Uma análise praxeológica das concepções do número fracionário no contexto de um livro didático de matemática do 6° ano do Ensino Fundamental

Abstract

This bibliographical, qualitative, and documentary research aims to analyze the praxeologies related to different conceptions of fractional numbers in a 6th-grade elementary school mathematics textbook and the relationships established between practical and theoretical blocks, focusing on the mathematical object of fractional numbers. The investigation is guided by the following question: Which praxeologies related to the different conceptions of fractional numbers are mobilized in a 6th-grade mathematics textbook? Given the conceptual complexity of this object, we delimited the study to different approaches/conceptions associated with fractional numbers, such as: part-whole, quotient, ratio, measure, and multiplicative operator. These multiple interpretations express the multifaceted epistemological nature of fractional numbers, whose adequate understanding constitutes a challenge for both students and teachers in the teaching and learning process. This analysis focuses on the textbook Mathematics and Reality, by Gelson Iezzi, Osvaldo Dolce, and Antônio Machado (2022), approved by the National Textbook Program (PNLD) and adopted by the Municipal Education Department of São Gonçalo dos Campos (BA). This teaching material, widely used as a reference for teaching practice, is the result of a process of external didactic transposition, in which knowledge produced and legitimized within the scientific domain is reorganized, selected, and adapted to the school context. Thus, the textbook reflects institutional choices that materialize specific ways of presenting and dealing with mathematical knowledge, thereby guiding teaching and learning practices.. The analysis is based on the theoretical and methodological contributions of the Anthropological Theory of Didactics (ATD), which provides a framework for understanding the organization and institutionalization of mathematical knowledge within the school environment. The ATD is mobilized in this research as a tool for didactic analysis, allowing the examination of praxeologies through the components: Task, technique, technology, and theory. Furthermore, the notion of didactic transposition is used to investigate how mathematical knowledge is reconfigured in the process of adapting to the demands of the school environment, transforming itself into knowledge to be taught and knowledge that is taught. The analysis is developed in three interdependent moments: (i) selection and contextualization of teaching materials; (ii) ecological analysis, focused on the institutional conditions that guide the organization of knowledge; and (iii) praxeological analysis, centered on the proposed tasks, the techniques required for their resolution, and the technological and theoretical discourses that support them. The results indicate a prevalence of traditional approaches in the treatment of fractional numbers, with emphasis on the part-whole conception, use of model examples, and activities guided by mechanical procedures, with limited exploration of the broader and multifaceted meanings of the conceptions. By articulating the fundamentals of Mathematics Didactics with the analytical structure of ATD, this research seeks to contribute to a critical understanding of the processes of didactization of mathematical knowledge and to reflection on the role of the textbook in the formation of more meaningful and contextualized teaching practices.