In the application of the principle of the synoptically play important role visualization and the creation and application of models to demonstrate and study the various objects and phenomena of the real world. Using the tools for visualization and modeling we are developing visual cognition student creative and logical thinking. The artist in the article dealing with the programming language Imagine and its options for the creation of various educational software suitable for visualizing.

We say that a function G is a control function of a map F when a norm of monomial difference of a function F is majorized by G. We present some applications of stability results concerning the conditions under which a map F with the control function G is uniformly approximated by monomial function as well as the conditions under which a norm of the values a map F is majorized by the sum depending on the control function and some non-negative constant common for all arguments of a map F and on the norm of the arguments of a function F.

This paper deals with the possibilities of using the Interactive Board SmartBoard at primary and pre-primary level of education. It introduces the education of teachers and its conception in mentioned area, it publishes the results of survey that was accomplished with students who used the Interactive Board at their teaching practice and it gives valuable illustrations of real use during the lessons.

We consider B-rings, a generalization of the notion of Boolean algebras, presenting their various properties. In particular, we discuss properties of differences. We extend Renyi’s theory of conditional probabilities to the case of B-rings.

Literature revealed that the most problematic content knowledge for the prospective elemantary teachers is geometry (Umay, Duatepe & Akkuş, 2005).The reason for this issue can be their failure in geometry and their negative attitude towards geometry (Cunningham & Roberts, 2010; Çetin & Dane, 2004, Sandt & Nieuwoudt 2003, Mayberry, 1983; Roberts, 1995). For the Turkish prospective teachers, it can be stemmed from their lack of knowledge on new changes in geometry curriculum. One of these changes have made on the topic of symmetry in elementary mathematics curriculum. The aim of this study is to answer the question of “how well preservice elementary teachers know and construct symmetry lines”. In other words this study is attempting to analyze preservice elementary teachers’ content knowledge which is very crucial for their career. This was a descriptive study in which 140 preservice elementary teachers were given a task on drawing symmetry lines of two figures. They were also asked to draw the symmetry of a shape considering the given symmetry line. Participants had already taken all the mathematics and teaching method courses in their program. The task was taken from a 5th grade mathematics textbook so preservice elementary teachers were supposed to answer it easily. Findings have showed that most of the participants were not successful on determining number of symmetry lines for the given geometric figures. They were better on drawing symmetric part of a given figure according to a given symmetry lines. Detailed results with suggestions will be displayed in the presentation.

In the talk we will survey the innovation of a basic mathematical course, Introduction to Mathematics. This subject is intended for students of the Faculty of Science of the University of Ostrava in bachelor programmes Mathematics, Applied Mathematics, Computer Science and Applied Computer Science in all forms of study. We particularly describe the innovation using modern ICT components that help students better understanding of the subject.

In probability theory, random events model the algebraic part of random experiments. In the classical probability, axiomatized by A. N. Kolmogorov, random events form a field (σ-field) of sets and correspond to the philosophy and logic of Aristotle and G. Boole. In our talk we present some ideas about classical random events and their generalization to fuzzy random events based on the multivalued logic.

By the degree of completeness of a logic we mean the cardinality of the set of all its axiomatic extensions. The cardinality of the set of all extensions of a logic is called its degree of maximality. Three-valued propositional logic W, introduced by Piróg-Rzepecka, belongs to the class of nonsense-logics. We denote by W^d the logic dual to the logic W in Wójcicki’s sense. It was shown that the matrices of this logic and of the logic W differ in their sets of distinguished values. We establish the degree of completeness and the degree of maximality of the logic W^d. We prove that the logic dual to the classical logic and the inconsitent logic are the only axiomatic extensions of this logic. In order to find its degree of maximality we consider 12 expresions built on one variable and 144 sequential rules. (This research was partially suppoorted by the Polish NSC grant 2011/01/B/HS1/00944.)

New Australian mathematics curriculum was developed in the years 2008 – 2010. Due to this, ICE-EM Mathematics series of textbooks for Year 5 – Year 10 were developed. This talk is focused on the topic Three-dimensional Objects. A short overview of its inclusion in Australian and Slovak mathematics curricula is given and the way of its different adaptation in Australian and Slovak mathematics textbooks is demonstrated.

A novel treatment of the canonical extension of a bounded lattice is presented which is in the spirit of the theory of natural dualities. At the level of objects, this can be achieved by exploiting the topological representation due to M.~Ploščica, and the canonical extension can be obtained in the same manner as can be done in the distributive case by exploiting Priestley duality. To encompass both objects and morphisms the Ploščica representation is replaced by a duality due to Allwein and Hartonas, recast in the style of Ploščica's paper. This leads to a construction of canonical extension valid for all bounded lattices, which is shown to be functorial, with the property that the canonical extension functor decomposes as the composite of two hom-functors.

This article focuses on the possible use of non-traditional methods for solving of mathematical problems. The main methods published in the work are those, which uses constancy of some quantities values and techniques which are based on keeping of certain but not always readily seen features of the observed objects. The first part of the work provides key definitions such as invariants, semiinvariants, Fermat’s Method of Infinite Descent. The second part of the work contains a set of didactic tests focusing on abilities relating to invariants and semiinvariants in the problem–solving.

This contribution describes the inclusion of selected mathematical games into the lessons as well as factors influencing these games. Last but not least, this contribution describes research tracking how you can affect an individual level of logical thinking through these games.

The paper presents the basic goals and strategies intended dissertation thesis named „Use of material didactic resources in teaching“. Also shows present results of preliminary research.

The aim of this thesis is to unify the findings about the general secondary education examination (GSEE) in mathematics until 2010 and the national general secondary education examination (NGSEE) emphasizing the exam in mathematics which took place in 2011, to find out and compare opinions of secondary school maths teachers as well as university students of mathematics on the introduction of the NGSEE, and, finally, compare and evaluate demands on students in selected secondary schools in Opava, success of students in the NGSEE in mathematics, etc. Results of researches are processed in this thesis. The first one deals with the information provided by selected secondary schools in Opava; the second research, which was carried out as a probe , includes information concerning opinions of secondary school maths teachers on the NGSEE. Finally, the third research is a kind of complement to the second one; it was carried out among university students of mathematics. Results of researches have been processed in the thesis and they have been completed with a detailed description of both the above mentioned general secondary education examinations including their progress as well as type examples and a brief history of the GSEE.

I will present the possibilites of using Lagrange interpolation formula for teaching especially in the last year of high school. There are good opportunities to use knowledge of the different chapters of mathematics, computer solving and preparation for dealing with competition problems.

Our research shaws that we can enregitrate new phenomena in learning process. There are new obstacles in learning process related with a new live style or better with family live style. We will describe these obstacles and present them at pupil´s school work in the social context. Casuistries apply to pupils from 5 to 14 years old.

Dual problem to the well known Černy´s conjecture on directable automata is formulated and studied. We will discuss the minimal size directable automata with the shortest reset word of the length n and the structure of the reset words for these minimal size automata.

Operational analysis is one of useful tools for modeling real systems, especially in decision making problem theories. In our contribution we deal with some its applications in economics, social sciences, and in transport problems, too. Moreover, we intend to present mathematics in situations such one would not expect it within.

Over the last 10 years teacher’s study fields were modified especially in connection with changes in the study programme structure. The aim of the contribution is global comparison of a conception of the undergraduate preparing teachers of mathematics at secondary schools at the University of Ostrava before 10 years and today.

Solve following problem: Consider a square grid on which a path can be tracen slong the horizontal and vertical lines. How many different routes are there from a given point S to a point F? Solution of this problem can lead to the famous Pascal´s triangle. In this article we want to show how to get others interesting numer triangles.

New hardware and software equipments in schools, such as interactive whiteboards, voting devices, notebooks and tablets bring to question their eective use. From the didactic point of view, teaching mathematics has a strange role. In this paper we want to point out the pros and cons of using interactive whiteboards, voting devices and tablets in teaching mathematics on elementary and secondary schools. We also want to point out some new suitable methods and forms of work in this digital environment. We present a few topics for such education.

The separation axioms T_0 to T_2 are equivalent in topological groups. This equivalence disappear if the algebraic structure is more general, the case of topological inverse semigroups or the relation between the topology and algebraic structure is weakened, the case of semitopological groups and semigroups. We will discuss the above cases, and consider order-separation axioms as well.

In the paper we study some approximation properties for Durrmeyer type operators. We prove a Voronovskaya type theorem and state a rate of convergence for these operators.

Parrondo's paradox is the counterintuitive result where mixing two or more losing games can surprisingly produce a winning outcome. In this paper we present some application of Parrondo's paradox in the Penney Ante games.

This article points at possibilities of using Open Source Software in education. It focuses mainly on possibilities of creation and spreading media and multimedia contents, such as pictures, animations, sounds and movies using open source software. It points on possibilities of using software in conversion of media files. Next, it shows how created contents can be spreaded by portal LMS Moodle and RSS channels.

The article presents the online international contest on informatics iBobor. We focus on the category for primary education (Bobrík) and on the development of mathematical skills of pupils. We describe the tasks of the competition according to The National educational program for primary education (ISCED 1). Then we analyze the tasks that had the smallest success rate and try to find out the cause.

The subject of this paper are selected issues concerning the pupils’ work on non-standard mathematical tasks. In particular, special observations were made on the pupils’ attitudes in response to mathematical tasks.

A random varible is one of important notions of the probability theory. In this work we will suggest a method of provingchosen theorems concerning continous random variables, with the use of the geometrical probability space.

In the Czech schools, there is the predomination of the integrated form of education for intellect gifted pupils. The teachers are trying to find the right and suitable content and form of the educational proposal. The character of mathematical tasks, especially its difficulty, are seemed to be very imporant in this matter. The article brings the findings from the empirical probe from the work with the pupil of a primary school, who shows the noticeable talent just in the mathematical sphere.

A pilot study spatial imagination is submitted to students 1st and 2nd year-class of secondary school and students of the same age at grammar school. Problems about spatial imagination allow teacher to specify level of student`s mathematical abilities. The aim of this pre-research is to compare knowledge of students of secondary school and students at grammar school.

The contribution presents current problem of contemporary society, which is a financial literacy and related skills. It presents the definition of financial literacy according to the document National Strategy of Financial Education and highlights the current possibilities to implement financial literacy according to the Framework Educational Programme for Elementary Education and the standard of financial literacy for primary and secondary level of elementary school defined in the document The building of financial literacy at elementary and secondary schools.

ID-probability concept represents one of attempts how to generalize classical probabilistic notions in order to carry out wider spectrum of situations in which uncertainty plays crucial role, especially those which are connected with quantum phenomena. We intend to give its basic ideas and constructions. Some fuzzy and quantum aspects of ID-probability would be indicated.

We deal with 6-face polyhedrons and dice as valuable sources of many interesting mathematical problems in geometry, probability, combinatorics, but also in set theory, arithmetics, calculus, and logic.

This contribution describes one of the general heuristic problem solving strategies – strategy reformulation of the problem. This strategy is illustrated by one historical excursion (Pythagorean construction of regular pentagon) and two examples of current educational practice.

1. Integral of a continuous function on a compact interval. 2. Corectness of the definition. 3. Elementary definition of Lebesgue integral on a comnpact interval. 4. Applications from the point of view of probability theory and functional analysis.

In the paper a method of calculating the day and the month of an event is presented. This method is also generalized to calculate the year of this event based on Diophantine equations.

The contribution reflects dissertation thesis outputs dealing with functional thinking of first grade undergraduate math students on Education Faculties in the Czech Republic. It describes the research targets, hypotheses and methodology of appropriate data collecting and their consecutive elaboration. At the same time it presents partial results of testing.

I will present some remarks on arithmetical and geometrical sequences of higher degrees in some "mod m" arithmetics.

The article describes the possibilities of development of inquiry skills of secondary school students during preparing and participating in mathematical competition B-Day. During the competition teams of three or four secondary school students work on open-ended tasks that require students’ investigation and original approach to the problem. The competition is not only about finding the winner, but it is suitable for identifying the level of students’ inquiry skills.

This is a report on tracking algorithms for an advanced photon-counting sensor with an array resolution of 32x32. The sensor has a FOV of 50 by 50 meters at 10 km, and the frame rate of 12 kHz. Each frame contains the energy returned from one laser pulse. We construct a 3D image from 200 consecutive frames by estimating the PDF of the range (depth). The voxel intensities are defined by the 3D density values. This sensor created the opportunity to develop algorithms for true 3D tracking. This presentation contains a look at the algorithm development for on-screen tracking, which uses data collected at different locations.

Galileo in 1638 published the book „Discorsi e Dimostrazioni matematiche intorno à due nuoue Scienze“. As suggested by prof. Peter Vopěnka, at the Faculty of Education at The Catholic University in Ružomberok, we have started to translate the Italian original. This article draws on the second chapter - Second day, which focuses on flexibility , and balanced strength of materials. This article describes a unique proof that Galilei invented to prove the law of the moment of forces.

The paper deals with generalisation of so called „Collector Problem“, a known problem from the theory of probability, when we are interested either in: 1. how many cards we should buy in order that an even with a given probability happens, or 2. how many cards we have to buy “on average” to make a collection complete. The generalisation is based on a more realistic background when we do not buy cards separately but in sets of several pieces.

At school we learned that there two naturally occurring allotropes of carbon - diamond and graphite. In 1985 another modification of carbon was discovered. It is formed by a group of molecules called fullerenes. The discovery of fullerenes was such an important achievement, that in 1996 the Nobel Prize in Chemistry was awarded for it. They were named after the noted American architectural modeler R. B. Fuller (1895-1983). The most famous fullerene molecule is the buckminsterfullerene. It is a spherical molecule with the formula C60, composed of 60 atoms of carbon. These atoms are positioned in the vertices of a polyhedron which is combinatorially isomorphic with the soccer ball. The diameter of this molecule is 7nm, which is 108 times less that the soccer ball. The 3-valent polyhedra consisting only of pentagons and hexagons have been intensively studied by a group of Slovak mathematicians from the year 1968. (See - E.Jucovič: Konvexné mnohosteny, Veda, Bratislava 1981) In our lecture, using the Euler polyhedron formula F-E+V=2 (where F,E,V are number of faces, edges, and vertices), we shall formulate some necessary conditions for the existence of the carbon molecules which play an important role in our lives.

We will present solutions of a functional equation for mappings defined on an Abelian group which is uniquely divisible by 2 and having values in the set of complex numbers. Inspired by Jean Dhombres’ paper Relations de dependance entre les equations fonctionnelles de Cauchy from 1988 we deal with the generalized Lagrange's equation.

The paper shows mathematical description of selected natural structures and phenomenon in nature from the view point of fractal geometry. Even though the fractal theory is rather complicated area, interfering into all natural sciences, we can introduce it to high-school students using only their present mathematical knowledge. We are going to mathematically describe fractal phenomenon which ruins usual principles of classical geometry in many cases and it can support the attractiveness of mathematics – the queen of natural sciences.

As G. Polya said, the best way to learn something is to discover it himself. In mathematics teaching, it is very important that the teacher encouraged the students to individual work to solve problems. We connect problem solving with guided discovery learning as a teaching method for finding all regular convex-polyhedrons. During the activity we use a teaching aid – Polydron. In our article we present inductive approach of learning about Platonic solids. Students are encouraged to investigate which regular polygons can form faces of a regular polyhedron. This leads to further investigation of inner angles of those polygons. Based on their findings, students supposed to formulate hypothesis of number of Platonic solids and formally proof their hypothesis. Then we create all regular convex polyhedrons and show that they are only five.

The paper is intended as an attempt to make learners acquainted with the fact that the existence of a discontinuous additive functions could be proved. The proof is based on the existence of a special set of elements called a Hamel basis of the set of real numbers; its existence follows from the Axiom of Choice. It means however that no concrete example of such base is known, i.e. effective examples of discontinuous additive functions do not exist. This learners'newly acquired knowledge strengthens insight into traditional concept of a function as such. Namely, they cannot base their considerations on concrete geometric figure - it appears as a very important observation from the didactic and methodological point of view within school mathematics.