Preparing prospective mathematics teachers to become teachers who recognize andrespond to students’ mathematical needs is challenging. In this study, we use the construct of critical incident as a tool to support prospective mathematics teachers’ reflection on their authentic fieldwork activities, notice students’ thinking, and link it to the complexity of mathematics teaching. Particularly, we aim to explore the characteristics and evolution of prospective mathematics teachers’ noticing of students’ mathematical thinking when critical incidents trigger reflective discussions. Critical incidents are moments in which students’ mathematical thinking becomes apparent and can provide teachers with opportunities to delve more deeply into the mathematics discussed in the lesson. In the study, twenty-two prospective mathematics teachers participated in fieldwork activities that included observing and teaching secondary school classrooms. The prospective teachers identified critical incidents from their observations and teaching, which were the foci for reflective discussion in university sessions. By characterizing the prospective teachers’ reflective talk in these discussions, we demonstrate the discussion’s evolution.In it, participants questioned learning and teaching mathematics and suggested alternate explanations. This characterization also shows that using critical incidents in the university discussions enabled the prospective teachers to link students’ thinking with the teacher’s teaching practices while supporting their reflection using classroom evidence. We emphasize the importance of descriptive talk in the discussion, which allows for deepening the prospective teachers’ reflections. Further, we explore the teacher educator’s contributions in those discussions, showing that the teacher educator mainly maintained the reflective talk by contextualizing the critical incidents and pressing the participants to explain further issues they raised in the discussions. Implications for mathematics teacher education are discussed.

Teachers face many issues when trying to integrate digital resources (DR) into mathematics classes. This article applies an identity-based perspective to understand teachers’ roles in the practices that evolve in such classes. We focus on Victor, a Greek mathematics teacher, who, when viewed from levels beyond the classroom, experienced to become a reform-oriented teacher and a designer of DR. We explore how these experiences of professional identity shapes and is shaped by his work with DR at the classroom level. We show how Victor’s identity changed from ‘being a Mathematics teacher who struggles with inquiry-based teaching’ to ‘becoming a mathematics Teacher who uses DR to support inquiry-based learning’ and outline what fuelled these changes. Our results suggest the importance of connecting an identity perspective to classroom interactions and mathematics.