This study aims to develop a methodical approach to optimize the structure of teaching techniques and approaches to distance education, as exemplified by Russian universities, to enhance students’ professional competencies. A pedagogical experiment was conducted with the participation of 746 second to fourth year students majoring in engineering at the Ural Federal University in Russia. The coefficients for the relative importance of professional competencies were obtained through expert assessment. The number of experts was 40 faculty members with teaching experience of more than five years. The pedagogical experiment revealed the regularities of the influence of the forms of distance learning on the formation of students’ professional competency levels. The values of students’ developmental competency levels have been determined. Non-linear regression models have been developed based on the values of the integral level of professional competencies and the percentage of distance-learning forms in the number of instructional hours. The proposed methodological approach will be useful for university professors and teachers in identifying the most effective forms of distance learning and in differentiating teaching methods not only in the context of the general pandemic but also in the natural conditions of the educational process.

The purpose of this study is to determine the methodology, develop a theory of construction, put into practice algorithmization and implement the functionality of a hybrid intelligent system for assessment of educational outcomes of trainees on the basis of the identified keyword parameters and structure of the artificial neural network using expert systems and fuzzy simulation; to develop a methodology for the construction of structural-logic, hierarchical, functional and fractal schemes for structuring databases of the didactic field of learning elements; to determine the content, structure of parameters and database components, selection criteria and the content of complexes of educational standards. The methodology of introducing intelligent systems into mathematical education is on the basis of the Hegelian triad: thesis (implementation of the coherence principle) – antithesis (implementation of principles of the fractality and historiogenesis) – synthesis (implementation of the principles of self-organization and reflection of the complex system inversion integrity). Requirements for the organization and construction of the artificial neural network for assessment of personal achievements on the basis of fuzzy simulation have been developed. In the direction of using elements of fractal geometry, the technological structures of clusters that constitute the basis of generalized structures have been developed. In particular, it is revealed that the didactic field of learning elements is equipped with a system of multi-level hierarchical databases of exercises, motivational-applied, research, practice-oriented tasks using expert systems and integration of mathematical, information, natural-science and humanities knowledge and procedures.

Modern researchers focus on spiritual and moral education only of children, particularly in the humanities. In this study, we develop and implement a program for the spiritual and moral education of 24 mathematics teachers aged between 30 and 65 years and assess its effectiveness. The program's effectiveness was assessed based on previously developed psychodiagnostic tools by expanding spiritual and moral development characteristics with new components. Diagnostics of the primary state of personal characteristics of spirituality was conducted before the experimental training and after implementing the program. We analyzed data using the Wilcoxon T-test, which revealed a significant increase in value-semantic, communicative, reflexive, motivational, and critical parameters. A qualitative analysis was conducted by evaluating the further research activities of 60 % of the students. Our program can be used in postgraduate education in a wide range of specialties and areas of training worldwide. The program can provide advanced training in solving spiritual and moral education issues for mathematics teachers.

The study is aimed to develop the open innovation model of student’s research activities based on the adaptation of modern scientific achievements in the context of transition to sustainable development of the education system. It advances a general approach to the design of the structure and implementation of main components of hybrid intelligent system to integrating a production-frame model of knowledge and neural network algorithm for the formation of an individual learning scenario. The software implemented as a hybrid intellectual environment for organizing student’s research activities is conventionally represented as a combination of blocks: a graphical user interface; technological core of hybrid learning environment, including a neural network classifier and a training data bank; a system for interaction and activation of third-party software and software and hardware. The technological basis for the individualization of the learning trajectory is the automated construction, forecasting of development and interactive correction of competence-oriented and associated models of a subject area. As a technological core, it uses a hybrid system of interaction between an artificial neural network and an expert system functioning based on a database of research projects. The presented innovation algorithm makes it possible to implement the technology of student’s research activities organizing as a process of setting up the generalized constructions of complex knowledge. Further research prospects consist of the program implementation of hybrid intellectual learning environment for the development of student’s research activities and its preparation for trial operation.

The article aims to reveal the didactic potential of modern educational technologies used within the framework of the fractal approach in teaching stochastics to learners, to show the effectiveness of fractal approach technologies in practice experimentally. In the course of the scientific research, the authors employed scientific analysis of literary sources on philosophical and methodological problems associated with the introduction of a fractal approach in teaching and informatization of education; systematization and generalization of the principles of fractal pedagogy; study, analysis, and concretization of advanced pedagogical experience in the use of modern educational technologies in the educational process; observation and analysis of the results of educational activities of seventh graders; and pedagogical experiment. This research allowed for identifying a group of modern educational technologies in the implementation of the fractal approach in mathematics lessons and identifying their didactic potential and possibilities of using, which is reflected in Table 1 of the main text of this publication. At the same time, it was found that the technologies of the fractal approach in teaching are quite useful: the experimental group received the best result.