Uma discussão sobre a relação entre isometria e simetria através da meta análise

Abstract

The actual assignment is based on the problem that there are divergences and confusions about the concepts of isometry and symmetry at Parâmetros Curriculares Nacionais– PCNs at Base Nacional Comum Curricular – BNCC, in textbook collections and in books intended mainly for mathematics teachers. Considering this problematic topic, it was conducted a meta-analysis research to investigate the issue: What is the relationship between isometry and symmetry of flat shapes? So, a sample of eleven works were analyzed. Seven of them are the most cited books in a sample of 74 dissertations, theses, or articles, which they have as main subject the isometry and/or symmetries. The other four works are related to the context of the New Mathematics Movement, because it was in this movement that the isometry and symmetries were inserted in the curriculum of Basic Education. The theoretical foundation was based on Raymond Duval's Theory of Semiotic Representation of Records. In special, it was performed treatments in the language records (natural and formal language) to compare definitions and concepts, and conversions were performed between representations in algebraic and figural records and vice versa, to compare and explain concepts and have demonstrations. The investigation of the relationship between isometry and symmetry involved discussions on the classification of isometries and symmetries, about symmetry and symmetrical figure, in addition to a resumption of the subjects of algebraic functions and structures, mainly dihedral and cyclic groups. The results of the analyzed works and the comparison of the data show us that there are no divergences on the concept of isometry because all the authors agree directly or indirectly that isometry is a geometric transformation which preserves distances. However, the same does not occur with the concept of symmetry and the classifications of isometry and symmetries, so these divergences implied a confusion about the relationship between isometry and symmetry. Thus, the symmetry is based on the concept of invariance because it is more formally accurate, because it is intrinsic to the study of algebraic structures and because all other interpretations about symmetry that omit or ignore the concept of invariance end up generating confusion.