The scientific journal "Physical and Mathematical Education" is an open scientific resource for highlighting new ideas and trends in the field of pedagogical science and practice and disseminating the results of scientific and pedagogical research. The journal applies double-blind peer review, follows publication ethics (COPE), and maintains policies on complaints, corrections, and retractions. The editors of the journal invite to publish articles by scientists dealing with education, theory and methods of teaching (physics, mathematics, computer science, technologies), theory and practice of vocational education, digital technologies in education. Key-title: Fìziko-matematična osvìta ISSN 2413-158X (online), 2413-1571 (print) Manuscript languages: Ukrainian, English. Publication frequency: 5 issues per year. Founded: 2013. Founder: Makarenko Sumy State Pedagogical University, Sumy, Ukraine. EDRPOU code: 02125510. Registration of Print media entity: Decision of the National Council of Television and Radio Broadcasting of Ukraine No. 479 as of 29.02.2024 Media ID: R30-02975 Research Organization Registry (ROR ID): 020kwj692 DOI: https://doi.org/10.31110/2413-1571 Cluster (scientific profile of the publication): “Human Capital Development, Social Sciences, and Journalism.” Specialties: A1 Educational Sciences, A4 Secondary Education, A5 Vocational Education. .The Ministry of Education and Science of Ukraine has conferred a category "B" to the professional edition “Physical and Mathematical Education” in the field of pedagogical sciences (13.00.02 - Mathematics, Physics, Informatics, 13.00.10) and specialties - 011, 014, 015 (Order of the Ministry of Education and Science of Ukraine №1412, December 18, 2018).
Сумський державний педагогічний університет імені А.С.Макаренка
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ELECTROMAGNETIC INTERFERENCE AND DATA TRANSMISSION RELIABILITY IN SDR-BASED EDUCATIONAL EXPERIMENTS
Formulation of the problem. This article explains how, within the course Physical Foundations of Information Systems, concepts of signal, interference, and communication channel can be connected to practical metrics of data transmission quality using instructional technical tools (software-defined radio, SDR, and simple radio modules). The aim is to systematize practices of (a) detecting and describing electromagnetic interference (EMI) and (b) assessing data transmission reliability, thereby providing a methodological bridge between physical manifestations in the radio environment and network-level outcomes (PER/PRR/throughput/latency/outages). Materials and methods. The review is conducted as a scoping review, which enables integration of heterogeneous sources: engineering studies on technology coexistence in the 2.4 GHz ISM band, educational publications on SDR laboratories, and standards used as reference points for measurement rigor. Results. The main results include: a taxonomy of EMI observation practices, a taxonomy of reliability metrics, and a correspondence matrix “EMI practice - reliability metric,” complemented by an implementation framework for laboratory work and a minimal package of reproducibility artifacts. The article delineates the limits of SDR use: the device is treated as an educational observation tool, while standards are used as a source of requirements for procedural validity rather than as a basis for certification measurements. Scientific novelty lies in a conceptual systematization of EMI detection/description practices and reliability assessment practices, in a form suitable for teaching the Physical Foundations of Information Systems course using instructional technical tools. Conclusions. The systematization of practices is presented as a correspondence matrix between types of EMI observations and reliability metrics, enabling the design of laboratory tasks with a predefined causal mechanism and transparent requirements for outcomes. Further research could focus on adapting the matrix for blended and distance-learning formats. It is important to expand the scope beyond a single range and a single class of devices: comparing learning outcomes and typical interpretation errors across different technologies (e.g., Wi-Fi, Bluetooth) and different environmental conditions will help refine the universal and specific elements of the proposed matrix and framework.
Actual problems of physics and methods of teaching physics
Computer sciences and methods of teaching them
Copyright (c) 2026 Володимир Шамоня, Владислав Беспалий https://creativecommons.org/licenses/by-nc-sa/4.0
PLANE DILATIONS IN THE SCHOOL MATHEMATICS COURSE AND THE THEORY OF GEOMETRIC TRANSFORMATIONS FOR FUTURE MATHEMATICS TEACHERS
Formulation of the problem. Geometric transformations constitute an important component of the school geometry curriculum and the professional training of future mathematics teachers. However, an analysis of school textbooks and educational literature reveals certain methodological inaccuracies in the interpretation of geometric transformations and their properties. This necessitates a more systematic, scientifically grounded presentation of similarity transformations and their particular subgroups in analytical geometry courses for pre-service mathematics teachers. Materials and methods. The study employs theoretical methods of scientific and pedagogical research, including the analysis and synthesis of scientific and educational literature, as well as mathematical methods for proving and substantiating geometric statements. Results. The article examines plane dilations as a subgroup of similarity transformations. Their fundamental properties and invariants — geometric properties of figures and relations between them that are preserved under the corresponding transformations — are analyzed. The connection between dilations, homotheties, and translations of the plane is established. It is shown that dilations form a group of geometric transformations. In addition, the paper presents theoretical statements and examples of problems illustrating the application of these transformations in the study of geometric objects. Conclusions. The conducted research enables the systematization of the concept of plane dilations as a subgroup of similarity transformations and clarifies their place within the structure of geometric transformations. The results can be used in teaching analytical geometry at higher education institutions and in school geometry courses, particularly in elective classes, mathematical circles, and extracurricular activities. Prospects for further research include expanding the analysis of subgroups of similarity transformations and their application in the methodology of teaching geometry.
Theory and methodology of professional education
Actual problems of mathematics and methods of teaching mathematics
Copyright (c) 2026 Микола Працьовитий, Наталя Правіцка, Софія Ратушняк https://creativecommons.org/licenses/by-nc-sa/4.0
A COGNITIVE-VERIFICATION APPROACH TO TRANSFORMING HOMEWORK IN MATHEMATICAL ANALYSIS IN THE CONTEXT OF ARTIFICIAL INTELLIGENCE USE
Formulation of the problem. The article addresses the problem of transforming homework in mathematical analysis in the context of the widespread use of artificial intelligence systems capable of automatically performing a significant portion of algorithmic mathematical operations. This creates a contradiction between the didactic purpose of homework and the possibility of its formal completion without engaging students’ cognitive activity. The aim of the study is to substantiate an approach to organizing homework based on the cognitive demand of tasks and the degree of their automation, as well as to experimentally verify its effectiveness. Materials and methods. The study was conducted as a pedagogical experiment in a distance learning environment with second-year students of a pedagogical university during the study of the topic “Functions of Several Variables.” The sample included 8 students. Assessment was carried out using three indicators: correctness of solution, explanation, and verification of the result (maximum score – 30 points per test). Subgroups were identified based on the nature of students’ learning activity. The proposed approach involves the inclusion of a mandatory explanatory and verification component and the use of tasks of different types according to their level of automation. Results. The results of the experiment showed that at the initial stage the students demonstrated a similar level of preparedness (48–50%). During the learning process, performance improved in both subgroups; however, a more pronounced increase was observed among students who systematically implemented the explanatory component, with final results of 77% compared to 62%. It was found that the main differences between the subgroups were related not to computational accuracy but to the level of development of explanation and verification skills. Conclusions. It is concluded that the inclusion of an explanatory component changes the nature of students’ cognitive activity, increases the cognitive demand of tasks, and reduces the possibility of their formal completion using artificial intelligence systems. The study substantiates the feasibility of transforming homework by combining different types of tasks considering their level of automation.
Copyright (c) 2026 Ярослав Чкана https://creativecommons.org/licenses/by-nc-sa/4.0
EXPERIMENTAL VISUALIZATION AND NUMERICAL SIMULATION OF FRAUNHOFER DIFFRACTION FROM ORDERED AND DISORDERED OPAQUE MICROSTRUCTURES FOR EDUCATION PRACTICE
Formulation of the Problem. In teaching Fraunhofer diffraction, experimental approaches that clearly reveal the relationship between real-space structures and their Fourier-space representation remain limited in accessibility and scope. This work presents a simple, low-cost, and visually effective method that enables students to observe diffraction patterns and systematically investigate the influence of fabrication-induced imperfections in printed optical masks. Materials and Methods. The approach combines experimental visualization using binary amplitude masks printed on transparent film with a standard office laser printer and illumination by a low-power semiconductor laser pointer, with numerical simulations based on two-dimensional fast Fourier transforms implemented in Python. Microstructures ranging from one-dimensional gratings to two-dimensional periodic lattices and structures with a basis (centered and honeycomb), as well as disordered ensembles and quasi-hyperuniform configurations, are directly compared between experiment and simulation. This comparison allows detailed analysis of how printer-induced deviations – including local distortions, positioning errors, toner spreading, and partial merging of elements – affect the resulting diffraction features. Results. The results demonstrate that in periodic structures the positions of diffraction maxima closely follow the designed geometry, while fabrication-induced imperfections primarily modulate intensities, partially restore nominally forbidden orders, and generate weak additional maxima. In disordered arrays of printed disks, the increased effective feature size and polydispersity in the printed geometry lead to strong smoothing and suppression of the higher-order diffraction rings, whereas imposed short-range correlations produce characteristic features such as a central dark ring. These findings illustrate how subtle systematic and stochastic deviations from idealized binary amplitude masks influence Fraunhofer diffraction patterns, thereby transforming typical printing artefacts into a valuable pedagogical tool. Conclusion. From an educational perspective, the method provides an affordable, hands-on platform for exploring Fraunhofer diffraction, Fourier optics, reciprocal-space concepts, and statistical correlations. By bridging idealized theoretical models, realistic fabrication limitations, and numerical simulation, it is particularly well suited for undergraduate laboratory courses in wave optics.
Copyright (c) 2026 Oleksii Voronkin, Sergiy Lushchin https://creativecommons.org/licenses/by-nc-sa/4.0
ALGORITHMIC SUPPORT FOR REVERSE GENERATION OF TRIGONOMETRIC TASKS USING OBJECT-ORIENTED MODELING
Formulation of the problem. The article presents the results of developing and implementing an algorithmic system to automate the preparation of trigonometry didactic materials. The core research idea is based on a transition from the traditional mechanical selection of problems to a model of reverse equation construction. The proposed approach implies that the software algorithm generates a mathematical condition based on a predetermined set of answers (roots), enabling the teacher to instantly create a vast number of variable tasks of the same complexity level to ensure a personalized learning trajectory. Materials and methods. The system's technical implementation is implemented in Python using the SymPy symbolic computation library. The paper details the developed class architecture, covering various types of trigonometric equations, from basic ones to homogeneous equations and those solved by the variable-substitution method. A crucial technical characteristic of the algorithm is the automatic verification of the domain of permissible values and the filtering of extraneous roots during problem condition synthesis, which guarantees the absolute mathematical accuracy of each generated task. Results. The described system is effective not only for rapid knowledge assessment but also for organizing students' independent work in distance and blended learning formats. The availability of automatically generated step-by-step solutions allows students to independently master algorithms for solving complex problems, minimizing knowledge gaps without constant teacher intervention. The effectiveness of the proposed methodology was verified during practical testing at a lyceum in the Chernivtsi region. Analysis of the experimental results showed that the systematic use of personalized tasks enabled students to better master trigonometric transformations, and the average academic performance in the experimental groups increased by 15–20%. Conclusions. The authors demonstrate that implementing these digital tools significantly reduces teachers' workload, freeing up time for creative pedagogical activities and individualized work with students. The conclusions identify opportunities to scale the reverse algorithm to other topics in the algebra and mathematical analysis course.
Copyright (c) 2026 Світлана Боднарук, Назарій Рибчинський https://creativecommons.org/licenses/by-nc-sa/4.0
CLASSIFICATION OF MOBILE LEARNING SUPPORT TOOLS IN GENERAL SECONDARY EDUCATION
Formulation of the problem. The contemporary development of digital technologies has led to the widespread implementation of mobile learning in general secondary education institutions, providing access to educational resources anytime, anywhere. At the same time, the growing number of mobile applications and digital services highlights the need for their justified selection and systematization. The scientific literature lacks a unified approach to classifying mobile learning support tools, which complicates their effective use in the educational process. Materials and methods. The study is based on the application of theoretical methods of scientific and pedagogical analysis. A review and generalization of scientific works by Ukrainian and foreign researchers were carried out, and publications indexed in Scopus and Web of Science databases were analyzed. Methods of systematization, comparison, and generalization were used to organize mobile learning support tools and determine their role in the educational process. Results. It has been established that mobile learning support tools should be considered as components of a digital educational environment that ensure the implementation of the main components of students’ learning activities. Their key didactic functions have been identified, including the presentation of learning content, organization of interaction, management of learning activities, implementation of practical and research tasks, control and assessment of learning outcomes, support of student motivation, and information search and processing. On this basis, a classification of mobile learning support tools has been developed, including informational-educational, communication, organizational-management, assessment, content creation, research and practice-oriented, gamified, and reference-information tools. Conclusions. The proposed classification ensures the systematization of mobile tools according to their leading role in the structure of learning activities and provides a basis for their pedagogically grounded use in general secondary education institutions. Its practical value lies in enabling the purposeful selection of digital tools in line with the didactic goals of learning and the structure of a lesson, thereby improving the effectiveness of the educational process in the context of digitalization.
Copyright (c) 2026 Юлія Носенко https://creativecommons.org/licenses/by-nc-sa/4.0
USE OF GAMIFICATION ELEMENTS IN TEACHING MATHEMATICAL ANALYSIS AT HIGHER EDUCATION INSTITUTIONS
The article presents the results of a study on the effectiveness of gamification elements in teaching Mathematical Analysis at the National Technical University “Kharkiv Polytechnic Institute”. The research focuses on assessing the impact of gamified approaches on students’ learning motivation, cognitive engagement, and the quality of their mastery of theoretical material among engineering students. Formulation of the problem. Modern teaching of mathematical disciplines is characterized by a high level of abstraction, which often leads to decreased student motivation and superficial knowledge acquisition. In the context of the digitalization of the educational environment, there is a growing need to identify effective didactic tools that activate students’ learning activities. This paper addresses the problem of the feasibility and effectiveness of implementing gamification in the teaching of mathematical analysis. Materials and methods. Within the study, an original didactic game, entitled Mathematical Associations, was developed and implemented during practical classes in mathematical analysis. The experimental base consisted of first-year undergraduate student groups. The research employed pedagogical observation, questionnaires, analysis of students’ learning activity, and assessment of task performance. Digital tools from Office 365 were used to organize feedback and collect responses. Results. The results indicate increased student engagement in the learning process, greater interest in the course, and improved understanding of the key topics in mathematical analysis. Enhanced communication within academic groups and increased learning activity among the majority of participants were observed. At the same time, a small proportion of students expressed a preference for traditional teaching methods. Conclusion. The results of the study confirm the effectiveness of gamification in teaching mathematical analysis to increase learning motivation, develop communication skills, and stimulate students’ cognitive activity. The findings may be used for further research and for implementing gamified approaches in teaching other mathematical disciplines.
Copyright (c) 2026 Ганна Лінник, Ірина Морачковська https://creativecommons.org/licenses/by-nc-sa/4.0
FOUR PROMPTING TECHNIQUES FOR ANALYTICAL WORK WITH TEACHING AND LEARNING RESOURCES
Formulation of the problem. Teachers are increasingly using large language models (LLMs) to enhance their teaching and learning resources (TLRs). However, most of them have no formal training in artificial intelligence and therefore rely on unsystematic approaches. The prompting techniques described in research literature can activate specific capabilities of LLMs, making interactions more predictable and manageable. Yet most existing taxonomies classify these techniques according to the model’s internal logic, whereas a teacher’s work is typically organized around the type of textual task being addressed. This highlights the need for a more purposeful selection of prompting techniques that support analytical work with teaching and learning resources (TLRs). Materials and methods. The research was conducted as an exploratory study, leading to the development of a practical toolkit for teachers. The study proceeded through three stages: a theoretical analysis and synthesis of the relevant literature, the development of a procedure for selecting appropriate prompting techniques, and pilot testing. The selection was based on the taxonomy by Schulhoff et al. (2025), which includes 58 textual prompting techniques. Results. The study produced a practical toolkit featuring four prompting techniques: Role Prompting, Plan-and-Solve, Chain-of-Verification, and Self-Refine, organized according to the logic of the Deming cycle (Plan → Do → Check → Act). The toolkit is implemented through standardized prompt templates adapted to four dimensions of analytical work with TLRs: content, structural and logical organization, language and style, and evaluation. It can be used in two ways: a procedural mode, in which all four techniques are applied sequentially, and an autonomous mode, in which a teacher selects a single technique based on the specific task at hand. Conclusion. The four selected techniques transform interaction with LLMs into a clear and well-defined set of steps. They provide teachers with a structured approach to analytical work with TLRs, regardless of their level of technical expertise. At the same time, any results generated by the model should always be critically evaluated by the user.
Copyright (c) 2026 Таяна Дєордіца, Марина Вороніна, Раїса Гладушина, Ольга Єпіфанова, Олена Козьменко, Володимир Толмачов https://creativecommons.org/licenses/by-nc-sa/4.0
SOFTWARE IMPLEMENTATION OF ALGORITHMS FOR CALCULATING PEARSON, SPEARMAN AND KENDALL CORRELATIONS IN C#
Formulation of the problem. In scientific research in sociology, education, physics, chemistry and other sciences, methods of correlation analysis are used. Correlation coefficients are used to determine the existence of a relationship between two or more data samples. The use of correlation coefficients allows the researcher to determine whether there is a relationship between two data sets and the nature of that relationship: strong or weak, direct or inverse. Thanks to correlation methods, the researcher has an effective tool for confirming or disproving a hypothesis. Information technology tools play an important role in the application of correlation analysis in practice. These tools enable reducing the processing time for scientific research results by several times. Software tools used in correlation analysis vary widely, from MS Excel spreadsheets to STATISTICA and the R programming language. A common drawback of these IT tools is that the researcher using them must have a certain level of mathematical training (STATISTICA), programming skills (R), or manual data entry skills (MS Excel). There is a discrepancy between the capabilities of ICT for correlation analysis and how they are currently used. Materials and methods. During the research, we used the following methods: analysis and synthesis of scientific literature on correlation analysis, algorithm theory, object-oriented programming, and the foundations of scientific research; comparative analysis of algorithms for calculating Pearson, Spearman, and Kendall correlation coefficients; algorithmic method for constructing correlation calculation algorithms; object-oriented programming methods for software development of developed algorithms; correlation methods; computational complexity testing; accuracy testing; exception testing (handling missing values, different sample sizes); numerical stability testing. Results. Algorithms for calculating Pearson, Spearman, and Kendall correlations have been implemented in the C# language. During development, component-oriented and object-oriented approaches in programming were used. Algorithms for calculating Pearson, Spearman, and Kendall correlations are presented in the C# language in the form of a program library in DLL format, which consists of 5 classes. Classes and methods for calculating Pearson, Spearman, and Kendall correlations have been developed. The objective novelty is the software implementation of the algorithm for calculating Kendall correlations for the .NET platform. To calculate the specified correlation coefficients, the CorrAnalyzer application has been developed, which connects to the program library created. The developed library and application are recommended for use in pedagogical, psychological, and sociological research to determine the relationship between the two samples. Conclusions. Correlation methods are used to determine the relationship between two samples. The most commonly used are Pearson, Spearman, and Kendall correlations. The Pearson, Spearman, and Kendall correlation coefficients differ in their principles and calculation algorithms. Based on testing results for accuracy, computational complexity, and exceptions, the suitability of the developed methods for processing experimental results was determined. To make the software library a powerful tool for determining relationships between samples, all three of the above correlation methods are implemented in it. The library for calculating correlation is implemented as a component (DLL) that is integrated into the CorrAnalyzer application, written in C#. The CorrAnalyzer application is designed to determine the relationship between small samples (up to 1000 values). In the future, it is advisable to supplement the developed software library with classes for calculating the Matthews correlation and concordance.
Copyright (c) 2026 Vitalii Bazurin, Sofia Bazurina, Svitlana Kolesnyk, Valerii Kolesnyk, Nina Dulenko https://creativecommons.org/licenses/by-nc-sa/4.0
SYSTEM OF APPLIED MATHEMATICAL PROBLEMS: CRITERIA FOR DEVELOPMENT AND FEATURES OF SOLUTION
Formulation of the problem. The State Standards of Basic Secondary Education (Ministry of Education and Science of Ukraine [MES], 2020) and Specialized Secondary Education (MES, 2024) define mandatory learning outcomes for students based on a competence-based approach, namely, the key competences (eleven in total). One of them is mathematical competence. These requirements as learning outcomes must be achieved during the study of every school subject, including mathematics. One of the most effective means of forming key competences in mathematics lessons is applied problems—problems that exist beyond mathematics but are solved using mathematical knowledge. They are commonly classified into real problems, which exist in practice, and imaginary (quasi-applied) problems, which arise in people’s imagination and may need to be solved in the future. Therefore, such problems are included in school mathematics textbooks, teaching aids, problem books, and didactic materials. These are special problems that differ from purely mathematical ones in their purpose, methods of solution, and the culture of mathematical communication. They help students develop skills in mathematical modelling, awareness, and the ability to apply mathematical knowledge in everyday life, in studying related subjects, and in acquiring professional knowledge in further education. A few isolated tasks cannot ensure the formation of the declared competences; therefore, systems of applied problems must be developed that correspond both to the content of mathematical training and to students’ age-related abilities and interests. To design such systems, clear selection criteria are required. This article addresses this issue and proposes a set of criteria for developing sets of applied mathematical problems to foster students’ key competences. Materials and methods. To develop the criteria for selecting applied mathematical problems, theoretical methods (analysis of regulatory documents, scientific papers, reference and teaching literature on mathematics education; synthesis and generalization of results), and empirical methods (teacher surveys, study of best teaching practices, analysis of students’ written work and test performance) were used. Results. The outcome is a system of criteria for constructing sets of applied mathematical problems, along with methodological recommendations for their solution. Conclusions. A system of applied mathematical problems is an effective means of forming key competencies among school students. Such problem sets should be an integral component of the task material in every topic of the school mathematics curriculum. Their solution makes a significant contribution to the development of students’ key competences.
Copyright (c) 2026 Василь Швець, Алла Прус https://creativecommons.org/licenses/by-nc-sa/4.0
OFFLINE-FIRST PWA WITH CONTROLLED GENERATIVE AI SUPPORT FOR TEACHING INFORMATICS IN A NEAR-FRONTLINE SCHOOL
Formulation of the problem. Near-frontline schooling faces a practical barrier that becomes methodological: learning continuity and assessment fairness are undermined when electricity and internet access are unstable, while generative AI can further weaken the validity of product-only grading in programming tasks. This study examines whether an offline-first learning design can preserve core learning actions and provide assessable evidence of work under disruption without normalizing unverified reliance on GenAI. Materials and methods. The intervention was an offline-first Progressive Web App (“Edu Survival Kit”) with local persistence and deferred synchronization, complemented by controlled GenAI tutoring and a process-oriented assessment approach based on artifact traces (versions, short reflections, and verification tests). The pilot involved three school cohorts (Grades 7, 10, and 11; 84 learners received access). Evidence included descriptive platform indicators, an anonymous learner survey (n = 33), teacher feedback (n = 2), and external expert appraisal (n = 1). Data were summarized descriptively and thematically. Results. The offline-first design sustained learning during planned outage conditions and markedly reduced task access time compared to an LMS route in a micro-comparison (sub-second versus tens of seconds). Learners reported high perceived usability and strong perceived support under unstable connectivity and stress; the GenAI hint layer received the highest usefulness ratings. Teachers and the expert confirmed crisis-fit and innovativeness but emphasized the main risk: over-trust in GenAI, which can mask shallow understanding. Conclusion. Offline-first architecture can function as instructional infrastructure in near-frontline settings when paired with an assessment that foregrounds process evidence rather than final products. GenAI support is pedagogically acceptable only under explicit transparency and verification rules and with data minimization.
Copyright (c) 2026 Dmytro Turchyn https://creativecommons.org/licenses/by-nc-sa/4.0
THREE STEPS FOR SOLVING PROBLEMS USING THE DEFINITION OF THE LIMIT OF A SEQUENCE
Formulation of the problem. An important component of mathematical training is learners’ command of the conceptual apparatus of mathematical disciplines, which, in most higher education institutions, is taught under the umbrella title Higher Mathematics, in particular Mathematical Analysis. The object of study in classical mathematical analysis is the function and various functional dependencies; its subject matter is the properties of functions, and the main tool for investigating these properties is the limit process. For courses in mathematical analysis, the central notion is that of a limit. This is explained by the fact that fundamental concepts such as the limit of a function, continuity, the derivative, and various types of integrals are introduced in terms of limits. Therefore, success in mastering these courses largely depends on the extent to which learners have mastered the notion of a limit, which makes the development of an effective strategy for forming the concept of the limit of a sequence relevant, including in tasks that require the practical use of the formal definitions of a limit. Materials and methods. The study employed an analysis of scientific and methodological literature and textbooks in higher mathematics and mathematical analysis; a systematization of domestic and international approaches to introducing the concept of a limit; and a generalization of the author’s experience in organizing practical classes and selecting exercises in which proofs are constructed without appealing to ready-made theorems, relying only on the definition. Results. A strategy was developed to develop learners’ understanding of the limit of a sequence and to consolidate this understanding in practical classes. An “algorithmic” approach to solving problems that apply the concept of the limit of a sequence is substantiated in the form of three consecutive steps based on the ε–n₀ definition of the limit of a sequence. The algorithm reduces learners’ cognitive load at the beginning of the topic, helps separate the heuristic search for estimates from the formal completion of the proof, develops skills in controlling error, and supports awareness of the dependence n0(ε). The algorithm serves as a basis for methodological recommendations for practical classes and strengthens skills in proving inequalities as analytical tools. Conclusions. A feature of the proposed method for applying the concept of the limit of a sequence in practical classes is that learners can independently grasp the meaning and importance of each detail in the definition of the limit of a sequence. Further research should be directed toward algorithmizing the application of definitions of the limit of a real-valued function of one variable at a point.
Copyright (c) 2026 Микола Білоцький https://creativecommons.org/licenses/by-nc-sa/4.0
ASSESSMENT OF THE LEVEL OF STUDENTS' AWARENESS REGARDING THE SAFE USE OF DIGITAL TECHNOLOGIES IN THE CONDITIONS OF TRANSFORMATION OF THE EDUCATIONAL SPACE
Formulation of the problem. The modern educational process is undergoing an irreversible digital transformation, requiring students to have a high level of digital competence. A critical unresolved problem is the insufficient awareness among primary school students (in particular, 8th graders) about the safe use of information and communication technologies (ICT). Insufficient knowledge in personal data protection, cyber threat prevention, and digital negligence creates a high educational risk, hindering the successful implementation of national strategies for the digital transformation of education. This necessitates an empirical assessment of the target group's actual level of awareness. Materials and methods. The study is based on an interdisciplinary approach, combining the concept of digital competence with the theory of educational risks. The empirical part was conducted at Chernihiv Lyceum No. 12, with 8th-grade students (N=77) surveyed during pedagogical practice. Methods of theoretical analysis and synthesis, as well as quantitative analysis with the calculation of descriptive statistical indicators, were used to assess the distribution of responses and the level of awareness of the components of digital security. The questionnaire was developed based on the security dimension of digital competence. Results. The study confirmed the high integration of digital technologies into the educational process (90.9% of students use digital technologies daily/several times a week) and an overall positive assessment of the tools (100% satisfaction). It was found that 37% of the surveyed students have insufficient knowledge and skills in the safe use of the Internet and in protecting their own data. It was found that 9.1% of students are not covered by targeted training for security issues. Infrastructure problems were also identified, in particular requests to improve Wi-Fi in protective structures, which are critical for ensuring the continuity and security of the educational process in wartime conditions. Conclusions. The study confirmed the problem's relevance and indicated the need to shift the emphasis from technological integration to the formation of a security culture. The key conclusion is the need for the systematic integration of digital security and ethics as a cross-cutting theme across all curricula. The development of integrated training modules focused on practical skills to counter cyber threats and on strengthening the lyceum's secure ICT infrastructure is recommended.
Copyright (c) 2026 Наталія Прядко, Дарія Любчикова https://creativecommons.org/licenses/by-nc-sa/4.0
MS EXCEL TEMPLATES FOR MODELING PROBLEMS ON INTEGRATION OF RATIONAL FUNCTIONS
Formulation of the Problem. In teaching mathematical analysis, regardless of the educational organization, training exercises play an essential role in developing and improving students’ practical skills and abilities. In particular, when studying the integration of rational functions, there is a need for a large number of similar tasks that are methodologically correct, computationally moderate, and focused on the formation of appropriate algorithmic skills. Practice shows that an arbitrary choice of numerical parameters in such tasks often leads to cumbersome calculations involving fractional or irrational expressions, which does not correspond to the didactic purpose of training exercises. This determines the relevance of developing formalized approaches to task modeling that meet predefined didactic requirements. Materials and Methods. To address the stated problem, an analysis of theoretical and practical results of Ukrainian and international researchers and practitioners concerning the design of mathematical tasks was conducted. MS Excel was used to create task templates, followed by formatting the problem statements and solutions using the LaTeX typesetting system. Results. A set of templates for the automated generation of collections of training tasks with controlled complexity and predefined methodological characteristics has been developed. It is shown that the use of such templates enables the avoidance of incorrect or didactically undesirable variants, increases the effectiveness of students’ independent work, and significantly simplifies the preparation of instructional materials. Conclusion. The development of templates in the MS Excel environment for modeling tasks on the integration of rational functions is an effective tool for creating educational content aimed at students’ independent practice. The proposed approach combines mathematical correctness, methodological appropriateness, and technological accessibility, making it suitable for teaching mathematical analysis.
Copyright (c) 2026 Сергій Радченко, Марія Астаф'єва, Анатолій Мазур https://creativecommons.org/licenses/by-nc-sa/4.0
VISUALIZATION OF KNOWLEDGE BY WEB TECHNOLOGY IN THE FORMATION OF PRE-SERVICE COMPUTER SCIENCE TEACHERS’ CRITICAL THINKING
Formulation of the Problem. The preparation of pre-service informatics teachers requires not only mastering programming languages and digital services but also developing critical thinking, which involves the ability to analyze information, identify assumptions, evaluate alternatives, build arguments, and reflect on one’s own judgments. Web-based knowledge visualization tools (mind maps, concept maps, diagrams, interactive whiteboards) create conditions for the external representation of learners’ reasoning; however, in practice, they are often used merely as auxiliary visual aids rather than as instruments for fostering critical thinking. Materials and Methods. The theoretical framework is based on the analysis of studies on critical thinking, teachers’ digital competence, knowledge visualization, and the use of web technologies in education. The empirical part relies on a comparative pedagogical experiment involving control and experimental groups of learners majoring in Secondary Education (Informatics). This involves analyzing their learning products (web-based maps, diagrams, and course models). Results. A generalized model for integrating web-based knowledge visualization tools into the preparation of pre-service informatics teachers was developed. It was established that the consistent implementation of the web-tool integration model ensures a steady, positive dynamic in all critical thinking indicators for the experimental group. A set of pedagogical conditions under which knowledge visualization tools fully realize their potential was empirically substantiated: integrated. It was shown that under these conditions, visual models cease to function as mere “illustrations” and become a means of analysis, critical revision, and self-correction of reasoning. Conclusion. The generalized model of integrating web-based knowledge visualization tools into the preparation of pre-service informatics teachers enables the purposeful development of critical thinking through work with professionally relevant objects, such as algorithms, digital resources, models of educational situations, and course projects. The identified pedagogical conditions can be used to improve the content and instructional design of courses related to methods of teaching informatics and educational technologies, and can be adapted for the preparation of other professional categories for whom knowledge visualization is a central means of understanding complex systems and processes.
Copyright (c) 2025 Артем Юрченко, Роман Момот, Марія Острога, Олена Семеніхіна https://creativecommons.org/licenses/by-nc-sa/4.0
OPTIMIZATION OF FACTOR VARIABLE SELECTION IN ECONOMETRIC MODELING: PRACTICAL RECOMMENDATIONS
Formulation of the problem. The correct selection of factor variables in modern econometric studies is a key condition for ensuring the accuracy, stability, and predictive power of models. However, student papers often contain typical errors related to an excessive number of factors, weak theoretical justification for their selection, or the presence of multicollinearity. The article substantiates the need to create practical recommendations that combines scientific validity with ease of use and ensures the correct selection of factor variables in the process of modeling economic phenomena. Materials and methods. The study uses a set of theoretical, statistical, and econometric methods, among which correlation and regression analysis played a key role, allowing us to identify and quantitatively assess the relationships between variables. To increase the reliability of the results and the stability of the model, a multicollinearity check was performed using the Farrar–Glauber algorithm, which involves the use of χ², F, and t criteria to assess the interdependencies between independent variables. Results. As a result of the study, a step-by-step algorithm for selecting factor variables is proposed, which involves: (1) calculating correlation coefficients to form a vector and correlation matrix; (2) selecting variables that have a strong relationship with the outcome variable and weak mutual correlation; (3) verification of the informativeness of variables using the method of information capacity indicators and the multiple correlation coefficient; (4) additional analysis for multicollinearity to identify redundant factors. The algorithm ensures the validity of the predictor choice, enhances the accuracy and stability of models, and is convenient for educational purposes. Conclusion. The proposed approach enables the systematic organization of the process of selecting factor variables in student econometric research, combining clarity, accessibility, and scientific rigor. Its application contributes to the formation of students' analytical thinking skills, statistical interpretation of results, and a conscious approach to model building.
Copyright (c) 2025 Оксана Глушак, Світлана Семеняка, Надія Зінченко https://creativecommons.org/licenses/by-nc-sa/4.0
SOLVING MATHEMATICAL PROBLEMS BY HIGH SCHOOL STUDENTS AS AN EFFECTIVE TOOL FOR DEVELOPING THE COMPETENCE OF «ENTREPRENEURSHIP AND FINANCIAL LITERACY»
The article highlights the methodological aspects of solving applied problems in mathematics lessons to form the competence «entrepreneurship and financial literacy»; the stages of solving such problems are given. Formulation of the problem. Considering the current trends in technology development, the digitalization of Ukraine's population, and the desire to enhance financial literacy, changes are occurring in educational trends. Today, the State Standard of Basic Secondary Education and the State Standard of Specialized Secondary Education are built on a competency-based approach. One of the key competencies is «entrepreneurship and financial literacy», which implies a person's ability to initiate, plan, and implement ideas, take responsibility, find opportunities and resources to achieve their goals, be able to effectively manage their own finances, use banking products, and make financial decisions. Considering regulatory documents, a modern teacher must find effective means and methods to implement the assigned tasks. Materials and methods. A combination of methods was employed in this study: theoretical, for analyzing regulatory documents, and empirical, involving pedagogical observation of the educational activities of 10th–11th grade students while they solved problems. Results. The study presents an example of a competency-based problem scenario tailored to the age-specific characteristics of high school students and designed to motivate them to solve the problem and independently seek information. The stages of solving applied problems are identified to effectively develop the competence of «entrepreneurship and financial literacy». Conclusions. Solving applied problems with financial content has proven effective in developing the competence of «entrepreneurship and financial literacy» among high school students. The use of relevant data, QR codes, and educational discussions enhances student motivation, fosters critical thinking, and transforms mathematics into a valuable tool for making informed financial and entrepreneurial decisions in real life.
Copyright (c) 2025 Олена Ткач https://creativecommons.org/licenses/by-nc-sa/4.0
CONSTRUCTION OF MATHEMATICAL MODELS WHEN SOLVING APPLIED PROBLEMS OF ECONOMIC CONTENT
Formulation of the problem. The article aims to systematize and generalize the main types of mathematical models used in solving applied economic problems within the higher mathematics course, as well as to determine the pedagogical conditions that foster the development of mathematical modeling skills among future economists. Materials and methods. The research employed a combination of theoretical and empirical methods, including an analysis of scientific and methodological literature on mathematical modeling, the synthesis and comparison of national and international studies, the systematization of typical economic–mathematical models, and pedagogical observations. Results. The study substantiates the role of mathematical modelling as a core method in forming the professional competence of future economists. The main types of mathematical models used in economic problem-solving —matrix, vector, linear, integral, differential, and derivative-based models — were systematized. A generalized structural scheme of a mathematical model of an applied problem is proposed, integrating the stages of formalization, analysis, and interpretation, and reflecting the relationship between theoretical and quantitative components. It was found that the implementation of this sequence contributes to the development of research-oriented thinking and analytical skills of learners. Conclusion. The study reveals a contradiction between the growing need for economists capable of modelling and analyzing economic processes and the insufficient level of students’ mathematical training. It is proven that the systematic use of applied economic problems in the higher mathematics course promotes the formation of professional competences and the development of economic thinking. Further research should focus on the design of methodological tools and digital resources for teaching mathematical modeling, as well as on integrating modeling techniques into interdisciplinary economics-related courses.
Copyright (c) 2025 Тетяна Грицик https://creativecommons.org/licenses/by-nc-sa/4.0
FEATURES OF TESTS AND TEST CONTROL OF ACADEMIC ACHIEVEMENTS IN PHYSICS
Formulation of the problem. The assessment of students’ academic performance constitutes an integral component of the educational process, with testing being one of the most widely applied forms of assessment. The principal research objective of this study is to address specific challenges arising in the development of test items in physics education, particularly in certain domains that necessitate deviations from conventional testing approaches. The analysis of such deviations, combined with the demonstration of their practical implementation, represents the core problem explored in this work. Materials and methods. The materials underpinning this research comprise both the scholarly contributions of colleagues and the authors’ own pedagogical experience in teaching physics across several higher education institutions in Kharkiv. A significant portion of the authors’ materials was subjected to statistical analysis. Additionally, the careful examination of selected chapters from seminal works authored by eminent physicists — such as Albert Einstein, Max Planck, Erwin Schrödinger, and Louis de Broglie — proved highly valuable. The primary methodological framework employed in this study involves a comprehensive analysis of research on pedagogical testing in physics, as well as the systematic application of the authors’ conclusions, derived from extensive teaching practice. Results. The distinctive features encountered in the construction of physics test items are intrinsically linked to the nature of the discipline itself. These features are determined by two principal factors. The first concerns the fact that within many branches of physics, numerous phenomena exhibit dual or even multiple interpretations (e.g., wave–particle duality, non-relativistic and relativistic formulations). Such cases pose significant difficulties for test design and necessitate the adoption of non-standard approaches. The second factor lies in the absence of rigorously established theories for certain experimentally discovered phenomena, although well-founded hypotheses exist to account for them. Accordingly, formulating test tasks in such instances requires innovative strategies. Drawing upon the authors’ experience, the most effective approach involves presenting each test item as a structured sequence of smaller, interrelated sub-tasks. This technique has demonstrated considerable pedagogical efficacy. Conclusions. In situations where physical objects or phenomena are described by multiple, mathematically consistent theoretical frameworks, both the structure and the content of test items depend on the number of theoretical approaches considered and the complexity of the mathematical relationships between them. Conversely, when physical objects or phenomena are characterized not by established theories but by provisional hypotheses, the structure of test items should reflect the number of such hypotheses and the experimental evidence underlying them. In this case, the content must be designed to enable the assessment of the relative plausibility of each hypothesis. As indicated by the authors’ teaching experience, this type of testing provides a highly accurate measure of students’ comprehension of complex physical phenomena. Furthermore, analyzing test results often helps instructors refine and improve their instructional methodologies.
Copyright (c) 2025 Марк Босін, Лариса Рикова, Галина Брославська https://creativecommons.org/licenses/by-nc-sa/4.0
AUGMENTED REALITY IN PHYSICS LESSONS: EXPERIENCE USING PHYSICS LAB AR IN BLENDED LEARNING
Formulation of the problem. Changes in the world of technology, particularly in the fields of augmented and virtual reality, breakthroughs in artificial intelligence, and the transition to blended learning, are forcing educational institutions to adapt their methods and approaches to the educational process. Recent studies have demonstrated that the application of augmented reality (AR) in education presents numerous advantages that positively influence the educational process. Materials and methods. A set of methods was employed, including analysis, systematization, and synthesis of scholarly sources; a review of scientific publications by both domestic and international researchers; examination of regulatory and policy documents; and reflection on our own professional experience. In particular, a student survey was conducted regarding the application of AR technologies in the educational process, complemented by an analysis of learners’ academic performance before and after the implementation of the study. Results. Blended learning is a modern pedagogical model that combines classroom teaching with digital technologies, allowing students to study material independently at a time convenient for them. This form of learning facilitates the integration of traditional methods with interactive tools, making the educational process more flexible and allowing students to work at their own pace, thereby increasing their level of independence. For natural science and mathematics subjects, conducting experiments is a key aspect of learning. In blended learning, this is made possible by augmented reality technologies and interactive simulations, particularly for students who are unable to attend face-to-face classes due to various reasons. Conclusions. The integration of AR into education fosters interactivity, increases learner motivation, and stimulates interest in the learning process. Visualization tools enhance comprehension of complex phenomena, while personalized pathways allow students to work at a pace aligned with their individual needs. Nevertheless, it must be acknowledged that modern students often display a strong dependency on digital devices. To mitigate potential negative impacts on the development of essential soft skills and social competencies, the use of AR and VR should be carefully balanced and applied in moderation.
Copyright (c) 2025 Ольга Слободяник https://creativecommons.org/licenses/by-nc-sa/4.0
STUDYING COORDINATES AND VECTORS IN 8TH GRADE OF NUS ACCORDING TO THE AUTHOR'S TEXTBOOK OF THE INTEGRATED COURSE OF MATHEMATICS
Formulation of the problem. The New Ukrainian School (NUS) reform is aimed at developing key competencies in students that will facilitate their further education in higher education and self-realization in future professional activities. According to this reform, the learning process in Ukrainian schools is organized as follows: according to the approved State Education Standards and the basic educational plan, model educational programs have been developed in all educational fields, in particular, in mathematics, and textbooks and other teaching aids are created based on these model programs. The goal of the article is to highlight the methodological features of studying coordinates and vectors in the 8th grade of the New Ukrainian School according to the textbook of the author team, Oleksandr Shkolnyi, Yevhen Nelin, Andrii Mylianyk, and Yulia Prostakova, created according to the model program of the author team led by Maria Vasylyshyn. According to this program and textbook, vectors and coordinates are studied in the 8th grade. This will make it possible to study the topic “Geometric transformations of the plane” at the beginning of the 9th grade, and future study of this topic will contribute to a more accurate presentation of the material on constructing a graph of a quadratic function using geometric transformations. In addition, coordinates and vectors are needed in the 8th grade, particularly for physics teachers to solve many problems in this field. Materials and methods. Theoretical analysis of methodological literature, comparative analysis, systematization, and generalization of existing theoretical research on the topic of the article were used, as well as pedagogical observation and generalization of one's own pedagogical experience in teaching mathematics in the 8th grade of the New Ukrainian School. Results. In the work, we describe in detail the methodology for studying coordinates and vectors, which involves studying the following subsections: “Coordinate system on the plane. Coordinates of the midpoint of a segment. Distance between two points with given coordinates. Equation of a circle”, “Vector. Absolute value of a vector. Collinear and equal vectors. Operations on vectors”, “Vector coordinates. Operation on vectors in coordinate form”, “Equation of a line. Mutual location of a circle and a line and two circles”, “Dot product of vectors and its application.” In particular, it is important for studying vector coordinates to introduce the concept of a basis, which corresponds to a given coordinate system on the plane. This makes it possible to introduce the concept of vector coordinates as coefficients in an expansion using basis vectors in a fairly simple, traditional way for a mathematics course. Conclusions. Testing of the textbook of the aforementioned author team during the 2024-2025 academic year in Ukrainian schools shows that students quite positively perceive the place and method of studying coordinates and vectors on the plane and properly use them in their educational activities. However, the proposed method for students in 8th grade at NUS to master this material requires further research. However, we are convinced of the feasibility and prospects of the described approach and will continue to use it to develop students' key competencies.
Copyright (c) 2026 Oleksandr Shkolnyi https://creativecommons.org/licenses/by-nc-sa/4.0
CONCEPTUAL MODEL OF GENERATIVE ARTIFICIAL INTELLIGENCE INTEGRATION INTO THE PROCESS OF DEVELOPING INFORMATION AND DIGITAL COMPETENCE IN STUDENTS AT THE BASIC SECONDARY EDUCATION LEVEL
The article considers the integration of generative artificial intelligence (GAI) into developing students' information and digital competence (IDC) in basic secondary education. The relevance of creating a conceptual model that combines pedagogical, cognitive, technological, and ethical dimensions based on the DigComp 2.2 framework and UNESCO recommendations is emphasized. Formulation of the problem. IDC is becoming increasingly important in the digital society, but traditional approaches in schools do not provide the proper level of its formation. Students often face difficulties in searching and critically analyzing information, and the educational process requires individualization and a broader use of digital tools. The spread of GSI highlights the need for its conceptual implementation, considering pedagogical feasibility, ethics, and security. Materials and methods. The analysis of scientific and methodological literature, international documents on digital literacy, the results of pedagogical observations, and content analysis of educational practices were employed. The modeling method and structural-functional analysis were employed to construct the model, enabling it to identify the relationships between its components. Results. A conceptual model of integrating GSI into the development of students' ICT has been developed, with a multi-level implementation. At the student level - personalization of educational trajectories, chat bots, recommendation systems, development of critical thinking; at the teacher level - optimization of routine tasks, analysis of achievements, creation of individual plans; at the management level - analytics for predicting results and correcting programs; at the social level - solving issues of accessibility, data protection, overcoming algorithmic bias. The model encompasses target, subject-object, content, technological, methodological, organizational, ethical, and result-evaluation components, forming a holistic system. Conclusions. Integrating GSI into teaching opens up new opportunities for developing critical thinking, digital creativity, and student autonomy. The model has both theoretical and practical significance, and can be used to update educational standards, create methodological materials, and inform teacher training programs. Its implementation contributes to the formation of students' readiness for conscious interaction with intelligent technologies in the context of digital transformation.
Copyright (c) 2025 Світлана Литвинова, Юлія Носенко, Катерина Осадча, Ольга Пінчук, Наталя Рашевська, Аліса Сухіх https://creativecommons.org/licenses/by-nc-sa/4.0
FORMATION OF CIVIC IDENTITY IN SCHOOLCHILDREN DURING MATHEMATICS EDUCATION IN GRADES 5-6
Formulation of the problem. The State Standard of Basic Secondary Education of Ukraine states that the formation of civic identity is a key competency that should be integrated into the educational process. In the fifth and sixth grades, when students form an initial understanding of social roles and responsibilities, it is important not only to impart subject knowledge but also to develop civic awareness. Mathematics as an academic discipline has a significant potential to form civic identity through tasks that model social situations, require logical reasoning, assessing the fairness of decisions, comparing alternatives, and making responsible decisions. Materials and methods. To obtain the results, we used theoretical (analysis of regulatory documents, reference and educational literature on the theory and methods of teaching mathematics, synthesis of the data obtained, and their generalization) and empirical (survey of teachers, familiarization with the best practices of teaching mathematics to students, and survey of students) methods of scientific knowledge. Results. The article proposes methodological approaches to the formation of civic identity in students of grades 5-6 in the course of mathematics, and provides examples of applied problems with an analysis of their solutions and relevant methodological comments to them. Conclusions. One of the ways to form civic identity in students of grades 5-6 is to solve practice-oriented tasks and applied problems whose plots reflect civic life in Ukraine. It is important that this process is accompanied by meaningful information content and that the educational programs for future mathematics teachers continue to adapt to modern challenges, providing students with the knowledge and skills necessary to actively participate in the life of Ukraine.
Copyright (c) 2025 Василь Швець, Катерина Мартинюк https://creativecommons.org/licenses/by-nc-sa/4.0
STEM-ORIENTED TEACHING OF COMPUTER ARCHITECTURE IN DEVELOPING ENGINEERING THINKING OF FUTURE COMPUTER SCIENCE TEACHERS
Formulation of the problem. The article substantiates the pedagogical feasibility of using the Proteus simulation environment in the process of STEM-oriented teaching of the "Computer Architecture" course with the aim of developing students' engineering thinking. The relevance of the study is determined by the need to train specialists capable of designing and analyzing digital devices, making technically sound decisions, and integrating knowledge from computer science, physics, mathematics, and electronics. Within the framework of the study, a pedagogical experiment was conducted with students who worked in different digital environments. Materials and methods. Research methods include: analysis of literature on STEM education, comparative analysis of the functionality of Proteus and Multisim environments, pedagogical experiment, and quantitative methods of analyzing the results. The experiment involved students of a pedagogical university studying hardware-oriented disciplines. Quantitative and qualitative analysis of the data made it possible to assess the level of content mastery, the dynamics of digital modeling skills development, the depth of reflection, and the manifestations of engineering thinking. Results. The results indicate that different simulation environments provide the acquisition of the basic content of the course; however, it is the Proteus environment that more actively stimulates technical creativity, flexibility in building models, and the ability to analyze and optimize circuits. Students working in Proteus demonstrated more diverse approaches to task completion, showed initiative in improving design solutions, and provided well-founded justifications for their choice of tools. Conclusions. The conclusion is made about the advisability of using Proteus in courses aimed at developing engineering thinking within the STEM paradigm. Methodological recommendations are offered for the optimal use of both environments depending on the didactic purpose. The obtained results have practical significance for the development of hardware-oriented academic courses, updating teaching methods, and implementing an interdisciplinary approach in the technical training of future computer science teachers.
Copyright (c) 2025 Володимир Шамоня, Максим Сорока, Олена Семеніхіна https://creativecommons.org/licenses/by-nc-sa/4.0
DISTANCE LEARNING PRACTICAL COURSE «3D GRAPHICS»: CHALLENGES AND PROSPECTS FOR IMPLEMENTATION
Formulation of the problem. Future computer science teachers and IT specialists studying at higher education institutions must acquire competencies that reveal their ability and readiness to effectively use computer graphics tools in their profession. It is advisable to include educational components in the content of educational and professional programs, the study of which is aimed at developing skills in working with three-dimensional graphics, modelling, animation and design tools. The implementation of an activity-based approach in combination with the updating of software tools necessitates the development of a distance learning course as a practice-oriented resource. Materials and methods. Analysis of scientific and methodological works on computer graphics; identification of key recommendations for the development and implementation of a distance learning course as a practice-oriented resource; observation, questioning, and presentation of personal teaching experience. Results. Ways to overcome the main challenges in studying 3D graphics are proposed, and individual aspects of developing and implementing a distance practice-oriented course that combines instructional, informational, control, and communicative components are analyzed. The course modules are presented as a system of laboratory (practical) classes, each of which contains: task formulation, instructional materials, an activity part, and a final reflection. Visualization of the survey results and generalization of our own pedagogical experience enabled us to identify the challenges and prospects of implementing a distance learning course in the professional training of future specialists. Conclusions. The key challenges of the educational process are issues of technical and software support, adherence to the principles of an individual and differentiated approach, and the introduction of active and practice-oriented learning. The training modules of the distance learning course comprise a set of laboratory (practical) works that contain variable, competence-based tasks of a professional nature. The effectiveness of the educational process is enhanced by the use of BYOD technology. Learning the Blender 3D software environment is advisable given its versatility, accessibility, and advantages in the process of working with it.
Copyright (c) 2025 Наталія Павлова, Катерина Дубич https://creativecommons.org/licenses/by-nc-sa/4.0
MODELS OF IMMERSIVE LEARNING ENVIRONMENTS
The article discusses innovative models developed and implemented to organize learning in an immersive environment. The structure of such models and their advantages in the learning process are highlighted. Formulation of the problem. The development of immersive technologies in education raises the question of designing a comprehensive model of an immersive educational environment that integrates pedagogical, psychological, technological, and developmental factors and meets the conditions of learning in general secondary education institutions. Currently, immersive learning models are fragmented, which complicates their systematic implementation. Materials and methods. The study uses a set of theoretical methods: analysis, systematization, generalization of scientific and pedagogical literature, and structural-functional and comparative analysis of pedagogical models. Results. Five models of immersive learning have been systematized and analyzed: cognitive-affective, research-based, evolutionary, flow state, and game-based learning based on artificial intelligence. Their main characteristics, advantages, and limitations have been identified. It has been established that existing models do not fully account for the specific nature of natural science and mathematics disciplines, as well as the educational needs of high school students. This necessitates further theoretical and practical justification of the integrative approach. Conclusion. The fragmented nature of the immersive learning models discussed underscores the need to develop a comprehensive model that ensures pedagogical expediency, technological flexibility, and the relevance of learning content in academic lyceums. Further research will focus on creating and testing such a model in the context of studying natural and mathematical disciplines using virtual and augmented reality tools.
Copyright (c) 2026 Наталя Рашевська https://creativecommons.org/licenses/by-nc-sa/4.0
INTEGRATION OF VIDEO COMMUNICATION AND AUGMENTED REALITY PLATFORMS INTO THE MODEL OF BLENDED AND DISTANCE LEARNING LESSONS IN GENERAL SECONDARY EDUCATION INSTITUTIONS
Formulation of the problem. The current transformation of the educational process has highlighted the need to find effective solutions for organizing blended and distance learning in secondary schools. Despite the widespread use of video conferencing platforms (such as Zoom, Meet, Teams), the educational process in digital format is often accompanied by a decrease in student engagement, monotony in the presentation of material, information overload, and a lack of flexible didactic solutions capable of taking into account the age, cognitive, and motivational characteristics of modern generations of students. In this context, immersive technologies, especially augmented reality, have significant potential for building a visually rich, interactive, and emotionally engaging educational environment. At the same time, within the framework of general secondary education in Ukraine, the possibilities of integrating AR content with synchronous interaction platforms into the structure of school lessons have not been sufficiently researched and methodologically standardized. This creates a demand for the development of models that would ensure the pedagogical integrity of such a combination and contribute to the more effective use of EdTech in the learning process. Materials and methods. The study uses theoretical methods, including analysis of scientific, methodological, psychological, and pedagogical literature on blended and distance learning, digital pedagogy, and pedagogical design, taking into account the age characteristics of different generations. Results. The main problems of distance and blended learning in secondary schools have been identified, in particular, reduced attention, limited visualization, and content overload. Based on an analysis of the age characteristics of the Zumer and Alpha generations, the feasibility of integrating AR content in combination with video communication platforms has been substantiated. A typical selection of lesson models with immersive elements has been developed, adapted to the subject area, age of students, and learning format. Conclusions. The integration of video communication platforms with AR content helps to increase the effectiveness of short-distance and blended lessons, engages students, and supports their cognitive needs. The proposed models and the catalog of immersive services can be used by teachers as a tool for updating lessons. Full implementation requires digital training for teachers and institutional support for immersive approaches in schools.
Copyright (c) 2025 Аліса Сухіх https://creativecommons.org/licenses/by-nc-sa/4.0
MANIPULATIVES USE IN MATHEMATICS CLASSROOMS OF INCLUSIVE BASIC SCHOOLS IN GHANA
Formulation of the problem. Mathematics is essential in every individual's life, including that of Learners with Visual Impairment (LVI), as it helps them acquire the competencies required in their daily life and profession. There is a growing emphasis on inclusive basic schools to avoid discrimination among people with disability, Such as Learners with Visual Impairment (LVI). One subject that LVI finds difficult to understand is mathematics because it utilizes many graphics that are challenging for LVI to comprehend. Manipulative materials help reduce the abstract nature of mathematical concepts by making them more visible to learners, including those with learning difficulties. Despite the benefits of manipulatives in inclusive primary classrooms, there is a lack of research on how manipulatives are used in inclusive primary schools in Ghana to enhance learners' understanding of mathematical concepts. Guided by cognitive and social constru­ctivism theory, this study explores how basic school mathematics teachers make manipulatives accessible to Learners with Visual Impairment (LVI), the support they provide, the assessment strategies they use, and the difficulties they encounter when using manipulatives to teach mathematics to LVI. Materials and methods. This study employed a case study research design. Six inclusive basic mathematics teachers were interviewed, and the results were analysed thematically. Results. Findings from the study indicated that most teachers do not have manipulatives in their schools and do not employ them during mathematics instruction. The results also showed that, since most teachers do not use manipulatives in teaching mathematical concepts, they were unable to identify the challenges they face when teaching LVI mathematical concepts with manipulatives. Conclusions. Based on the outcome of this study, manipulatives for teaching mathematics, especially to LIVs, should be made available in inclusive basic schools in Ghana. Teachers in inclusive basic schools in Ghana should be given workshops and seminars on the use of manipulatives in teaching mathematics to LVI.
Copyright (c) 2025 Augustina Adu, Jones Apawu, William Obeng-Denteh https://creativecommons.org/licenses/by-nc-sa/4.0
ORGANIZATION OF REPETITION AND SYSTEMATIZATION OF THE SCHOOL MATHEMATICS COURSE IN A UKRAINIAN UPPER SECONDARY PROFILE SCHOOL
Formulation of the problem. The article raises the problem of inconsistency between the existing state regulations on the final repetition and systematization of the school mathematics course in the context of a competency-based approach on the one hand and its practical implementation according to the current programs and textbooks for senior grades on the other. Taking into account cognitive theories of memory development and challenges during the final testing of senior students in mathematics, the relevance of researching effective approaches to the repetition of educational material, which would contribute to the reduction of educational losses and the proper formation of mathematical competence and other competencies in accordance with the concept of the New Ukrainian School (NUS), is substantiated. Materials and methods. Theoretical analysis of methodological literature, comparative analysis, systematization, and generalization of existing theoretical research on the topic of the article, as well as pedagogical observation and generalization of one's own pedagogical experience in repeating and systematizing the school mathematics course, were used. Results. As a result of the analysis of current textbooks and collections for preparation for external independent assessment (EIA) in the form of a national multi-subject test (NMT) in mathematics, a variety of approaches to the implementation of current and final repetition were revealed. It was found that textbooks differ in the presence, structure, and content of review rubrics, and theoretical materials for systematizing knowledge are practically absent. Textbooks, in particular, comprehensive publications for preparation for EIA (NMT), partially compensate for these gaps, but do not provide sufficient differentiation and require the active participation of the teacher in the process of selecting material to ensure the proper quality of final repetition. Conclusions. It has been established that the existing textbooks and manuals on mathematics only partially provide the conditions for high-quality summative repetition, not covering the need for differentiation and feedback. The effectiveness of this stage of learning largely depends on the professional skills of the teacher and the ability to combine different sources and approaches. The development of adaptive repetition technologies using digital tools has been identified as a promising direction.
Copyright (c) 2025 Yuliia Tykhonenko, Oleksandr Shkolnyi https://creativecommons.org/licenses/by-nc-sa/4.0
ADDITIONAL METHODS FOR TEACHING THE QUANTUM THEORY OF THE HYDROGEN ATOM
This paper proposes the possibility of applying integral equations corresponding to the Schrödinger equation to find the radial component of the wave function of a quantum system of two particles and the permissible values of the principal quantum number of the system. The relevance of the work is due to the need to supplement the methodology for presenting such an important key problem of quantum mechanics as the quantum theory of the hydrogen atom, which consists of expanding the set of methods for finding solutions to the Schrödinger equation for the hydrogen atom and their systematization, which will contribute to a better understanding of this topic. The proposed sequence of presenting the material takes into account the latest trends in the development of methods for finding solutions to the stationary Schrödinger equation in analytical form, demonstrating the possibility of finding the energy spectrum of a quantum system of particles using integral equations. Formulation of the problem. The quantum theory of the hydrogen atom in physics textbooks and manuals is most often taught based on the single-particle Schrödinger equation for the electron, switching to a reference frame associated with the atomic nucleus, neglecting the motion of the nucleus, or part of the material is taught when considering the motion of the electron in a central force field. To generalize the teaching methodology, it is better to immediately consider a quantum system of two particles: an electron and a hydrogen atom nucleus. Materials and methods. The main methods for solving the problem are: means of finding solutions to ordinary, linear, and partial differential equations, the theory of multidimensional integral equations, and the theory of Green's functions. Results. It was found that the radial components of the wave functions of the hydrogen atom, which determine the energy spectrum, can be obtained without transitioning to a spherical system, which simplifies the presentation of the topic material. A generalization of the sequence of presentation of the material on the quantum theory of the hydrogen atom is proposed. Conclusions. The proposed generalization of the method for finding solutions to the Schrödinger equation for the hydrogen atom allows for a more complete presentation of the material on the topic "Quantum Mechanics", increases the degree of provability of the presentation of the material, and facilitates its understanding.
Copyright (c) 2025 Костянтин Авдонін https://creativecommons.org/licenses/by-nc-sa/4.0