# VISUALIZATION OF NON-OSTENSIVE MATHEMATICAL OBJECTS AS GEOMETRIC INTERPRETATIONS OF SEMIOTIC REGISTERS IN ALGEBRAIC SETTING

The concept of the determinants naturally emerges in the prosess of solving of linear equations. Being involved in different contexts and situation problems, the traditional algebraic representation of a system of linear equations takes different semiotic representations such as so called matrix form or the form of linear combination of vectors. Interpretation of every one of these semiotic registers usually depends on the contexts, for example, a vector form register can be treated as the linear dependency of the elements of a vector space or as the representation of the image of an element under a linear transformation (Grossman, 1987, p.273), as well, as a representation of the coordinates with respect to different bases. The theory of OSA allows to study difficulties related to the semiotic function (Font, et al. 2007), which students encounter in their independent studies

The study of determinants in the theoretical aspects presents serious problems for the students of the Linear Algebra course. Being aware of the crucial role which the visualization plays in learning demand to achieve the comprehension of a concept, we suggest some geometric interpretation which may be used to produce images that accompany the mathematical ideas employed in different context.

In order to describe the various processes involved in the competencies development we employ the Ontosemiotic Approach (OSA), which provide us a generalized point of view both of mathematical objects and their relations, in terms of institutional and personal practices, leaving aside the prospect conceptualist/formalist in that mathematical objects are reduced to their definitions and logical relationships with other objects (Godino & Font, 2007), and representation, considering the mathematical ontology and semiotic functions (Font, et al., 2007).

The mathematical ontology that is proposed in the Ontosemiotic Approach, along with the notion of semiotic function, can contribute to achieve the main goal of mathematics instruction. Ecology of meanings provides an appropriate way to realize different types of analysis on the different didactic levels in the production and communication of mathematics.