The Harmonic Series is one of the best known infinite series in mathematics. It plays an essential role in creation of students' conception about convergence or divergence of series and also about the speed of divergence. The paper presents and analyses some practical examples leading to the harmonic series.

Several theorems, including characterizations, concerning generated sets in d-σ-posets are proved. These results generalize some theorems of W. Sierpinski on the σ-rings and the monotone families generated by a family of subsets of a given set.

The article describes the influence of supporting materials on university students and their study results. Four hypotheses that were tested on first-year students in the course of Basic Mathematics will be introduced.

Let us consider a logic with a strongly adequate matrix and whose set of tautologies is empty. In such a logic the logical reasoning is performed only on the base of inferential rules, since the logic does not contain any axioms. From the most general point of view, we can identify the logic with the structural operation of consequence based on propositional calculus.

What’s the current attitude of secondary school teachers to geometric terminology? Are you primarily interested in answering the following questions: Which basic symbols of plane geometry do teachers prefer? Which geometrical definitions are used in their practical teaching? Are they willig to accept any potencial changes of terminology? In addition to this, which sources are used for their maths lesson’s preparation.

A tolerance relation of a lattice L is a reflexive and symmetric relation compatible with the operations of L. It is clear that every congruence of a lattice L is a tolerance L. A tolerance T of a lattice L is called a glued tolerance if its transitive closure T^∗ is the total relation L^2. We give the number of all tolerances and all glued tolerances on an n-element chain. We also estimate the density of congruences and glued tolerances among all tolerances in case of chains. Then we discuss more general questions.

We consider properties of the graphs that arise as duals of bounded lattices in Ploscica's representation via maximal partial maps into the two-element set. We introduce TiRS graphs which abstract those duals of bounded lattices. We demonstrate their one-to-one correspondence with so-called TiRS frames which are a subclass of the class of RS frames introduced by Gehrke to represent perfect lattices. This yields a dual representation of finite lattices via finite TiRS frames, or equivalently finite TiRS graphs, which generalises the well-known Birkhoff dual representation of finite distributive lattices via finite posets. By using both Ploscica's and Gehrke's representations in tandem we present a new construction of the canonical extension of a bounded lattice. We present two open problems that can be of interest to researchers working in this area.

In my presentation, I would like to introduce the use of computers in teaching mathematics. Specifically, creating a simple algorithms and programming of some basic mathematical terms.

Paper outlines a brief pilot study of financial literacy and deals with the issue of financial literacy from the viewpoint of teachers at secondary school. In an increasingly risky and globalized marketplace, people must be able to make well-informed financial decisions. Many studies show the increasing indebtedness of Czech pupils (vulnerable group consists people with mental disabilities) and the insufficient of education in the area of financial literacy.

Use of modern technologies in education is not only a question of support and development of the existing teaching methods, but also the question of how to innovate the content taught, how to set new types of problems to pupils, how to achieve that pupils gain new types of knowledge. The paper focuses on one specific way of the use of computers in mathematics classrooms - dynamic models. What is characteristic for dynamic models is the use of a new dimension - movement. The use of movement and its recording enables not only to present new problems to pupils but also to deepen and extend their existing knowledge. While working with pupils and students, the author tried to identify and subject to further study the areas where the use of dynamic models is of benefit to pupils and the kind of new knowledge pupils may gain thanks to the use of dynamic models. The author of this presentation focuses primarily on knowledge that pupils would hardly gain without the use of computer technology. The presentation characterizes four types of problems in which dynamic models efficiently introduce new knowledge that could be hard to visualize using traditional means and that would probably have to be derived analytically.

Theory of prototypes is part of cognitive science which describes certain representation of concepts in mind. Specifically it introduces prototypes, central instances of concept. We will introduce this theory in general, summarize previous research of prototypes in mathematics education and present results of our pilot study on prototypes of functions.

Vectors have several meanings in the science. Mathematicians, physicists and other scientists make use of the notion of a vector. The notions they make use of are different in different subjects. Here we present different meanings of vectors and give some hints how to understand them in different subjects of science.

In this contribution will be presented the concept of a new seminar intended for students of teaching study programmes. The aim of the seminar is, to students of teaching study programmes be able to find and create resources and tools for appropriate and effective motivation of pupils. They can use for this work less conventional forms and methods. We will present chosen proposals that has arisen on the seminar.

One of the most important skills we need to develop in future teachers of mathematics is the skill of problem-solving. Strategies are tools for solving problems and since antiquity, mathematicians have devised strategies to help them solve problems. It is obvious from their work that even then, mathematicians knew the strategy of generalization and that of reformulation of the problem. We will show you examples of the use of these strategies dating from the time of ancient Greece.

At the moment, several projects in Slovakia are taking place, that are focused on using tablets and smartphones in education. Interconnecting new educational forms within mobile learning and the GeoGebra software, brings new possibilities to teaching elementary and secondary school mathematics. This kind of teaching can motivate students and benefit in active learning, as well as raise effectiveness. This paper is focused on the possibilities of using GeoGebra in teaching specific mathematical topics in a mobile learning environment.

Similar to the symmetry of the quadratic parabola there is a kind of symmetry in the case of the parabola order three and order four, whihc can serve to discuss the nature of the solutions of the respective equations too. We will use GeoGebra to study and to visualise the given symmetries.

The aim of the talk is to investigate the approximation properties of Gauss-Weierstrass operators in the space $L^p$. In particular, the Voronovskaya type formula will be proved and some boundary value problems will be presented.

This paper describes Horner´s method for calculated values of a polynomial for pre-selected “x”. Focused on calculation with using of computer program “Microsoft Excel”. All procedures are based on formulas and their interconnections. This method is very simple, fast and transparent. Never used any Macros. The conclusions of this paper can be used as educational material for high school students to repeat (or. extension) knowledge about polynomials.

The space of strictly monotonic functions is defined as a set of all continuous real-valued functions of a real variable x from R which map one-to-one the interval R on the interval (a,b), where a and b are real or extended numbers. In this space the theory of Abel functional equations is studied. The Abel functional model reduces under specialization to a linear functional or difference equations. In the paper we introduce a new definition of a difference of a function which treats all classes of strictly monotonic functions.

In this paper we introduce some linear positive operators of the Baskakov-Durrmeyer type in the space of continuous functions of two variables. The theorems on convergence and the degree of convergence are established.

The article points out the possibilities of Cabri in the future mathematics teachers training with the constructivist approach known as an inquiry-based education. Very suitable area of mathematics for independent inquiry and students discovering is geometry. This approach is more enhanced by the potential of dynamic geometry such as e.g. Cabri Geometry. In this paper we describe on the example of circular inversion experiences with guided own discovering the principles and features of this transformation by students themselves.

Several tasks solved by the graphical method will be presented. Often, this method provides a very simple way to a quick solution of the task. Pupils do not always like computational solutions. The graphical solution also stimulates the imagination of a pupil.

The idea of an angle has several meanings at school mathematics. We propose the survey of different understandings of this notion with respect to the age of pupils, the level of school. We discuss the difficulties in defining and understanding of the notion of an angle. Of course, we discuss the topic with respect to Polish programmes of school mathematics.

J. Borsík and J. Holos defined three classes of porouscontinuous functions. We propose to study similar notion of symmetrically porouscontinuous functions. We investigate some properties of these functions and we present their connections with porouscontinuous functions.

Geometry is difficult for pupils at every stage of education. Because of this reason is important to develop space imagination from the early age and to use proper didactic tools for this. One of them is puzzle Tangram that is ideal connection of game and didactic tool.

In the last years, the natural sciences are not very popular, although the institutions of non-formal education have been founded in Europe (with the support of the EU), offering insight into these subjects in a popular way. The success is based on an excellent concept of a programme, which must meet a number of seemingly unimportant parameters. In the paper there will be discussed not only the creation of the overall concept of activities but there will be also introduced the particular activities which were created specifically for the newly opened Science and technology center in Ostrava - creative science, science boxes, lessons and more.

Our paper is dealing with the results of research on pupils' attitudes towards mathematics. We study changes in the attitudes between pupils in different grades of lower secondary education. Analysis of these changes is the source of information beneficial for potential improvement of these attitudes.

The historical textbook “Geometrical notions for primary school” offers very interesting instructions for every teacher that explains geometry. The author Franz Močnik (1814-1892) introduces us teaching, where students discovers new knowledge by themselves. Everything is very demonstrative and students can better understand context in lesson.

The approach CLIL is a relatively new trend in education. It is a combination of content and language learning. The article is focused on using the CLIL in the Czech Republic and it presents research results of the influence on students' motivation in mathematics lessons with CLIL.

The paper deals with the fact that in the certain phase of high school students’ mathematical development their teachers are forced to make the formalization of trigonometric functions, for the geometrical definition, which students usually meet, is intuitive - the existence of the bijection of the interval [0, 2π] on the set of all points of the unitary circle is assumed. There is a cosine function defined by means of d’Alembert functional equation in this article.