Ontological Framework in an Integrated SOI (Structure of Intellect)-Turing Machine-Kant Knowledge System

A ‘how to use’ framework for the integration of the concepts according to Piaget’s operating cognitive schemes, along with the Intellect Model Structure (SOI), the Diagram of States of the Touring Machine and Kant’s Cognitive System, is the result of the research presented as a course in: Arithmetic-Algebra, Set Theory-Symbolic Logic, Combinatory Geometry-Euclidean Geometry. The building of Piaget’s operational schemes is the learning objective. These three schemes are the basic foundations of the overall mathematical cognitive system; which from the psycho-pedagogical perspective, in order to be assimilated them, they have to be introduced not as symbolic contents but as figurative ones, that is, the students have to construct a set of mathematical objects for each scheme. The comparison of this process with Kant’s Cognitive System matches up within its initial phase; therefore the student is able to build up the set of objects based on their description, showing that the sole description of characteristics of the mathematical objects it is the necessary guidance for an intuitive process. Starting on the Intellect Structure Model’s perspective, the virtual learning process starts with figurative contents (the objects) following the SOI’s product sequence. When the student is building up a product it is in a state of a Touring-Machine (a knowledge state), making transitions in order to building up the next product in the sequence. Every transition in the Touring Machine is a transformation or a translation to a different mathematical representation, which represents the mathematical thinking to be developed in the student.