O Algoritmo de Euclides e os feijões: uma proposta de sequência didática

Abstract

This dissertation investigates the articulation between the structural rigor of the Euclidean Algorithm and a didactic transposition proposal grounded in the use of beans as a manipulable resource for teaching divisibility and the greatest common divisor in the 7th grade of Elementary Education. Based on the historical-epistemological understanding of the algorithm, systematized in Books VII to IX of Elements by Euclid of Alexandria, the study seeks to overcome the traditionally mechanized approach to teaching arithmetic procedures by promoting a pedagogical reconstruction of its logical structure and invariant properties.The theoretical framework is grounded in the historical tradition of Mathematics and in references from meaningful learning theory and pedagogical mediation, which support the progressive construction of knowledge through cognitively structured experiences. The research is characterized as a qualitative study, involving the implementation of a Didactic Sequence organized into three articulated stages: empirical investigation of divisors through concrete groupings; identification and analysis of common divisors; and formalization of the algorithm as a conceptual generalization of the explored process.The findings indicate that concrete manipulation enhances the understanding of the iterative dynamics inherent in Euclidean division, allowing symbolic formalization to emerge as a conceptual synthesis of a previously structured experience. The Educational Product demonstrates feasibility for implementation in diferente regional contexts, contributing to structurally grounded mathematical learning.