Artifacts and Utilization Schemes in Mathematics Teacher Education: the Case of Positional Representation of Numbers in Primary School

In this article I discuss a case of primary school teacher education concerning an important piece of mathematics knowledge, that is the positional representation of numbers. Traditionally the topic is introduced by means of different artifacts (for instance abaci; base ten blocks). But it is well known that no artifact is transparent for the mathematical meaning, unless the social practices of the mathematics classroom, started by the teacher, are effective. The aim of this article is to describe and discuss a four session (16 hours) laboratory for 30 prospective primary school teachers, where the topic of the use of artifacts for positional representation of numbers in base ten is tackled, within a suitable theoretical framework. In the article the different sessions are analysed, focusing the differences between the tasks related to different components of the knowledge needed by mathematics teachers.