Contexts of Divisibility as an alternative to the development of Elementary Algebraic Reasoning Contextos de divisibilidade como uma alternativa para o desenvolvimento do Raciocínio Algébrico Elementar

Background: Nowadays, the importance of developing elementary algebraic reasoning in the mathematical education of students and teachers is indisputable. It is necessary to identify what other elementary arithmetic content can contribute to the development of elementary algebraic reasoning (EAR) and what the relationship is between the mathematical practices that involve them and the levels of algebraic reasoning. Objective: This paper presents a study on the potential of divisibility contexts as a means of promoting elementary algebraic reasoning. Design: This study was conducted using a qualitative, theoretical-analytical, and documentary approach. Analysis: Using theoretical and methodological notions from the Onto-Semiotic Approach to Mathematical Knowledge and Instruction and, in particular, the proposal for elementary algebraic reasoning levels developed within this theoretical approach, a systematic review and a didactic analysis of research on divisibility was carried out to organize and analyze the situations that are usually proposed in basic education. Results: Based on that, partial meanings of divisibility are identified, which in turn can be associated with mathematical practices with features of different levels of algebraic reasoning. It is evident that divisibility, traditionally approached as arithmetic content, can promote processes that are relevant to elementary algebraic reasoning, such as generalization, argumentation, and algebraic symbolization. Conclusions: Divisibility is a useful context as an alternative to pattern contexts for the development of elementary algebraic reasoning, confirming that algebraic reasoning can be developed from various elementary arithmetic contents.