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SYNERGETIC EFFECTS MANIFESTATION BY FOUNDING COMPLEXES DEPLOYMENT

OF MATHEMATICAL TASKS ON THE CHESSBOARD

Dr. Svetlana N. Dvoryatkina, Department of Mathematics and Methods of Teaching, Bunin Yelets

State University, Yelets, Russia

E-mail: sobdvor@yelets.lipetsk.ru

Dr.Vladimir S.Karapetyan, Department of Pre-school Pedagogy and Methods, Armenian State

Pedagogical University after Kh. Abovyan, Yerevan, Armenian

E-mail: vskarapetyan@mail.ru

Dr.Svetlana A. Rozanova, Department of Higher Mathematics, Russian Technological

University, Moscow, Russia

E-mail: srozanova@mail.ru

Dr.Eugeny I. Smirnov, Department of Mathematical Analysis, Yaroslavl State Pedagogical University

after K.D. Ushinsky, Yaroslavl, Russia

E-mail: smiei@mail.ru

ABSTRACT

The relevance of synergetic effects by solving mathematical tasks on the chessboard as the educational problem is

conditioned by the insufficiently development of theoretical, content-technological and scientific-methodological aspects of

the interdisciplinary integration of mathematical knowledge and chess skills in the context of setting and selecting

technologies and goals. It supposes the student's comprehension of their own existential essence with subsequent creative

and volitional actions and the goal-setting as a choice or solution of uncertainty in conditions of alternatives variety.In this

article the description and assessment of synergetic effects manifestations by solving mathematical tasks on the chessboard

with creative activity of students are described. The methods of the research are leading methodological position of teaching

mathematics technology by founding complexes implementation on the basis of solving tasks on the chessboard and the

synergy attributes actualization and manifestation of creative effects of student’s research activity. It became the ideas and

principles of synergetic approach and foundation of personal experience on the basis of visual modeling. The peculiarity of

the approach, chosen by the authors, is the possibility of obtaining and further diagnosing of creative synergetic effects,

which is naturally manifested during the solution of uncertainty on the chessboard in the conditions of the alternatives

variety. The results of the research present the innovative technology of teaching mathematics on the basis of solving

problems on the chessboard with the actualization of synergy attributes (variety of goal setting, generalized construction of

problem zone, dialogue of mathematical, information, humanitarian and cultures of natural sciences). The complete and

hierarchical complex of multi-stage mathematical tasks on the chessboard are developed and realized on the basis of

founding conception, which allows to master not only different methods (combinatorial, probabilistic, graph theory,

mathematical and computer modeling), but also promotes the development of basic qualities of personality – creativity,

reflection of own choice, creative self-independence. To analyze the phenomenon of creativity and its structural

components, the authors have developed the psycho-diagnostical tools that include diagnostics of the intellectually-

heuristic, intellectual-logical, motivational and reflexive aspects realized in the experiment. Practical significance of the

article have a practical value for the research in the field of methodologies of teaching mathematics, psychology, pedagogy.

The information noted in the article will be interested to a wide range of readers: scientists, methodologists, lecturers,

postgraduates, students.

Keywords: founding complexes of tasks, mathematical tasks on the chessboard, student’s creativity, synergetic

effects, variety of goal setting, visual modeling.

1. INTRODUCTION

The psychological interconnection of interdisciplinary integration of mathematical knowledge

and chess skills is not sufficiently considered, in particular, the psychological mechanisms of goal-

setting and manifestation of synergetic effects are not described. However, long experience of

pedagogical practice of teachers and chess trainers, shows that chess, in addition to influencing

intellectual development, promotes both creative activity and the development of reflexive abilities of

the learners. Chess fantasy expands the player's view abilities in the process of playing activity and

promotes the choice of only correct move in the position. In psychology the creative activity of this

aspect has not been studied enough. In researches (Karapetyan, Mirzakhanyan & Gevorgyan, 2016;

Karapetyan & Gevorgyan, 2017) learned the problems between the determination of choosing chess

moves appropriateness by children and typical manifestations of cognitive dissonance and consonance,

which naturally arise in the process of argumentation. According to the reseacher, the psychological

phenomenon of cognitive dissonance and consonance in the sphere of argumentation in the process of

playing chess is transformed into appropriate oriantational situations. The results of emotional and

logical comparison with logical and internal conflicts are considered. This is evidenced by the

subsequent choice adequately or inadequately expressed on the players' emotional and behavioral

manifestations. However, the psychological mechanisms for solving uncertainty under conditions of

alternatives variety have not been practically studied. A special approach is also needed for the

assessment of manifestations of synergetic effects of student's creative activity in the context of setting

and selecting goals for both in solving mathematical tasks and in choosing the only correct step in the

chess position. Mathematical tasks on the chessboard belong to the class of algorithms that are not

solvable, the algorithm, which consists in a complete search of all variants, i.e. to the class of "NP-

complete tasks" (nondeterministically polynomial) in the theory of algorithms (McConnell, 2001). In

the researches of recent decades, the integration of chess teaching with mathematical education aimed

at the development of learners' logical thinking and intellectual abilities have been sufficiently

described (Brestel, 2011; Geek, 2009, 2010; Dvoryatkina & Loskutov, 2016; Karapetyan, 2014;

Polоudin, 2017; Sukhin, 2012). European researchers (Burgoyne & Sala, 2016; Sala & Gobet, 2016;

Sala, Foley & Gobet, 2017) considered that chess teaching improves the mathematical ability of

learners at primary and basic general education. Thus, mathematical tasks on the chessboard are

traditional creative tasks because, firstly, the search for their solution is carried out on an infinite,

unlimited space; secondly, the variety of goal setting creates the possibility of making an unlimited

number of decisions. So the skill of goal-setting, that is, the choice of goals in the process of

mathematical tasks solving on the chessboard, can contribute to the development of student's creative

thinking and the reflection of own choice. The description of synergistic effects manifestations and

student's goal-setting activity help determine the essence of synergy in the context of creative activity

and assess the magnitude of synergistic effects from the integration of mathematical knowledge and the

ability to choose a solution in conditions of different alternatives. So the main problem of the research

is to define the technology and realize the description and assessment of synergetic effects

manifestations of student's creative activity in the process of setting and selecting goals for solving

mathematical tasks on the chessboard.

2. METHODOLOGY

In methodology description the synergetic approach has been chosen as a starting point (Haken,

2004; Knyazeva & Kurdyumov, 1994; Malinetskii, 2013). The peculiarity of synergetic approach is in

the possibility of obtaining and further diagnosing of creative synergetic effect, which naturally

manifests itself in the solution of uncertainty on the chessboard in conditions of the alternatives variety.

Integration of chess playing and mathematical education of students contains the enormous

opportunities for self-organization and development of personal and creative potential leading to self-

regulation and reflection. The material for implementation of technology was the mathematics of the

chessboard, which makes it possible to solve mathematical tasks of different levels by combinatorics,

graph theory, number theory, probability theory etc.

Quantitative assessment of synergetic effect of student's creative activity in the context of the

integration of chess instruction in the mathematical education was carried out on the basis of a holistic

methodical complex developed by the authors, including the diagnosis of the following components of

creativity: intellectual-heuristic, intellectual-logical, motivational and reflexive. It is noteworthy that

the psychologist D.B. Bogoyavlenskaya (2017) uses a variety of chess tasks with non-standard figures

on non-standard boards for the diagnosis of creativity. The unusualness of the situation, according to

the researcher, should smooth out the differences between experienced chess players and those who did

not know how to play chess before. The development of personality is impossible to imagine without

the presence of situations of intellectual tension in the conditions of uncertainty and lack of future

activities prediction, opportunities to overcome the problem areas that arise in integration of

mathematical education and chess playing. The presence of such problem areas and situations of

overcoming is directly related to the development of complex knowledge and is an important attribute

of qualitative changes in the development of personality.

The methodology and guide for the study of such education content are the functional

characteristics of the state and order parameters in the transitions from chaos to the equilibrium setting

of systems of different types (fractal objects and processes, the theory of dissipative structures,

nonlinear dynamics, etc.). It is quite significant that in this case, the transition from chaos to order (and

vice versa) occur through the universal mechanisms of system complexity. The possibility of setting

and solving the individualization problems of mathematical education and motivation to study

mathematics can be based on the actualization of leading pedagogical technologies of founding

procedures (Smirnov, 2012) and the adaptation processes of complex knowledge (including modern

achievements in science) to the development of mathematical methods by most accessible for these

purposes means, methods and selection (Smirnov, Sekovanov & Mironkin, 2014). At the same time,

the effectiveness achieving of teaching mathematics is possible on the basis of synergetic effects

manifestation and research approaches in the development of educational activities of students in the

creation of rich information and educational environment of mathematics and chess integration.

Thus, it is possible to potentially, in a generalized sense, to implement in the process of learning

mathematics the integration and mathematical dialogue of humanitarian and information science

cultures at different levels of actualization of forms, methods and means of multifunctional and multi-

stage cognitive activity. Mathematics education as a complex and open social system carries a huge

potential of self-organization and positive manifestation of synergetic effects in different directions: the

development and education of the individual, the orderliness of the content and structure of cognitive

experiences of communication and social interaction of subjects on the basis of cultures dialogue. The

student should technologically comprehend a series of specific mathematics problems or "problem

zones" on the chessboard (as well as infoermation and humanitarian problems) as integrative points of

relevant information, solved by mathematical and information methods and technologies. The content-

technological stage of mathematical education synergy is aimed at development of the adaptation

stages of generalized construct of «problem zone" of school mathematics to the current state of

mathematical knowledge and methods of students educational and chess playing activity. It is

necessary to design techniques and methods of reflection and research of technological parameters of

generalized construct against the background of the adaptation system and obtaining new results.

3. MATERIALS AND METHODS

Experimental base of the research has been the Scientific Research Institute of Chess of

Armenian State Pedagogical University after Khachatur Abovyan, Chess Academy of Armenia,

Schools № 57, 194, 33 in Yerevan, secondary school №.5 in Yelets, Lipetsk Region. The research of

the problem was carried out in three stages:

- in the first stage, a theoretical analysis of synergetic approach and founding methodology,

dissertation works on the problem, as well as theory and methodology of pedagogical research; the

problem, purpose, and research methods are set out. So, the experimental research plan is worked out;

- in the second stage, the implementation of a four-stage integrative technology for teaching

mathematics has been presented on the basis of solving tasks on the chessboard with the actualization

of synergy attributes (variety of goal setting) and the manifestation of creative synergetic effect; a

holistic methodical complex has been worked out, including diagnostics of the components of

creativity (intellectual-heuristic, intellectual-logical, motivational, reflexive). An experimental work

has been carried out; the conclusions obtained during the experimental work have been analyzed, tested

and clarified;

- in the third stage, the experimental work has been completed, theoretical and practical

conclusions have been refined, statistical processing of the data has been carried out, obtained results

have been summarized and systematized.

The researchers have developed the aproarch of integrative learning technology for mathematics

on the basis of solving tasks on the chessboard with the actualization of synergy attributes (the variety

of goal setting) and the identification of creative synergetic effect. As part of the technology of using

the holistic and hierarchical complex of multi-stage mathematical tasks on the chessboard on basis of

the founding conception (Smirnov, 2015) is described. It should be noted that the concept of founding

is considered in a broader sense, as a process of becoming an individual based on a phased expansion

and deepening of knowledge and experience necessary and sufficient for mastering the subject content.

It has been aimed on the development of learners' personal qualities, including creativity and creative

independence. Founding procedures for the transition from the current state of student’s mentality and

its actual representation to the potential development of personal qualities are multi-stage and

multifunctional. Many-stage mathematical assignments for the purpose of forming creative activity of

students were previously used by the Polish mathematician M. Klyakla (2003) in schools with in-depth

study of mathematics, Russian researchers V.S. Secovanov and E.I. Smirnov (2014) in the teaching of

fractal geometry etc. According to the mentioned authors, the introduction of innovative teaching

means in the form of multi-stage mathematical assignments as part of the substantiating complexes

ensures the formation of creativity, including: the development of intellectual operations and reflexive

abilities, convergent and divergent thinking, tolerance to innovation, the development of skills to

predict the results of mathematical activity. Each cycle is a logical chain of tasks connected with a

single supporting idea, with the gradual accumulation and complication of information on the

implementation of that idea. It will be illustrated the technological chain of solving mathematical tasks

on the chessboard with the actualization of synergy attributes (the variety of goal setting, the

generalized construct of the “problem zone” of mathematics, the dialogue of the mathematical,

information, humanitarian and natural science cultures) and the manifestation of creative synergetic

effect.

The motivational stage is manifested in the expression of the value and personal characteristics

of cognitive and creative activity of the learners in mastering the standards and models of

phenomenology of visual-intuitive modeling of applied mathematical tasks. In this stage, it will be

expedient to offer students by solving the most common geometric tasks - the tasks of cutting a

chessboard, allowing to obtain geometric forms of different complexity and accordingly, consciously

(by choice, rather than pattern) solve geometric tasks for symmetry, parallel lines, coordinate system,

equality of figures, the properties of a square, a triangle etc. The main method for solving tasks of this

class will be a problem situation or a problematic dialogue that establishes the possibility of

formulating the ultimate goal for finding the potential for the choice of solutions.

1. The legend about four diamonds. One oriental lord was a very good chess player, that in his

entire life he lost only four games. In honor of his winners, he ordered four pieces of diamond to be

inserted in the board, one for each field, where he was put mat (Figure 1, where knights are depicted

instead of diamonds). After the death of the lord, his son, a weak player and cruel despot, decided to

take revenge on them. He ordered the players to cut the board into identical pieces, so that in each of

them there was one diamond. It is assumed that the cuts pass only along the boundaries between the

verticals and horizontal contours of the board.

Decision. One of the correct solutions is shown in Figure 1.

Having four knights in different fields of the board, we can get many tasks connected with

cutting. The interest is not only finding one necessary cut, but also counting the number of all methods

to cut the board into four identical parts containing one knight each. It is noted that the greatest number

of solutions (800) is the problem with the arrangement of the knights in the corners of the board.

Fig. 1. The legend about four diamonds

2. The paradox of cutting the chessboard. Cut the chessboard into four parts, as shown in Picture

3, and draw a rectangle from them (Figure 2). The chessboard consists of 64 cells, but the resulting

rectangle is made up of 65. Question: where was one extra field from?

Fig. 2. The paradox of cutting the chessboard

The solution of paradox lies in the drawing fact is not entirely accurate. If you make it more

accurate, you can see a barely noticeable parallelogram. Its area is equal to the area of the extra cell.

3. The Proof of Pythagoras theorem on the chessboard. Cut the board as shown in Figure 3. The

board is divided into a square and four equal right-angled triangles. In both cases, the areas of the

triangles are equal, which means that the areas of the remaining figures are also equal. Since a large

square is built on the hypotenuse of a right-angled triangle, and small ones are on its cathete, hence the

square of the hypotenuse is equal to the sum of the squares of the cathete.

Fig. 3. Theorem of Pythagoras on the chessboard

4. How many cuts are needed on the board to cross all of its fields?

The stage of process-activity is manifested in the design and organization of procedures for

mastering innovative manifestations of the essence of a mathematical construct on the basis of

actualizing the techniques of creative cognitive self-activity. To improve the effectiveness of goal-

setting at the stage of learning activity of students organizing it is necessary to develop the creativity of

thinking to optimally overcome the situation of uncertainty in the conditions of alternatives variety. It

forms such information contexts where the development of goal-setting is carried out in the logic,

posing and solving both mathematical and chess problems.

At this stage, tasks can be recommended that illustrate the unusual geometry of the board and the

unexpected properties of the figures. For example, according to Euclidean geometry, the shortest

distance between two points is a straight line. However, in chess this is not always the same. On this

property, such techniques as "repulsion by the shoulder" and others are based. One of the most famous

chess e'tudes by R. Reti can illustrate this property (Figure 4).

Fig. 4. E'tude by Rikhard Reti

It seems absolutely incredible that in this position the white king is able to cope with a black

pawn. However, this becomes possible if he follows her not by the usual straight line, but by the

"royal" one. To the field of h2, the white king can reach in different ways. The route of h8-g7-f6-e5-f4-

g3-h2 is equal in length, i.e. in number of turns to the route of h8-h7-h6-h5-h4-h3-h2. On the

chessboard, the two cathete turned out to be equal to the hypotenuse. The method of solution, proposed

by Ye. Ignatov (2018), will be innovative and unexpected, for the derivation of algebraic formulas with

the help of the chessboard. For example, to prove the equality:

a);

2)1(

...21

nn

n

; b).

2

)12(1)...21(8 nn

.

The proof of equality a): let's find the sum of the n of the first natural numbers "by the method of

the chessboard". To do this, we black out all the fields of the first vertical on the board (n + 1) × n

(Figure 5, where n = 8), all the fields of the second vertical (except for the top), the third vertical

(except for the two tops), etc., finally - the bottom field of the n-th vertical. As a result, white and black

fields on the board will be equally divided, especially 1 + 2 + ... + n. Since the board area is n (n + 1),

we get 2 (1 + 2 + ... + n) = n (n + 1).

The control-corrective stage is manifested in monitoring and diagnostics of procedures for

measuring the state and expansion of experience, in manifesting of synergetic effects and

characteristics of learners' personal qualities, in determining and optimizing technological procedures

and the content of mathematical education.

At this stage, it is possible to recommend assignments with different variations of the conditions

and data of the tasks, an estimate of the choice of optimal method for solving the problem, tasks with

incomplete data etc.

Task. How many ways are there to place 8 rooks on the chessboard so that neither of them can

take the other?

Fig. 5. To place 8 rooks on the chessboard

In "mathematical" presentation the task can be formulated in several ways : "Fill the 8 × 8 matrix

with zeros and ones so that the sum of all elements of the matrix is 8, the sum of elements in any

column , row or diagonal array did not exceed one "; or to obtain other variants of its solution by

means of following transformations: by turning the chessboard by 90, 180, 270 degrees , respectively;

axial symmetry with respect to main diagonals: horizontal and vertical axes of symmetry of the

chessboard. As a result, we obtain equivalence classes of possible arrangements that pass into each

other, that is, combinatorial orbits. The ultimate goal set before decisive this task can also be

formulated in several variants:

1. Construct one solution of the task;

2. Analytically prove that the solution exists;

3. Determine the possible number of solutions;

4. Construct all possible solutions.

To solve a similar problem, learners can use the following types of solutions: combinatorial

method, graph theory, the method of mathematical induction, and also use the arithmetic progression,

which was successfully used by the French mathematician E. Luke. Mastering a wide arsenal of

methods for solving the mathematical tasks, students are flexible in the formulation and selection of

goals, maximum self-independence and, as a result, the ability of self-realization. The most productive

at this stage will be combinatorial tasks on the chessboard (Marcuson, 1989).

The general-transformative stage is characterized by the content and characteristics of

innovations transfer into the mass practice of school mathematics mastering, the integration of the

individual and the social in the design of innovative mathematical constructs, exchange of information

and verification of the innovative activity of students. In this stage, it is also possible to form several

goals. The main efforts of the learners are concentrated on a multipurpose approach. There is a strong

correlation between the capacity for goal-setting and reflection, which provides a productive analysis of

the goal-setting process; creative thinking, which allows the effective usage of creative goal-setting

strategies; motivation, which activates and regulates the self-development of the learners as an

individual of goal-setting.

It is possible to offer research tasks to junior schoolchildren on modern scientific problems, for

example, on combinatorics: architectural combinatorics (studying the problems of architectural shaping

on the basis of various combinations), combinatorics in programming (research and study of various

combinatorial algorithms for EBM), combinatorics of orbits (obtaining new solutions of combinatorial

tasks by converting and separating classes of equivalence), etc.

4. RESULTS

This integrative learning technology of mathematics teaching on the basis of solving tasks on the

chessboard with the actualization of synergy attributes (varieties of goal setting) and the manifestation

of a creative synergetic effect have been realized in the schools of Armenia (№№57, 194, 33 of

Yerevan) and Russia (Lipetsk region). Students at the age of 9 to 16 years took part in the formative

experiment. For the students of the experimental group (n1 = 180), the methodology of teaching

mathematics was based on the introduction of chess in the educational process. In the control group (n2

= 180), the teaching of mathematics was taught using traditional teaching methods. On the basis of

theoretical analysis and generalization of scientific research in the studied direction have been

identified the following criteria for the manifestation of creative effect: intellectual-heuristic

(Karapetyan & Gevorgyan, 2017); intellectual-logical (Sala, Foley & Gobet, 2017; Dvoryatkina &

Loskutov, 2016); motivational (Rozanova, Karapetyan, Smirnov et al., 2015); reflexive (Hartkens,

2017), and also search of corresponding diagnostics techniques allowing to fix this quality was carried

out (Table 1). The results of the interview with PARLA method and the analysis of the video game of

chess revealed that the usefulness of the learned move does not correspond to its future

implementation. The PARLA method was also used in the process of argumentation with the aim of:

a) Combining or correlating the data of the trainer and the school psychologist with the results

obtained with PARLA method;

b) Consideration of chess game from beginning to the end in chain context: "problem - action -

result - training - implimentation";

c) Clarification of those personal qualities of chess players, which are manifested especially in

the chain of "teaching- implementation".

In addition to the above-mentioned, during the interview we were interested in how the students

learned to approve their own moves and how to implement them. It turned out that application of

PARLA method is more expedient in the course of the interview, after the joint work of the trainer and

the school psychologist, as the number of new actions, which are previously unfamiliar to the child

(training and implementation), increases.

Table 1. Components, criteria of creativity and diagnostic means

Structure of

creativity, key

components

Diagnostic characteristics of creativity

(criteria for synergy manifestation)

Diagnostic Means

Intellectual-heuristic

- ability to produce a large number of ideas

based on internal cognitive consonance;

- ability to overcome emotional instability

(dissonance) with the strengthening of the

logical component of argumentation;

- ability to produce unusual, non-standard

ideas (originality)

methodology of PARLA with

the purpose of determining

the validity of the course of

problems solving on the

following chain: problem-

action-result-

teaching/learning the usage of

experience in solving of

similar tasks

Intellectual-logical

- ability to analyze and synthesize;

- divergence and associativity of thinking;

- flexibility and fluency of mental

operations;

test of the structure of

intelligence of R. Amtkhauer;

Motivational

- self-determination and self-improvement,

creative self-realization;

- cognitive activity, motivation to

achievement.

methodology of Ch. D.

Spielberger (the modification

of A. D. Andreev for Russia)

Reflexive

-self-evaluation of the person;

-reflexion.

methodology for research of

self-evaluation of the person

by S. A.Budassey;

methodology for diagnosing

the level of reflexivity by A.

B. Karpov

For the convenience of the comparative analysis, all diagnostic data were distributed according to

levels - low, average, high. On the basis of the frequency distribution of the experimented subjects by

levels, the average level indicator (ALI) of each quality was calculated on a three-level scale by the

formula:

100 32 cba

ALI

where a, b, c is the percentage of the number of experimented subjects

with low (a), average (b), high (c) levels of developmental characteristics, according to the used

diagnostic methodologies. Table 2 presents the comparative diagnostic data for all structural

components and their characteristic properties, including the indicator of average level and the integral

indicator in the control and experimental groups.

Table 2. The diagnostics for creative effect

Components of diagnostics

Levels of the development of creative

activities

ALI

low

average

high

contr.

exper.

contr.

exper.

contr.

exper.

contr.

exper.

< 93

≥ 100

15 - 33

Intellectual-

heuristic

Problem

(argumentation

of own moves)

80

52

95

103

5

25

1,58

1,85

Action (ability to

argumented

actions)

92

54

73

93

15

33

1,57

1,88

Result (the

number of new

creative actions)

79

44

101

111

0

25

1,56

1,89

Implementation

(the number of

usage of

originally

learned moves)

125

83

55

74

0

17

1,30

1,60

Intellectual-logical

<80

80-100

>100

1,61

1,84

82

54

86

100

12

26

Motivational

<28

29-44

45-60

1,61

1,88

78

44

94

113

8

23

Reflexive

self-

evaluation

of the

person

-1<r<-0,39

-0,38<r<0,37

0,38<r<1

1,58

1,85

86

57

82

93

12

30

reflexion

80-122

123-147

148-172

1,53

1,80

86

57

92

101

3

22

Integral coefficient

1,54

1,82

A statistics using Pearson's criterion has been established the level of creative activity

development of students for all components in the control and experimental groups (for the absolute

frequencies of the trait). The main tested hypothesis, which is that there is no difference in the level of

the development of the cognitive sphere for individual components between the control and

experimental groups, was rejected (

,26.7

2

exp

p<0.05 for the intellectual-heuristic component,

,82.6

2

exp

p<0.05 for the intellectual-logical component,

,22.11

2

exp

p<0.05 for the motivational

component,

;57.6

2

exp

,30.6

2

exp

p<0.05 for the reflexive component).

Fig. 6. Visual representation of empirical data

Positive dynamics in the change in the ALI of trainees, characterizing the level of cognitive

processes development for the selected components (Figure 6), as well as significant changes in the

integral indicator (positive dynamics by 28%) have been revealed. All this made it possible to reliably

assert that the introduction of chess in teaching mathematics had a positive impact on the level of

development of all diagnosed indicators.

5. DISCUSSION

V.S. Secovanov, E.I. Smirnov ― the researchers of the problem connected with the formation of

creativity in the process of mathematical education, view it in the context of the introduction of the

technology for solving multi-stage mathematical tasks, including: the development of intellectual

operations and reflexive abilities, the formation of tolerance for innovation and the ability to predict the

results of mathematical activity. However, the authors do not cleary propose a mechanism or an

algorithm of actions that leads to the creativity of thinking and the reflexion of own decisions.

The core of logical and divergent thinking is the ability of argumentation. Argumentation of

solutions to mathematical tasks or moves on the chessboard stimulates not only creative independence,

but also the activity of structural and functional components of synergy. In particular, non-standard

original ideas are revealed based on the internal cognitive consonance; by overcoming emotional

instability (cognitive dissonance), the logical component of the argumentation is strengthened.

According to the results of the latest research of V.S. Karapetyan, the ability of argumentation

easily develops in the process of playing chess, since the playing pupil is put in conditions for a

permanent comparison of the emotional and logical components of cognitive consonance, as it is

evidenced by the subsequent choice of adequately or inadequately expressed emotional and behavioral

manifestations of the players. A deeper study of the psychological mechanisms of argumentation in the

process of integrating chess in mathematical education will allow to expand the possibilities for the

development of personal and creative potential, self-organization and self-regulation. Due to

overcoming cognitive dissonance in solving mathematical problems on the chessboard, the divergence

of thinking is formed, the reflexion of own judgments is manifested and the argumentational actions

are selected. Manifestations of creativity and synergy in solving of mathematical tasks on the

chessboard can be promoted by stimulating the motivational aspect of cognition with PARLA method.

6. CONCLUSIONS

The modern concept of the development of learner's personality is based on the actualization of

such personal qualities and abilities as self-organization and self-expression, self-esteem and reflexion.

This reaearch have shown that integration of thinking and creativity elements, emotional and volitional

manifestations, typical to chess play, the harmonious construction of synergy can become a

psychological mechanism not only for the intellectual, socially-personal but creative development of

the child. The technology of teaching mathematics based on solving mathematical tasks on the

chessboard with the actualization of synergy attributes (a variety of goal setting, generalized construct

of “problem zone”, dialogue of mathematical, information, humanitarian and natural science cultures)

has been worked out. According to the presented technology, the effective development of creativity is

possible with the consistent organization of the educational process by including multi-stage

mathematical assignments on the chessboard as part of founding complexes. Each cycle is a logical

chain of tasks connected with a single supporting idea, with the gradual accumulation and complication

of information on the implementation of the idea. The effectiveness of designed methodology and

multidimensional qualimetric tools has been worked out, including diagnostics of main components of

creative sphere (intellectual-heuristic, intellectual-logical, motivational and reflexive). The obtained

statistical results allow drawing a conclusion about the effectiveness of designed technology. In the

future, it is possible to modernize the methodological teaching material for a system of inclusive

mathematical education based on a chess game. The potential of chess, as an intellectual sport with

small motor activity of individuals, combined with digital and mathematical education, will provide

favorable conditions for improving the inclusive process. The synergy of inclusive mathematical,

digital teaching and chess as an innovative multifactorial tool in the complex system of rehabilitation of

persons with disabilities in Russia can bring significant changes. In particular, it will increase the

degree of formation of educational-cognitive, informative, communicative competencies of learners,

actualize the intellectual activity, and creative potential of students with disabilities, will give an

opportunity to get high-quality education in accordance with modern federal state educational

standards.

ACKNOWLEDGMENTS

The research was supported by Russian Science Foundation (project No.16–18–10304).

Conflict of interests

The authors declare no conflict of interest.

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