Maria Alice Gravina

Tese de doutorado: defendida no Programa de Pós-Graduação em Informática na Educação para obtenção do título de Doutor em Informática na Educação, na Universidade Federal do RGS, Brasil, em setembro de 2001.

Orientação: Prof ª. Dra. Lucila M. Costi Santarosa e Prof ª. Dra. Liane Tarouco

Banca de Tese: Profa. Dra. Colette Laborde, Prof. Dr Hermínio Borges Neto, Prof. Dr. Jaime Ripoll, Prof. Dr. Sergio Franco.

Abstract: The proof process is central in the construction of knowledge in Mathematics. In Euclidean geometry it is one of the aspects that is a source of obstacles for the students. One of the difficulties lies in the necessary transition from empirical knowledge, already acquired, to more advanced knowledge: the geometry as a theoretical model, organized in axioms, theorems and proofs. Computer technology nowadays available incites an investigation towards the pedagogical interventions that can aid the knowledge construction. The understanding of its potential becomes an investigation subject: what happens with the cognitive process when the students work in a computer environment, where a dynamical representations of their mental actions is possible, and where in face of the retroaction provided by the immediate feedback they are provoked to new actions? Are these interactions substantial for ascending to new knowledge? The aim of this thesis is to present a didactic engineering in a dynamical geometry environment that favors the students knowledge ascension in geometry - from empirical to deductive. The theoretical frame is Piaget's theory, as well the theory of didactical situations in Mathematics, developed by the French school. The engineering is developed in three levels: in the first level the purpose is to understand the meaning of a mathematical proof and its necessity; in the second level the purpose is the development of elementary abilities for producing a proof; the final one provokes a high level of cognitive operations for accomplishing the proper treatment of geometrical figures - the design extensions / reconstructions and operative apprehensions that will produce the identifications of sub-configurations to support a deductive reasoning. An analysis a posteriori of the student's production confirms the anticipations made a priori, in the phase where the activities to be developed were delineated: the student's progress towards a geometrical knowledge as a mathematical model were expressive.