This paper aims to analyze the structure and quality of prospective mathematics teachers’ (PMTs)’ argumentation when identifying and interpreting critical incidents from their initial field experiences. We use Toulmin’s model and recent elaborations of it to analyze the discussions that took place at the university where PMTs reflected on their recent classroom experiences. Our aim is to identify the structure of the argumentation and characterize the emerging warrants, backings, and rebuttals. Results indicate different argumentation structures and types of warrants, backings, and rebuttals in the process of PMTs’ interpretations of students’ mathematical activity. We discuss these findings from the perspective of noticing to identify shifts at the level of PMTs’ interpretations.

In this paper, we explore the process of becoming a teacher educator in the pedagogical use of digital tools in mathematics teaching. The study took place in the context of an in-service program during the trainees’ engagement in their practicum fieldwork activities including the process observation–reflection–design–implementation–reflection. We explored the features of this context that facilitated the trainees’ transition from the level of trainee educator to the level of teacher educator as well as the nature of the trainees’ documentation work for teachers. The results showed that observation of other teacher educators’ teaching in conjunction with reflection during the program’s respective sessions facilitated the trainees’ transition to the professional level. The identified operational invariants underlying the trainees’ designs concerned the focus of their observation in teacher education classrooms, the importance they attributed to the constraints and opportunities provided by the wider educational context and epistemological issues regarding the teaching and learning of mathematics with technology. The analysis of trainees’ designs revealed three kinds of documents (‘‘explanatory,’’ ‘‘instructive’’ and ‘‘facilitative’’) and corresponding roles of trainees during the implementation. These documents targeted different aspects of TPACK depending on the trainees’ conceptualizations of teachers’ roles either ‘‘as students’’ or ‘‘of students.’’