Instruments for semiotic mediation in primary school classrooms

Instruments have been used for centuries in the mathematical experience and in the teaching tradition as well. We may quote several examples:1)the concrete materials, artificially designed by educators to reflect underlying mathematical ideas (e. g. Dienes' blocks) that were from the sixties (and still are) very popular among teachers2)the cultural instruments inherited from tradition (e. g. rulers, abaci, compasses, curve drawing devices), that have followed or anticipated the theoretical development of mathematics;3)the technological objects taken from everyday life (e. g. scales, gears, coins), that witness, in their functioning, hidden pieces of mathematics knowledge;4)the softwares developed in the information technology (e. g. CAS or dynamic geometry systems), that allow a quick approach and solution to complex problems.The first and the last case are meaningful: the teaching aids in the past and the new softwares today have been ofted advertised as useful, powerful and effective instruments to solve the problems of teaching. The expectations have been often frustrated. The research in didactics of mathematics has shown that artefacts become efficient, relevant and transparent through their use in specific activities, in the context of specific types of social interactions, and in relation to the transformations that they undergo in the hands of users. In this paper, we shall analyse two cases of instruments from the above class 2 (the compass and the abacus), very common in primary school classes, by inserting them in a Vygotskian framework, that allows to precise the quality of social interactions (individual and group tasks; discussions orchestrated by the teacher), realised under the teacher's guidance, to foster the individual construction of mathematical meanings. The two examples concerns two artefacts inherited from tradition and describe the shift from the 'concrete' to the 'mental' instrument in primary school. The users are young pupils, to emphasise also the need of starting quite early to nurture a theoretical attitude towards Mathematics.