Algebraic reasoning through justification in the context of generalised arithmetic and equivalences in primary education

In this article, we characterise the algebraic reasoning of a group of 17 primary education students (aged 10–11) when solving and justifying tasks on generalised arithmetic and equivalence. A design-based research approach was adopted. Two activities involving properties of addition were designed to reduce the operational focus. In the first, students were asked to recognise and justify the equality of operations containing unknown digits; in the second, to analyse and justify the (in)validity of identities involving literal symbols. Through students’ individual written justifications and contributions during classroom discussions, we analyse levels of generality, structural reasoning and symbol use and interpretation, considering different types of justification. Overall, students recognised equality between sums but struggled with combined operations and correctly identified when identities were true. Their written justifications mainly relied on generality, with the literal symbol often interpreted as a placeholder rather than a generalised number. Structural reasoning varied by task, appearing more frequently when justifying equal sums with unknown digits and identities with a single literal symbol. In other cases, students tended to rely on examples or element comparisons without referring to properties of operations. During whole-class discussion, structural reasoning improved, highlighting the importance of combining written and oral justification.