# FROM LINEAR ALGEBRA TO GEOMETRIES: DIDACTIC PROPOSAL BASED ON THE ONTOSEMIOTIC APPROACH

L. Sbitneva1, N. Moreno Martínez2, R. Valdez1

2Cinvestav (MEXICO)
The main objective of our educational project is to provide the undergraduate students in the modality of Blended Learning with the didactic materials as a tool for friendly introduction to geometric interpretations of mathematical models which involves linear equations, so that students, without previous fundamental knowledge, could be able to develop gradually the geometric intuition and visualization of 2 and 3 dimensional geometry.

The majority of the undergraduate mathematics courses deals with the models which require coordinate description of the mathematical objects of different nature useful in applications in other branches of sciences, which suppose the familiarity with the basic concepts of numbers as well as relative concepts of geometric nature, in order to give a graphic representation of phenomena of linear algebra limited to two and tree dimensions, because the nature has provided us with a ready-made “demarcation line” in endowing us with geometric intuition for spaces of these dimensions.

Nowadays, coordinates appear as a result of the axiomatic method in teaching Linear Algebra courses, nevertheless almost all learning books suppose the previous knowledge on geometry, which is not the case due to educational innovations during last decades. Mathematicians have been long aware of such phenomena where the replacement of one axiomatic system by another equivalent but more suitable system, sometime leads to considerable simplifications. However, this does not always produces the desirable effect on student’s learning. Considering our experience at various Universities in Mexico, we have seen that many students are not capable to relate proper meaning to many mathematical objects in analytic geometry and solve problems operatively, nevertheless this situation can be improved in rather short time while the discursive practice on the historical development of geometries in past centuries.

This educational project is based on our experience to teach analytic geometry using the advantages of linear algebra because it allows all the development of “elementary geometry” to be set forth quite impeccably and even effortlessly” according to the Jean Dieudonné ´proposal (Dieudonné, J. 1969).

The sequence of the didactic materials for b-learning modality starts from a system of linear equations on two and three variables so that the so called “Analytic geometry” and is developed rather naturally. Our main objective is to introduce the fundamental geometric concepts like straight line, planes and interrelation between them, emphasizing different types of geometric properties.

The methodology for implementation of our innovations is based on some theoretical and methodological elements developed in the frames of the Ontosemiotic (OSA) approach (Malaspina, 2007; Godino, Contreras, & Font, 2007). In OSA approach the mathematical activity plays a central role and is modeled in terms of systems of operative and discursive practices. From these practices, the different types of related mathematical objects (language, arguments, concepts, propositions, procedures, and problems) emerge, building cognitive or epistemic configurations among them.

The comparison between cognitive and epistemic configurations demonstrated the success of the teaching sequence.
The viability of the proposal was verified through the analysis of responses to a questionnaire applied to our students.