Teacher education is central to the development of the professional knowledge of pre-service teachers. The main goal of this paper is to reflect on the development that the analysis (done by a group of pre-service secondary teachers) of a set of tasks, based on elements related to domains of KTMT-Knowledge for Teaching Mathematics with Technology-can bring to the knowledge of pre-service teachers of mathematics. Specifically, the goal was to investigate the following questions: (1) What are the factors that guide the pre-service teachers' task discussion? (2) Which KTMT domains are emphasized by pre-service teachers during task discussion? The elements taken into account are the characteristics of the tasks (focus on cognitive level, structuring level and technology role), the use of representations (focus on balance and articulation of representations), and the equilibrium between experimentation (focus on digital technology affordances) and justification (focus on argumentation and proof). The methodology of this case study involves a qualitative approach. The main conclusions suggest that influences in the pre-service teachers' discussion of tasks fell into the following categories: the potentialities of technology, the type of tasks, and the prospective teachers' experience with a set of tasks, and analysis of some real students' reports. With regard to KTMT, although it was possible to identify some global development, Teaching and Learning and Technology Knowledge was the domain in which stronger development took place.

Proof plays a central role in developing, establishing and communicating mathematical knowledge. Nevertheless, it is not such a central element in school mathematics. This article discusses some issues involving mathematical proof in school, intending to characterize the understanding of mathematical proof in school, its function and the meaning and relevance attributed to the notion of simple proof. The main conclusions suggest that the idea of addressing mathematical proof at all levels of school is a recent idea that is not yet fully implemented in schools. It requires an adaptation of the understanding of proof to the age of the students, reducing the level of formality and allowing the students to experience the different functions of proof and not only the function of verification. Among the different functions of proof, the function of explanation deserves special attention due to the illumination and empowerment that it can bring to the students and their learning. The way this function of proof relates to the notion of simple proof (and the related aesthetic issues) seems relevant enough to make it, in the future, a focus of attention for the teachers who address mathematical proof in the classroom. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

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Using technology to get a graphical representation of functions requires the choice of a suitable viewing window. The literature suggests teachers’ knowledge as being central in this process, but does not tell us much about how the teachers should manage this. This study aims to improve understanding about the teachers’ choices related to the viewing window on the study of functions. It also intends to illustrate how this analysis can be a useful way to access aspects of teachers’ knowledge based on the framework of Knowledge for Teaching Mathematics with Technology (KTMT). The study adopts a qualitative methodological approach, undertaking case studies of two teachers. The findings recognize the importance of carefully sequencing the students’ contact with this kind of situation and the impact of the viewing window over the teachers’ choices of tasks, pointing to the relevance of considering situations where the students actually face the difficulties. This highlights the relation of teachers’ choices of viewing window to their knowledge of different KTMT’s domains, and specifically the relevance of knowledge of different ways to change the window; of the difficulties faced by the students; of the most suitable tasks (and required viewing window) at each moment; of the impact of the viewing window on the appearance of a graph and of the different situations associated, from the mathematical point of view.

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