Difficulties encountered by pupils in gaining competencies in algebraic calculation are well-known, and they have been addressed many times and from different point of views (Freudenthal, 1983, Tall & Thomas 1991, Kieran 1992).

The introduction of algebra requires in fact the development of a different way of thinking, which cannot be considered as a pure generalization of arithmetics (Mariotti & Cerulli, 2003). Differently from arithmetics, whose main goal is to do calculations so as to have a result, algebra offers an operational language for representing, analyzing and manipulating relations which contain both numbers and letters.

This diversity is explained by Sfard (1991) in terms of two different perspectives from which mathematical objects can be conceived: in a structural way (as objects) or in an operational way (as procedures or processes).

"There is a deep ontological gap between operational and structural conceptions [...] Seeing a mathematical
entity as an object means being capable of referring to it as if it was a real thing-static structure, existing somewhere in space and time. It also means being able to recognize the idea 'at a glance' and to manipulate it as a whole, without going into details [...] In contrast, interpreting a notion as a process implies regarding it as a potential rather than actual entity, which comes into existence upon request in a sequence of actions. Thus, whereas the
structural conception is static, instantaneous, and integrative, the operational is dynamic, sequential, and detailed." (Sfard, 1991).

We address the problem of gaining a structural approach to algebraic expressions, by adopting a cognitive-semiotic view-point,
provided by the theory of **Semiotic Mediation**. The theory draws from a Vygotskijan paradigm, and frames in particular processes of mediation carried out by means of artefacts in classroom context. In this perspective, the use of artefacts for accomplishing a task leads the individual to the construction of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a **semiotic mediator.** The **semiotic mediation** function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the
awareness of the **semiotic potential** of the artefact both in terms of mathematical meanings and in terms of personal
meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

The evolution of signs, central in the perspective of semiotic mediation, can be described in terms of transformations of registers of representation (Duval, 1995). Furthermore, the notion of semiotic register could reveal a useful tool both in the analysis of the semiotic potential of the artefact, as well as in the design of tasks both within Aplusix and in paper and pencil environment. Specifically, Aplusix provides two different kinds of representation systems for algebraic expressions: the “standard” representation (SR), and the “tree” representation (TR). They can be considered two different *registers of representation *in the sense of Duval. According to Duval (1995), a "register of representation" is a semiotic system (i.e. a system of signs) that provides for specific possibilities of transformation of the representations. In our case, natural language allows transformations such as paraphrases and reformulation, and standard and tree representation allow transformations that express operations among numbers and symbols. Beyond this kind of transformations, that pertain to a specific register and are called "treatments", there are transformations that occur between different registers: they are called "conversions". Treatments and conversions are central in mathematics:%2