“Comparing without Calculating”: the ArAl project’s research on verifying the equivalence of mathematical expressions

Abstract

This article presents the research project “Comparing without Calculating” (CsC), developed within the ArAl project. Its aim is to investigate to what extent and in what ways a relational approach to teaching arithmetic can foster students’ argumentation about the truth of equivalences between numerical representations, overcoming their tendency to interpret them merely as calculations. The theoretical and methodological framework of the project is outlined, together with the competencies promoted by the proposed CsC activities. Several episodes drawn from classroom activities in primary and lower secondary school are analysed with the aim of identifying teaching strategies – together with the corresponding teacher competences – capable of fostering the development of students’ relational competences. The findings summarize such strategies, which include enhancing and supporting intuitive insight, distinguishing between local and global calculations, identifying premises and conclusions within an argument, and avoiding the so‑called “relational cage”. At the same time, the results highlight the need for specific teacher competences to implement these strategies effectively.