Contribuições da teoria do ensino desenvolvimental de Davydov: proposta de construção de plano de ensino para a formação do conceito de números reais

Abstract

In elementary education in Brazilian schools, the orientation of official documents for teaching the concept of number occurs in the following sequence: natural, whole, rational, culminating in the irrational ones, usually in the ninth year of elementary school. These numbers put together are called real numbers. In this reality, its teaching contemplates external nexuses, in which the student is encouraged to identify and classify them, according to their decimal representation, to the detriment of internal and conceptual nexuses, their essence. From a methodology that investigated theoretical aspects and legal documents, involving the reading and analysis of theses, dissertations and scientific articles focusing on the teaching of real numbers, in all teaching modalities, from 2010 to 2018, official documents of the Ministry of Education (MEC), such as the National Curriculum Parameters (PCNs), the National Textbook Plan (PNLD) and the Common National Curriculum Base (BNCC) and theoretical contributions related to cultural-historical theory, we find a contradiction, the namely, an insufficient argument to address the irrational numbers in depth. Some related productions emphasize the concept of number, based on davydovian's proposal, in the relationship between measurable quantities. In fact, this is understood as plausible for rational numbers, using measurable quantities, but not feasible for irrational numbers, before clarifying another type of quantity, the incommensurable. This refers to the first epistemological obstacle of mathematical science, conceived in the Pythagorean school and overcome by Eudoxo de Cnido. Without understanding this event and others that have appeared throughout the history of mathematics, as social contexts demanded new scientific formulations to explain the emerging contradictions, it is not possible to carry out a teaching focused on the essence of the object in question. Strictly speaking, we understand that the elaboration of a proposal for teaching real numbers requires a movement from the abstract to the long-lived, gradual concrete, finalized by the appropriation of the concept of irrational number in the ninth grade of elementary school. Thus, we present the research problem: What are the contributions of Davydov's developmental teaching to the planning of a study activity for the development of the concept of real number in the last year of elementary school? In search of an answer, based on a historical-bibliographic and documental methodology, based on a dialectical perspective, we carried out a logical-historical study of real numbers in a contextualized way, in order to obtain the contradictions and overcoming¿s of the community of mathematicians to a synthesis of the general concept of number; we deepened the reading of premises of the cultural-historical theory to study the object, from its historical and dialectical perspective, conceiving the concept through its syntheses, with emphasis on the understanding of irrational numbers and from the developmental premise that this is possible by the movement from the abstract to the concrete, from the most general aspect of the concept. The result obtained from this investigative incursion was the elaboration of a Davydovian teaching plan for the concept of real numbers, for the ninth year of elementary school, in which the need to understand the incommensurable quantities that should emerge was emphasized of study tasks. The argument emphasizes that the formation of the concept of number, in all its generality, is only possible from the student's substantive abstraction of incommensurable quantities. In any case, the need for experimental studies to evaluate this proposal is highlighted