Introduction. The most important concepts underlying beauty are the mathematical concepts of symmetry and fractality. These categories are fundamental for modern mathematics, science and culture in general. However, in mathematical education and pedagogical literature, the ratio of these main categories has not been considered yet. Of special interest is the fact that the concepts 
of fractals, fractality and fractal geometry and fractal graphics are not included in the vast majority of high school programmes, although they have become commonly used among mathematicians and graphic designers.
The aims of the article were the following: to demonstrate intersectionality and correlations of the basic concepts of symmetry and fractals from the point of view of synergetics, to establish the relevance of studying these concepts in the course of mathematics for aesthetic education of students and development of their worldview.
Methodology and research methods. A significant role in the study is given to post-non-classical methodology based on synergetic worldview. The author employed the provisions of trinitarian methodology: in addition to two binary oppositions, the third element is necessary to solve the problem of contradiction of these
oppositions and integration into one coherent whole as the onditions of their coexistence. In the course of the research, analysis and generalisation of pedagogical and methodical literature, methods of comparative, historical and logical types of analysis were used.
Results and scientific novelty. For centuries, beauty has been understood as a stable order and symmetry. The synergetrics as a general scientific theory about self-organisation of complex systems allows us to give another interpretation of beauty – as a kind of attractor, the result of self-organisation of nature or the
flight of human thought. In the most general view, symmetry can be considered as transformation of similarity, which is also the core of another concept – fractality. On the one hand, fractality can be considered as one of the manifestations of symmetry in the broad sense. On the other hand, symmetry can be considered as a manifestation of fractality with a finite number of iterations. Thus, the concepts of symmetry and fractality are closely interrelated. Symmetry and fractality are two opposites, mutually complementing each other, aesthetically and mathematically  mutually passing into each other. Symmetry reveals the beauty of a sustainable order and fractality reflects the beauty of the result of self-organisation of the chaos of nature or the freedom of the human mind. Therefore, symmetry and fractals are the most important concepts for the disclosure of the beauty of the universe, which determines their importance for mathematical learning and for aesthetic education of students.
Practical significance. Taking into account the fact that the concepts of symmetry and fractals are directly related to each other, they should be jointly-taught. This will contribute to the development concept of mathematics education: to increase motivation for mathematical studies, to develop cognitive interests and activities, to narrow the gap between education and research processes, to overcome the problems with aesthetic education of students.

Introduction. The urgency of the problem of training future teachers to implement the ideas of multicultural education consists in the need to create a new school and a new person, for whom the interiorisation of human and national heritage is a unifying beginning and a stimulus for the development of society, culture, education and personality. The implementation of the resources of dialogue educational space allows this problem to be solved. A dialogue in its various modifications acts as a pedagogical tool, by which various cultural spheres are integrated and creative activity of pupils as citizens of the multicultural world is stimulated.
The aim of the present publication was to present the options for the organisation of theoretical and practical training of future teachers for effective and creative professional activity in the dialogue educational space, which operates at the level of interaction of different cultures.
Methodology and research methods. In the course of the research, the scientific ideas of humanistic, cultural and dialogical approaches to modern education were used and revised. The humanistic approach is the key approach in the study, since it determines the basis of multicultural education, emphasises the central position of a personality in the interethnic dialogue. The culturological approach in education is focused on the socio-cultural processes, valuable attitudes, unique traditions and customs of diverse ethnic groups, and on the adaptation of students to the available multiethnic environment when preserving national identity. The dialogical approach combines humanistic and culturological approaches, it justifies and creates interaction between different cultures based on the principles of pluralism and tolerance.

Results and scientific novelty. The article reveals the purpose of multicultural education – the versatile development of a personality on the basis of universal and national values, aimed at preserving the ethnic and cultural diversity of the peoples of Russia. The authors formulated theoretical foundations of multicultural education, in which the development of multicultural dialogue of tolerance and respect for a different way of life and mental (intellectual) values of other ethnic groups are dominant aspects. The prerequisites for the development of the humanistic potential of the teacher and student are shown in the context of modelling and implementation of ethno-cultural component of education, development of intercultural interaction and cultural experience. The principles of multicultural education are characterised: continuity, differentiation and diversity, creativity, cultural integrity, three-dimensional (stereoscopic) picture of the world, multilinguality and variability. The definition of the concept of dialogue educational space is given. Dialogue educational space is considered as specially designed environment of positive multicultural cooperation between students and teachers.
Practical significance. The research materials can be used in the practice of higher pedagogical education to improve the quality of training and development of multicultural competency of future teachers.

Introduction. Nowadays, substantial intensification of inclusive processes in educational system (especially in higher education) identifies the need for thorough research of psycho-pedagogical foundations for the development of inclusive education culture.
The aim of this research was to identify and compare the levels and factors of social-psychological well-being of healthy students, students with disabilities and students of inclusive groups.
Methodology and research methods. At the methodological level, the research was based on the proposition that inclusive education assumes the formation and support of such an integrative socio-psychological space, in which neither students with disabilities nor healthy students feel anxiety in the course of interaction in the classroom or out-of-class situation. As the main methodological tool,
a specially designed questionnaire was used for fixing the socio-emographic characteristics of the respondents, as well as for clarifying the problematic experiences of students and their attitudes to using social support resources, for identifying the characteristics of students’ assessments of the conditions of higher education, their satisfaction and involvement in student life and attitudes towards inclusive education. The received results of monitoring were processed by means of the following qualitative and quantitative methods of the analysis: the content analysis; the analysis of percentage with the use of the Pearson’s chi-squared test (χ2); comparative analysis with the use of Student’s t-test; the median test and one-factorial dispersive analysis; correlation and factorial types of the analysis. The statistical software packages Statistica 7.0 were employed for the calculation of the results.
Results and scientific novelty. The authors clarified the patterns of formation of the barrier-free socio-psychological environment of the university, which implements the concept of inclusive education. It was established that there are no global differences in the parameters of socio-psychological well-being between students with disabilities and students without disabilities. This determines the favourable social and psychological prerequisites for development of inclusive higher education. Belonging to the same socio-typological and age group and the same leading activity cause the similar experience (e.g. similar problems and difficulties) of both groups of students. The type of disabling health condition, i.e. a disabled student with the defined particular nosological group determines only the specifics of the difficulties, which might be faced in higher school. The conclusion was drawn that successful inclusion of students with disabilities in educational process and social space of university contributes to positive students’ perception 
of the high school environment.
Practical significance. The research outcomes might be used to define the content, directions and forms of work to support students in the conditions of inclusive education. Thus, it is recommended to further develop the system of inclusive education through the following actions: to optimise social conditions of disabled students; to hold rehabilitational and psychological training sessions focused on communication; to organise educational work in order to raise students’ awareness of special technical means of inclusive education; to involve inner circle of people (parents, friends, fellow students) close to disabled students in the process of socio-psychological support. 

Introduction. The need for increasing the investment attractiveness of Russian regions and meeting the needs of territories for qualified personnel updates the search for infrastructure and coordination mechanisms and tools of the solution of these problems. At the same time, the diversity of interests of participants in dual education and their various structural positions do not allow to solve this task by directive methods, both on the part of state structures and corporate sectors. One of the solutions might be to form an effective management system for dual education, which assumes the organisation of practical vocational training according to inquiries of employers, who are involved in implementation of educational process in order to promote labour market with in-demand experts. 
The aim of the present research was the scientific substantiation of the choice of the institutional model for management of dual education system at the regional level, taking into account the interests of all its members: enterprises, government agencies, students and educational institutions.
Methodology and research methods. The research was based on the systematic approach. The methods of modelling and comparative analysis were employed. The provisions and principles of utility theory served as the methodological basis of institutional model formation.
Results and scientific novelty. The main target reference points of regional participants of dual education were identified and formalised in the form of certain indicators. Two types of models of management of these subjects were considered: models with and without the usage of institutional superstructures. As a key criterion for model selection, maximization of a complex indicator of its utility for interested parties was offered. In particular, it was demonstrated that supervision of graduates by a certain employer is caused by parameters of employment attractiveness, such as a wide range of horizontal and vertical career cards and a possibility of continuing vocational education. The conducted quantitative research on utility indicators allowed the authors to design the institutional model for management of regional participants within the system of dual education and to recommend it for further implementation.
Practical significance. The institutional model presented in the publication is directed to the solution of tasks faced by partner companies and educational establishments within the system of dual education. When applying this model, it is possible to forecast personnel requirements – job availability in the future of jobs, resulting to a considerable simplification of vocational guidance  work, which can be carried out in the course of applicants’ preparation.

Introduction. In recent decades, diverse scientific directions have been actively developed at the interface between various fields of knowledge, including social sciences and humanities. Interdisciplinary research provides a high-quality “unification” of achievements of different sciences and gains fundamental value, since they allow full information on this or that studied subject or phenomenon to be received and options of optimum solutions for difficult research tasks to be obtained. However, there are two problem zones in an integrative trend of knowledge – ontological and epistemological origin.
The aim of the research was to reveal the potential of interdisciplinary research and integration of diverse directions from several disciplines by the example of musical psychology.
Methodology and research methods. The provisions of the system-integrative approach and the concept of “meta-individual world” (MIW) were employed. The MIW theory justifies the heterogeneous, multi-qualitative, multi-determined nature of poly-systems.
Results and scientific novelty. The subject field of musical psychology is outlined and characterised. It is shown that now it is shattered into multi-directional components from the field of psychology and musicology, which links are poorly articulated. The reason is not only in heterogeneity of basic psychological and musicological concepts, but also in the discrepancy of ontological status of objects and objects of two key structural components of the sphere under discussion. Nevertheless, there is urgent necessity for their cross-disciplinary integration and elimination of epistemological dualism in the development of scientific industry. The theoretical and logical grounds for the introduction of the concepts of  “musicological” and “psychological” musical psychology are obviously provided. Differences between them are shown: if the first is turned to a piece of music, then the second – to mentality of a composer and a musician. However, they generate unequal phenomenology and lie outside to each other, although they move in the opposite direction. Therefore, the need for creation of integrative musical psychology (IMP) as psychogenetics, psychophysiology, ethnopsychology or behavioural geography is recognised. As the integrative prototype, it is proposed to use the concept of MIW, which supports the pluralistic view on IMP, i.e. IMP considers individuality (personality, mentality, consciousness) and world (external realities, social groups, culture, art) in a coherent manner. The authors proposed the idea of musical psychology transcendence – bidirectional transitions from one its subject field to another. The main point of existence of such transitions consists in the emergence of the phenomena of otherness as the form (way) of overcoming gaps between subject areas of “psychological” and “musicological” musical psychology. The present research concretises and justifies the ways of formation and development of IMP and contributes to the methodology of social sciences and humanities.
Practical significance. The research materials are of practical importance for education of graduates of musical and art specialties. In the teaching and educational process, it is necessary to use the potential of two aspects of musical psychology, but not separately, in order to form and develop in students the skills of transcendental perception and analysis of pieces of music. 

Introduction. Recently, a professional activity and development of a teacher has been carried out in the context of permanent changes in the education system. The necessity to adapt to innovations is accompanied by an increase in tension, in resistance and in the probability of destructive tendencies in teacher’s personality and professional crisis that can face the teacher in the process of professionalisation. A constructive resolution of crisis leads the teacher to the transition to a new stage of professional development and enrichment of his or her professional and personal potential. The non-constructive resolution of the crisis leads to stagnation of the teacher and increased destruction of the personality. In light of this, it is relevant to study the peculiarities of the teacher’s professional personality crisis and to identify the mechanisms for successful recovery from the crisis.
The aim of the present article was to disclose the research results on current sense states, which teachers face when experiencing professional personal crisis. 
Methodology and research methods. The methodological basis of the work was the systemic and phenomenological approaches. The method of cross-sections was used as an organisational method. The collection of empirical data was carried out by means of survey methods: a phenomenological questionnaire “Experiencing a professional personality crisis” and an adapted version of the test of life-sense orientations by D. A. Leontyev. The research sample was made by teachers of secondary schools. The analysis of the results was carried out through content analysis, comparative and frequency analysis.
Results and scientific novelty. Theoretical and methodological analysis of the works of foreign and domestic scientists allowed authors to form a hypothesis that the situation of crisis is followed by disagreement in consciousness of the person’s professional past, present and future. The conducted research showed that only 19%  of teachers in the state of crisis possessed the balanced temporary loci. Apparently, these respondents were at the stage of recovery from the crisis and the transformation of values and meanings in their consciousness that already occurred. Sense-bearing desynchronosis was observed among vast majority of teachers (81%), expressed in “leaving” into any of temporary loci (in the past, present or future), i.e. a kind of escape from a crisis situation, which, in turn, gave rise to decrease in meaning of life in general. The authors concluded that fragmentary perception of life timelines leads to strengthening of destructive trends in teacher’s professional development and their stagnation. 
Practical significance. The research materials can provide a framework for design and realisation of the programmes of psycho-pedagogical support for teachers, taking into account their feelings during professional identity crisis. 

Introduction. The competitiveness between people covers different areas of human life, including vocational education. The increased competitions in labour market and special attention of both organisations and certain experts to the aspects of competitiveness have given rise to the emergence of such scientific direction as competitionology. Competitionology is focused on identifying and understanding existing laws, principles and rules, which should be used in the course of education of a competitive person and directly in competition in order to achieve goals under legal and faithful conditions.
The aims of the present sociological research were the following: to clarify the impact of competition on personal development, to determine the positive and negative impact of competition on personal development and to compare two types of competitiveness – personal (individual) and corporate (group).
Methodology and research methods. At the theoretical stage, various scientific and expert publications within the framework of the research were analysed. At the empirical stage of work, the survey was conducted through the website ianketa.ru, followed by processing and analysis of the collected data. The survey (n = 210 respondents, average age – 30 years) consisted of close-type and open-type questions.
Results and scientific novelty. Social characteristics are formulated; similarities and differences in personal and corporate competitiveness of future experts are demonstrated. Personal competitiveness is entirely obtained by own abilities in the course of competition with other individuals or groups of colleagues. Corporate competitiveness is reflected in the personal recognition of the staff priority interests to fully satisfy own needs in the future.
In the course of the sociological survey, it was revealed that two thirds of the respondents (76.2%) show personal competitiveness and over half of the participants (56%) feel more comfortable when working individually. To better understand respondents’ perception, the following aspects were analysed: degree of competitiveness, understanding of the concept of competitiveness, possible gendered nature of this phenomenon, competitive effects on  personal traits and behaviour of examinees and their level of preparedness to be competitive. The results of the survey led to the conclusion that competitiveness is a natural form of relationships between people. Building personal competitiveness on today’s world is the urgent need, which will have to be given priority in vocational education of the 21st century. It is expected that scientific knowledge in the field of competitionology will be actively accumulated in the coming years. Therefore, interdisciplinary sciences such as pedagogy and psychology will gain a new impetus to further development through practical application of competitionology.
Practical significance. The research materials might be useful to researchers, who are interested in the problems of development and formation of personal and corporate competitiveness of future specialists; to teachers of secondary and higher vocational education; to students interested in self-development and personal career planning.

Introduction. Nowadays, patriotic education is a strategic reference point of the Russian education system. However, in the modern sociocultural situation, characterised by diversity, discrepancy, multi-vector nature of development of society and polystylicity of culture, it is necessary to revise the forms of patriotic education in order to adjust them in accordance with the  requirements and mental features of new young generation – so-called generation Z, which is considerably different from the previous generation of peers.
The aims of the research were the following: to identify promising areas and vectors of patriotic education, to demonstrate its resources and to highlight the most relevant means and possibilities, which are appropriate to present-day realities.
Methodology and research methods. The work was based on the method of discursive analysis, which allows the dependence of patriotic education practices on modern realities and socio-psychological characteristics of adolescents and youth to be revealed. In the course of the research, the methods of comparative analysis, systematisation and classification were employed. 

Results and scientific novelty. The conceptual and category-based framework of patriotic education and its structure are specified. Patriotism is considered as the most important component of the value-normative system of a personality. Patriotism is based on a positive cultural identity. Patriotism is distinguished by an active character-readiness and ability of a person to transform his or her  feelings towards homeland into deeds and acts, which are necessary for the good of the country. The analysis of practices of patriotic education in different countries allowed the authors to allocate two basic directions – civil-patriotic and military-patriotic education, which act as the complementary components of integral system. Civil-patriotic education is a purposeful, consciously carried out pedagogical process of organisation and management of learners’ activities for mastering socio-cultural, socio-political, economic, moral and ethical experience of democratic relations through media education and participation in socially approved projects, civil-law campaigns and charitable events, which foster tolerance for various categories of fellow citizens. Military-patriotic education implies the formation of necessary qualities and skills for performance of a military duty in ranks of the Russian Federation’s Armed Forces and, furthermore, for understanding and awareness of the personal role and place in service to the Fatherland. This direction involves education towards military traditions, museum pedagogics, participation in historical reconstructions and specially designed computer games related to military-patriotic issues.
The list of educational practices, outlined in the present publication, does not purport to be complete and exhaustive. The task of authors consisted in the search for and choice of those, which, on the one hand, fill an ideological construct “patriotism” with live contents, attract teenagers and youth, and, on the other hand, answer the requests of society for education of a socially responsible personality with sustainable value system. 

Practical significance of the research is determined by the appropriateness and efficiency of applying the described forms and methods of educational work into teaching practice.

Introduction. The increasing amount of knowledge, skills and competencies to be mastered inevitably lead to the need for more intensive classroom and independent studies. Providing favourable psychological background and ensuring active working condition of a student are the most important factors for increasing the effectiveness of the educational process (EEP). This fact has special relevance in the organisation of the educational process for students with physical disabilities, foreign students, as well as students, who receive education under individual educational trajectories.
The aim of the present research was to analyse the possibilities and prospects for increasing the EEP through continuous monitoring and control of the current functional and psycho-emotional state (FPES) of students directly in the process of conducting training sessions.
Methodology and research methods. The research was based on: a systematic approach to the accumulation, systematization and analysis of individual biometric information that allows a reliable assessment and forecast of changes in the current FPES of students; implementation of the principle of biofeedback (BF) and passive remote non-contact measurements of current human biological parameters that exclude any impact on it. Information technologies were used for digital signal processing in the optical and acoustic spectra of natural human radiation.
Results and scientific novelty. The work substantiates an integrated approach to planning, organising and conducting training sessions, as well as control measures, which allow increasing the EEP by taking into account the individual characteristics and capabilities of students using modern digital information technologies to monitor their current FPES. It is shown that the system parameters characterising heart rate variability (HRV) can be used to assess the level of psycho-physiological adaptation of students to the learning process in general, as well as their responses to the complexity of training and control activities. Systemic parameters include the vegetative equilibrium index (VBI), the vegetative index of rhythm (VPR), the indicator of the adequacy of regulatory processes (PAID) and the stress index of regulatory systems (IN). The parameter IN is highlighted as the most informative. The possibility of using the results of the analysis of the change dynamics in the parameter IN for the evaluation of the current FPES of trainees is demonstrated. The possibility and expediency of using embedded monitoring systems has been experimentally confirmed. The bio-mouse and biometric chair are identified as universal means of monitoring the current FPES of students both in the  educational institution and at home (in a hostel environment).
Practical significance. The developed methodical and technical means are of interest to a wide range of educational workers, including such levels as primary, basic and secondary general education, secondary professional and higher education, as well as special professional training and retraining.