In our talk we would like to present some results of application of the Ontosemiotic Approach (EOS Theory), considered as a unified framework to the study of the cognitive and instructional phenomena which we have encountered while some experiments in teaching Geometry.

The Geometry 1 course has been designed as a short course for the first year students at the Faculty of Sciences (Morelos State University, Mexico) in order to introduce the basic knowledge of vector algebra and to demonstrate the power of the methods of vector algebra in applications to teaching analytic geometry as well to Euclidean geometry.

Thus, searching for an adequate instrument we have found this EOS theory (Ontosemiotic approach) which is proposed by the authors Juan D. Godino, Vicenc Font, Angel Contreras y Miguel R. Wilhelmi as a coherent articulation to the well known theories such as Theory of Didactic Situations (TSD) Brousseau, Anthropological Theory of Didactic (TAD) Chevallard, the Semiotic Representation Registers (RRS) Duval, and the theory of Conceptual Fields (TCC) Vergnaud to study the process of mathematical cognition.

In this talk we shall give some examples demonstrating the effectiveness of this approach (EOS theory) in order to classify the numerous situations emerged due to intervention of the mediator and the resulting innovations produced as a result of the interaction between the professor, students and the accompanying professor, during the experiment (investigation-action) in teaching the Geometry 1 course.

The idea of teaching the Geometry 1 course has been based on the Jean Deudonné’s point of view proposed in his book Linear Algebra and Geometry, that requires no preliminary knowledge of Euclidean geometry, at least from the purely logical point of view.

It happens, that at present time, in Mexico there is found an adequate situation to undertake an experiment with these innovative ideas, because the traditional Euclidian geometry, which is the base of all the learning books on analytic geometry, is not included in the mathematical programs at the higher school level. (In particular, the students have no knowledge of space geometry, only some fragmentary aspects of plane geometric figures).

Thus, the main objective of this basic course is to introduce the fundamental geometric concepts (straight lines, planes and interrelation between them emphasizing “affine” type of geometric properties and “metric” type of properties) and to work with the models that admits coordinates, demonstrating how the algebraic and analytic results interplay with geometry.

Nevertheless, apparently convincing arguments of J. Deidonne have encountered serious obstacles in the process of realizing his programming ideas; some relevant principal points of Jean Deudonné are the following:

Starting with extremely simply stated axioms (in contrast with those of Euclid-Hilbert) everything can be obtained in a very straightforward manner by a few lines of trivial calculations.

It should be a relatively simple task since there are few mathematical concepts simpler to define than those of vector space and linear mapping: fortunately, nature has provided us with a ready-made “demarcation line” in endowing us with geometric intuition for spaces of two and tree dimensions. It is therefore possible to give a graphic representation of all phenomena by introducing these concept in some experimental form.