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.

This article is centred in university mathematics education. It reports the results of an extension to a research literature study identifying the competencies that mathematics majors acquire as they learn mathematics within the context of designing, programming and using Exploratory Objects, a requirement in a core mathematics programme called Mathematics Integrated with Computers and Applications (MICA) at Brock University. Some of the identified competencies are used to ascertain parallels between Exploratory Objects and Microworlds. The MICA programme was instituted in 2001 and it remains a core programme for mathematics majors. This study suggests that the research on Exploratory Objects can benefit from the rich knowledge about Microworlds, and reversely the Microworld community could benefit from research on the sustained, systemic implementation of MICA.