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Resumo A pesquisa atual sobre aprendizagem da Matemática reconhece que os indivíduos atribuem significado a objetos de seu pensamento. No entanto, alguns processos dinâmicos e interativos envolvidos na atribuição de significado não estão suficientemente especificados. Aqui, o foco é direcionado a três desses processos: contextualizar, complementarizar e complexificar. Os objetivos do artigo são estender as perspectivas existentes sobre tais processos e destacar aspectos epistemológico, conceitual e cognitivo que são significativos para a aprendizagem da Matemática. Para tal, uma agenda de pesquisa é elaborada colocando em diálogo diferentes perspectivas teóricas e posições. Argumentamos que os três processos – contextualizar, complementarizar e complexificar –, juntos, constituem um modelo interpretativo para a aprendizagem da Matemática de uma perspectiva da atribuição de significados. Abstract Recent research on Mathematics learning recognises that individuals ascribe meaning to objects of their thinking. However, some dynamic and interactive processes involved in ascribing meaning are not yet sufficiently specified. Here, the focus is directed at three such processes: contextualising, complementizing, and complexifying. The aims of this paper are to extend existing perspectives on these three processes and to highlight epistemological, conceptual, and cognitive aspects that are significant for mathematics learning. To this end, a research agenda is pursued that brings into dialogue different theoretical perspectives and positions. We argue that the three processes – contextualizing, complementizing, and complexifying – together constitute an interpretative model for learning mathematics from a perspective of ascribing meaning.

Geometrical constructions (whether with paper and pencil or with appropriate software) are widely considered to be a suitable vehicle for secondary school students to gain experience of proof and proving. It is also recognised, however, that there can be a tension between the practical aspect of physically carrying out a geometrical construction and the theoretical aspect of constructing the related proof. This paper reports on data from a teaching experiment in which Grade 9 students worked on a series of challenging geometrical construction problems. Analysing the data through the lens of cognitive unity, we report on whether, or to what extent, geometrical constructions in particular encourage the uniting of student conjecture production and proof construction. Our analysis suggests that this is not automatic.

This paper reports on ideas developed during an analysis of data from a project involving young children (aged 5-7 years) in a whole-class situation using dynamic geometry software. The focus is a classroom episode in which the children try to decide whether two lines that they know continue (but cannot see all of the continuation) will intersect, or not. The analysis illustrates how the children can move from an empirical, visual description of spatial relations to a more theoretical, abstract one.