of knowledge, focusing on the basic ideas of function. The research results show that all groups manifested the operational form of knowledge more easily than the predicative form of knowledge and that the more aware the student is of his operational form, more condition he will have to manifest the predicative form, either orally or in writing. The basic idea of function most manifested in both forms of knowledge was regularity. Regarding the manifestation of the operational form of knowledge, the basic idea in which the groups presented more mistakes in their resolutions was the correspondence, especially in the problems of Combination – part unknown. As for the predicative form of knowledge, the groups had greater difficulty in expressing the ideas of generalization, variable and correspondence, followed by the idea of dependence. The type of problems that allowed the manifestation of the greatest amount of ideas based on the same problem was the Simple Proportion – One for many. As a result the research indicates that the implementation carried out in the proposed manner contributed to the simultaneous manifestation of the operational and predicative forms of knowledge, promoting learning, for the construction of function base ideas, sometimes mobilized synchronously, supported by structural content multiplicative strategies and for mobil zing different schemes for solving problems.