The article deals with the problems of using adaptive cloud-based learning systems (ACLS) in the modern high-tech educational environment and expanding access to them as tools of educational and research activity at higher education pedagogical institutions in Ukraine. The conceptual apparatus of cloud-based adaptive learning systems application and design is considered; their main characteristics are revealed; the ways of their pedagogical application are described. The experience of Institute of Information Technologies and Learning Tools of NAES of Ukraine on designing and applying of the cloud-based learning and research environment is outlined. The results of the survey of 31 higher education pedagogical institutions on using ACLS are presented. It is established that in the near future ACLS will become the driving force behind the development of new pedagogy, new strategies for personalizing education, and expanding opportunities for active learning.

The role of neural network modeling in the learning сontent of special course "Foundations of Mathematic Informatics" was discussed. The course was developed for the students of technical universitiesfuture ITspecialists and directed to breaking the gap between theoretic computer science and it's applied applications: software, system and computing engineering. Co-Calc was justified as a learning tool of mathematical informatics in general and neural network modeling in particular. The elements of technique of using Co-Calc at studying topic "Neural network and pattern recognition" of the special course "Foundations of Mathematic Informatics" are shown. The program code was presented in a CofeeScript language, which implements the basic components of artificial neural network: neurons, synaptic connections, functions of activations (tangential, sigmoid, stepped) and their derivatives, methods of calculating the network`s weights, etc. The features of the Kolmogorov-Arnold representation theorem application were discussed for determination the architecture of multilayer neural networks. The implementation of the disjunctive logical element and approximation of an arbitrary function using a three-layer neural network were given as an examples. According to the simulation results, a conclusion was made as for the limits of the use of constructed networks, in which they retain their adequacy. The framework topics of individual research of the artificial neural networks is proposed.

The article discusses the prospects of the augmented reality using as a component of a cloud-based environment. The research goals are the next: to explore the possibility of the augmented reality using with the involvement of the cloud-based environment components. The research objectives are the next: to consider the notion of augmented reality; to analyze the experience the augmented reality using within the cloud environment / system; to outline the prospects of the augmented reality using in educational institutions; to consider the technical conditions of the augmented reality use. The object of research is: the educational process in educational institutions of Ukraine of different levels of accreditation. The subject of research is: the educational process in a cloud-based environment in educational institutions of Ukraine. The research methods used are the next: analysis of scientific publications, observations. The results of the research are the next: on the basis of the analysis of scientific works, it has been established that the experience of the augmented reality using in the systems based on cloud technologies already exists. However, the success of such a combination has not yet been proven. Currently, laboratory tests are known, while the experiment was not carried out under natural conditions in control and experimental groups. It is revealed that the attraction of the augmented reality for the educators requires the development of new methodologies, didactic materials, updating and updating of the curriculum. The main conclusions and recommendations: the main principles of augmented reality use in the learning process are: designing of the environment that is flexible enough, attention should be paid to the teaching and didactic issues; adjusting the educational content for mastering the material provided by the curriculum; the research methods that can be used in training along with the elements of augmented reality are to be elaborated; development of adaptive materials; training of teachers, which will include augmented reality in educational practice.

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.

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