The relevance of the research is explained by the necessity of developing digital competence of teachers of Humanitarian disciplines at the higher education institutions (HEIs) in the conditions of the quarantine measures to prevent the spread of COVID-19. It is particularly challenging for teachers of Humanitarian disciplines, who are not specialists in the digital sphere as their profession is focused on humans. The purpose of the paper is to define the influence of the quarantine on the work of teachers at HEIs, revealing the level of the development, challenges and peculiarities of enhancing the digital competence of teachers of Humanitarian disciplines, theirs needs in education and giving recommendations how to arrange education and make administrative decisions. The methods of the research are survey, theoretical analysis, others. The results are to define the peculiarities of the influence of the quarantine on the arrangement of distance learning, the peculiarities of developing digital competence of teachers during the quarantine, their needs in education, the challenges in conducting mass distance learning, the determined complex of scientific approaches to developing digital competence. The practical significance of the research refers to developing three internship programs, based on the results of diagnosing teachers’ digital competence.

The purpose of the research is to determine the effective tools for the critical thinking development for primary school pupils. Methodology. The retrospective analysis of the primary education systems development in Ukraine and in highly developed countries was used, the primary school is determined to be the initial stage of the pupils’ critical thinking development. The material of the study is primary school educational programmes, branches of education, and existing academic subjects, it indicates the necessity to develop and implement the particular subject, and its aim is to develop critical thinking and the abilities to express one’s own opinion, assess risks and solve problems. According to the State Standard of Primary Education the key competencies require the following skills: creativity; initiative; the abilities to justify one’s position logically, to manage emotions constructively, to assess risks, to make decisions, to solve problems, to cooperate with others. Results. The article clarifies that critical thinking is a type of thinking aimed at solving problems, namely studying the argument line (hypotheses, criteria, definitions, arguments, facts, etc.), analyzing alternative solutions; forecasting and assessing consequences. It is proved in practice that inventive problems serve as an effective way to develop critical thinking. The basis of such problems is the theory of inventive problem solving (TIPS). It is determined that an inventive problem contains a cognitive contradiction, its solving promotes the pupils’ critical thinking development. The authors’ subject ‘Eureka’ for pupils of 2-4 classes is offered. Its aim is to develop pupils’ critical thinking, creative and inventive abilities, and also primary skills to carry out natural science researches. The results of the subject approbation are given. Conclusions. Primary school pupils’ critical thinking development can be achieved through the introduction of a range of new subjects at the initial stages of education. The aim of such subjects is to teach pupils to solve inventive problems.

From the analysis of the main theoretical provisions of heuristic self-organization systems and logical models, it follows that according to O. G. Ivakhnenko's systems of heuristic self-organization, the first task is to determine the factors content “that determine the essence of different images”. These are the images that characterize the objects of a particular subject area. After determining the composition and content of these images, the next problem is solved, namely, the problem of “generating the new successful heuristic”, which in content is a solution that leads to increased accuracy. Note that we are talking about improving the accuracy of solving the problem of data processing. It follows from the above mentioned that heuristic self-organization systems are data processing systems. This allows the multiplicity of heuristics. Heuristics in content correspond to the logical rules applied in heuristic self-organization systems. The main provisions of the heuristic self-organization system theory were developed by O. G. Ivakhnenko in the eighties of the last century, but they remain unnoticed to this day. At this time, the task is to explain why the neural network makes such a decision and not another. Based on this, the concept of “explainability of artificial intelligence” was introduced for artificial intelligence. It is the content of heuristics that forms the structure of the neural network in the form of logical rules and determines the logic of the decision made. It is established that the derivation rule, which is the basis for constructing artificial neural networks, is an abductive rule, which, unfortunately, does not meet the fourth heuristic and does not meet the definition of intelligence: intelligence is the ability to measure things. Unfortunately, none of the neural networks can measure things. From the analysis of the basic rules content of inference, it follows that the dialectical method of inference is general (generating) for the basic logical methods of inference. The difference lies in the composition and content of the middle member of the triangular relationship, namely, in the form of the element combination of the relationship: the transition from one concept to another. The explainability of artificial intelligence refers to the laws of the structure and activity of artificial neural networks. But modern theories of artificial neural networks ignore the existence of logical rules (heuristics), which were established by O. G. Ivakhnenko. After all, only knowing the rules based on which problems are solved, it is possible to check the correctness of the decision, but not by searching for such rules. The three hypotheses about the explainability of artificial intelligence and the theory of machine identification can be further defined as statements or theorems and strictly proved.

The issue of "Internet Culture" is considered and the stages of its formation are customary, professional, pedagogical, and ideological. Peculiarities and risks of its functioning in the modern educational environment are determined. The relationship between the Internet and culture is analysed. The conditions and possible directions for the formation of Internet culture of the future teacher of musical art are offered. Positive results of studying the discipline "Innovative pedagogy" are given. Conclusions and proposals on improving the system of training future teachers of music in the educational environment of the Free Economic Zone are substantiated. Two components of Internet culture are substantiated: communicative and operational-technological. The communicative component includes the ability to find the necessary reliable information, to master the means of communication, and knowledge of the ethics of network communication. The operational and technological component consists of the ability to master software products; the ability to work with search engines; the ability to use e-mail; knowledge of the rules of safe use of the Internet, etc. The problems of formation of Internet culture of the future teacher of musical art are outlined: 1) formation of the system of information concepts; 2) the teacher's ability to carry out information activities; 3) readiness of the teacher for constant self-education in the field of Internet technologies and effective use of electronic resources and sources of information; 4) conscious regulation of their own behaviours in the network, moral and legal norms, compliance with network etiquette.

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