L. Sbitneva1, N. Moreno Martínez2, R. Valdez Delgado1

1Morelos State University (MEXICO)
2CINVESTAV-Matemática Educativa (MEXICO)
Blended Learning in Higher Education has shown to be perspective since it combines the traditional face-to-face teaching and the advantages of online learning. At the Morelos State University there has been created a flexible educational model in order to provide the favorable conditions and learning opportunities for the increasing number of aspirants.

The implementation of virtual and blended learning systems is a process both innovative and complex, as well, it requires some new approaches in the design of the resources for particular practices in distance education. There appear new aspects related to independent study of contents of the courses in the process of the construction of knowledge by the students without traditional intervention of a professor.

We would like to present some innovations suggested by the theory of Ontosemiotic Approach (OSA) in order to enhance the Linear Algebra course management and to create didactic materials (resources) for virtual and blended learning at the Morelos State University.

The main learning objective for students of the standard course of Linear Algebra and Applications is to grasp the main concept of linear space and to make systematic use of systems of linear equations as a tool in different contexts in a variety of applications.
A system of linear equations is a complicated object that appears in mathematical practices and is represented by different sets of symbols, which are put together according to certain rules of mathematical operations. Being involved in different contexts and situation problems, the traditional algebraic representation in the form 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, 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 which students encounter in their independent studies, for example, there has been introduced some cognition dualities, in particular, expression-content (semiotic function), which permits to detect so called semiotic conflict due to wrong meanings of an expression.

Thus in our opinion, 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. Ontology of meanings provides an appropriate way to realize different types of analysis on the different didactic levels in the production and communication of mathematics.