Historical Artefacts, Semiotic Mediation and Teaching Proof

This chapter presents two examples where physical artifacts have been introduced to encourage young children and secondary students to practice validation. The first involves toothed wheels linked together where the turning of one causes the turning of the other in the opposite direction; the other uses mechanical devices representing and constructing parabolas. The background theoretical framework, presented below, is based on activity theory (Vygotsky, 1978), which highlights the use of signs in a social context and is part of a much wider framework of mathematical thinking where artifacts and signs are in the foreground. Bartolini Bussi & Mariotti, 2008, presents details and additional examples. Signs include not only words and symbols but also gestures, facial expressions, drawings and other ways of communicating. When a learner is given a mathematical task, even if specific artifacts are called into play, it is not evident that the resulting signs are related to mathematical signs; however, a major aim of teaching is to foster the construction of this relationship.