This work provides a characterization of the learning of graph theory through the lens of the van Hiele model. For this purpose, we perform a theoretical analysis structured through the processes of reasoning that students activate when solving graph theory problems: recognition, use and formulation of definitions, classification, and proof. We thus obtain four levels of reasoning: an initial level of visual character in which students perceive graphs as a whole; a second level, analytical in nature in which students distinguish parts and properties of graphs; a pre-formal level in which students can interrelate properties; and a formal level in which graphs are handled as abstract mathematical objects. Our results, which are supported by a review of the literature on the teaching and learning of graph theory, might be very helpful to design efficient data collection instruments for empirical studies aiming to analyze students' thinking in this field of mathematics.

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From a commognitive approach, this article focuses on the discourse generated by preservice primary teachers who are solving didactic-mathematical tasks. Our aims are to study the characteristics of the aforementioned discourse and, through these characteristics, identify whether a discourse close to the one of primary teachers is beginning to emerge. The sources of data were audio-recordings of group discussions and group reports. Two different discourses were identified in our results. One is the discourse generated by pre-service teachers when adopt the role of students of any level who have to solve a task proposed in the classroom. The other discourse is linked to the adoption of a role close to their future professional work. If we consider that the acquisition of a specific discourse enables future teachers to integrate into the community of practice of primary teachers, the role of the different discourses becomes a relevant element in teacher education.

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