In the paper we consider questions related to the creation of pedagogical conditions necessary for the activation of the project-research activity of students in the classes on mathematical analysis during the studying the theory of limits. The project-research activity of students is the activity of designing your own research which involves the selection of goals and problems, principles for selecting techniques; planning the progress of the research; determining the expected results, evaluation of the feasibility of the study, identification of necessary resources. In addition, it is an important part of the learning process, a necessary means of increasing the motivation of students to study at a university. This activity is also a necessary means of quality professional training, and it promotes the successful adaptation of future young professionals to modern socio-economic conditions, the formation of students' need for knowledge, high professional motivation and the desire for self-education.
Introduction. High level of mathematical education is one of the most important conditions for scientific and technological progress and production development. According to some researchers, the modern period of education development is characterized by the fact that the traditional (knowledge) educational paradigm no longer satisfies the requirements of society to modern education [1,2].
Training future bachelors of mathematics should be focused not only on the acquisition the necessary amount of knowledge and skills on the mathematical disciplines, but also on the development ability of future experts to self-search and application of new knowledge and technologies in conditions of rapidly changing reality. It requires them creative thinking, quick orientation, creative approach to solving professional problems. In compliance with these requirements a necessary condition for preparation the future bachelors of mathematics is to increase the share of project-research activities in the course of their training special mathematical disciplines. Mathematical disciplines contribute to the formation the mathematical style of thinking and mathematical culture. A modern professional should be able to analyze particular phenomena and find general patterns, and exactly, Math is the best way to these promotes.
In formation a professional bachelor of Mathematics the course of mathematical analysis plays an important role. This course contains a theoretical foundation for many of provisions; other mathematical disciplines, rationale of both theoretical and practical provisions of a number of fundamental questions of school mathematics. The main importance of the course of mathematical analysis for training the bachelor-mathematics is characterized by the following problems:
- teaching students the fundamentals of modern mathematics;
- formation the mathematical world;
- development scientific, logical thinking, necessary for further work by the specialty;
- masterya sufficient number of mathematical methods by students;
- development hard skills to construct mathematical models and skills to carry out computational calculation.
According to the academician N.N. Luzin [3], "problem of teaching mathematical analysis is one of the most difficult problems of science and education. All circumstances are complicating for the task: the growth of science with its continuous enrichment by new facts, and the related oscillating light, seemingly, firmly established principles, and, finally, changing the level of knowledge and the needs of those communities, which turned the word of a teacher".
However, many scientists and teachers note a decrease of mathematical education level at university, which is characterized by the fact that knowledge of the majority of graduates are formal; the required level of ability to use the "high school" mathematical knowledge to improve contents of school education and study of logical structure of the school mathematics is not reached; level of formation professional self-education skills of graduates does not corresponds to requirements that are now necessary for bachelor of mathematics.
That is why the mathematical analysis and in the aspect of professional activity, and in the aspect of teaching this course, most need to develop components of research and project creative activity in its structure. This, in its turn, requires modernization of traditional forms on teaching mathematical analysis, and suggests increase of importance of organizational and pedagogical support on activation independent project and research activity of students in the educational process.
Results of the analysis of scientific and methodological research leads to the conclusion that specifics of project-research activity of students in the process of learning mathematical analysis, and problem about activation this activity has not yet been investigated fully. And it is impossible to describe the scrutiny level of the problem, considered in our study, without comparing the results of studies on two problems, traditionally allocated in pedagogical science: the problem of organization and development student’s project activities, and the problem of organization and development their study and research activities.
Studies of E.S. Bulychev [4] and A.G. Podstrigich [5] found that learning mathematical disciplines by using projects is more efficiently than traditional, in particular, it contributes to formation mathematical concepts, improvement the quality of mathematical knowledge. Studies by Z.V. Toropova [6] and Zadorozhnaya [7] show that the use of training projects contributes to the quality of knowledge. However, potential of training projects on various disciplines, and especially in mathematical analysis, has been studied far from
By the majority of authors projection-research are considered as related but different in goals, meaning and content of types of activities. However, works of A.I. Savenkov [8], A.V. Leontovich [9],
A.S. Obukhov [10] and etc. justified the idea that at the present stage of development of these kinds of activeities they are increasingly considered in terms of their synthesis, integration, and especially in the modern educational process. This allows us to understand allocation of integrative way to organize learning activities-project and research activities, and to consider it as a new phenomenon in education, which has so far not received adequate theoretical justification, despite the fact that the term "project-research activity" more and with different semantic loading meets in the publications of innovative teachers.
Main part. Today there are no specific studies on organization of project-research activities of students during the study theory of limits in the course of mathematical analysis. In terms of modern educational process the project-research activity can be considered as a specially organized, informative creative activity of students, the purpose of which is to obtain new knowledge about the studied object, to form research abilities and cognitive motives. Project-research activity on its structure corresponds to the structure of scientific activity, and it has inherent purposefulness, activity, objectivity, motivation and awareness. Organization and carrying out such activities in the class to study the theory of limits are not reflected in studies, respectively, in publications, which summarize and present the experience of teachers-mathematicians. Importance of improving the content of the project and research activity of students in learning process of the theory of limits on mathematical analysis is not in doubt. Realization of this problem in education with a high probability leads to the emergence of innovative forms, methods and means to improve quality of mathematical education of future bachelor-mathematics.
In our studies project-research activities should be considered as a form of learning and cognitive activity, which combines project problems, necessary for solving, and research activities, aimed at study one of the fundamental concepts of mathematical analysis, limit of a numerical sequence or function. In this regard, under the pedagogical terms of the project research activities of students we understand the process of increasing the intensity of interaction of the basic concepts of mathematical education, aimed at development project-research activities, at improvement activity and initiative of students in finding and solving educational and research problems.
Project-research activity of students includes the following components: project, research, project and research.
Purpose of the project constituting learning the theory of limits isto form cognitive basis of students' project activity that allows students to organize students' activities from the idea to its practical application. Its realization is based on the principles, norms and rules of project. This includes organization IWS on the mastery of knowledge and practical skills in the field of research activity. And for this it is necessary to organize the process of solving specific educational problems and to provide methodically additional independent work of students on the problems. The defining moment of this component is to organize IWS on the isolation, comparing and combining different methods of solving the educational problem. This contributes to the fact that in the process of learning the theory of limits, students’ ability to discretion new approaches is formed to calculate the limits.
Research component is defined as an independent solution of new problems by students, using such elements of scientific research, as observation and independent analysis of the facts, hypotheses and its verification, formulation the conclusions and law. The use of research method is possible during the solving a complex problem, analysis of information from primary sources, permissions of a problem, assigned by teacher.
Aim of the project and research component of learning the theory of limits is integration in the process of abstract transformation the objects of the limits theory (i.e., in the process, implemented in structure of the project components) results, obtained in structure of the research component. This integration is done through scientific thinking and practical combination of known knowledge, which were obtained in the process of self-study. In addition, for students the project-research activity result is not only achievement of learning objectives (formation knowledge and skills), but also development a new approach to calculate limits.
Exactly, the presented structure of models of project-research activity, provided relationship of its components, is the basic guideline to organize educational process, ensuring achievement of the results described above. Therefore, implementation of this model becomes a prerequisite to organize and conduct project-research activity of students.
An equally important and related to the previous conditions is the system development of educational problems. Pedagogical organization the process of their solution is an essential component of the model proposed above.
This system includes the following interrelated problems:
- use of original methods to calculate limits (rationalizing substitution, standard asymptotic formula);
- historical background of development of the basic concepts, associated with the limit (numerical sequences, infinitesimals, and );
- applied aspects of the limits theory (economics, medicine, and );
- use of computer mathematics systems (CMS).
Problems of the first type are offered to students after studying the sections "Differentiation" and "Integration". They are aimed at organization independent work of students on study rationalized substitution and the standard asymptotic expansions. Learning mathematical analysis, it was historically developed so that the study by students the rationalized substitutions and the standard asymptotic expansions, as a rule, is in the "Integration" and "Differentiation" sections, and in the calculation limits they are not properly applied. Moreover, the practice of organization project-research activity of students on mathematical analysis has shown that the use of the rationalization method and standard asymptotic formulas to calculate limits of sequences and functions is very effective.
The tasks of the second type allow them to:
- develop research and cognitive abilities, a certain style of thinking, ideas about mathematical concepts, science literacy and scientific language;
- form ability to properly execute their ideas with the help of a certain scientific material, to carry out mathematical reasoning and research by analysis;
- improve understanding properties and theorems, universal geometric illustrations of limits for number sequences, and develop the historical and mathematical culture of the future bachelors mathematicians.
Concepts of limit and methods of its research are rather abstract. To form and develop abilities and skills on limit applications in study numerical sequences and functions, particular attention should be paid to the connection of studied concept of the limit with the specific interpretations of life, i.e., to show their practical application in specific examples.
In its turn, ability to distinguish the types of numerical sequences and limits in tasks of a certain level of complexity facilitates understanding of their content, and ability to use certain properties and theorems helps to write voluminous exercises compactly. Study theory of limits allows us to find out what stationary position the system takes, after running of all the fluctuations and transients (oscillation theory). Will the planet orbits a star, or fly away, or fall into (astronomy)? At what point will stop the stone in relief (construction)? What temperature distribution is established after turning the stove (the heating engineer)? What current will flow in the circuit (electrical engineering)? and etc. Answers to all these questions can be obtained in the process of disclosure the relationship of the limit theory with the outside world, other sciences and manufacturing. Realization of an applied orientation of mathematical concepts requires a high level of training from the student: he must possess a large margin of mathematical knowledge with applied content, and it allows expand the range of mathematical knowledge. Moreover, the solution of tasks with applied content contributes to implementation of important pedagogical purposes: formation of research skills and development of cognitive interest.
Nowadays introduction of information and communication technology (ICT) to the educational process opens up wide possibilities of a qualitative reorganization the principles and methods of classical mathematical disciplines teaching. To do it, it is necessary to create new methodical systems of training the future bachelors of natural science direction, focused on development intellectual potential of students, on formation their abilities to acquire knowledge independently and carry out various types of research activities. ICT opportunities in the calculation of limits contribute to: improving the quality of assimilation the studied material, improving knowledge, skills and speed of their production, development of space-graphic culture of the future bachelors of mathematics.
Mini, local, term, course and global projects can be identified by the content of projects. Mini-projects include specific issues of topics, which are set out in the part of the lecture; local projects include one or more topics of the course of mathematical analysis. Term and course projects include one or more sections of the course, one or more semesters. In selection a content of educational projects focus is on interconnection and interdependence of concepts, topics, sections of the course of mathematical analysis through analogy, generalization, hierarchy of different objects that provide the link between various educational projects [11].
Here is a concrete example for creation of a mini project on study mathematical analysis aimed at identifying the links between calculation and geometric illustration of the limits.
Mini-project 1. Geometric illustration of limits.
Before talking about sequence limit, a student must understand the question that the concept of a sequence limit is one of the basic concepts of mathematical analysis.
There are many approaches to calculate limits of numerical sequences. In this mini-project a student has to learn the material from the recommended sources [12,13,14], write, organize and analyze it.
Student’s analysis. Limit of sequence [12,13,14] is defined as: 1) infinitesimal number; 2) there exists an integer; 3) execution of inequality; 4) limit.
Student’s conclusion. Each definition reflects a certain facet of the concept of a sequence limit. This is due to the fact that 1) there are different methods to calculate numerical sequences (analytical, graphical and etc.); 2) sequence limit refers to the primary concepts of mathematical analysis; 3) limit of a sequence reflects existence of interdependence of processes and phenomena in the real world.
Conclusion. Summarizing, we note that the project-research activity of students in study the theory of limits allows students not only to apply the knowledge and skills obtained in the classroom, but also to actively join in cognitive activities, to intensify his creation, to apply mathematical knowledge in practice, to produce new knowledge in dependently, to analyze unusual situations, to organize searching of solutions, to consolidate the acquired knowledge. The results of study are personal characteristics that students take in the process of project-research activities, and specific quantitative assessment received from the teacher, which advocates as a result of the work.
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