Aesthetic education: the Process of teaching mathematics with the Application of Information-Communicative

Technologies

A. Dautov1 [0000-0002-7785-8799], Al. Aktayeva2 [0000 – 0002 – 4308 - 4553], 

K. Kozhabaev3 [0000-0002-****-****]

¹                      Doctorate, Sh.Ualikhanov Kokshetau State University, Kokshetau, 020000 Kazakhstan, [email protected]

²                      Dr. PhD., Dept. IS&CE, Sh.Ualikhanov Kokshetau State University, Kokshetau,

020000 Kazakhstan, [email protected]

³                      Dr. of Pedagogy, Dept. Mathematics and Methods of Teaching, Sh.Ualikhanov Kokshetau State University, Kokshetau, 020000, Kazakhstan, [email protected]

Abstract. In the article one of leading aims of educating to mathematics is examined is aesthetic education of student facilities of mathematics. Presentation of aesthetic beauty at her decisions possibility of students is investigated, specifying them on the decision of one problem in several ways that assists the detailed consideration of idea of aesthetic education, through programmatic foods. The article will describe understanding of beauty the decision of problem, methods of decisions that are accompanied by the use informatively-communicative technologies in a mathematical model. Such sort of activity assists aesthetic education, allowing to develop a culture and logical thinking, forming at students a different choice, grace of decision of problems

Keywords: personality, aesthetic education, knowledge, mathematical education, competence, software products, ICT.

1  Introduction

Among the main directions of the modernization of education, one can single out the personal orientation of its content and activity-oriented nature, the focus of the content on the development of generalized methods of various types of activity, as well as the development of key competencies, which in turn are aimed at the aesthetic education of the student’s personality, and his cognitive abilities.

Developing the beauty of mathematics has great potential in educating a student (person).

The aesthetic education of students is of paramount importance when developing a harmoniously balanced personality. It is very difficult for modern youth to appreciate aesthetic beauty, because it has recently lost its significance. It is impossible to generalize, because there are true connoisseurs of the beautiful among the students. Since mathematics is one of the most important subjects at school, its role in aesthetic education is hard to miss. The potential of mathematics in this respect is enormous.

Mathematics is very rich in beautiful formulas, proofs, and various methods for solving problems. Here one can specify entire sections where the multifaceted elements of beauty and grandeur are hidden. It is necessary to develop a sense of beauty, and to form high aesthetic tastes when choosing a solution to a problem. ICT can be used for the purpose of locating, understanding and appreciating the quality of the result [1].

Most studies have neglected the vast possibilities of modern information technology in solving the problem of developing a sense of beauty and aesthetics. Research on this issue has appeared only in recent years due to the rapid development of computer and information technology. However, almost all of these studies are related only to the study of fractal geometry elements [2].

Such an application will give a leap to the improvement of the subject, especially aesthetics, which have great cognitive and educational power. Note that the “beauty of mathematics” means the fundamental “types of beautiful solutions”. It is necessary to search for the basics of aesthetic education of students not only in nature, music, and architecture, but also in the content and technologies of teaching mathematics. The beauty of mathematics becomes visible through various types of educational activities performed by students while teaching mathematics. This aesthetic beauty must go through such stages as visual-effective, visual-figurative, and written beauty.

The effective development of the wide potential of mathematics involves the fully fledged perception of mathematical literacy, the cultivation of aesthetic feelings, taste and ideals through the figurative perception of the beautiful. A person’s value orientation is formed through his quest for the beauty of original problem solving.

2                    Methodology of introducing elements of aesthetic education in the educational process

Aesthetic education is part of the ideological and moral education which has as purpose the combining of the student’s intellectual excellence with the high culture of the modern world. An aesthetic attitude to the world is, of course, not only the contemplation of beauty, but, above all, the desire for creativity according to the laws of beauty. In the pedagogical literature, aesthetic education is considered as a system of measures for the development of a person’s good artistic tastes and the ability to correctly and truly appreciate the beautiful in art.

So, D. von Neumann noted that mathematics, like art, is driven almost exclusively by aesthetic motives. J. Hadamard argued that a scientist, seeing a structurally imperfect, asymmetric, “crooked” mathematical construction, begins to feel the need for active work to harmoniously supplement (correct) it [3].

In the words of V.G. Boltyansky, the beauty of a mathematical object can be expressed through isomorphism between the object and its visual model, the simplicity of the model and the unexpectedness of its appearance. This statement can be supported by the formula of “mathematical aesthetics” [4]:

beauty = visibility + surprise = isomorphism + simplicity + surprise

It is clear from the above quotes that one of the most important tasks in developing a person’s worldview is the creating of an aesthetic attitude towards mathematics. When teaching mathematics, school students can and must learn to perceive, and feel the beauty of mathematical expressions and theoretical constructions. They must learn to evaluate the wide possibilities of a mathematical culture from an aesthetic point of view.

The development of modern information technologies does not eradicate the need for creativity, but, on the contrary, demands an ever higher level of general cultural development, creativity and action from a person. Modern information technologies open up additional didactic opportunities in realizing the goals of aesthetic education in mathematics, which should be used to familiarize students with beauty, and teach them aesthetic tastes and experiences through integrative courses related to web visualization, computer graphics, and the development of multimedia tools, etc. To reveal the beauty of the content of mathematics to the students, use the capabilities of applied programs to identify the students' creative approach to mathematical culture. This will contribute to the development of creative potential in the classroom, and prepare them for life in modern conditions - an important factor in aesthetic education.

The use of modern software tools (computer graphics, application programs) in mathematics when considering historical and mathematical material, introducing students to outstanding works of art and architecture, creating computer-aided mathematical and artistic compositions (fractal, symmetrical), using demonstration on-line and off-line programs illustrating various phenomena and processes of reality, and working with colour all contribute to the development of figurative thinking and imagination of the students. These different approaches contribute to their aesthetic intuition and increase the level of aesthetic perception of the material. 

The development of logical thinking of the students is facilitated by programming and building patterns, ornaments, fractal sets, etc. on the computer. Based on the above, the aesthetic teaching and education of students should be done in such a way that it harmoniously becomes an integral part in the process of teaching mathematics. The aesthetic education of schoolchildren in mathematics by means of the knowledge of beauty with the effective use of ICT is achieved if:

1.          a solution to problems in order to develop creative activity is provided, striving for the beauty of the originality of the tasks to be solved;

2.          various tasks, including exercises, that require the use of several methods in their solution are incorporated;

3.          the integration of mathematics learning processes together with development of the cognitive interest of the students is ensured by providing information and communication technologies. 

3  The main part: formation of aesthetic features

The innovative activity of a teacher is a comprehensive integrative approach to teaching. Ensuring innovative development and improving the quality of professional education through the development and application of technical means in the process of teaching mathematics is a focused activity on the use of various innovations.

Innovation is understood as an idea of the field of education. Information technology training, according to the dictates of time, leads to greater efficiency, communication activities, and pedagogical competence.

The quality of changing the system of pedagogical mastery in education also becomes a prerequisite for personal and professional growth. An example of this is the elimination of stereotypes that go beyond existing mechanisms by searching for hi-tech ways to solve impossible problems in the process of solving.

The use of technological tools in teaching different types of tasks can have a positive impact on the formation of aesthetic features, on the growth of interest in the study of mathematics, as well as on the improvement of the level of fundamental knowledge using digital technologies, while the traditional methods of various levels of education and social development of students of middle and senior levels are improved.

The use of ICT opens up new didactic opportunities to realize the aims of esthetic education in mathematics classes, which should be used to connect to beauty, to educate such actions, which according to the taxonomy of B. Bloom should be subject to analysis and synthesis in order to select the most optimal, rational ways of solving the problem. After these actions, the trainee begins to find out the conformity of the conclusions with the available data [5,6].

For example, due to the built-in Wolfram programming language, you can interactively manage created objects, changing any of their parameters. The whole collection of special functions serves for mathematical calculations, the necessity in which often arises when working with graphics. Built-in interface monitoring and easy-to-use debugging tools can help you find errors in a problem-solving scenario.

Students are then asked to perform some practical exercises using Wolfram's software capabilities: Build different objects and their compositions, draw from these objects of graphics. Students can use any of the program's features (graphic, animation, algebraic, etc.) while performing these tasks. After receiving a symmetrical image, it is possible to emphasize its beauty with various special effects, provided by the program's capabilities, to transform it into an even more interesting form (see Fig. 1).

Fig. 1. Wolfram software. [16]

The above approach will allow students to fully understand the subject connection of mathematics and computer science, to form a view on technological possibilities, to feel the attractiveness of the solution of problems, thus knowing the aesthetic appeal of mathematics (see Fig. 2)

Fig. 2. The solution of one problem on the Wolfram software platform. [17]

Performing several practical exercises with different methods of solution will influence the versatile consideration of the opportunities of students at any level, including the process of esthetic education through the beauty of illustration.

Let us show an example of a problem where students use different methods and, justifying the methods, form the essence of the same answer of the same task.

The task: Two cars left a city in the same direction. The first travelled at a speed of

60 km/h, and the second at 90 km/h. The second car departed 2 hours after the first.

After how many hours and at what distance from the city will the car with the highest speed catch up with the car with the lowest speed?

The first method: an arithmetic method. We give a brief record of the problem (see Fig. 3).

Fig. 3. The scheme of movement of two cars.

1)       602 120 km – the distance the first car will have travelled in 2 hours;

2)       9060  30 km/h – it is the difference in speed of the two cars;

3)       120:30  4^h – after so many hours after its departure, the second car will catch up with the first one;

4)       904  360 km – this is the distance at which the second car will catch up with the first one.

The second method: a geometrical method. Let's consider the solution of the task on schedule, that is, we will describe a visual schedule of the movement of the cars.

When solving the problem in a geometric way, we choose the vertical axis of a coordinate system with one calibrated length for 60km, and the calibrated length of the horizontal axis for 1 hour. Let us set the horizontal line from 0 to 6 hours, and vertically we will indicate the distance traveled by each car in 1 hour, 2 hours, 3 hours, etc. We will also show the solution using the Wolfram software product to demonstrate the visual intersection of two vehicles, depending on the time (see Fig. 4).

Fig. 4. A chart of the movement of the vehicles.

To determine the path from the beginning of the time, first indicate the distance traveled by the first car until the departure of the second vehicle, and then the distance traveled by the first and second vehicles on the chart (see Fig. 4). Complete table 1:

Table 1 – Distance traveled by vehicles according to the time of departure

The segments of the journeys are equal, meaning that the second car will catch up with the first in 4 hours after the first car departed. As you can see, this will happen at a distance of 360 km from the city (see Fig. 4).

The third method: geometric construction of the problem by splitting it into intervals according to the difference in speed of the two vehicles. Assuming that the length of one segment is 30 km, let us measure segments of the path traveled by each of the cars. The interval of the path corresponds to 360 km from the beginning of their movement (see Fig. 5).

Fig. 5. The distance traveled in a certain time.

The fourth method: algebraic method. We compose a linear equation, the solution of which is an ordinary linear homogeneous equation and a solution using ICT - Wolfram:

                                             S      S

                                                       2                                         (1)

60 90

Fig. 6. Unknown path segment.

So S120x , where x is the path to go. Substituting it into equation (1) we obtain the following equality (see Fig. 6):

120x 120x

                                                                  2                                  (2)

                                         60            90

From equality (2) it is easy to calculate that x240km, this is the distance traveled by the first car in 4 hours after 120 km/h.

The fifth method: algebraic method.

1)         Finding the distance traveled with the help of the least common multiple (LCM) speeds of the two cars: In two hours of time;

2)         Then, to find the full path, we          multiply the result by two: 2LCM(60, 90)  360km - from the beginning of their movement;

3)         360:90 4h - after so many hours after its departure, the second car will catch up with the first.

In each of the proposed solution methods, the same result was obtained.

In solving above problem in several ways, the students' knowledge in mathematics, their ability to solve problems, and their ability to build mathematical models were tested.

4  Conclusions and future work

The process of computer modeling, based on modern graphic tools, increases the motivation of students, creates an atmosphere of creativity and allows one to look at the world through the eyes of the Creator. Moreover, the computer is not only a powerful cognitive tool for studying world culture, but also a tool for mastering the theoretical foundations of painting, the development of artistic taste, and the development of imagination and creative abilities. 

The use of technological tools in training in solving various types of problems can have a positive impact on the development of aesthetic features, and the growth of interest in the study of mathematics, as well as the increase in the level of fundamental knowledge using digital technologies, while improving traditional methods of different levels of training and social development of students. Solving the same problem in various ways (if necessary, using ICT) increases the level of general development of students, contributes to their intellectual growth, that is, affects the development of mental actions in the development of personality. By forming aesthetic forms of learning, a link can be constructed to generalize and consolidate knowledge, skills and abilities by the example of solving one problem.

Thus, according to the practical activities of students, it is possible to identify certain links in understanding mathematical simplicity, that is, the aesthetic potential in solving one problem in various ways.

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Note on the authors:

Dautov A., doctorate, Sh. Ualikhanov Kokshetau State University, Kokshetau, Kazakhstan

Research Flied: math is research areas, [email protected];

Aktayeva Al., doctor Ph.D., Dept. IS&CE, Sh. Ualikhanov Kokshetau State University, Kokshetau, Kazakhstan 

Research Flied: IS and Privacy, Social network analysis, Quantum Computing, [email protected];

Kozhabaev K.G., Dr. of Pedagogy, Dept. Mathematics and Methods of Teaching,

Sh.Ualikhanov Kokshetau State University, Kokshetau, 020000, Kazakhstan  Research Flied: Theory and methodology of teaching mathematics. Psycho-pedagogical and didactic-methodical foundations, [email protected]