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INTRODUCTION TO PROOF: THE MEDIATION OF ... - IngentaConnect

50 MARIA ALESSANDRA MARIOTTI supported by the mediation of new external signs (offered by the software), that pupils autonomously create and use with that specific pupose. The evolution of the pupils’ internal context is rooted in the construction activities, where different processes such as conjecturing, arguing, proving, systematizing proofs as formal deduction are given sense and value. Our experiments also show that the contradiction highlighted by Duval between everyday argumentation and deductive reasoning, between empirical and geometrical knowledge, can be managed in dialectic terms within the evolution of classroom culture. In our experiment classroom culture is strongly determined by the recourse to mathematical discussion orchestrated by the teacher to change the spontaneous attitude of students towards theoretical validation. The introduction of computer environments in school mathematical practice has been debated for years, in particular the potentialities of Cabri, and dynamic geometry in general, have been described and discussed. With this experimental research study we hope to offer a contribution in illustrating the cognitive counterpart of classroom activities with Cabri that allows an approach to theoretical thinking. ACKNOWLEDGEMENTS The research project has received financial support from C.N.R., M.U.R.S.T. (National Project ‘Ricerche di Matematica e Informatica per la Didattica’, coordinator: F. Arzarello). I wish to thank the teachers of the research team, M.P. Galli, D. Venturi e P. Nardini, who carried out the classroom experiments and discussed with me the data on which this paper is based. I would also like to thank the many colleagues who in the past years have generously contributed to making me develop these ideas. NOTES 1. The following remark concerns types of software which share with Cabri the general feature of a ‘drag mode’; I mean for instance Sketchpad or Geometric Supposer. 2. In this section I will employ the term ‘tools’ as used in the current English translations of Vygotskij’s works. In current literature the terminology is rather confused, as different authors use the same words with different meanings. 3. This unusual construction, quite different from those which can be found in the textbooks, seems to provide a strong support to our interpretation. 4. After the construction of the angle bisector and the inclusion of the corresponding command in the available menu, the definition of perpendicularity was introduced.

**THE** **MEDIATION** **OF** A DYNAMIC S**OF**TWARE ENVIRONMENT 51 REFERENCES Balacheff, N.: 1987, ‘Processus de preuve et situations de validation’, Ed. Stud. in Math. 18, 147–176. Balacheff, N.: 1998, ‘Construction of meaning and teacher control of learning’, in D. Tinsley and D.C. Johnson (eds.), Information and Communications Technologies in School Mathematics, Chapman & Hall, IFIP, London, pp. 111–120. Balacheff, N. and Sutherland, R.: 1994, ‘Epistemological domain of validity of microworlds, the case of Logo and Cabri-géom`tre’, in R. Lewis and P. Mendelson (eds.), Proceedings of the IFIP TC3/WG3.3: Lessons from Learning, Amsterdam, North-Holland, pp. 137–150. Bartolini Bussi, M.G.: 1991, ‘Social interaction and mathematical knowledge’, in F. Furenghetti (ed.), Proc. of the 15th PME Int. Conf., 1, Assisi, Italia, pp. 1–16. Bartolini Bussi, M.G.: 1994, ‘Theoretical and empirical approaches to classroom interaction’, in Biehler, Scholz, Strässer and Winckelmann (eds.), Didactic of Mathematics as a Scientific Discipline, Kluwer, Dordrecht, pp. 121–132. Bartolini Bussi, M.G.: 1996, ‘Mathematical discussion and perspective drawing in primary school’, Educational Studies in Mathematics 31(1-2), 11–41. Bartolini Bussi, M.G.: 1999, ‘Verbal interaction in mathematics classroom: a Vygotskian analysis’, in A. Sierpinska et al. (eds.), Language and Communication in Mathematics Classroom, NCTM. Bartolini Bussi, M.G. and Mariotti, M.A.: 1999, ‘Instruments for perspective drawing: Historic, epistemological and didactic issues’, in G. Goldschmidt, W. Porter and M. Ozkar (eds.), Proc. of the 4th Int. Design Thinking Res. Symp. on Design Representation, Massachusetts Institute of Technology & Technion – Israel Institute of Technology, III, pp. 175–185, Bartolini Bussi, M.G. and Mariotti, M.A.: 1999, ‘Semiotic mediation: from history to mathematics classroom’, For the Learning of Mathematics 19(2), 27–35. Bartolini Bussi, M.G., Boni, M., Ferri, F. and Garuti, R.: 1999, ‘Early approach to theoretical thinking: Gears in primary school’, Educational Studies in Mathematics 39(1–3), 67–87. Bartolini Bussi, M.G., Boni, M. and Ferri, F. (to appear), ‘Construction problems in primary school: A case from the geometry of circle’, in P. Boero (ed.), Theorems in School: From history and epistemology to cognitive and educational issues, Kluwer Academic Publishers. Baulac, Y., Bellemain, F. and Laborde, J.M.: 1988, Cabri-géomètre, un logiciel d’aide à l’enseignement de la géométrie, logiciel et manuel d’utilisation, Cedic-Nathan, Paris. Boero, P., Dapueto, C., Ferrari, P., Ferrero, E., Garuti, R., Lemut, E., Parenti, L., Scali, E.: 1995, ‘Aspects of the mathematics-culture relationship in mathematics teaching-learning in compulsory school’, Proc. of PME-XIX, Recife. De Villiers, M.: 1990, ‘The role and function of proof in mathematics’, Pythagoras 24, 17–24. Duval, R.: 1992–93, ‘Argumenter, demontrer, expliquer: continuité ou rupture cognitive?’, Petit x 31, 37–61. Fischbein, E.: 1987, Intuition in Science and Mathematics, Reidel, Dordrecht. Halmos, P.: 1980, ‘The heart of mathematics’, Mathematical monthly 87, 519–524. Hanna, G.: 1990, ‘Some pedagogical aspects of proof’, Interchange 21, 6–13. Heath, T.: 1956, The Thirteen Books of Euclid’s Elements, Dover, vol. 1.

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