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The relevance of the research: For the fourth decade, Russian education system has undergone constant changes at various levels: from preschool to higher education. Accordingly, the process of teacher's professional training is being modernized, including state educational standards, which assume significant strengthening of the role of practical training. In this regard, it becomes urgent to find a balance between the theoretical and practical training of the future mathematics teacher. The purpose of the study is to identify the optimal balance between theoretical and practical teaching in the process of professional training of the future mathematics teacher. The research was carried out based on the regularities of the methodological problems of pedagogical forecasting, including the expert evaluation method, historical analysis, practical methods (questionnaire surveying).

This paper addresses the problem of improving the quality of education for children attending schools located in rural areas, including under-filled rural schools and branches of rural schools in Russia. The relevance of this study is determined by the lack of scientifically substantiated, practically approved models of interaction of these branches with the basic school, the absence of a sufficient regulatory framework for their operation, and the scientific and methodological support of the educational process therein. It was found that there is a strong correlation dependence between the share of rural schools and the share of rural population in the region, and there is a weak correlation dependence between the number of rural school branches and the share of the rural population, the population density, and the number of rural schools. On the basis of the results obtained, provisions on the subjectivity of branch establishment process in rural schools are formulated.

The ideas and principles of synergistic and fractal approaches make up methodological provision for the research. An important role is also given to methods of system analysis, mathematical and visual modeling, experimental and statistical methods. The methodological basis of the article is a synthesis of synergistic and fractal approaches. The purpose of the article is theoretical underpinning and development of effective didactic mechanisms for enhancing pedagogical synergy in mathematical education by improving the key components of pedagogical process on the basis of the fractal approach. The article presents all stages of a technology for enhancing pedagogical synergy in mathematical education on the basis of the fractal approach. As a didactic mechanism of pedagogical synergy, the authors propose a set of teaching objectives and professionally oriented tasks correlated with an interdisciplinary fractal-organized base of key mathematical concepts, as well as a system for monitoring/assessment of students' knowledge on the basis of fractal methods.