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Euclid's geometry is well-known for its theorems concerning triangles and circles. Less popular are the contents of the tenth book, in which geometry is a means to study quantity in general. Commensurability and rational quantities are first principles, and from them are derived at least eight species of irrationals. A recently republished work by Johannes Kepler contains examples using polygons to illustrate these species. In addition, figures having these quantities in their construction form solid shapes (polyhedra) having origins though Platonic philosophy and Archimedean works. Kepler gives two additional polyhedra, and a simple means for constructing the “divine” proportion is given.

In the introductory part of the dissertation I deal with mathematics for the first classes of elementary schools and applicability for computers in education. The main part contains the methodical manual which helps teachers with orientation in the interactive textbook that I wrote and which describes the work with particular exercises. The next part is aimed for material verification in practice. In conclusion I evaluate results and analyze observations from the experiment.

This diploma thesis is focused on issues connected to possibilities of developement of mathematically talented pupils from the first grade. In the theoretical part there basic terms related to the education of talented pupils in th Czech Republic are summarised and explained. In next part motivation, processes of motivatiom and the most frequent usage of Math competitions as a part of the education in the Czech Republic are presented. The base of the practical part of this diploma thesis is a set of non standardized tasks and problems, whose target is to activate the cognitive developement of talented pupils. Part of the practical part of this thesis is a protocol about an experiment and analysis of the particular tasks and activities for talented pupils. The main aim of this thesis is to emphasise the importance of the developement of talented pupils and compile/form enriching tasks for talented pupils during Maths lessons. KEYWORDS Talent, talent models, motivation, cognitive process, non standardized tasks, aktivity.