Research has shown that the tools provided by dynamic geometry systems (DGSs) impact students’ approach to investigating open problems in Euclidean geometry. We particularly focus on cognitive processes that might be induced by certain ways of dragging in Cabri. Building on the work of Arzarello, Olivero and other researchers, we have conceived a model describing some cognitive processes that can occur during the production of conjectures in dynamic geometry and that seem to be related to the use of specific dragging modalities. While describing such cognitive processes, our model introduces key elements and describes how these are developed during the exploratory phase and how they evolve into the basic components of the statement of the conjecture (premise, conclusion, and conditional link between them). In this paper we present our model and use it to analyze students’ explorations of open problems. The description of the model and the data presented are part of a more general qualitative study aimed at investigating cognitive processes during conjecture-generation in a DGS, in relation to specific dragging modalities. During the study the participants were introduced to certain ways of dragging and then interviewed while working on open problem activities

Assuming that dynamic features of Dynamic Geometry Software may provide a basic representation of both variation and functional dependency, and taking the Vygotskian perspective of semiotic mediation, a teaching experiment was designed with the aim of introducing students to the idea of function. This paper focuses on the use of the Trace tool and its potentialities for constructing the meaning of function. In particular, starting from a dynamic approach aimed at grounding the meaning of function in the experience of covariation, the Trace tool can be used to introduce the twofold meaning of trajectory, at the same time global and pointwise, and leads students to grasp the notion of function.

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The notion of mediation, widely used in the current mathematics education literature, has been elaborated into a pedagogical model describing the contribution of integrating tools to the human activity, and to teaching and learning mathematics in particular. Following the seminal idea of Vygotsky, and elaborating on it, we postulate that an artifact can be exploited by the teacher as a tool of semiotic mediation to develop genuine mathematical signs, that are detached from the use of the artifact, but that nevertheless maintain with it a deep semiotic link. The teaching organization proposed in this paper is modeled by what we have called the didactical cycle. Starting from assuming the centrality of semiotic activities, collective mathematical discussion plays a crucial role: during a mathematical discussion the intentional action of the teacher is focused on guiding the process of semiotic mediation leading to the expected evolution of signs. The focus of the paper is on the role of the teacher in the teachinglearning process centered on the use of artifacts and in particular a dynamic geometry environment. Some examples will be discussed, drawn from a long-term teaching experiment, carried out over the past years as part of a National project. The analysis is accomplished through a Vygotskian perspective, and it mainly focuses on the process of semiotic mediation centered on the use of artifacts and on the role of the teacher in this process.