The focus of this research is how Sicilian state university mathematics professors faced the challenge of teaching via distance education during the first wave of the COVID-19 pandemic. Since the pandemic entered our lives suddenly, the professors found themselves having to lecture using an e-learning platform that they had never used before, and for which they could not receive training due to the health emergency. In addition to the emotional aspects related to the particular situation of the pandemic, there are two aspects to consider when teaching mathematics at a distance. The first is related to the fact that at university level, lecturers generally teach mathematics in a formal way, using many symbols and formulas that they are used to writing. The second aspect is that the way mathematics is taught is also related to the students to whom the teaching is addressed. In fact, not only online, but also in face-to-face modality, the teaching of mathematics to students on the mathematics degree course involves a different approach to lessons (as well as to the choice of topics to explain) than teaching mathematics in another degree course. In order to investigate how the Sicilian State university mathematics professors taught mathematics at distance, a questionnaire was prepared and administered one month after the beginning of the lockdown in Italy. Both quantitative and qualitative analyses were made, which allowed us to observe the way that university professors have adapted to the new teaching modality: they started to appropriate new artifacts (writing tablets, mathematical software, e-learning platform) to replicate their face-to-face teaching modality, mostly maintaining their blackboard teacher status. Their answers also reveal their beliefs related to teaching mathematics at university level, noting what has been an advantageous or disadvantageous for them in distance teaching.

This paper addresses two examples of MOOCs aimed at developing mathematics teachers' professional learning. The programme, named Math MOOC UniTO, was developed under the guidance of the three authors, in collaboration with some researcher-teachers from the University of Turin. This paper analyses the development of teachers' learning while attending the virtual environment of a MOOC, where all resources are available online and where peer interactions take place in asynchronous mode thanks to specific communication message boards. To analyse the data, two theoretical lenses were considered, namely, Meta-Didactical Transposition and Connectivism. The first lens allowed us to describe teachers' improvements at macro-level (praxeologies) and micro-level (agents); the second lens made possible the utmost consideration of the network of knowledge (learning is interpreted in the light of how nodes and connections within the network are determined dynamically). Using these theoretical lenses, we observed two different teachers' learning processes, namely, one that evolved dramatically because of the interventions (we call it an explosion), the other less proactively (we call it linear). We discuss them, presenting two different emblematic examples of data.

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