Цифрлық технологияларды қолданып болашақ математика мұғалімдерін геометриялық фигураларды кескіндеуге оқыту әдістері

The article examines the problem of improving the teaching of geometric construction problems using information and communication technology (ICT) tools in the process of training future mathematics teachers at pedagogical universities. During the clarifying experiment, methodological problems in teaching geometric construction problems were identified, and as one of the ways to solve them, the need for methodological training of future mathematics teachers to use applied information and computer resources in teaching geometric construction problems was based. Analysis of information and computer technologies and programs intended for use in teaching geometry courses was performed, and the effectiveness of using the “Live Geometry” program in creating geometric construction problems was shown. Learners are more interested in geometry and construction problems, and can save all work in electronic format for future use. While studying the topic, the tools in the program will help to draw figures and use these figures in solving problems, formulate the properties of geometric figures, and use them to model geometric bodies. In the given article, the authors analyzed the state programs and concepts, the works of domestic and foreign scientists, and the results of an online survey and interviews with students. 54 students of the mathematics education program were involved in a pedagogical experiment, and the results were shown in detail. As the results of the experiment show, after using the “Living Geometry” program in the lesson, students were recognized from a certain personal subjective point of view, which regulates their cognitive and practical activities, and an increase in theoretical knowledge and practical skills was determined. It has been proven by mathematical and statistical methods that the use of the “Living Geometry” program in the teaching of structural problems allows effective and efficient organization of education and increases the methodological training of future mathematics teachers.

The article highlights the ways of the use of digital technologies in the higher education system of Ukraine in the professional training of future teachers in the context of modern social challenges. The importance of digital resources in the organization of distance and blended learning under martial law is substantiated. It is emphasized that the informatization of society and the activities of educational institutions significantly affects the formation of content, organizational forms, teaching and management methods. It is emphasized that the digital culture of a teacher implies a high level of digital competencies, i.e. the ability to work with modern digital technology and master modern information and communication technologies. The article presents the results of an experimental study on “Use of digital technologies in the educational process of a higher education institution” conducted at the Vasyl Stefanyk Precarpathian National University. It was conducted with the aim of identifying the level of students’ awareness of the possibilities of using modern digital tools in the course of bachelor’s education, substantiating and developing methods for using digital resources in the process of distance and blended learning, promoting the formation of digital competence of future teachers and improving the preparation of students for professional activities. It is emphasized that the experimental study made it possible to identify and specify the didactic and practical prospects for the use of digital technologies in the process of students’ professional development. It is concluded that in the process of digitalizing the professional training of future teachers, special attention should be paid to the creation of interactive educational environments, the use of digital methods and innovative pedagogical teaching technologies, the formation of digital competence of students, as well as the development of networking between participants in the educational process in the context of global challenges.

Relevance. The relevance of the study is explained by the need to explore the personal approach and computer educational technologies for the development of imagination in students over 18 years of age who do not have experience working with pictorial materials. The work is based on classical methods of teaching fine arts combined with modern digital technologies. The article describes the main conditions and methods that develop the imagination of students over 18 years of age in fine arts in institutions of additional education.Aim. To develop a curriculum that includes elements of interactive cooperation with students, the use of digital technologies (neural networks) to develop spatial thinking and coloristic and compositional imagination of students.Methodology. Analysis of scientific literature, conducting classes on the subject «Painting» with students over 18 years old, analysis of their educational and creative activities in the classroom, monitoring the development of the group’s creative skills are the methods for the research into the ways to develop the imagination of students.Scientific novelty/theoretical and/or practical significance. The theoretical and practical significance of the study is its contribution to the process of teaching art disciplines, the possibility of using the methods of developing imagination in the methodology of teaching painting, and improving the fine arts teaching programs for students of different ages. Based on the tested results, it is planned to write methodological manuals on fine arts. The students who completed the course in the discipline «Painting» with the use of digital educational technologies developed exhibition and competitive activities, increased interest in world artistic culture and art, and developed desire to improve their skills. The features of the influence of digital technologies on the development of imagination for the implementation of creative ideas are shown.Research results. A program with elements of digital technologies was introduced into the educational process, the skill and creative level of students increased: they successfully transferred three-dimensional objects to the plane of canvas or paper in the form of well-built coloristic and compositional solutions.Conclusions. Digital and information technologies, as well as technical means, help to achieve good results in the modern educational process. With the introduction of technology to the process of teaching fine arts, pedagogical activity is realized with obvious advantages.

Introduction. Most mountainous areas of Russia have an agricultural orientation and are characterized by the presence of complex development problems: low level of arable land use, low crop yields. Despite this, mountainous areas have significant resource potential for growing vegetables, grain and fruit crops. The use of modern digital technologies, explainable artificial intelligence in agriculture will help preserve ecosystems and efficiently use natural resources, increase the competitiveness of the agricultural sector, and create conditions for the economic and social revival of rural regions in mountainous areas. Research objective. To study the theoretical foundations and modern applied aspects of the use of digital technologies in agriculture in mountainous areas, and to propose a concept for integrating explainable artificial intelligence into digital agriculture to ensure sustainable development. Research methodology. The justification of proposals and recommendations was carried out using a logical analysis of issues and problems of implementing modern digital transformation technologies by agricultural enterprises; an illustrative method was used to clearly present a structural description of the integration of explainable artificial intelligence into digital agriculture in mountainous areas, emphasizing its impact on the transparency of the model and user trust. Research results. The article presents the results of a study on the application of digital technologies for the development of the agro-industrial complex of mountainous territories. The possibilities of smart farming, farmers’ attitudes towards new technologies, and existing barriers in mountainous territories are described. The study systematizes modern digital technologies in the agricultural sector according to certain criteria: space technologies, Internet technologies, information and communication and end-to-end technologies, sensors and detectors. Each individual block of digital technologies is analyzed in terms of the possibilities and advantages of practical use. The key role of explainable artificial intelligence in the formation of precision farming is determined. Discussion of results. The needs of small-scale mountain agriculture can be met using the so-called “appropriate technologies” or “intermediate technologies” approach, which is aimed at developing relatively simple and energy-efficient small-scale tools using materials that are readily available in their intended environment. It has been established that the application of technologies requires a “useroriented” and systematic approach that hybridizes practical and technical knowledge to solve user problems. When designing and implementing digital platform solutions in mountain agriculture, it is advisable to involve different actors: end users, machinery and equipment manufacturers, university research centers and other local stakeholders. Conclusion. The article presents the results of research on the application of digital technologies in mountain agriculture. A conceptual framework for integrating explainable artificial intelligence into digital mountain agriculture is proposed, which will serve as a practical guide for making informed decisions. The results of the study can be useful for researchers, practitioners, and educators in the field of smart agriculture.

The article highlights the relevance of the problem of preparing future mathematics teachers for health-saving learning, preserving and promoting the health of students, and forming a healthy lifestyle. It has been found that changing the forms, methods and teaching aids in the training of future teachers is important for the realization of this problem. It is noted that one of the most effective ways to achieve a high level of professionalism and their compliance with these requirements is the introduction of training technologies as a form of organization of training in the process of professional training of future professionals. The content of the concept of "training" is disclosed. It is established that in accordance with the urgency of the problem of health saving students in the learning process and the requirements of modern teacher training, training technologies are quite effective in preparing future mathematics teachers for students' health education. We have developed a training course "We for a healthy lifestyle through the eyes of mathematics", which aims to broaden and deepen students' knowledge about health, healthy lifestyles and factors that influence health formation; develop the ability of future mathematics teachers to develop healthy lifestyles in the younger generation; update students' understanding of the importance of health, the complexity of achieving and preserving it, as well as the responsibility for maintaining the health of student youth; demonstrate the health of mathematics. The structure of the training is defined, and the individual techniques that should be used at different stages of the training are described. Particularly noteworthy are the techniques associated with the assimilation of the material and practical orientation. These include: Lotto Health, Researcher, and Inverted Learning. For each of the proposed methods, specific examples of tasks that should be offered to future mathematics teachers are given, in order to develop in them the skills and skills of providing a valeological support for teaching mathematics students. Key words: future mathematics teachers, training of future mathematics teachers, training, health care, valeological support.

The article deals with the peculiarities of using modern technologies in the educational process of primary school. It is noted that in addition to the use of various digital technologies, Internet resources and interactive tools in STEAM education, it is effective to use special software tools to visualize, demonstrate and reproduce various phenomena. After conducting an input monitoring, the authors identified an insufficient level of students' theoretical knowledge about the use of STEAM projects, which proves the relevance of the chosen research topic. The authors demonstrate the advantages of using virtual laboratories, immersive technologies (virtual and augmented reality) and relevant mobile applications in the application of STEAM projects for future primary school teachers. The article presents a selection of available modern digital technologies that educators most often use in project activities. The authors analyze modern online resources and offer examples of their use in a higher education institution in the training of future primary school teachers. The study was conducted during the educational process at Borys Grinchenko Kyiv Metropolitan University. This approach allowed us to identify digital applications for STEAM education. The article describes the use of digital technologies such as virtual tours, 360° museums, simulation programs, experiments (Viola the Bird and Paint with music), PhET Sound Simulation, AI (SYNO AI), NoiSee, sketch.metademolab.com, Chrome Music Lab, etc. The introduction of software tools for the implementation of the 5E project method allows students to gain valuable experience in organizing and implementing project-based learning with the involvement of STEAM educational elements.

Introduction. With the increasing number of students with special educational needs (SEN), the Ukrainian system of mathematics teacher education faces significant challenges, particularly the insufficient knowledge and practical skills required for effective teaching in inclusive classrooms. The study of international models and practices is of particular importance for the Ukrainian education system. It not only enables the identification of effective strategies and approaches for organizing inclusive mathematics education but also supports the modernization of professional training for future mathematics teachers in the context of inclusive education. Purpose. The purpose of the study is to analyze contemporary international models and practices of inclusive mathematics education in general secondary schools, with a focus on improving the professional preparation of future mathematics teachers in higher education institutions. Methods. To achieve this aim, general scientific methods (analysis, synthesis, concretization, comparison, classification, generalization) were applied, allowing for a comprehensive examination of international models and practices in inclusive mathematics education. Results. The findings indicate that mathematics teacher training should explicitly incorporate issues related to the implementation of inclusive practices. The role of mathematics and mathematics education in the formation and development of personality has been substantiated. It has been emphasized that new approaches, models, and effective practices should be introduced into mathematics teaching within inclusive education. Several models for implementing inclusive education, including mathematics instruction, have been presented and compared. Such international experience can be adapted and partially integrated into Ukrainian practice, as well as serve as a foundation for modernizing the professional preparation of future mathematics teachers. It is concluded that international experience reveals a wide spectrum of models and practices in inclusive mathematics education, which combine adapted teaching methods, differentiated approaches, extensive use of digital technologies and platforms, and strategies for collaboration with specialists in psychological and pedagogical support. Originality. The scientific novelty of the study lies in the analysis, comparison, systematization, and conceptual justification of international experience in inclusive mathematics education within the context of modernizing mathematics teacher preparation. It also lies in a transformational approach that translates international practices into specific, well-founded, and ready-to-implement recommendations for the Ukrainian education system. Conclusion. International practice demonstrates a systematic and multi-component approach to inclusive mathematics education, offering diverse models and effective strategies. Among them, the European multi-track model is particularly relevant for Ukraine, as its flexibility allows for the combination of different forms of learning while addressing individual student needs. The most effective framework for implementing inclusive education is Universal Design for Learning (UDL), which emphasizes flexibility and the adaptation of both content and teaching methods. The international models highlighted in this study can be adapted to Ukrainian realities. Some models, such as the “pull-out” and “cluster” approaches, are already partially implemented in Ukraine through the organization of resource rooms. Effective inclusive mathematics education is impossible without close collaboration between teachers and assistants, as well as the continuous professional development of educators. The process of professional training for future mathematics teachers is crucial, as it shapes their inclusive competencies and readiness for teaching in inclusive classrooms. In this context, modernizing teacher education programs requires the inclusion of specific components and specialized modules on inclusive education. These should address not only theoretical knowledge but also practical skills, such as task differentiation, the use of digital tools, and effective collaboration with representatives of psychological and pedagogical support for children with SEN. Future research should focus on developing methodological materials for university lecturers that integrate inclusion-related issues into courses on general methods of teaching mathematics in secondary schools.

Digital health technology use among people aged 55years and over: Findings from the 45 and up study.

The article highlights the international and Ukrainian experience of incorporating elements of digitization into the preparation of future modern mathematics and physics teachers. The article analyzes international experience, which includes advanced approaches to using digital technologies in the educational process. It also examines Ukrainian experience, focusing on the implementation of various programs to improve the accessibility of educational materials and communication between teachers and students.The significance of preparing teachers for the effective use of digital technologies in the educational process is emphasized. Furthermore, the article emphasizes the advantages that digitization brings to the preparation of future mathematics and physics teachers. A detailed analysis of international experience demonstrates successful examples of using digital technologies in the educational process. The Ukrainian experience also offers interesting solutions for integrating digital technologies into the preparation of mathematics and physics teachers.The overall conclusion of the article underscores the necessity of integrating digital technologies into the professional preparation of future mathematics and physics teachers, which will contribute to improving the quality of education and preparing graduates for the modern digital world.It has been analyzed that the implementation of programs increases studentsʼ interest, improves the accessibility of educational materials, fosters creativity and analytical skills, and prepares students for the demands of the modern digital world. Keywords: digital modernization; professional training; future math teachers; future physics teachers; digital technologies; computer programs; online courses; innovative approaches; effectiveness of the learning process; improvement of education quality.

Problem formulation. Problem formulation. At the present stage of the educational system development, information and communication technologies (ICT) play one of the most important roles in the didactic system of the future teacher of mathematics. In the process of preparing a future mathematics teacher it is necessary from the very beginning to prepare students for future professional activities at the same time with professional competencies in terms of educational informatization. One of the effective ways of informatization of mathematics education can be the using of the Internet during the process of teaching mathematical disciplines. Because of the fact that every student cannot imagine his life without the Internet, it is possible to move from passive presentation of material to active through the tools of the Internet, which will significantly increase the interest in learning the discipline and improve learning outcomes. The aim of the article is to analyze the issues of didactic possibilities of ICT tools in mathematical education of future mathematics teachers. Materials and methods. The following methods were used to conduct this research: analysis of psychological and pedagogical and methodological literature, systematization and generalization of different views on determining the main ideas, types and methods of construction, as well as the use of Wiki technology in training future mathematics teachers. Results. Based on the analysis of the didactic possibilities of ICT tools in mathematics education of future mathematics teachers, it was determined that among the large number of Internet resources, the Web 2.0 service deserves special attention. This service is a block of social services (blogs and microblogs (Twitter, Blog, Live Jornal), social networks and social presentation systems (viber, Telegram, Instagram, Facebook), wiki projects (Wikipedia, Wikiversity, Wikibooks, Wiktionary) and management systems content (Moodle, Jomla, etc.); social media repositories (YouTube, etc.)), which allow students not only to receive but also to create information, information resources. For example, a blog in mathematics education can be used as a platform for information exchange, consultation, research and more. An equally interesting resource is the Wiki service, the use of which achieves the greatest effect in the independent work of students in the process of creating educational and research projects. Using Web 2.0 tools it is possible to create small training modules on a specific problem, illustrated with visualized examples with animation. The set of such modules can be a distance learning course that students can access at any time. The author also considers the directions and some possible types of use of Internet services in the training of mathematics teachers. Conclusions. An analysis of the didactic possibilities of ICT in mathematical education of future mathematics teachers showed that among all possible Internet services, Wiki has all main features and capabilities of Web 2.0 technology, and therefore is well suited for organizing an educational network of future maths teachers. Thus, the active use of various Internet services in the process of preparing future mathematics teachers turns the usual learning process into a joint creative activity in which all participants actively cooperate in the process of learning and acquiring new knowledge, which significantly increases the efficiency of education.

Developing professional competence among future teachers is a key aspect of the modern education system, especially in the context of the rapid advancement of digital technologies. Since the early 1990s, the European Union has paid special attention to the education and training of highly qualified educators capable of effectively utilizing modern teaching methods. The report “Europe and the Global Information Society” emphasized the need for integrating new technologies into the educational process. This research project focuses on a relatively unexplored area—the use of digital technologies to develop future chemistry teachers’ competencies in Kazakhstan. The research aims to enhance understanding of the effective application of digital technologies in chemistry education and provide practical recommendations to improve the quality of education. The study utilized various methods, including theoretical analysis, online surveys, modeling, experiments, and the analysis of test results. Survey results among students confirmed the importance of using digital technologies to develop the professional competence of future chemistry teachers. A pedagogical experiment using digital technologies revealed the effectiveness of educational and methodological support specifically designed to enhance the professional skills of future chemistry teachers. Monitoring results indicated a significant increase in the professional competence of experiment participants. These findings are crucial for the developing curriculum aimed at preparing future chemistry teachers and contribute to the improvement of education quality in this field. The conclusions drawn can be used to further enhance teaching methods in chemistry using digital technologies in Kazakhstan and other countries.

The content of the article presents an analysis of the problems and trends in the development of modern digital technologies of inclusive education in the field of professional education, taking into account the development of relevant guidelines for customers of educational services. Current problems of digital inclusion in vocational education institutions are considered, namely: high-quality digital Ukrainian-language content that is accessible to various nosologies of education seekers with special educational needs, and the problem of training pedagogical workers in educational institutions to implement the concept of digital inclusion. It was determined that the use of digital technologies means a paradigm shift in approaches to teaching and learning, which is a potential for the transformation of professional education and an opportunity for customers of educational services to succeed in the global economy, ensuring the personalization of the educational process and overcoming the distance between the academic environment and production. At the same time, realizing this potential requires concerted action to address issues such as the digital divide, technology obsolescence, and quality and trust issues. In the context of inclusive educational innovation, digital accessibility is an important aspect of ensuring that all people, including persons with disabilities, have equal access to digital technologies and services. During the study of trends in the development of modern learning technologies in the field of professional and technical education, the staff of the department conducted an analysis of the interaction of inclusive learning technologies and modern digital technologies, focused on understanding how these two elements can interact to create an inclusive educational environment. The ways and possibilities of using digital technologies in the process of inclusive education are determined. The implementation and integration of digital technologies requires significant investment in infrastructure and ongoing support to ensure their sustainability and effectiveness. A successful example of such an infrastructural solution is the single educational-digital ecosystem of the Bilotserk Institute of Continuing Professional Education, which combines the intellectual, scientific, professional, educational, and technological potential of the subjects of the network educational-digital environment, provides for the use of digital, andragogic and interactive technologies in virtual space. The findings of the study indicate that although digital technologies have the potential to significantly improve the quality of vocational education, their successful implementation depends on addressing the associated challenges. It is important to constantly evaluate, study and introduce new initiatives in the field of digital learning and digital inclusion into the educational process, constantly update the educational process through the integration of digital technologies, ensure the appropriate quality of educational programs and resources.

In the context of the digitalization of the economy, fundamentally new approaches are required in the training of engineers based on the capabilities of both modern digital technologies and the digital environment. The modern volume and quality of digital technologies create conditions for the development of new ways of their use in education, as well as the application of world experience in innovative solutions in the field of digitalization of education. One of the key peculiarities of open education is the widespread use of modern digital technologies in the learning process, open educational resources (OER), massive open online courses (MOOCs). There are two trends in the implementation of the concept of open education. The first trend is to use the advantages of OER and MOOCs to maximize access to the world’s highest quality educational services. To solve this problem, it is necessary to develop methods for the use of OER materials in education. The second trend is to create conditions for the design and constant updating of the personal educational environment of teachers and students based on modern digital tools. The use of OER and online learning technologies in the educational process have not become widespread yet. In fact, there is not enough attention which is paid to methodological support for the integration of both educational and digital technologies. These problems can be solved using blended learning technology, which allows combining the advantages of digital and pedagogical technologies, overcoming their disadvantages. Thus depending on the tools used in the digital educational environment, the nature of their application in the educational process, as well as on the degree of restructuring of the educational process, various options for the design and organization of blended learning can be developed. The obtained results of the study allowed us to propose the following recommendations: conducting an information campaign to reveal the prospects, advantages, strategies in the use of OER and MOOCs in higher education; creation of conditions for the development of innovative approaches to the use of digital technologies, the creation of methodological support for blended learning models, the processes of integrating both educational and digital technologies; for the continuous development and renewal of the competencies of teachers and students in the field of using digital resources.KeywordsIntegration of digital and pedagogical technologiesBlended learningOnline trainingMassive open online-courses

The article reveals one of the acute aspects of higher education – the use of modern digital technologies. The article describes the possibilities of modern service programs used in the learning process, methods of presentation of educational material and knowledge control. The interpenetration of the content of different academic disciplines as a factor of creating a unified educational space through modern digital technologies is analyzed in the article. The use of modern digital technologies is considered as a factor of successful mastering of educational material. The authors come to the conclusion that the productive use of digital technologies, the involving students in an independent search, the selection of information forms students' competencies that meet modern requirements. Using test system developed at MPEI for grammar assignments is presented in the article.

The effective use of digital technologies in the organization of distance education plays an important role in the rapid assimilation of new information introduced into science. It will also be possible to quickly enter additional information within the virtual subject being studied. Therefore, the study of methods for introducing digital technologies and their effective use in distance learning processes, the creation of distance learning courses belongs to the group of topical scientific and technical problems. At present, the importance of using digital technologies in improving the quality of education is very high. The purpose of the research work is the effective use of digital technologies in distance education and the theoretical basis for testing its effectiveness. The article mentions the possibilities of the Moodle system, the advantages of course elements, effective methods of teaching science. The use of modern digital technologies and means of communication in distance education will also be discussed. The article outlines the stages of development of digital technologies, defines the meaning and content of the concepts “Digital technologies”, “Digital educational technologies”. Description of information about the Kahoot and Quizizz software platforms for an interesting organization of the educational process. The advantages of this platform in developing the intellectual and creative abilities of students and increasing their enthusiasm for learning are emphasized. Regardless of whether the educational process is traditional or distance learning, it is necessary to increase students’ interest in the lesson, encourage them to work on themselves, and form independent learning skills. Organization of the educational process on gaming platforms, creation of a set of electronic tests and its use in teaching, increasing the effectiveness of the lesson and increasing the independence of thinking of students, interest in the lesson, and a high level of mastery of the subject being taught. It is advisable to use digital technologies in the educational process to develop students’ interest, to develop the skills and abilities to use additional literature outside the textbook, to direct students to scientific research, as well as to constantly transfer knowledge to students.