INTRODUCTION TO PROOF: THE MEDIATION OF ... - IngentaConnect
26 MARIA ALESSANDRA MARIOTTI We shall limit ourselves to the description of some elements of the software, in order to present an analysis of the process of semiotic mediation (Vygotskij, 1978) that can be realised in classroom activities. The following sections (§2-§3) are devoted to clarifying the theoretical frame; in particular, the notion of ‘Field of Experience’, and ‘Mathematical Discussion’ will be discussed in relation to their utilisation both in the design and analysis of the experimental project. The results will be presented in terms of educational goals and analysis of the teaching-learning process, since the research project is to be considered a ‘research for innovation’, in which action in the classroom is both a means and a result of the evolution of research analysis (Bartolini Bussi, 1994, p. 1) On the one hand pupils achieved a theoretical perspective in the solution of construction problems; the theoretical meaning of geometrical construction provided the key to accessing the general meaning of Theory. On the other hand, the study made it possible to clarify the role of a particular software in the teaching/learning process; the functioning of the specific elements of the software will be described and discussed according to the theoretical reference frame. The software utilised is Cabri-géomètre (Baulac et al., 1988). 2. THE THEORETICAL FRAMEWORK: ANOVERVIEW Geometrical Constructions constitute the field of experience in which classroom activities are organised. According to Boero et al. (1995, p. 153), the term ‘field of experience’ is used to intend the system of three evolutive components (external context; student internal context; teacher internal context), referred to a sector of human culture which the teacher and students can recognise and consider as unitary and homogeneous. The development of the field of experience is realised through the social activities of the class; in particular, verbal interaction is realised in collective activities aimed at a social construction of knowledge: i.e. ‘Mathematical Discussions’, that is polyphony of articulated voices on a mathematical object, that is one of the objects – motives of the teaching – learning activity (Bartolini Bussi, 1996, p. 16). Polyphony occurs between the voice of practice and the voice of theory. The practice of the pupils consists in the experience of drawing, evoked by: • concrete objects, such as drawings, realised by paper and pencil, ruler and compass. • computational objects such as Cabri figures or Cabri commands.
THE MEDIATION OF A DYNAMIC SOFTWARE ENVIRONMENT 27 Geometry theory, imbedded in the Cabri microworld, is evoked by the observable phenomena and the commands available in the Cabri menu. Figures and commands may be considered external signs of the Geometric theory, and as such they may become instruments of semiotic mediation (Vygotskij, 1978), as long they are used by the teacher in the concrete realisation of classroom activity to introduce pupils to theoretical thinking. 3. THE FIELD OF EXPERIENCE OF GEOMETRICAL CONSTRUCTIONS IN THE CABRI ENVIRONMENT 3.1. Reference culture The reference culture is that of classic Euclidean Geometry. Euclidean Geometry is often referred to as ‘ruler and compass geometry’, because of the centrality of construction problems in Euclid’s work. The fundamental theoretical importance of the notion of construction (Heath, 1956, p. 124 segg.) is clearly illustrated by the history of the classic impossible problems, which so much puzzled the Greek geometers (Henry, 1994). Despite the apparent practical objective, i.e. the drawing which can be realised on a sheet of paper, geometrical constructions have a theoretical meaning. The tools and rules of their use have a counterpart in the axioms and theorems of a theoretical system, so that any construction corresponds to a specific theorem. Within a system of this type, the theorem validates the correctness of the construction: the relationship between the elements of the drawing produced by the construction are stated by a theorem regarding the geometrical figure represented by the drawing. The world of Geometrical Construction has a new revival in the dynamic software Cabri-géomètre. As a microworld (Hoyles, 1993), it embodies Euclidean Geometry; in particular, as it is based on the intersections between straight lines and circles, Cabri refers to the classic world of ‘ruler and compass’ constructions. However, compared to the classic world of paper and pencil figures, the novelty of a dynamic environment consists in the possibility of direct manipulation of its figures and, in the case of Cabri, such manipulation is conceived in terms of the logic system of Euclidean Geometry. The dynamics of the Cabri-figures, realized by the dragging function, preserves its intrinsic logic, i.e. the logic of its construction; the elements of a figure are related in a hierarchy of properties, and this hierarchy corresponds to a relationship of logic conditionality. Because of the intrinsic relation to Euclidean geometry, it is possibile to interpret the control ‘by dragging’ as corresponding to theoretical control