Obstáculos que influenciam o processo de compreensão dos números racionais em sua representação fracionária presentes em dissertações e teses

Abstract

This dissertation investigates the genesis and nature of epistemological and didactic obstacles that permeate the process of understanding rational numbers in their fractional representation, grounded in the theoretical framework of the Theory of Didactical Situations (TDS) developed by Guy Brousseau. Its primary objective is to characterize, within stricto sensu academic works, the obstacles that underlie the comprehension of rational numbers expressed in fractional form. To this end, the study adopts the premise that the difficulties inherent in learning fractions do not constitute individual deficiencies or isolated procedural gaps; rather, they manifest as forms of knowledge endowed with local efficacy. As such knowledge stabilizes within specific didactic configurations, it tends to resist the conceptual reorganization required by the domain of rational numbers. Within this analytical scope, the investigation examines how these obstacles emerge from the intricate articulation among the meanings attributed to fractions, the typology of the quantities involved, and the architecture of the didactic system. Methodologically, the study is grounded in a qualitative approach of a bibliographic-analytical nature. The documentary corpus comprises master’s theses and doctoral dissertations defended between 2014 and 2024, indexed in the databases of the Coordination for the Improvement of Higher Education Personnel (CAPES) and the Brazilian Digital Library of Theses and Dissertations (BDTD). The findings corroborate the recurrence of epistemological obstacles associated with the improper transposition of properties of natural numbers to the domain of fractions. This hegemony is evidenced in the discretization of continuous quantities, in the fragility of the conservation of the unit of reference, and in the resistance to recognizing fractions as rational numbers. At the didactic level, the predominance of the part-whole meaning is identified, frequently associated with static figural representations. This configuration fosters an undue homogenization of the meanings of fractions and the stabilization of action schemes that are functionally limited yet validated by the prevailing didactic contract. Regarding strategies for overcoming such obstacles, the study identifies convergences in the literature pointing to the intentional diversification of meanings, the integration of multiple types of quantities, and the mobilization of a didactic situations grounded in intellectual autonomy. From the perspective of the TDS, such strategies prove particularly effective when linked to the structuring of an antagonistic milieu. It follows that overcoming the obstacles to learning fractions does not depend on adherence to isolated pedagogical resources, but rather on a reconfiguration of the didactic architecture. This architecture must be capable of articulating meanings, quantities, and validation mechanisms, thereby ensuring the effective circulation of the learner through the dialectics of action, formulation, validation, and institutionalization of mathematical knowledge. It is concluded that students’ difficulties in understanding rational numbers in their fractional form are intrinsically related to epistemological and didactic obstacles.