ΚΑΤΑΝΟΗΣΗ ΣΤΑ ΜΑΘΗΜΑΤΙΚΑ

1. ΚΑΤΑΝΟΗΣΗ ΣΤΑ ΜΑΘΗΜΑΤΙΚΑ R.Duval

22. Music, traffic signs, cooking, the way we dress up, natural language etc. All of them are activities or structured systems, which could be called as human systems. Despite these systems may seem to be completely apart from each other; they have something in common: they use signs in one way or another.

23. A sign is more than a simple notation, it is a sort of a complex relation in which diverse factors play. First we have to consider that every time we find a sign there is an implicit object involved. An object is a material or abstract 'thing' with certain content which is represented in the sign, then when we look at the sign: • it has content/meaning −of the object represented in a particular sign− called the signified and • there is a way in which the object is represented i.e. the material expression of the sign called the signifier.

24. We follow the definition of sign as a representation which is in constant changing depending on the semiotic system where it is being used (cf. Duval, 1998, p.1). Semiotics is the study, from a structural linguistic point of view, of a broad field of human sign systems (cf. WinslΩw, 2004, p.1).

25. In mathematics we deal with mathematical objects (e.g. graphs, numbers, fields, relations) only through the signs, i.e. only through their representations. Representations in mathematics are dynamic, are in constant changing because they experience transformations which go hand by hand with the mathematical activity, i.e. calculating, proving, or solving problems. This dealing, mobilisation and creation of signs, is what in semiotics is called semiosis.

26. The fundamental law of cognitive functioning of thought: (Duval 1995, p.4-5, 22) Semiosis is necessary for noesis, i.e. there is no noesis without semiosis. Noesis is the intentional act of intellect, in other words noesis can be defined as the action and the effect of understanding. Therefore we have that a subject is able to understand the signified of a sign only if he or she is able to mobilise it and create other representations from the same object. The main problem about the relation between semiosis and noesis concerns only to the semiotic registers of representation and not to all the semiotic systems. (Duval ,1995, p.22).

27. Semiotic representations 'There is no knowledge that someone can mobilise without a representation activity’ (Duval, 1995, p.15). What we need to mobilise knowledge? An intuitive answer can be: we need a medium (i.e. a way to “describe”, “treat” or "pack" information) to “contain” knowledge in order to be able to mobilise it.

28. Semiotic registers of representation We defined semiosis as the transformation and the creation of signs. Three cognitive activities play a main role in the activity of representation and at the same time are the fundamental cognitive activities of semiosis. These activities are: 1) Formation - of representations in a particular semiotic register either to express a mental representations or to recall a 'real' object. 2) Treatment - a transformation within the register. 3) Conversion - a transformation that results in a representation in another register.

29. Duval (1995, p.21) says a semiotic system of representation is called a register of semiotic representation when it satisfies the three cognitive activities which are inherent to all representations: 1. Constitutes a set of perceptible mark(s) that allow one to identify it (them) as a representation of something on a given system. 2. The representations can be transformed within the semiotic system (according to the rules in it) such that the obtained representation constitutes a gain in knowledge in comparison with the initial representation. 3. The representations can be converted, from one system into another, such that (the resulting representation) allows one to make explicit other meanings related to that which is represented. It is important to notice that not all semiotic systems are registers of semiotic representation.(cf. Duval, 1995, p.22)