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.

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.

This paper investigates the introduction of the derivative notion and, specifically, the introduction of the derivative function, as a significant moment in the development of mathematical work on functions. In particular, we analyse the process of genericization that two Italian teachers conducted with their grade 13 students, in order to make them shift from the derivative at a specific point x 0 to the derivative as a global function in the x variable. Specifically, we analyse the role of the teacher in the semiotic genesis of this process and investigate the role of semiotic resources therein. As a result, we highlight the importance of conducting carefully this shift from the pointwise x 0 sign to the global x sign, in order to gain an actual shift in the perceived properties of the derivative function, which depends on the x sign as a variable. In conclusion, we connect our findings to the model of the Mathematical Working Space of functions, with particular regard to the "visualisation" process and the semiotic axis.Keywords: Derivative Function. Generic. Perspectives on Functions. Calculus. ResumenEn este artículo se estudia la introducción de la noción de derivada, especialmente de la función derivada, como un momento significativo en el desarrollo del trabajo matemático relativo a las funciones. Particularmente, se analiza el procesos de genericización que dos profesores italianos conducen con sus estudiantes de último año de liceo, con el objetivo de de hacerlos pasar de la derivada en un punto específico x 0 a la derivada como una función global de la variable x. Específicamente, se analiza el rol del profesor en le génesis semiótica del proceso y se investiga el rol de los recursos semióticos implicados. Como resultado, se resalta la importancia de conducir cuidadosamente el paso desde el símbolo puntual x 0 al símbolo global x, para ganar un cambio real en las propiedades percibidas de la función derivada, que depende del símbolo x como variable. En conclusión, conectamos los resultados al modelo del espacio de trabajo matemático de funciones, con un particular énfasis en el proceso de "visualización" y el eje semiótico.Palabras claves: Función Derivada. Generico. Perspectivas en funciones. Cálculo.

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