Semiotic mediation in the mathematics classroom: artifacts and signs after a Vygotskian perspective

Abstract Elaborating on previous research studies carried on at different school levels and involving different artefacts, the authors present a synthesis of their theoretical reflection as a model of the teaching and learning process based on the notion of Semiotic Mediation. The notion of mediation, widely used in the current mathematic education literature, is here elaborated in a more complex way that overcomes the basic assumption about the contribution of integrating tools in the human activity, and in mathematics activity in particular. Following the seminal idea of Vygotsky and elaborating on it the author postulate that an artefact (either an abacus or a pocket calculator, either a compass or a Dynamic Geometry command) can be exploited by the teacher as a tool of semiotic mediation to develop genuine mathematical meanings, detached from the use of the artefact, but maintaining with it a deep semiotic link. The evolution of meanings, corresponding to the move from personal meanings rooted in the context of the artefact to conscious mathematical meanings, is a long term process that, according to our assumption, is neither spontaneous nor granted. The teaching organization that is proposed in this paper is modelled by the iteration of what has called a didactical cycle. In particular, the authors assume the centrality of the design of semiotic activities, such as production and interpretation of texts, and, in particular, collective mathematical discussion where the intentional action of the teacher is focussed on guiding the evolution of signs. The model is illustrated by examples drawn form different teaching experiments, carried out at different school levels and involving ancient and modern artefacts.