The urgency of the studied problem is because in conditions of significant socio-economic changes, the system of human resource management technologies becomes significant, especially relevant is the use of the potential of leadership influence. In this regard, this article is aimed at revealing the problem of developing leadership qualities of heads of preschool education institutions. The article highlights current issues of leadership development of future heads of preschool education. The leading method of the research of this problem is the analysis of the researched problem in pedagogical theory and practice and experimental check of pedagogical conditions of formation of leadership qualities of future heads of preschool educational institutions that allows to reveal and characterize features of formation of leading qualities of future heads of preschool educational institutions. The article identifies the components, criteria, indicators and describes the levels of leadership development of future leaders of preschool education. The basic concepts of the problem are defined: leader, leadership, professional qualities of the leader, leadership qualities of future heads of preschool educational institutions, pedagogical conditions of formation of leadership qualities of future heads of preschool educational institutions. Relevant requirements to the personal and professional qualities of a leader are revealed. The structural components of leadership qualities of future heads of preschool education institutions are characterized, namely: motivational-value, integrative-activity, directional-activity, emotional-reflexive. The pedagogical conditions of the formation of leadership qualities of future heads of preschool education institutions are substantiated. The offered materials of the article represent practical value for heads of preschool education institutions, educators-methodologists, and future teachers of preschool education.

The article highlights and theoretically substantiates the problem of methodical support for introducing innovations in preschool education institutions. Such basic research concepts as methodical support, innovations, innovative technologies, and stages of innovations (diagnostic, consultative-educational, corrective-developmental) are revealed. An important methodical aspect of the introduction of innovations is the readiness of the preschool teacher for innovative activities, who must be motivated to participate in innovative activities, be able to self-assess their capabilities, be a creative person, a teacher-researcher, and be able to reflect. The article discloses modern innovative technologies, the main goal of the methodical work of the state preschool education institution. Preschool education is the first link in the formation and development of a child as an individual. That is why preschool education is the foundation of the educational system of the country and has great socio-economic importance, which is determined by the need for comprehensive implementation of the development program for preschool children. Presented forms of methodical support (consulting, seminars, workshops, coaching, master classes, «round tables», pedagogical readings, business games -these forms of methodical support reach a new stage with the use of ICT. Such platforms as Zoom, Google Meet, Skype, etc. allow one to organize an informative methodical event not only individually for one teacher but also for a group of educators who need help in solving pedagogical problems in online conference mode). In the activities of preschool educators, their initiative and ingenuity, creative approach and desire to develop and implement changes, interesting ideas aimed at the educational process and their practical implementation, as well as the conditions which the head of the preschool educational institution creates for his employees to carry out innovative activities.

The urgency of the topic is due to the need to study the methodological foundations of the development of acquaintance with the number of young children using play activities. The game is essential for the formation of basic mathematical concepts at the initial stage of development of a child of this age. In the article we reveal the problem of development of acquaintance with the number of young children through play activities, we present several features of perception and reflection of sets by young children.The functions of assistance to preschool teachers and young children in mastering the number in the process of play activities are determined. It is in play activities that the child acquires new knowledge, skills, abilities, develops, and improves their abilities. The essence of the concept of «set» is clarified and several features regarding the perception and reflection of sets by young children are identified.There are proposed methods and techniques of work in the process of acquainting young children with the number, such as: visual (showing illustrations and actions with objects); verbal (conversations, use of verbal artistic image); game (game situations, game exercises, songs).The authors identify the following functions of assistance to preschool teachers and young children in learning the number: the implementation of the principle of clarity; adaptation of abstract mathematical concepts in a form accessible to children; assistance to preschool children in mastering the methods of action necessary for the emergence of basic mathematical concepts; promoting the accumulation in children of the experience of sensory perception of properties, relationships, connections and dependencies, its constant expansion and enrichment.The main types of innovative technologies used in preschool educational establishments for the mathematical development of young children to increase interest in educational activities and promote better learning of new material are proposed.

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Improving the quality of education in preschool education institution is a socially significant problem of nowadays, which faces pedagogical science and practice. Today, pedagogical education is a means of self-development andself-realization, whichchangesthegoalsofeducation, itsmotives, forms, methods, andtheroleofthepreschool teacher. And the purpose of pedagogical education is high-quality training of highly qualified and competitive pedagogical staff for the entire field of education, able to masterfully carry out professional activities and be able to use modern information technologies. One of the tasks of modern pedagogical education is the formation of a person with an innovative type of thinking, pedagogical culture, and developed readiness for innovative activities, professional in his field. Only the active life position of a preschool teacher, improving his professional skills will help him ensure one of the most important rights of students — the right to quality education. There are many facts that affect the quality of pedagogical education: material and technical support of the educational process, information and methodological, financial, etc., but the most important thing is the staffing of teachers in the educational process, who have professional competencies, information technology and apply them in the educational process of the preschool education institutions. The article analyzes the problem of motivation of professional development of preschool teachers as a condition for improving the quality of preschool education in the context of European integration. The concepts of «postgraduate education», «continuing preschool education», and«pedagogicalexcellence» are defined. The article substantiates the need of a comprehensive study of the experience of professional training of preschool teachers in the UK and its use to create an effective system of pedagogical education in Ukraine, also presents the features of the model of pedagogical education in the UK.