Noción de número racional en grado tercero: construcción de objetos abstractos a partir de acciones concretas

Abstract

Obando (2003) argues that there are many questions that arise in the teaching-learning processes of mathematics around the subject of fractions. In this paper we present a cognitive analysis that allows us to determine a path of construction of the notion of rational number from the perspective of the APOE Theory (Acronym of Action, Process, Object, Scheme) (Arnon et al., 2014). Given the complexity that the construction of rational number presents for students, the following question arises: What structures and mental mechanisms on the notion of rational number show third-grade students? In particular, the construction of abstract objects is analyzed. from the application of Actions on specific Objects. Analyzing the study of the notion of rational number, in different researches in Mathematics Didactics, as well as the study of textbooks, a Genetic Decomposition (DG) is proposed - a cognitive model by means of which a student can build a concept (Dubinsky , 1991) - that allows to explain those Constructions and Mental Mechanisms, that hypothetically a student makes visible, when constructing the Notion of Rational Number as Mathematical Object. This research is supported methodologically in the development of the first and third components of the APOE theory Research Cycle. In the first component Theoretical Analysis, a cognitive model of construction of the Notion of Rational Number for children of third grade of primary school is proposed, starting from the application of Actions on concrete Objects. The second component, design and application of instruments, begins with the design of a Didactic Unit composed of a set of tasks that are analyzed based on the cognitive model presented in the first component. This investigation is of qualitative approach and experimental empirical cut, taking the case study, founded, with 6 students that participate in the development of the investigation. The results of the work are analyzed in light of the preliminary genetic decomposition and are fed back by the evidence obtained through the interview. Although initially the APOE theory was designed and implemented for advanced mathematical concepts and notions, the application of this in the present investigation, allows to extend this vision and apply it in a mathematical context where relationships can be established between the Structures and Mental Mechanisms developed by students of primary on the Notion of Rational Number. Taking the theoretical elements that explain how students construct abstract objects from the application of Actions on Concrete Objects. Finally, this research presents the analysis of the results to design a Didactic Unit; which can be taken by other primary school teachers to support the design of their classes and assessment models. In addition, the use of concrete material such as strips, fractional cakes, among others, from a cognitive perspective, can point out to the teacher of mathematics, the need to support its use and the impact that it can have on teaching processes and learning of mathematics in the medium and long term.