Multiplicative structure of natural numbers: an analysis of mathematics textbooks used in the early years of elementary school

Abstract

This article presents the results of a study that analyzed the conceptual approach of the multiplicative structure in didactic manuals, because in our view, concepts should be formed and developed from a set of problem situations that consider the representation and the concept itself. With this premise, this study aimed to classify different problem situations of the multiplicative structure of natural numbers in a collection of textbooks used in the early years of elementary school from the perspective of the Theory of Conceptual Fields developed by Gérard Vergnaud. This study is linked to the Research Group on Science and Mathematics Teaching (Grupo de Pesquisa em Ensino de Ciências e Matemática - GPECMA). This is a qualitative study with characteristics of documentary analysis and is methodologically based on content analysis. This analysis considered selecting, exploring, and treating data as necessary steps to meet the objective of the study and allowed us to identify and analyze the mathematical and didactic organization adopted in teaching the multiplicative structure in the collection of textbooks analyzed. When preparing and analyzing data, five types of multiplicative situations emerged. The algorithm of multiplication and division was formalized in the last volume of the collection. Among the problem situations identified were multiplication as the sum of equal parcels, rectangular arrangement, combination of possibilities, and dividing equally and determining how many fit. These problem situations were presented in three recorded units that differed by the representation used, in natural and figural language, and manipulable materials. It should be noted that most of the problem situations presented mobilized concepts of isomorphism of measures. The reasoning involved in the product of measures class was contemplated in multiplicative situations, excluding division situations, which indicated the need to propose to students more situations that organized the multiplicative thinking in different aspects, addressing multiplication and division in the product of measures class. It also implied the need to approach several multiplicative situations within different contexts and degrees of difficulty, since textbooks are often the main material used by teachers to prepare their lessons.

Keywords

• isomorfismo de medidas
• produto de medidas
• campos conceituais
• livro didático de matemática.