This paper investigates upper secondary school students’ use of algebraic language as a thinking tool for constructing proofs in arithmetic. Drawing on research in algebraic thinking and proof, the study examines three key components involved in this process: coordination between verbal and algebraic registers, coordination between conceptual frames, and the activation of anticipating thoughts. Small-group discussions and written productions were analysed through these lenses while students tackled a proving task. The analysis identifies four categories of approaches, ranging from limited to effective use of algebraic language, each characterised by a specific configuration of the three components. These categories highlight both the breakdowns that occur when components are only partially mobilised and the productive interactions that support successful proofs. The study offers a fine-grained characterisation of students’ proving behaviours and contributes to understanding how algebra can function as a tool for reasoning and proving in arithmetic.
Annalisa Cusi (Sun,) studied this question.