The research described in this paper focused on the issue of describing and understanding how mathematical discussion develops in a hybrid learning environment, and how students participate in it. The experimental plan involved several classes working in parallel, with pupils and teachers interacting both in their real classrooms and in a digital environment with other pupils and teachers. The research was based on a rich set of data collected from the M@t.abel 2020 project, which was developed in Italy during the Covid health crisis. Based on Complementary Accounts Methodology, the data analysis presented in this paper involved specialists from the fields of mathematics education and inclusive education. In the study we considered the complexity of learning and the different elements that have an impact on students’ activity and participation, when they are engaged in mathematical discussions within the multilevel-digital environment that emerged due to the pandemic. These parallel analyses showed that ‘mathematical discussion in the classroom’ is a complex (and sometimes chaotic) phenomenon wherein different factors interweave. A complementary approach assists in developing a global vision for this dynamic phenomenon and in highlighting local episodes that are crucial in this interplay of factors. It is precisely in these episodes that the role of the teacher is fundamental: these episodes appear as catalysts for the different variables, with the teacher acting as mediator.

The issue of students’ difficulties in processing operations with percentages has been addressed in several international research studies from a qualitative perspective. In this study, we analyze students’ difficulties on this topic, focusing on the transition from middle school to high school with a mixed methods research design. We focus on students’ responses in a specific task belonging to the Italian large-scale assessment analyzed through the Rasch model, and we deepen the task analysis thanks to interviews, which enlightened image schemas and metaphors underlying students’ reasoning. From the qualitative point of view, the Rasch model shows that students’ difficulties in dealing with percentages is a macrophenomenon that involves the higher levels of competences. From the qualitative point of view, the metaphoric approach outlines the image schemas that foster the correct conceptualization of percentage and those that hinder their correct learning and can be one of the possible causes of the emerging aforementioned macrophenomenon.

Whole-class mathematical discussion in a problem-solving activity is recognized as a powerful pedagogical activity but also a challenge for teachers who must consider several difficulties that learners might face, particularly in terms of an overload of Working Memory and Executive Functions. This study investigates how the use of a digital platform (Padlet) can support participatory and inclusive mathematical classroom discussion. We proposed a teaching experiment based on graphical tasks anticipating integral calculus to grade 13 students, and we examined how the use of the digital platform plays a role in the construction and interpretation of new mathematical objects emerging from the activity. The use of Instrumental Genesis and Double Instrumental Genesis frameworks allowed us to make the affordances of the tool emerge. As a result, we got evidence of how mathematical discussion may develop as a network of interactions, feedback, and connection of input and discuss examples of how active participation and inclusion are enhanced by the tool affordances. Indeed, the digital platform allowed easy interaction, with many ways to represent and express the ongoing evolution of personal and shared meanings and the possibility to manage the time of the activity. This fostered students’ participation and students which did not participate in previous discussions were actively engaged in it.

In this study we analyse results of Italian standardized tests in mathematics integrating quantitative analysis based on the Rasch Model and didactical interpretation. We use specific graphs to analyse the trend of each answer as function of the students’ math ability. This approach led us to...