On teaching of bases of real variable functions theory

The main aim of the work is one of the mathematics disciplines teaching methodology improving, namely the real variable functions theory. The article outlines the theoretical material basic concepts, statements in their logical sequence. For practical lesson learning task is chosen, which implies the need for the student to identify a particular something in common, as discussed in the lecture. The teacher guides the work of students c through questions to encourage them to self-reasoning and active search for the right solutions. In this article we present the test questions and tasks to determine the rate of formation of the competence of the student. Thus, we believe that the study is transformed from a traditional discipline of study in cooperation and conscious acquisition of knowledge. The paper presents one of the methods of presentation of theoretical material basic concepts, statements in their logical sequence. For the practical sessions as the theme chosen this learning task, which involves the need for the student to "touch" material, in particular to identify common, as discussed in the lecture. In the practical classes the teacher directs the work of students c through questions to encourage them to self-reasoning and active search for the right solutions. In this regard, we give control questions and tasks to determine the rate of formation of the competence of the student. Thus, we believe that the study is transformed from a traditional discipline of study in cooperation and conscious acquisition of knowledge.

Elementary knowledge of real variable functions theory is the necessary part of mathematical culture of future teacher. Discipline "Theory of real variable functions" refers to the cycle of fundamental mathematical disciplines. The study of this discipline is based on the "Mathematician" knowledge the students in frames of high school, and also such mathematical disciplines as: "Analytical geometry", "Linear algebra", "Calculus". Free possession is assumed also the basic concepts of Calculus, such as a limit, derivative, integrals and series. However, knowledge of these concepts in the volume of course of Calculus is not always enough for the modern applied and theoretical tasks solution.

Therefore there is a necessity of knowledge expansion for the following: students’ structural thinking development, mathematical knowledge forming for a successful capture by professional skills at necessary scientific level. In addition, substantive provision of discipline "Theory of real variable functions" is foundation of mathematical formation of the applied mathematician education, important for the successful study of general mathematical and special disciplines.

The content of discipline supposes the study of the following questions: «Capacity of set», «Countable and countless sets», «Structure of the closed and open sets on a numerical line», «Concept of metrical  space», «Complete metrical spaces», «Measure of Lebesgue», «Sets and functions», «Measurable on Lebesgue», «Lebesgue integral», «Fourier series».

For better understanding of course of real variable functions theory , for students’ creative independence development, stressful analysis of basic concepts, methods and theorems of mathematical analysis  may serve the tasks of studying and scientific research character, because they have the direct exit on serious mathematical research.

The solution of these tasks requires the independent mathematical reasoning, acquaintance and working of scientific methodological literature, ability to process scientific information, make independent conclusions.

The theory of real variable functions is one of the most important studied subjects at physics and mathematics faculties in pedagogical universities. A teacher constantly meets concepts of a set, real number, function, limit, continuity, measurement of sets which form the maintenance of this subject during work. It is impossible to conduct teaching any school course in mathematics at a relevant scientific level, without knowing the bases of the real variable functions theory, the ideas of which cover the whole areas of mathematics.

Psychologists assert that the source of the creative thinking and its beginning is the difficult situation, both theoretical and practical, that requires the search of solution and, certainly, research. The educating problem is a method, during that the serve of new material takes place through creation of problem situation that is for a student intellectual difficulty. 

Traditionally, the lecturer delivers already prepared material, thus, giving the students the supposed answers. Knowledge got in the process of lecture listening are percept almost always in the appropriate manner. But that’s definitely the mechanical knowledge acquisition. Therefore the question: is there student’s initiative in such knowledge receipt? Practically both methods (problem and traditional) educating are used by teachers together, they complement each other. However the teacher‘s role here is not diminished, it is increased instead.

Pedagogy gives an opportunity for enormous amount of methods and problem situation introduction variants in the process of educating. They assist the variety in educating, i.e. possibilities to choose the variant of material giving out.

In this article we try to review another method of teaching to basic concepts of the point set theory. Meantime, we want to examine various point sets, the sets and elements of which points either a numerical straight line or a point of any n-dimensional Euclidean space.

As one-to-one correspondence between set of real numbers and all point sets of numerical straight line, studying of linear dot sets that are point sets of a straight line determined is identical to studying of the sets consisting of real numbers. While teaching this course, it is essential to introduce the definitions of the simplest and most common point sets, segment, interval, semi-interval.

It is also necessary that students clearly understand due to one-to-one correspondence between the set of all real numbers and the set of all points of the real line the segment definition, the interval and interspaces for them are identical to the definitions of numerical sets. Then the definition of contracting sequence of segments and intervals are introduced [1].

The concepts of a segment and an interval extend in multidimensional spaces. Thus, under the segment of two-dimensional space, i.e. in the plane we mean the set of all points (x, y) of plane, where each of the coordinate values form a line segment, for example

The change of social role of knowledge (in particular, mathematical) and creative possibilities of personality in a modern period of development of society inevitably puts questions about optimal correlation of technological and humanistic orientations in organization of educating to mathematics in pedagogical institution of higher education, conditioning for the independent mastering of new experience.

Thus, it is necessary substantially to reconstruct a structure and maintenance of mathematical preparation of students on the basis of psychology-pedagogical analysis and integration near an innovative pedagogical process taking into account experience of preceding researches. 

References

1. Iskakova A., Khanzharova B. About another method of teaching to basic concepts of point sets theory. International Conference: Science and Education in XXI December 1, 2014. Bozeman, Montana, USA. — P. 117–119.
2. Натансон И.П. Теория действительных переменных. — М.: Наука, 1974. — С.
3. Гурвиц A., Курант Р. Теория функций. — M.: Наука, 1968. — С.