Various studies highlight the role of linguistic and argumentative competences in teaching and learning of mathematics (e.g., Ferrari, 2004). In the historical-cultural perspective, language built through interpersonal interaction is internalized, becoming a powerful tool for controlling cognitive activity. Vygotsky (1934) argued that higher cognitive functions involve manipulating psychological tools (Friedrich, 2014), with language being a particularly sophisticated one. These tools act as a bridge between the individual and the social context, influencing and transforming the structure of higher mental functions. The mind is inherently social and children internalize the public meanings of artifacts —both material and symbolic, such as language — through communication. This occurs as they act on objects and adapt to them.
The Theory of Semiotic Mediation (Bartolini, Bussi & Mariotti, 2008) offers a framework for analysing artifacts used in mathematics education. It examines the relationships between artifacts, tasks, signs and the mathematical knowledge being mediated.
Our studies analyse mathematical activities where students work within a dialogic space for thinking (Iannaccone & Zittoun, 2014; Perret-Clermont, 2004), which supports social interactions and the manipulation of linguistic objects, creating the conditions for knowledge appropriation through a dynamic process of constructing and negotiating meanings (Iannaccone, 2010).
Argumentation — viewed as a verbal description of the logical relationship between a set of premises and a conclusion — becomes a psychological tool, allowing for a transition from knowledge acquired at an interindividual level to individual appropriation through internalization. The comparison of different viewpoints fosters a cognitive decentralization, essential for understanding a task from multiple perspectives (Coppola et al., 2022).
We investigated if and how the use of different types of artifacts in a dialogic space for thinking, designed according to TSM, can support mathematical knowledge appropriation — for example, both procedural and conceptual understanding of mathematical operations (Pacelli et al., 2022) — by means a cognitive decentralization.