Abstract:

The EOS-Theory (Ontosemiotic Approach) is well known as a unified framework to the study the cognitive and instructional phenomena (J. Godino, V. Font, A. Contreras y M. Wilhelmi, 2006, Godino, Batanero & Font, 2006, 2007). In our talk we shall demonstrate the effectiveness of the Ontosemiotic approach while some experiments in the process of construction of mathematical knowledge corresponding to 3D-Geometry.
The 3D-Geometry course has been initially designed as a short course for the first year students at the Faculty of Sciences (Morelos State University, Mexico) . The main idea has been based on the Jean Deudonné’s point of view proposed in his book Linear Algebra and Geometry (1969), which requires no preliminary knowledge of Euclidean geometry, at least from the purely logical point of view, i.e. “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.” Nevertheless, apparently convincing arguments of J. Deudonné have encountered the series obstacles in the process of realizing his programming ideas.
Fortunately, the guiding ideas provided by EOS-theory allowed the authors to implement two introductory courses for the 1st semester concerned to 2D-geometry and to elements of linear algebra, before the 3D-Geometry course, which is taught now at the second semester. These two introductory courses embrace the representative system (mostly the Cartesian plane, taught at a school level) as well as a traditional semiotic representation of plane geometric objects, resulting in reaffirmation of a process of visualization that further allows articulations between representations. This modification of the program facilitated to achieve the main objective of our 3D-Geometry course: that 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. While working with these objectives especial attention has been given to the dialectic between ostensive and non-ostensive objects, since one of the EOS-argument states that the external representations (symbolic, graphical, linguistic, ostensive objects) are inevitable and dialectically accompanied by other non-ostensive mathematical objects and processes.
It allows to develop the teaching 3D-geometry starting with the consideration of the ostensive objects : points, base-vectors for the straight line and for the plane, then constructing their semiotic representations as vector and parametric equations, which are transformed by means of algebraic operations to the well established, so-called, canonical equations.
Furtheremore, these non-ostensive objects should be interpreted as the geometric objects by means of the reverse process, revealing the ostensive objects: points and base-vectors, although different from the initial ones.
Analyzing these canonical equations in the 2D-setting the students recover the well known geometric objects of plane geometry. But viewing the same equations as the semiotic representation of the objects in the space, it turns out that one has 2 forms of interpretation of the same non-ostensive object described by the same semiotic representation.

Keywords:

Ontosemiotic Approach, 3D-Geometry, semiotic representations.