The COVID-19 crisis has strongly affected the school system. In Italy, at-distance forms of didactics have been activated, changing the physiognomy of schools in terms of social interaction, practices and the identity of the individuals. In this paper, we address the issue of how teachers are facing the crisis: our focus is on assessment, as a key variable catalyzing personal history; beliefs; the interface between students; teachers and the school system. We study teachers’ beliefs as part of their identities and assessment as a fundamental variable of beliefs. A qualitative content analysis of the open-ended answers to an online questionnaire is carried out to understand the main characteristics associated with assessment by teachers and the obstacles to overcome in the context of long distance learning (LDL). The data show that teachers did not identify valid assessment methods for LDL during the lockdown, especially due to the lack of control over the students. A misconception emerges concerning the definition of formative assessment together with a new awareness of the possibilities offered by digital technologies regarding the individualization of didactics. This study helps to understand which teachers’ beliefs are related to assessment are and how they are shaped.

In this article, we present the first results of the project OPEN-MATH. The research project aims at acknowledging the need for learning environments with differentiation strategies for all. We developed a model for inclusive mathematics learning, based on the Theory of Objectification and a broad idea of differentiation realized through Open Learning. It poses an interdisciplinary research issue that requires the collaboration of two sub-disciplines pertaining to the area of educational studies: Inclusive Education and Mathematics Education. The results we present here are related to the dialogue between theory and practice, whose outcome is a teaching and learning model for inclusion in mathematics. The construction of the teaching and learning model moves along two complementary paths: (1) concerning the theoretical point of view, we implemented connecting theory strategies to network Open Education and the Theory of Objectification; (2) concerning the methodological point of view, we implemented Educational Design Research. The new teaching–learning model is the result of theoretical and methodological validation in real contexts according to an interdisciplinary approach. This study shows the strengths of interdisciplinary research for the pursuit of inclusive mathematics and high standards of learning.

The GeomBot is a drawing robot that combines the well-known strengths and opportunities offered by Scratch with those of Papert’s original robotic drawing-turtle. In this study we look at the GeomBot as a physical programmable artifact around which action research with a group of teachers and a researcher was carried out with the aim of designing, implementing and discussing geometry activities for primary school classes. The aim of this article is to investigate teachers’ positioning and perspectives with respect to the activities and the educational environment emerging around the Geombot in the action research. The action research meetings between September 2018 and June 2019 included nine primary school teachers from seven different Italian schools, who met regularly with the first author. After the design and experimentation sessions, in June 2019, the teachers shared their experiences during a final meeting and group discussion guided by the first author. The data collected from the teachers were analyzed using cultural categories from the Semiotic Systems of Cultural Signification, theorized by the Theory of Objectification, to identify the most significant features defining the teachers’ perspectives and identity. The seven emergent features cover teachers’ positioning with respect to: forms of rationality and language and languages: sensuous cognition and the use of ideal and material semiotic resources; accepted teaching practices, problems and situations; mathematical knowledge; the conception of the student; social interaction and forms of rationality; ethical issues; technology.

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.

In this article, grounded in the concept of interpretative knowledge (IK), which is well known in the literature, we introduce and discuss the construct of semiotic interpretative knowledge (SIK). This theoretical tool unfolds the interpretation of conceptual knowledge into a broader construct that intertwines the use of semiotic representations with the manifold aspects of mathematical learning. In the first part of the article, we first introduce Duval’s semio-cognitive approach, in which semiotics is the element characterizing the specific cognitive functioning of mathematics. On the basis of some classic examples from the relevant literature, we then show the necessity of introducing the semiotic component into IK to expand the interpretive power of the teacher. In the second part of the article, through the analysis of six episodes involving students with a specific learning disorder (SLD) engaged in mathematical activities, we show how the development of the teacher’s SIK emerges as a necessary condition for the implementation of inclusive teaching practices. This permits us to face the research question focused on how SIK allows us to understand the student’s behavior in a special needs educational context and provide effective feedback. In this sense, SIK is shown to be an intrinsic prerequisite of inclusive teaching, thus extending the concept of IK to the components of pedagogical content knowledge (PCK).

We present a theoretical study that allows us to attempt framing in an embodied perspective the effectiveness of the drawing robot GGBot in the learning of geometry. The aim of the article is to set the intertwining of activity, semiotics, perception, and knowledge at the crossover of Radford’s theory of objectification (TO) and Borba and Villareal’s notion of humans-with-media. Such a crossover is articulated in four building blocks: (1) processes of objectification and the role of semiotic means of objectification, where we state that digital artifacts such as the GGBot change the topology of the semiotic means of objectification; (2) cognition is sensuous and learning is a process of domestication of the eye, where perception is theoretically shaped by the interaction with GGBot; (3) GGBot and humans-with-media, where we outline new thinking collectives and their modes of activity; (4) domestication of the eye triggered by transitions between domains of activity. Each building block of our theoretical discussion is empirically anchored to four episodes involving primary school students’ learning geometrical figures using the GGBot. To conclude, we focus on two basic concepts of geometrical thinking that unfold in the shift between domains.