SUMMARY
This paper deals with the problem of theoretical justification and development of scientific and methodological support for using the cloud service CoCalc as a tool for the formation of mathematics teachers’ professional competencies. The following contradictions exist concerning the processes of forming these competencies: between the level of abstraction of mathematical objects and the possibilities of providing their visualization by computer interpretation; between the expediency of widespread use of ICT services for mathematical purposes in the training of pre-service teachers of mathematics and the lack of opportunities for their provision to ICT subdivisions and pedagogical universities; between the feasibility of using ICT outsourcing of cloud infrastructure for the pre-service training of mathematics teachers in Ukraine and the non-adaptability of foreign cloud-based mathematical services to the requirements of national educational standards; between the possibilities of application of cloud mathematical services in the process of formation of professional competencies of mathematics teachers and the absence of an adequate method for their implementation. The paper describes the professional training of mathematics teachers in universities of Ukraine, and considers the national and foreign experience of using the cloud-based services in mathematics teachers’ pre-service training and also the tendencies and prospects of using CoCalc in teaching mathematical disciplines. The process of system design of mathematics teachers’ professional competencies is characterized, and the model of using the cloud service CoCalc as a tool for forming mathematics teachers’ professional competencies is developed. The indicators and levels (high, sufficient, medium, low) were identified for each component of the pre-service mathematics teachers’ professional competence system within the proposed model. The method of using CoCalc as a tool for forming professional competencies of mathematics teachers is developed and its basic components such as purpose, content, tools, methods and results are elaborated. Information regarding the stages of research and also the experimental work objectives and content are presented; the quantitative and qualitative analysis of the main stages (ascertaining, formative) of the pedagogical experiment is performed, confirming the hypothesis of the study.