Este artigo analisa os processos de raciocínio em tarefas matemáticas de dois alunos do 9º ano do Ensino Básico e dois alunos do 2º ano do Ensino Superior. Debruçamo-nos, também, sobre a representação e a significação, dada a sua estreita relação com o raciocínio matemático. Os resultados apresentados têm por base dois estudos de natureza qualitativa e interpretativa que usam diversas fontes de dados. Estes resultados mostram que o domínio da linguagem algébrica por parte dos alunos do 9º ano é ainda insuficiente para resolverem de forma expedita os problemas propostos, o que já não ocorre com os alunos do Ensino Superior. As estratégias iniciais de todos alunos são de natureza indutiva. No entanto, uma das alunas do 9º ano e os dois alunos do Ensino Superior mostram uma capacidade já bem visível para raciocinar dedutivamente. Os níveis de significação variam consideravelmente, tendo diversos alunos mostrado capacidade de construir ou mobilizar significados relevantes. O modelo de análise que apresentamos, articulando raciocínio, representações e significação, revelou-se um instrumento promissor para estudar os processos de raciocínio dos alunos. Palavras-chave: Raciocínio matemático. Representações. Significação. AbstractMathematical reasoning in Elementary School and Higher Education This paper analyzes the reasoning processes in mathematical tasks of two students in the 9th year of elementary school and two students in the second year of college. It also focuses the representation and meaningfulness, given their close relation with the mathematical reasoning. Results presented are based on two qualitative and interpretive studies which resort to several data sources. These results show that mastering of the algebraic language by the students in the 9th year is still insufficient to promptly solve the problems proposed, which does not occur with the college students though. All students use inductive initial strategies. However, one of the students in the 9th year and both college students revealed clearly their capability to reason deductively. The signification levels vary considerably, and several students have shown skills to build or mobilize relevant meanings. The model of analysis presented, articulating reasoning, representations and meaningfulness proved itself a promising tool to study the students’ reasoning processes. Keywords: Mathematical reasoning. Representations. Meaningfulness

The literature emphasizes the interest and need to learn the concept of continuity with understanding, although this meaning it is not consensual nor specified how it may emerge and be developed in the learning process. In this paper we report the results of a study aiming to analyse Brazilian preservice mathematics teachers’ understanding of the continuity of a function concept, in a context of a teaching experiment which design is based on a theoretical model for learning with understanding that consider three dimensions: the meanings of continuity concept, the use of its different representations and their application in problem solving that involves it. Data collection includes participant observation with audio and video recording of the lessons, students’ written and digital work on the tasks proposed during the teaching experiment, and interviews applied to the students. The qualitative and interpretative data analysis showed that students, in general, assigned different meanings to the continuity, which emerged from their concept image. At the end of the teaching experiment, they show an adequate conception of this concept, being able to articulate different representations and perform treatments and conversions to interpret and represent the (dis)continuity of a function, and to correctly apply the criteria of existence of the concept of continuity and the intermediate value theorem in the analysis of errors and in the proving of mathematical propositions, and in solving problems that call for mathematical modelling. Thus, as they revealed a learning with understanding of the continuity concept, the study also provides information to propose innovative mathematics teaching and learning methodologies aimed at improving that learning.