This paper presents a study aimed at investigating the didactic potentiality of the combined use of two different kinds of artefacts for the purpose of constructing and conceptualizing mathematical meanings related to the notion of axial symmetry. In our view, the process of meanings construction can be fostered by the use of adequate artefacts, but it requires a teaching/learning model, which explicitly takes care of the evolution of meanings, from those personal, emerging through the activities, to the mathematical ones, aims of the teaching intervention. The main hypothesis of this study is that a potential synergy may occur between the use of different artefacts, synergy that can foster the integration of different and complementary meanings providing a rich support to the development of the expected mathematical meaning. The Theory of Semiotic Mediation offers the theoretical framework suitable to design the teaching sequence and to analyze the collected data. Specifically, the construct of semiotic potential provides the tool for describing the potentialities of the two artefacts, while that of didactic cycle offers a model for the organization of the different activities. The paper reports on a teaching sequence and its implementation in a teaching experiment, involving pupils at fourth grade level. We describe them, within the chosen theoretical framework, and provide the analysis of key episodes of the teaching sequence. We show evidence supporting our main hypothesis about the combined use of an artefact that can be manipulated (paper and pin), and a digital artefact (Dynamic Geometry Environment) in the development of the notion of axial symmetry and its properties: the combined, intentional and controlled use of the two artefacts may develop a synergy, so that each activity enhances the potential of the other.

On the basis of the hypothesis that has emerged from recent research studies, this article discusses the new pedagogical training model for primary school pre-service teachers (PSTs), namely the Meta Discussion on a Pedagogical model (MDPm). The study was conducted with 400 PSTs enrolled in a primary education mathematics course. The MDPm is a collective discussion involving PSTs and their facilitator about the prior teaching experience based on a Mathematical Discussion (MD) model. In this article we try to give a more detailed definition of MDPm by analyzing different examples drawn from a collection of data gathered in the frame of different teacher training experiments. The results indicate that engaging PSTs in both a direct MD and a subsequent reflective meta-discussion leads to enhanced awareness of the teacher’s role and a more robust conceptualization of the MD model. These experiments aimed to help future teachers recognize the characteristics of their instructional actions within a pedagogical model, engaging them actively in their training, in and from practice. In this setting, PSTs assumed a dual role as both students and future teachers. The research has been conducted in a technological environment, which assumed a fundamental role in supporting the processes involved in the teaching experiment. The examples are analyzed through a specific lens aimed at identifying key elements supporting and refining the MDPm, with the aim of more precisely characterizing it.