The word ‘mathematics' is said to go back to Pythagoras, who called his most advanced disciples mathematikoi (mathmatikh). In the present context, this word will be interpreted in the following way:
Within the language of itself, mathematics can be described as the study of homomorphisms (= structure-preserving transformations) between the brain and its environment - including the brain itself.
Life is structure, and since mathematics is the language of structure, it is the language of life, the ultimate ruler of the subspace within which life can be talked and reasoned about.
Mathematics is a feeling, a sensitivity and an awareness of structure, which is planted in each and every one of us, like a seed of the cosmic consciousness. To practice the art of mathematics is to be involved in a purely mental process, which has surprisingly strong connections with the surrounding physical reality.
When one is developing ones mathematical understanding, both halves of the brain work together in the process of constructing combinations of mental fantasies that are tested for logical functionality. The right half of the brain is fantasizing, and the left part is analyzing and testing the logic of the suggested ideas. Only the ideas that survive the logical examination are elevated to the status of mathematical truths. Mathematics can therefore be described as logically tested fantasies. It offers a powerful means for its devotees to overcome some of their sensory limitations and contemplate the inner profundity of the structure of the universe.
The First Class Mathematics Project aims to convey an image of mathematics as such a double-brained mental activity. In order to develop the ability for structural thinking, it is of the utmost importance that all students are encouraged and supported in their attempts to develop their own mathematical fantasies during their entire learning process. This is only possible if they are confronted with interesting examples of ‘good mathematics' during every stage of their mathematical education. An illustrative example of the effects of such a confrontation is given by the so called Rubrik´s cube (known simply as “the cube“), which was such a fascinating mental torment to the kids a couple of years ago. They got into doing their own advanced forms of algorithmic mathematics - but only during their breaks and time of leasure from a school-system that was totally incapable to realize what was going on, let alone to make use of it in order to promote the mathematical interest of the students.
By the aid of modern computers, many exiting mathematical structures can be animated and brought to interactive life in ways which open up new and exiting pedagogical possibilities. Today there is a large number of computer-based mathematical tools that empower a student to explore mathematical concepts in interactive and dialectical ways, i.e. in an ongoing dialogue with the computer program (and of course with the teacher). It is the aim of the First Class Mathematics project to contribute to the developments of such tools, as well as to promote a pedagogical strategy that makes it possible to integrate them in the educational process in a way that stimulates interest to learn more about the underlying mathematical structures.