ACTIVATION OF STUDENTS COGNITIVE ACTIVITY IN SOLVING BOUNDARY VALUE PROBLEMS FOR THE DOOFING EQUATION

ACTIVATION OF STUDENTS COGNITIVE ACTIVITY IN SOLVING BOUNDARY VALUE PROBLEMS FOR THE DOOFING EQUATION

АКТИВИЗАЦИЯ ПОЗНАВАТЕЛЬНОЙ ДЕЯТЕЛЬНОСТИ У СТУДЕНТОВ ПРИ РЕШЕНИИ КРАЕВЫХ ЗАДАЧ ДЛЯ УРАВНЕНИЯ ДУФФИНГА

Aruzhan Nurgali

Master's student, Kazakh National Women Pedagogical University,

Kazakhstan, Almaty

ABSTRACT

This article deals with the activation of cognitive activity in students when solving boundary value problems for the Duffing equation. The author considered the role of cognitive activity activation in students when solving mathematical problems. It is concluded that the activation of cognitive activity in students is more active in solving such problems as boundary value problems for the Duffing's equation.

Keywords: activation, cognitive activity, students, solution, equation, problems, Duffing.

Nowadays, the most important task of civilization is to teach people to think. This statement is suggested by the fact that from the right thoughts come rational actions. In the world, the development of a competitive economy imposes new, higher requirements in the field of professional training of specialists, who must be able to adapt quickly to the constantly changing working conditions, have professional mobility and competence. The problem of correlation of a large volume of information with the limited time for sufficiently in-depth training arises in the process of training any profession. The way out of this contradiction can be seen in the unusual presentation of training material in the form of modular blocks, and then practicing it at seminars and practical classes, in the intensification of students' independent cognition. Since it is obvious that it is impossible to teach everything in a lifetime, it means that a future specialist must learn to acquire knowledge himself and apply it creatively in practice. This skill can be mastered only within the walls of the educational institution [1, p.62].

In our society is changing the view of the learning process, there is a search for new forms and methods that would most fully and correctly help to solve the problems facing the teacher. The modern educational institution is required to significantly improve the quality of education, ensure a high level of teaching subjects, improve the entire educational process. The deeper this process develops, the more pronounced the individual differences in student learning, the more obvious it becomes that it is impossible to create a single educational system that is equally optimal for each student. When the strategy of change in public education proclaims the idea of humanization of the educational process, putting the child at the head of the whole system of education, from the teacher requires a reorientation to the assignments and natural talents of each student, helping him to show independence.

Nowadays, society is more interested in its citizens being able to act independently, be active, make non-superficial decisions, flexibly adapt to changes in living conditions. In mathematics, physics and similar disciplines of natural science, medicine, engineering, economics and other areas of human activity we have to deal with certain values that change over time. These changes are called fluctuations. There are regularities common to fluctuations of different kinds. In this case we speak of isomorphism (identity of form) of various oscillatory phenomena (chemical, mechanical, biological, economic). This fact is the basis of the theory of vibrations - a special scientific discipline with a unified mathematical apparatus and universal terminology. The development of such disciplines as quantum electronics, solid state physics, astrophysics and biophysics has significantly expanded the scope of the theory of vibrations [2, p.141].

With the activity-based approach to learning, the formation of learning activities begins with the formulation of a learning task. At the same time, each learning task is solved on the basis of learning activities, which should represent purposeful preparation in the process of student's actions when solving the tasks. The experience of mathematics teachers shows that at first the students are not able to set learning goals and implement actions to solve learning tasks independently. At first they are helped by the teacher, but gradually the students themselves acquire the relevant skills, while practicing their independence. We know that the same disciplinary task can be used to achieve several specific learning objectives. Therefore, it can be an integral part of several learning objectives. On the other hand, a particular educational goal can be accomplished by more than one disciplinary task. At the same time, learning tasks help students to achieve the goals of learning activities, so they influence the formation of its positive motives.

Significant changes are also associated with the emergence and spread of computers, which allowed to define and solve previously inaccessible tasks. This has led to a great deal of interest in developing non-linear systems that can exhibit complex behaviour, including chaotic behaviour. One of the well-known equations in the theory of non-linear oscillations is the Duffing equation. It is used to describe the oscillations of a mathematical pendulum at small deflection angles, the motion of a particle in a potential field and other systems with irregular oscillations. In addition, with the help of the Duffing equation it is convenient to study such effects of oscillating systems as anharmonicity, non-isochronousness and multistability [3].

In vibration theory, mathematical models are used to describe physical processes. The Duffing oscillator, or cubic nonlinear oscillator, is one of the most widely used models in vibration theory. The Duffing oscillator is considered the simplest one-dimensional nonlinear system. A feature of the Duffing oscillator is the possibility of obtaining chaotic dynamics. The equation was first studied by the German engineer Georg Duffing in 1918.

As early as 1950, the Duffing equation was considered in a book together with the van der Pol equation as the classical nonlinear oscillation equation. In 1987, a paper by Yoshisuke Ueda, significant for the entire history of research in this field, was published. It discussed the chaotic behavior of the cubic Duffing equation. In the same year, 1979, a paper devoted to the study of the nonlinear Duffing oscillator in the context of its bifurcation behavior appeared. It is worth noting that the greatest number of mentions of the word 'Duffing' in the title refers to engineering articles (40%), followed by physicists (25%) and only later by mathematicians (19%). An example is the Duffing equation, a dynamical system exhibiting chaotic behaviour. The equation is as follows:

                                                                                                             (1)

Where   is the unknown function that is shifted by time  or ,   is the first derivative of  at time , while  is the second derivative of  at timet , and  are constants. Quite familiar notations for a student.

In addition, the Duffing system introduces the phenomenon of jump resonance into the frequency response, which is a kind of frequency hysteresis [4, p.424].

This equation can be obtained by considering the oscillation of a mathematical pendulum at small deflection angles, the oscillation of a weight on a spring with a non-linear restoring force placed on a flat horizontal surface. The equation can also be obtained by describing the motion of a particle at a potential equal to two. Due to the impossibility of finding exact analytical solutions to the Duffing equation, the question of finding approximate analytical solutions and numerical solutions to this equation arises. In order to solve such boundary value problems for the Duffing equation it is necessary to stimulate the cognitive activity of the students.

The development of mathematical theories has always been based on overcoming the contradiction between well-established notions and new facts, experimental data that do not fit into the framework of these notions. Bringing students to the realisation of the solution of mathematical problems, involving them in reflection, also in the search for solutions to these problems is a reliable way for students to fully understand the experimental basis on which the new theory and its main points were built. In this case, even if the solution of the problems is revealed by the teacher himself (problem statement), the emergence of new ideas is in some way "experienced" by the students, and the emergence of these ideas is perceived by them as a natural and inevitable result of the development of science [5, p.125].

An example is the study of finding the non-standard volume of a body using an integral.

After demonstrating different bodies, they are asked to find the volume of these bodies using the previously obtained knowledge, which leads to a complex solution, after which the formula for finding the volume through an integral is introduced. Such tasks may be creative, requiring students to be independent in their judgement and to find solutions that have not been tested before. Problem tasks are effective when pupils have already acquired the skills and abilities to solve problems using a ready-made model, and there comes a stage when it is necessary to activate this knowledge. Thus, the tasks are mainly used at the final stage of consolidation and in repetition. Tasks of a problematic character may be used as homework and solved in groups. In the latter case, experimental problem tasks are particularly effective, especially if they cover a wide range of issues on the topic. Work in the classroom is inevitably limited in time. This often does not allow students to propose sufficiently complex tasks. Also, not all types of problem tasks can be used in the classroom.

For example, tasks for constructing and making three-dimensional figures, conducting experiments that require long observations or repeated control, etc. Therefore, tasks of a problematic nature are no less important than the work done in class.

Cognitive activity characterises the attitude of students to the learning process. Cognitive activity is a striving for independent thinking, discovering one's own approach to solving the problem set, striving for independent acquisition of knowledge and formation of critical approach to judgement of others and independence of own judgement. Student activity may disappear if the necessary conditions for its realisation do not exist. Cognitive activity contributes to revealing and increasing the degree of involvement, concentration and focus of the student on the subject of his/her activity. Cognitive activity is inextricably linked with independence in thinking and practical activities. Which, in turn, are necessary to increase cognitive activity of students in solving Duffing equation problems.

The management of students' cognitive activity is traditionally referred to as activation. Activation is defined as a continuous process of inducing students to intensive, goal-oriented learning, overcoming passive and stereotypical activity, decay and stagnation in scientific work. The main aim of activation is to generate activity, improve the quality and effectiveness of the educational process. Complete learning of mathematics is not possible without solving mathematical problems. The problem with the traditional method of problem solving is that neither the content of the problems nor the process of solving them usually arouse cognitive interest in students. Different techniques could be used to stimulate pupils' intellectual activity and their interest in solving mathematical problems.

On mathematics lessons problem solving plays an important didactic role. Because calculation, proof and construction tasks contribute to the development of spatial representations of pupils. Computational problems are often also proof problems, because justifications are required, and the design problem is always related to the proof. In solving these types of problems, construction, measurement and calculation are often combined. Solving each type of problem requires cognitive activity on the part of the students, and this activity depends on the degree of development and spatial representation of the students.

Iterative processes can be ordered in a similar way to the way the work is done. This will increase the efficiency of the processes, due to the fact that regularized processes 'work' with positively defined matrices, which greatly extends the scope of iterative methods. Since the Duffing problem is one of the most important problems in the theory of nonlinear oscillations, the ideas outlined above can be successfully extended to general second order nonlinear boundary value differential problems.

To summarize, it should be noted that all modalities (techniques, methods, rules, recommendations) of cognitive activity activation in classes depending on their type and place can be presented as activation modalities: lectures, seminars, groups, master classes and practical classes; during students' independent work with study material; methods of raising emotional mood and behavior regulation. Thus, methods of motivation formation for active learning and cognitive activity of students should be in the compulsory arsenal of every teacher along with methods of learning activity activation in relation to the specifics of each type of teaching and solving various tasks, including edge ones [6, p.163].

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4. Chureeva L.A. Activation of cognitive activity of medical college students in practical classes of professional module as a condition for learning success // Scientific and methodical electronic journal "Concept". - 2016. - Т. 46. - p. 424-433.
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