This study aims to explore how teachers from four countries—France, Israel, Italy, and Germany—manage their teaching–learning activity in the context of lockdown due to the COVID-19 pandemic. About 700 teachers from the four countries participated in this study. They were given an online questionnaire that involved 22 open-ended items, in which they were requested to complete the items that were structured taking into account the relationships between teacher, students, mathematics and resources. The qualitative analysis of teachers’ answers was carried out, referring to both the meta-didactical transposition model and Bishop’s framework on values to investigate teachers’ teaching–learning activities and the reasons underlying their choices. The empirical analysis suggests four tasks corresponding to the main challenges that teachers had to face during the time of lockdown: (a) managing distance learning to support students’ learning through specific methodologies; (b) managing distance learning to develop assessment; (c) managing distance learning to support those students that face difficulties and/or are living a difficult situation/developing inclusive teaching; and (d) managing distance learning to exploit its potentialities for fostering typical mathematical processes. The values that motivated the teachers to change their teaching–learning activities are discussed, conclusions of how the time of lockdown affects the mathematics teaching is drawn, and finally, recommendations and insights from this study are shared.

In this chapter we report on an ongoing international research about MOOCs for inservice mathematics teacher professional development. We describe and analyse two different experiences of this kind: two seasons of the Italian Math MOOC UniTo (Geometria MOOC and Numeri MOOC) and two seasons of the French eFAN Maths MOOC. Both MOOCs aimed at supporting teachers' professional development through a suitable mediation of technology. They also aimed at triggering as much as possible the teachers' engagement in order to develop one or more teachers' communities of practice. Starting from our methodological choices, as trainers, we propose some reflections about design principles of MOOCs for mathematics teacher education using the theoretical lens of the Meta-Didactical Transposition (Arzarello et al. 2014). These reflections focus on trainers' practices to foster participation and collaboration among trainees and to assess efficiently this kind of engagement.

Meaning Equivalence Reusable Learning Objects (MERLO) items are a new kind of didactical tool that can be designed by teachers and used in classes, in tasks aimed at engaging students in deep reasoning, exploring and arguing about mathematical concepts. The interactions across communities of mathematics teachers and educators in on-line professional development were studied with attention to the phenomenon of boundary crossing of MERLO items, viewed as boundary objects in this article. The study analyseda first (international) crossing of the object that passedinstitutional boundaries (between Australian and Italian school systems), and a second crossing of the same object –inside the Australian institutional community – that passed boundaries (from static to dynamic representations) that we interpret in terms of method both for teachers and for researchers. This passage is possible due to the use of dynamic geometry software (GeoGebra) that supported a modification in the design of the item. The analysis shows boundary crossing as a process of transformation that can influence a modification (more or less stable) in the practices of the teachers involved and provides a deep research insight in relation to existing theoretical frameworks.

Formative assessment is a process that can inform both teachers and students of their understanding of knowledge at stake. Technology can enable data about student understanding to be collected, organised and shared in novel ways. The FaSMEd project aims to study how technology can play an effective role within a formative assessment process. This article presents a case study that allows us to understand better how a teacher processes data from student use of technology (e.g., tablets, a student response system, interactive whiteboards) and how he uses it to inform his teaching. Our observations in a grade 9 tablet-using classroom show that technology can be considered as an element of the classroom milieu facilitating the process of formative assessment. Both the teacher and the students took advantage of the interpretation of data: the teacher modified his teaching regarding students' responses, while the students improved their learning in response to the teacher's feedback.