The quality of education is a key factor in the social development of the individual, society, and the state. This article is aimed at identifying the theoretical essence of the concept, the “quality of education” in the context of the international PISA comparative study and the ongoing education of math teachers as a factor in the development of students’ mathematical literacy. The diagnostics for interviewing and questioning teachers, students and parents revealed problems and difficulties in teaching and studying mathematics. A key tool was context analysis of the GGTU Center for teachers’ professional development activities. The Ongoing Education Program “Schoolchildren Practical Math Literacy Formation” presented in the article and the author’s teaching manual “Life as a Math Plot” reveal their practical focus on the development of math teachers’ professional subject competence, on schoolchildren’s math literacy development, and, as a result, a general improvement in the quality of education according to the PISA results.

The article reveals the problem of education quality improvement, which acquires an important social significance nowadays. The priorities of education quality in Russia are formulated by the President Vladimir Putin who emphasizes that education quality is a system-forming and dominant factor in the social development of the individual, society and state. Analyzing philosophical, social and pedagogical approaches to the notion "quality of education'', it is revealed that this phenomenon determines the conditions and effectiveness of educational process in society. Education, corresponding to the needs of society is considered in the context of a comparative study of PISA general education quality, emphasizing that the results of PISA correlate with the economic and social development of the country. We suggest practical ways to improve the significance of Russian general education system quality in the definition of 15-year-olds mathematical literacy based on the identification of math students' problems. The levels of mathematical literacy achievements and their special characteristics are given in the table of the indicators. As a practical significance of the research results presented in this paper, is the Program "Formation of practical mathematical literacy of schoolchildren", used in the process of math teachers training and aimed at improving their professional competence to form schoolchildren mathematical literacy. The article represents the training manual ''Life as a mathematical plot'' to master the formation of students mathematical competence in grades 7-8. It is directed to making the transition from abstract mathematical tasks to solving practical social problems.

This article actualizes the problem of future teacher professional training in the terms of new pedagogical education; it reveals the methods of dynamic and contextual educational system analysis and theoretical analysis of scientific and special literature, which formulates social challenges to education, its new sociocultural meanings; it also gives normative constructs of national educational policy, regulating and approving main directions of further national education development. The review presents the experience of foreign countries that achieved success in education (France, Finland, Germany, Poland) and the methodological bases of socio-cultural interaction between State and Society, according to the sociocultural partnership principle and methodological approaches (culturological and humanitarianculturological) for the formulated problem solution. It formulates conceptually and justifies essentially the definition of "socially active school" and "socially oriented education", which highlights the interrelated componentssocial and educational, that allows these phenomena to solve social and educational problems successfully and effectively. The article gives the analysis of future teacher professional competence and formulates the following labor functions: necessary skills and knowledge for the professional standard "Teacher", labor actions, it also gives the normative basis for new generation pedagogical shots training and intends for unified requirements establishment to the professional pedagogical activity maintenance and quality , for teachers' qualification level assessment; it presents different resources for new pedagogical education taking into account contemporary socio-cultural meanings (senses). The data broaden the theoretical idea of the teacher of the future professional training in the Russian education contemporary development and stimulate projecting new pedagogical education.

The relevance of the research stems from the fact that the subject of public and professional discussions is indicated by the competitiveness of pedagogical universities, searching for new solutions and opportunities of social partnership based on international experience, using the methods of context, dynamic and theoretical analysis. The theoretical background of this problem made it possible to formulate the concepts of "competitiveness" and "social partnership", as a system of civilized social relations, highlighting universal joint activities of social subjects to achieve overlapping interests, based on the principles of democracy, publicity, openness, respect for the rights and freedoms of all parties of the international social interaction. The article reveals the processes of the international social partnership of a regional university with its foreign partners. It determines the commitment to achieving the goals and solving problems associated with the formulation of international priorities, increasing the university international reputation, the confidence of foreign social partners, ensuring international effects. Successful international educational partnership and practice, implemented by GGTU, presents the project " Spanish educational content .'' It included two significant events: "Educational tour to universities of Spain" and organization of the international Russian-Spanish Forum "Education without Borders: science, experience, technology." The practical significance of this study is aimed at enhancing GGTU quality of pedagogical education and increasing international competitiveness of a regional pedagogical University, and the development of professional skills of teachers, the intensification of international sociocultural partnership and communication of scientists and practitioners in the world.

The problem of continuous pedagogical education in the modern university cluster, representing now theoretical and practical interest for involved communities, is actualized in the article. The necessity of cardinal renewal of organizational, content and technological components of the university education, dictated by the digital economy, is grounded. The possible reasons for the failures of Russian education, influenced by the crisis in the world educational trends, are revealed. The difficulties for the implementation of professional educational activities, associated with the socio-economic uncertainty of the teacher's professional status, the high probability of unpredictable situations, the contradictions between public needs, time challenges and educational activities, are determined. The authors present the implementation experience of the practice-oriented model of teacher's continuous education, actualizing such aspects as professional selection of school graduates showing an inclination for teaching in pedagogical universities; updating the system of pedagogical vocational education in accordance with the personal, social and state requirements; young teacher's multi-faceted accompaniment in the postgraduate period. The results of the project "Continuous pedagogical education: from vocational guidance to postgraduate support" at State University of Humanities and Technology (GGTU) are summarized: pedagogical classes as a component of the system service and tools of future applicants' pedagogical motivation; innovative competence-oriented model of continuous pedagogical practice using the digital environment capabilities; the development of psychological, pedagogical and methodological support system for young teachers -graduates from GGTU -at the professionalization stage. The authors focus on the modern university educational space digitalization, which gives new opportunities for continuous pedagogical education.