In the sections that follow, I shall present three examples of instrumental learning trajectories, in which the interdependencies among the tools and meanings are analyzed. Dynamic Geometry allows for the creation of interdependencies and intra-dependencies between mathematical objects, diagrams and tools. Instrumental trajectories can show us the interdependencies/intra-dependencies that exist or can be created between different tools.

Instrumental trajectories are not just construction instructions, or a set of information which provides the properties of the figure as the figure is constructed. I point out the role of instrumental decoding in a static or dynamic environment, and how the competence of the participants (students-researcher/ teacher) can influence the holistic result of the learning process by creating interdependencies/intra-dependencies during the construction of instrumental learning trajectories.

In the current study the theoretical notion of instrumental learning path or trajectory is analyzed through examples based on a research synthesis. The central idea is the following: Do the tools of Dynamic Euclidean Geometry determine a new kind of Geometry? Is Dynamic Euclidean Geometry a new kind of geometry? Does it have its own axiomatic system or its own undefined terms? In the paper, the notion of an instrumental learning path/trajectory is introduced as the interdependence/intra-dependence between dynamic tools, diagrams and mathematical objects during an instrumental decoding process. These considerations provide a theoretical basis for the idea that, in order to solve a mathematical construction problem in Dynamic Euclidean Geometry, we have to build up the interdependencies of tools in various sequential steps (based on theorems and definitions and the competence in using tools) which can be linked to the level of our conceptualization. Dynamic Euclidean constructions will be considered using pseudo-Toulmin' diagrams. The dynamic notions (e.g., dynamic point, dynamic segment, instrumental decoding, hybrid-dynamic objects, active/ 'alive' representations etc.), are taken as given and form the specific /particular theoretical basis for the constructive processes. Competence in the DGS environment depends on the competence of the cognitive analysis as students seek to decode their ideas using the tools provided by the software. The present paper attempts to bridge the world of DGS technology with the world Euclid bequeathed to us in his 'Elements'.

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