Um estudo sobre as provas e demonstrações na licenciatura em matemática : articulações entre os níveis de pensamento geométrico de Van Hiele e os tipos de prova de Balacheff

Abstract

This doctoral research aimed to establish articulations between van Hiele's levels of geometric thinking and Balacheff's types of proof, based on the discussions brought by Jaime and Gutiérrez and Balacheff, and the argumentations/justifications produced by undergraduates in Mathematics. The research carried out is characterized as quali-quantitative, with aspects of a case study. The data collection procedures were: questionnaire, activities with mathematical proofs, field notes, participant observation, video recordings and semi-structured interviews, carried out after the application of the activities. Eleven undergraduates in mathematics from a public university in the state of Paraíba participated in the research, who were between the 6th and 10th period of the course. As a theoretical framework, we used the approach of mathematical proofs and demonstrations under the perspective of Balacheff and considered the approach of the levels of geometric thinking under the discussions of van Hiele, De Villiers, Nasser, Kaleff et al., Dall'Alba, Ontario, Vargas and Araya, Jaime and Gutiérrez, among others. The analysis of the results showed that: (1) the undergraduates did not know how to differentiate the words proof and demonstration, they did not have experience with the proofs and demonstrations in Basic Education and the work with them in the Mathematics Degree was not satisfactory, because they still have a lot of difficulty in writing and understanding them, and do not identify with the area; (2) the undergraduate students, in their majority, presented arguments/justifications within the pragmatic proofs, without an adequate mathematical basis, only validating the statements through experimentation. Only one pair was able to construct mental experience proofs in six out of seven activities, validating their strategies generically; (3) the undergraduate students fluctuated a lot from one level of thinking to another, mainly in activities that involved the same concepts. Because of this, they built different types of proof; (4) the majority of undergraduates understand the difference between particular cases and generic cases in mathematics when analyzing their arguments, justifications and proof of mathematical statements. We check out experimentally that the undergraduates who were at level 4 of van Hiele, managed to elaborate proofs of the mental experience type and those who were at level 3, elaborated proofs of the generic example type. The undergraduates who were at level 2, on the other hand, managed to elaborate two types of pragmatic proofs: naive empiricism and crucial experience, while those who were at level 1, did not take proofs, as they did not feel the need to justify their ideas. Therefore, we can say that the research contributes to Mathematics Education by establishing more specific articulations between van Hiele's levels of geometric thinking and the types of proof proposed by Balacheff, which have not yet been discussed in the literature. In addition, we believe that the results can lead to a reflection on the possibility of differentiating the words proof and demonstration, the teaching of Mathematics in order to develop the mathematical reasoning of students and the teaching of demonstrations in the Mathematics Degree, enabling them to do mathematics.