A lot of research has indicated that the way students are taught has a strong influence and impact on their ideas about the subject and how the subject should be taught. In 1908 at the International Congress of Mathematicians in Rome the great mathematician Felix Klein (1849Klein ( -1925 talked about the paradox of double forgetting. The essence of the paradox is the life experience of young teachers who have to forget about a lot of university training and its scientific thinking to successfully teach mathematics. This phenomenon currently persists at Slovak primary and secondary schools. The problem is especially striking in teaching geometry. Many Slovak and Czech teachers agree that geometry has many applications in everyday life, but there is not enough "real-life" everyday problems in Slovak mathematics textbooks which are structured according to the deductive approach to teaching and learning. Based on the concept of realistic mathematics education, we deal with the implementation of special geometrical tasks called topographical tasks in teacher training. Our findings are based on real experiences teaching through workshops devoted to topographical tasks which were realized during two school lessons. In the conclusion we analysed the benefits of this new approach to active teaching. The students´ solutions to the tasks and their methodological analysis are included, too.

Managed broadleaf deciduous forests are an important type of forest vegetation in Central Europe, also in the Western Carpathians. These forests are both economically and environmentally valuable. However, little is known about ecological species groups and the inter-specific associations of dominant species in temperate deciduous managed forests in Central Europe. Since the forest stands are in a managed landscape, they are not consistent with the traditionally recognized and used vegetation associations in the Western Carpathians. For these reasons our research contributes to understanding the consequences of broadleaf deciduous forest management. The aim of this research was the determination of ecological species groups and an investigation into the main environmental drivers, in order to explain the distribution of ecological species groups. The numerical TWINSPAN classification was selected to distribute 146 relevés to the five ecological species groups. Of these, 77 relevés were divided into two groups with Fagus sylvatica dominant, while 63 relevés were Quercus petraea dominant. Carpinus betulus, Tilia cordata and Fraxinus excelsior were dominant in 19 relevés. Constrained Analysis of Principal Components was used to explain the vegetation–environment relationship on three transects in the Male Karpaty Mountains. Altitude, pH, Ca, C, K and Mg were selected as the significant environmental drivers responsible for a large part of the species group variability (31.8%). The main requirement for sustainable forest management is knowledge of the vegetation–environment relationship and this research was focused on gaining such understanding. This knowledge can be used as a decision support tool for sustainable management in managed deciduous forests.

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