Francesca Ceragioli’s research while affiliated with Polytechnic University of Turin and other places

In teaching Maths, it is often difficult to combine the discovery dimension with that of training, even though both are needed for the acquisition of mathematical knowledge. For this reason, as teachers of the Maths Teaching Laboratory of Politecnico di Torino, we propose some activities focused on experimentation and discovery, with the aims of creating interest in Maths and of stimulating its study. The "Mathematical journey on a desert island" is an activity we have experimented with groups of students (from primary school to university) and families, as part of the Biennale Tecnologia event organised by the Politecnico di Torino. In a time of about two hours, we propose a path involving definitions, examples, and conjectures. The point of arrival is a special case of a theorem by the American mathematician Marston Morse: the activity we describe is inspired by one of his lectures, the video of which was very kindly made available to us by the Library of Institute of Advanced Studies in Princeton (New Jersey, USA). Depending on our interlocutors, the problem has been posed with different levels of formalism: here we will present a version of the activity suitable for children and families. The starting point is the description of points of maximum, minimum and saddle points of functions of two variables described as mountain peaks, lakes and passes. The formula we get to, links the numbers of the different types of points and involves only sums and subtractions. Participants in the activity are guided to the discovery of the formula through different steps in the form of observation and manipulation games. The proposed path is intended to exemplify what the mathematical approach consists of: posing a problem, visualising, studying examples and particular cases, formulating conjectures and translating them into an appropriate language, proving and asking new questions.

Per accompagnare gli studenti del primo anno dei corsi di ingegneria nell'apprendimento critico e consapevole della matematica, il Laboratorio di Matematica del Politecnico di Torino ha realizzato delle esperienze di apprendimento laboratoriali progettate intorno ad applicazioni ingegneristiche. I ragazzi e le ragazze sono coinvolti, tramite l'impiego di oggetti tangibili e il confronto tra pari, in attività che stimolano il processo di astrazione, oltre che di consolidamento e contestualizzazione dei saperi.

Social dynamics models may present discontinuities in the right-hand side of the dynamics for multiple reasons, including topology changes and quantization. Several concepts of generalized solutions for discontinuous equations are available in the literature and are useful to analyze these models. In this chapter, we study Caratheodory and Krasovsky generalized solutions for discontinuous models of opinion dynamics with state dependent interactions. We consider two definitions of “bounded confidence” interactions, which we, respectively, call metric and topological: in the former, individuals interact if their opinions are closer than a threshold; in the latter, individuals interact with a fixed number of nearest neighbors. We compare the dynamics produced by the two kinds of interactions in terms of existence, uniqueness, and asymptotic behavior of different types of solutions.

This work explores models of opinion dynamics with opinion-dependent connectivity. Our starting point is that individuals have limited capabilities to engage in interactions with their peers. Motivated by this observation, we propose an opinion dynamics model such that interactions take place with a limited number of peers: we refer to these interactions as topological, as opposed to metric interactions that are postulated in classical bounded-confidence models.

A learning experience that is concentrated on sight and hearing is one of the most evident features of online teaching. In this paper we describe an activity which is aimed to include touch in learning, even for high school and undergraduate students.

Social dynamics models may present discontinuities in the right-hand side of the dynamics for multiple reasons, including topology changes and quantization. Several concepts of generalized solutions for discontinuous equations are available in the literature and are useful to analyze these models. In this chapter, we study Caratheodory and Krasovsky generalized solutions for discontinuous models of opinion dynamics with state dependent interactions. We consider two definitions of "bounded confidence" interactions, which we respectively call metric and topological: in the former, individuals interact if their opinions are closer than a threshold; in the latter, individuals interact with a fixed number of nearest neighbors. We compare the dynamics produced by the two kinds of interactions, in terms of existence, uniqueness and asymptotic behavior of different types of solutions.

In the article, we present the tasks of the Maths lab at Politecnico di Torino that we tested for the first time in the academic year 2018–19, which were proposed for undergraduate Engineering students in their first year with the aim to integrate active learning into our traditional courses. The student activity in response to the tasks took place in classes of at most 25 individuals. The work was organized in small groups, mainly following a problem-solving approach and a central role was given to artefacts and technological applications. Here, we discuss the main motivations of our lab project, two specific tasks and some data collected by means of a questionnaire, in order to examine the interplay between tangible and virtual tools.

This work explores models of opinion dynamics with opinion-dependent connectivity. Our starting point is that individuals have limited capabilities to engage in interactions with their peers. Motivated by this observation, we propose a continuous-time opinion dynamics model such that interactions take place with a limited number of peers: we refer to these interactions as topological, as opposed to metric interactions that are postulated in classical bounded-confidence models. We observe that topological interactions produce equilibria that are very robust to perturbations.