Pedagogical science has always faced the issue of finding effective means for achieving educational results. This problem is especially urgent today, when in the rapidly changing world the tools, which yesterday could be used to support the interest of schoolchildren in study of mathematics and could provide an opportunity for the development of their mathematical abilities, quickly become obsolete. Today it is very important to search for new means that foster the development of students with the help of mathematics and mechanisms for including mathematics in the educational process. Thus, the aim of the article is to analyze puzzles as a didactic tool and study the possibilities of using puzzles in the process of teaching junior schoolchildren mathematics, both in the classroom and extra-curricular activities. The leading method here is the modeling of the methodical training system in general and additional mathematical education of schoolchildren, with the inclusion of a new didactic tool that fosters the students' interest to the subject, develops individual mathematical abilities: logical thinking, abstraction, combining, operating spatial images, critical thinking, mathematical memory, etc. As a result of the research, the authors have determined the place, features and methodological aspects of the inclusion of puzzles in the process of teaching mathematics in general and additional school education. They can be used in the system of classical and creative math lessons and in extra-curricular activities of students: a mathematical club, a system of mathematical competitions, a mathematical camp, etc. The practical use of this model makes it possible to reduce the lack of tools in teaching for the development of students' mathematical abilities, which in its turn, makes it possible to speak of purposefully high results in students' mathematical activities, which is confirmed by the conducted experimental research.

To study STEM and STEAM education as an innovative technology, the authors of the paper conducted a comprehensive theoretical and methodological study on the stated problem. A comprehensive theoretical and methodological study was directed towards the solution to the problem of developing a new model for the STEM and STEAM education system to make correct management decisions in the framework of the country's socio-economic development based on objective data. The object of the study was STEM and STEAM education as an innovative technology. The subject of this study is the mechanisms of STEM and STEAM-education in the system of higher professional education of the Russian Federation. The methodological foundation of the study was the general scientific ideas for holistic, systemic, and integrative approaches to the study, which gave an opportunity to assess a set of interrelated and interacting elements that form a certain integrity. The use of these approaches provided a comprehensive analysis and formulation of the research problem and determined the strategy for its study. Both theoretical and empirical methods of studying the indicated problem were used in the comprehensive theoretical and methodological study presented in the paper. The leading methods of scientific knowledge were the following: the method of comparison; phenomenological data analysis; discourse analysis of the text; theoretical analysis; empirical description (expert assessment method and in-depth interviews). The result of the study is the STEM education model developed by the authors as an innovative technology in the system of higher professional education of the Russian Federation. The materials of the paper will be useful for specialists in the field of higher professional education, teachers of natural sciences and humanities, the teaching method of which is based on the STEM approach.

Among the tools used in teaching and learning, computers are very im-portant. Determining what kind of computer-related studies are carried out in the learning and teaching processes for university students will contribute to future research. In this context, the aim of the study is to examine university students’ documents related to computers in teaching and learning, scanned in the Web of Science database, thematically and methodologically. The study was designed by adopting the case study method from the qualitative research models. A sample was not determined for the articles to be included in the study but it was aimed to reach the whole population. In this context, all documents in the Web of Science database have been accessed through the library system of the university. In the document search, the keywords ‘learning and teaching’ and ‘computer’ were searched in all areas, and the studies were found by searching the keyword ‘university student’ within the title. All the studies were analysed by year, document type, Web of Science category, country and language of publication. The findings obtained from the research are explained in detail by content analysis method. A total of 305 documents were reached in the research findings. The findings obtained as a result of the research carried out are given in detail in the findings and results section.

The characteristics of the competitive specialist and requirements for their training served the basis for the development of the generalized specialist model. A new imitative tutorial The Simulative Company, which is a systemized complex of operation teaching methods, is worked out and implemented. The research provides the algorithm of the teaching method The Simulative Company development realized and implemented by the authors in student groups on the specialties Management and Economy and Bookkeeping. The algorithm comprises the following stages: the launch of The Simulative Company (presenting The Simulative Company, the analysis of the environment); the functioning of The Simulative Company (recruitment, professional activity appraisals, partnering); the creation of an automated workplace. It is proved that in secondary professional educational institutions, active teaching methods should be given precedence to, which will increase the student's learning activity, facilitate the development of their initiative and personal potential, generate their creative approach, as well as will allow solving important teaching and organizational tasks. The established specifics of imitative teaching methods usage enable: to put into practice the models of the competitive specialist within the educational process; to realize the teacher’s potential in improving his/her pedagogical excellence; to implement the system of new imitative teaching methods which is presented by the new technology The Simulative Company.

Problems of developing subject education and science have become more acute during different social changes. The most problematic is the teaching of exact disciplines, including mathematical disciplines (mathematics, algebra, geometry), which form knowledge for working in high-tech industries. It is worth noting that the methodology of teaching mathematics has accumulated a large unsystematized portfolio of new techniques and methods. Thus, open problems aimed at developing students' creativity are used more actively in the lessons. Tasks that require students to use non-standard approaches to the solution affect academic success. But a teacher of mathematics needs to have a classification of problems in the educational process to choose tasks of an open type correctly. In this regard, the article deals with substantiating the classification of open problems for teaching mathematical disciplines and developing methodological guidelines for lessons that help increase learning motivation and academic achievement of students. The leading research methods are the following: observation of teachers' methodological work, talks with teachers, analysis of methodological developments, educational documentation and teachers' surveys. Statistical processing of the results using Wilcoxon W test. In 2020-2021 the authors carried out an experiment with 358 students from 16 classes. Based on its results, the authors of the article have identified a classification of open problems for teaching mathematical disciplines; developed and implemented methodological guidelines for lessons of mathematics that contribute to developing students' creativity in the academic progress. The authors evaluated the effectiveness of using the selected classification of open problems for teaching mathematical disciplines. They concluded about the increase of group motivation among students and academic performance in mathematical disciplines. The practical use of the classification of open problems allows the teacher to differentiate educational problems in mathematics, taking into account the relationship between their logical structure and the level of ability achieved by the student to solve problems in mathematics. Methodological guidelines developed by the authors can be used to help increase the learning motivation and academic achievement of students while preparing for mathematics lessons.