Proofs and demonstrations and levels of geometric thinking: concepts, epistemological basis and relationships

Abstract

The ideas presented by Balacheff show the importance of working with mathematical proofs and demonstrations in Basic Education, because in his study he was interested in knowing the nature of the proofs, if it is possible to elucidate a hierarchy of the demonstration genesis and what are their evolution. Balacheff had as an epistemological basis the method of proofs and refutations of Lakatos, which describes Mathematics as a fallible, semi-empirical science that grows through the criticism and correction of theories, thus stimulating the work by means of the search for regularities, testing, formulation, justification, refutation, reformulation, reflection and generalization. Already the ideas defended by van Hiele show the importance of understanding the levels of geometric thinking of the students in order to elaborate materials and use the appropriate language for each level. For this, van Hiele receives some influences from the Gestalt psychology on the concept of insight and the laws of the theory of apperception, but also brings some concepts of the rational mental process of Selz and the ideas of Van Parreren about intentional and autonomous thinking. We also noticed some similarities and differences with Piaget's theory of cognitive development. Therefore, in this article, the authors sought to carry out some reflections on the research advocated by Balacheff and van Hiele, highlighting some important aspects cited by them, briefly highlighting their epistemological bases and establishing some relationships between the levels of geometric thinking and the types of proof and mathematical demonstrations.