Mathematical Reasoning in Algebraic and Geometric Contexts: an analysis with grade 9 medalist students

Abstract The purpose of this article is to understand the reasoning processes of grade 9 students employed in solving two tasks: one of an arithmetic/algebraic nature and another of flat geometry. The methodology used is qualitative, inscribed in an interpretive paradigm using document analysis of six written productions by four students: medalist students at the Brazilian Mathematics Olympiad in Public Schools (OBMEP) and at the Regional Mathematics Olympiad (ORM) in the State of Santa Catarina. The analysis of the students’ solutions was carried out according to the following categories: (i) processes and types of reasoning and (ii) representations. The results point to a great potential of the use of Algebra (with mastery of algebraic language) in the realization of the students’ justifications. Such use seems to have allowed students to diversify their semiotic representation records and to coordinate them. Furthermore, and in a different way from what is reported in the literature, the students were able to carry out their justifications without the need to be questioned by the teacher. This question seems to be associated with the fact that these students are medalists in Mathematics Olympiads, events in which the construction of a plausible justification assumes a central role.