Physical and Mathematical Education
type: channel
description: 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 Journal “Physical and Mathematical Education” is included in the List of scientific professional editions of Ukraine (Category "B") (the order of the Ministry of Education and Science of Ukraine from June 11, 2026, No. 928)
publisher: Сумський державний педагогічний університет імені А.С.Макаренка
language: en
publication name: Physical and Mathematical Education
issn: 2413-1571
copyright: Authors grant the journal a right of the first publication of the work under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (CC BY-NC-SA 4.0)that allows others freely to use (read, copy and print) submissions, search content and link to published articles, disseminate their full text and use them for any legitimate non-commercial purposes (i.e. educational or scientific) with the mandatory reference to the article’s authors and initial publication in this journal. Original published articles cannot be used by users (exept authors) for commercial purposes or distributed by third-party intermediary organizations for a fee.
items: DIGITAL TOOLS IN A MULTILINGUAL EDUCATIONAL ENVIRONMENT: RATIONALE FOR AN INTEGRATED APPROACH TO THE TRAINING OF A MODERN PROFESSIONAL
items: GRANT WRITING SKILLS IN THE STRUCTURE OF PROFESSIONAL COMPETENCE OF FUTURE STEM EDUCATION SPECIALISTS
items: FROM SELF-ASSESSMENT TO PEER REVIEW: THE DEVELOPMENT OF A COGNITIVE-VERIFICATION APPROACH TO THE EVALUATION OF MATHEMATICAL PROBLEMS IN THE CONTEXT OF GENERATIVE ARTIFICIAL INTELLIGENCE
items: CASCADE APPLIED PROBLEMS IN MATHEMATICS AS A MEANS OF DEVELOPING KEY COMPETENCIES OF STUDENTS
items: CONCEPTUAL FOUNDATIONS OF USING PROBLEM-ORIENTED SITUATIONS IN THE PROFESSIONAL TRAINING OF MATHEMATICS TEACHERS
items: STEM-MAGNETIC LEVITATION EXPERIMENT
items: ELECTROMAGNETIC INTERFERENCE AND DATA TRANSMISSION RELIABILITY IN SDR-BASED EDUCATIONAL EXPERIMENTS
items: PLANE DILATIONS IN THE SCHOOL MATHEMATICS COURSE AND THE THEORY OF GEOMETRIC TRANSFORMATIONS FOR FUTURE MATHEMATICS TEACHERS
items: DEVELOPING PREDICTIVE COMPETENCE IN FUTURE COMPUTER SCIENCE TEACHERS THROUGH THE INTEGRATION OF TEACHING PRACTICE AND MATHEMATICAL STATISTICS ASSIGNMENTS
items: A COGNITIVE-VERIFICATION APPROACH TO TRANSFORMING HOMEWORK IN MATHEMATICAL ANALYSIS IN THE CONTEXT OF ARTIFICIAL INTELLIGENCE USE
items: EXPERIMENTAL VISUALIZATION AND NUMERICAL SIMULATION OF FRAUNHOFER DIFFRACTION FROM ORDERED AND DISORDERED OPAQUE MICROSTRUCTURES FOR EDUCATION PRACTICE
items: ALGORITHMIC SUPPORT FOR REVERSE GENERATION OF TRIGONOMETRIC TASKS USING OBJECT-ORIENTED MODELING
items: CLASSIFICATION OF MOBILE LEARNING SUPPORT TOOLS IN GENERAL SECONDARY EDUCATION
items: USE OF GAMIFICATION ELEMENTS IN TEACHING MATHEMATICAL ANALYSIS AT HIGHER EDUCATION INSTITUTIONS
items: FOUR PROMPTING TECHNIQUES FOR ANALYTICAL WORK WITH TEACHING AND LEARNING RESOURCES
items: SOFTWARE IMPLEMENTATION OF ALGORITHMS FOR CALCULATING PEARSON, SPEARMAN AND KENDALL CORRELATIONS IN C#
items: SYSTEM OF APPLIED MATHEMATICAL PROBLEMS: CRITERIA FOR DEVELOPMENT AND FEATURES OF SOLUTION
items: OFFLINE-FIRST PWA WITH CONTROLLED GENERATIVE AI SUPPORT FOR TEACHING INFORMATICS IN A NEAR-FRONTLINE SCHOOL
items: THREE STEPS FOR SOLVING PROBLEMS USING THE DEFINITION OF THE LIMIT OF A SEQUENCE
items: ASSESSMENT OF THE LEVEL OF STUDENTS' AWARENESS REGARDING THE SAFE USE OF DIGITAL TECHNOLOGIES IN THE CONDITIONS OF TRANSFORMATION OF THE EDUCATIONAL SPACE
items: MS EXCEL TEMPLATES FOR MODELING PROBLEMS ON INTEGRATION OF RATIONAL FUNCTIONS
items: VISUALIZATION OF KNOWLEDGE BY WEB TECHNOLOGY IN THE FORMATION OF PRE-SERVICE COMPUTER SCIENCE TEACHERS’ CRITICAL THINKING
items: OPTIMIZATION OF FACTOR VARIABLE SELECTION IN ECONOMETRIC MODELING: PRACTICAL RECOMMENDATIONS
items: SOLVING MATHEMATICAL PROBLEMS BY HIGH SCHOOL STUDENTS AS AN EFFECTIVE TOOL FOR DEVELOPING THE COMPETENCE OF «ENTREPRENEURSHIP AND FINANCIAL LITERACY»
items: CONSTRUCTION OF MATHEMATICAL MODELS WHEN SOLVING APPLIED PROBLEMS OF ECONOMIC CONTENT
items: FEATURES OF TESTS AND TEST CONTROL OF ACADEMIC ACHIEVEMENTS IN PHYSICS
items: AUGMENTED REALITY IN PHYSICS LESSONS: EXPERIENCE USING PHYSICS LAB AR IN BLENDED LEARNING
items: STUDYING COORDINATES AND VECTORS IN 8TH GRADE OF NUS ACCORDING TO THE AUTHOR'S TEXTBOOK OF THE INTEGRATED COURSE OF MATHEMATICS
items: CONCEPTUAL MODEL OF GENERATIVE ARTIFICIAL INTELLIGENCE INTEGRATION INTO THE PROCESS OF DEVELOPING INFORMATION AND DIGITAL COMPETENCE IN STUDENTS AT THE BASIC SECONDARY EDUCATION LEVEL
items: FORMATION OF CIVIC IDENTITY IN SCHOOLCHILDREN DURING MATHEMATICS EDUCATION IN GRADES 5-6

DIGITAL TOOLS IN A MULTILINGUAL EDUCATIONAL ENVIRONMENT: RATIONALE FOR AN INTEGRATED APPROACH TO THE TRAINING OF A MODERN PROFESSIONAL
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description: Problem statement. The professional training of modern specialists increasingly takes place in digitally mediated, multilingual, and intercultural environments. Specialists work with digital sources, cloud services, collaborative platforms, machine translation systems, generative artificial intelligence, and multimodal content. At the same time, digitalization and multilingualism in higher education often develop in parallel rather than in an integrated way: digital courses tend to focus on technological operations, whereas language training is frequently separated from professional tasks. This creates a need to justify an integrated approach in which digital tools and multilingual practices are considered interdependent foundations of professional training. Materials and methods. The study is theoretical and analytical. It employs the analysis of scholarly and policy-related sources, comparison of approaches to digital and plurilingual competences, categorization of digital tools, conceptual synthesis, and pedagogical modeling. Results. The article clarifies the concept of a multilingual educational environment as a system of resources, interactions, and activities in which several languages are functionally used for cognition, professional communication, mediation, collaboration, and the creation of a digital product. The functions of digital tools in such an environment are systematized as information-search, communication-collaboration, translation-mediation, productive-creative, adaptive-support, analytical-assessment, and reflective-ethical functions. The study proposes a matrix for the use of digital tools, linking groups of tools with multilingual practices, professional tasks, and learning outcomes. The duality of digitalization and multilingualism is substantiated as a functional interdependence of a digital tool, multilingual activity, a professional task, and reflective verification of the result. The article also identifies pedagogical conditions for im…
subject: Articles
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subject: Theory and methodology of professional education
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subject: Information technology in education
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subject: Educational, pedagogical sciences
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creator: Діана Смужаниця
rights: Copyright (c) 2026 Діана Смужаниця https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-06-30
publication date: 2026-06-30
starting page: 20
ending page: 28
doi: 10.31110/fmo2026.v41i3-03

GRANT WRITING SKILLS IN THE STRUCTURE OF PROFESSIONAL COMPETENCE OF FUTURE STEM EDUCATION SPECIALISTS
type: item
description: Problem statement. The development of STEM education requires specialists who cannot only organize learning in line with interdisciplinary principles but also initiate educational and research projects, substantiate their relevance, attract resources, and responsibly present expected outcomes. In the training of master’s and doctoral students in the field of education, these skills are often regarded as additional, although they actually determine the future specialist’s ability to act in the conditions of project-based, competitive, and international academic activity. Therefore, there is a need to provide a theoretical justification for grant writing as a component of the professional competence of future STEM education specialists. Materials and methods. The study is theoretical and analytical. The materials included scholarly works on the professional competence of STEM educators, technological pedagogical content knowledge, pedagogical content knowledge, professional development, academic writing, research self-efficacy, mentoring, coaching, and grant proposal preparation. The methods of analysis, synthesis, comparison, source content grouping, generalization, and conceptual modeling were used. Results. The article shows that grant writing cannot be reduced to either the technique of writing a proposal or a means of obtaining funding. It integrates research thinking, the design of educational innovation, evidence-based work, communication with diverse stakeholders, resource planning, risk assessment, digital literacy, and ethical responsibility. The expediency of considering grant writing competence as a special professional competence of a future STEM education specialist is substantiated. Its structure is proposed and includes analytical and research, project-design, communicative and argumentative, financial and organizational, digital, and ethical-reflective components. Conclusions. Grant writing competence is in demand among master’s and doctoral students …
subject: Articles
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subject: Theory and methodology of professional education
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subject: Educational, pedagogical sciences
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creator: Mykola Kapliienko
rights: Copyright (c) 2026 Микола Каплієнко https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-06-30
publication date: 2026-06-30
starting page: 6
ending page: 13
doi: 10.31110/fmo2026.v41i3-01

FROM SELF-ASSESSMENT TO PEER REVIEW: THE DEVELOPMENT OF A COGNITIVE-VERIFICATION APPROACH TO THE EVALUATION OF MATHEMATICAL PROBLEMS IN THE CONTEXT OF GENERATIVE ARTIFICIAL INTELLIGENCE
type: item
description: Problem statement. Generative artificial intelligence undermines the diagnostic reliability of homework assignments in distance mathematics education: an externally correct result is no longer sufficient evidence of a student’s independent achievement of learning outcomes. The key risk is students’ metacognitive passivity, understood as delegating to technology the analysis of the problem conditions, the choice of method, and the verification of the result. This article extends the author’s cognitive-verification approach by shifting verification activity from individual self-checking to the analysis of another student’s mathematical reasoning through peer review. Materials and methods. The study is theoretical, with a pilot trial. The theoretical framework is based on Messick’s concept of assessment validity, the notion of metacognitive passivity, and the concept of assessment twins. On this basis, a five-component risk-oriented assessment model was constructed. The pilot involved one academic group of 10 students, one mathematical analysis task, and a qualitative descriptive analysis of the results. Results. A five-component model was developed: a cognitive-verification solution with checkpoints; structured review of an analogous work according to a rubric; a mandatory author’s response to the review; brief self-reflection; and selective instructor audit of 20–30% of the submitted works. Examples of model implementation for a mathematical analysis course were designed, including the limit of a function of two variables, extrema, and a double integral, as well as three variants of final assessment. Conclusions. The model shifts the problem from detecting the use of artificial intelligence to ensuring assessment validity by generating several complementary sources of evidence of competence. Selective audit is a more scalable alternative to the comprehensive individual defence of all submitted works. The model is particularly promising for the training of mathematics…
subject: Articles
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subject: Theory and methodology of professional education
  topic: category
subject: Actual problems of mathematics and methods of teaching mathematics
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subject: Information technology in education
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creator: Yaroslav Chkana
rights: Copyright (c) 2026 Ярослав Чкана https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-06-30
publication date: 2026-06-30
starting page: 37
ending page: 44
doi: 10.31110/fmo2026.v41i3-05

CASCADE APPLIED PROBLEMS IN MATHEMATICS AS A MEANS OF DEVELOPING KEY COMPETENCIES OF STUDENTS
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description: Problem statement. Contemporary educational reforms in Ukraine emphasize the development of key competencies as mandatory learning outcomes at all levels of secondary education. State regulatory documents — the New Ukrainian School (2016), the State Standard of Basic Secondary Education (2020), and the State Standard of Profile Secondary Education (2024) — identify a broad spectrum of competencies that students are expected to acquire, including mathematical, civic, environmental, entrepreneurial, and digital competencies, among others. The school mathematics course has significant potential for integrated development, particularly through purposefully designed systems of applied problems that embed mathematical content in authentic real-world contexts. However, existing mathematics textbooks rarely include problems of this kind, creating a clear methodological gap between normative expectations and classroom practice. This article addresses that gap by focusing on a specific type of applied problem — the cascade applied problem — and on approaches to its effective use in teaching students at the basic and profile secondary education levels. Materials and methods. The study employs theoretical research methods: analysis of regulatory documents; reference and educational literature on the theory and methodology of mathematics teaching; and synthesis, comparison, and generalization of the findings. The theoretical foundations of the study draw on international research in mathematical modeling and competency-based education. Results. The article introduces and substantiates the concept of applied problems in mathematics as a cascade, an original didactic construct. A cascade-applied problem is defined as a problem that combines an authentic real-world context with a cascade of interrelated requirements of increasing complexity, mandatory interdisciplinary connections, a discussion question requiring reasoned judgment, and a purposeful orientation towards specific key …
subject: Articles
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subject: Actual problems of mathematics and methods of teaching mathematics
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creator: Vasyl Shvets
creator: Alla Prus
rights: Copyright (c) 2026 Василь Швець, Алла Прус https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-06-30
publication date: 2026-06-30
starting page: 45
ending page: 54
doi: 10.31110/fmo2026.v41i3-06

CONCEPTUAL FOUNDATIONS OF USING PROBLEM-ORIENTED SITUATIONS IN THE PROFESSIONAL TRAINING OF MATHEMATICS TEACHERS
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description: The article defines the conceptual foundations of using problem-oriented situations as an innovative technology in the professional training of mathematics teachers, outlines the authors’ methodology for designing such situations in teaching mathematical disciplines and methods of teaching mathematics, and provides examples from the authors’ generalized pedagogical experience. Formulation of the problem. Problem-oriented learning (POL) contributes to the development of subject-specific and methodological competence in future mathematics teachers. However, existing studies focus primarily on individual problem-based tasks rather than on a systematic methodology for designing problem-oriented situations. Thus, the issue of how to prepare future teachers not only to solve such situations but also to design them independently remains unresolved. The authors’ methodology addresses this gap by proposing a typology of problem-oriented situations and a step-by-step implementation algorithm that progresses from students’ subjective experiences of mathematical problems within mathematical disciplines to reflective design in the course on methods of teaching mathematics. Materials and methods. The methodological basis of the research is formed by systemic and activity-based approaches to the professional training of mathematics teachers. The methods of analysis, synthesis, systematization, comparison, and generalization of the authors' own pedagogical experience in implementing educational innovations were used. Results. The conceptual foundations for implementing problem-based learning (PBL) as an innovative technology have been defined. The author’s methodology for constructing problem-based situations (“algorithmic impossibility,” “time constraints,” “methodological irrationality,” etc.) that stimulate students’ independent exploration is presented. The article outlines the specific features of implementing problem-oriented learning in the teaching of mathematical disciplin…
subject: Articles
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subject: Actual problems of mathematics and methods of teaching mathematics
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creator: Olga Yakovlieva
creator: Olha Papach
creator: Anastasiia Drots
creator: Maria Parlikova
rights: Copyright (c) 2026 Ольга Яковлєва, Ольга Папач, Анастасія Дроць, Марія Парлікова https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-06-30
publication date: 2026-06-30
starting page: 55
ending page: 63
doi: 10.31110/fmo2026.v41i3-07

STEM-MAGNETIC LEVITATION EXPERIMENT
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description: Problem formulation. The phenomenon of magnetic levitation using high-temperature superconductors (HTSCs) allows integrating knowledge of electrodynamics, solid-state physics, and cryogenic engineering. However, its pedagogical potential as a means of forming experimental competencies of students of physics and mathematics specialties remains insufficiently realized. The purpose of the article is to develop and test STEM-oriented laboratory work in physics based on the phenomenon of magnetic levitation with volumetric HTSCs and empirically verify its impact on the formation of competencies of students. Materials and methods. The educational experiment was implemented in the form of laboratory work on the study of magnetic levitation between a permanent magnet and HTSCs at liquid nitrogen temperatures. The study used theoretical and empirical methods: analysis of scientific and methodological literature, generalization of pedagogical experience, educational physical experiment, observation of students' activities, and analysis of experimental measurement results. The operationalization of experimental competencies was carried out according to four indicators: planning, operational-measuring, analytical, and interpretative-reflective competence. Each component was operationalized through a system of indicators and quantitative assessments. The study was conducted in the format of a quasi-experimental pedagogical study with a “before/after” design without a control group. The pedagogical influence was STEM-oriented laboratory work on magnetic levitation. The structure of the study included the following stages: initial diagnostics of the level of formation of experimental competencies; pedagogical influence - performance of laboratory work on magnetic levitation; final diagnostics according to the same indicators. Results. A comparative analysis of the results before and after the laboratory work showed positive dynamics according to all indicators. The share of studen…
subject: Articles
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subject: Actual problems of physics and methods of teaching physics
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creator: Igor Linchevskyi
creator: Vitalii Peklun
rights: Copyright (c) 2026 Ігор Лінчевський, Віталій Пеклун https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-06-30
publication date: 2026-06-30
starting page: 14
ending page: 19
doi: 10.31110/fmo2026.v41i3-02

ELECTROMAGNETIC INTERFERENCE AND DATA TRANSMISSION RELIABILITY IN SDR-BASED EDUCATIONAL EXPERIMENTS
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description: 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.…
subject: Articles
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subject: Actual problems of physics and methods of teaching physics
  topic: category
subject: Computer sciences and methods of teaching them
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creator: Volodymyr Shamonia
creator: Vladyslav Bespalyi
rights: Copyright (c) 2026 Володимир Шамоня, Владислав Беспалий https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 94
ending page: 100
doi: 10.31110/fmo2026.v41i2-09

PLANE DILATIONS IN THE SCHOOL MATHEMATICS COURSE AND THE THEORY OF GEOMETRIC TRANSFORMATIONS FOR FUTURE MATHEMATICS TEACHERS
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description: 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 t…
subject: Articles
  topic: section
subject: Theory and methodology of professional education
  topic: category
subject: Actual problems of mathematics and methods of teaching mathematics
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creator: Mykola Pratsiovytyi
creator: Natalia Pravitska
creator: Sofiia Ratushniak
rights: Copyright (c) 2026 Микола Працьовитий, Наталя Правіцка, Софія Ратушняк https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 71
ending page: 85
doi: 10.31110/fmo2026.v41i2-07

DEVELOPING PREDICTIVE COMPETENCE IN FUTURE COMPUTER SCIENCE TEACHERS THROUGH THE INTEGRATION OF TEACHING PRACTICE AND MATHEMATICAL STATISTICS ASSIGNMENTS
type: item
description: Formulation of the problem. The article addresses the development of preservice computer science teachers' prognostic competence in the course Probability Theory and Mathematical Statistics. The relevance of the study is determined by the need to foster future educators’ ability to predict educational outcomes in the modern, digitalized, and uncertain educational environment. The article presents the experience of implementing project-oriented learning, which involves students’ collection and processing of research data during pedagogical practice. The impact of practice-oriented projects on the level of prognostic competence formation is analyzed. Materials and methods. The study employed analysis and synthesis of scientific and pedagogical literature, modeling of a methodological approach to teaching, and a pedagogical experiment in the format of the case method, conducted with second-year students majoring in 014.04 Secondary Education (Computer Science). The level of predictive competence development was assessed using the “Forecasting Ability” questionnaire. Statistical analysis of the obtained data was conducted using Student’s t-test. Results. The analysis of scientific sources clarified the essence of prognostic competence and identified its main structural components. The results of the experimental work demonstrated that implementing the developed methodological approach contributes to the development of the cognitive-informational, intellectual-mental, instrumental-technological, and worldview-professional components of prognostic competence. Based on survey results conducted before and after the implementation of practice-oriented projects, an increase in the level of prognostic competence was observed, confirming the effectiveness of the proposed methodological approach. Conclusions. The effectiveness of integrating the course Probability Theory and Mathematical Statistics with pedagogical practice for the purposeful development of prognostic thinking o…
subject: Articles
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subject: Theory and methodology of professional education
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subject: Actual problems of mathematics and methods of teaching mathematics
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creator: Daria Tinkova
creator: Galyna Lutsenko
rights: Copyright (c) 2026 Дарія Тінькова, Галина Луценко https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-06-30
publication date: 2026-06-30
starting page: 29
ending page: 36
doi: 10.31110/fmo2026.v41i3-04

A COGNITIVE-VERIFICATION APPROACH TO TRANSFORMING HOMEWORK IN MATHEMATICAL ANALYSIS IN THE CONTEXT OF ARTIFICIAL INTELLIGENCE USE
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description: 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…
subject: Articles
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subject: Theory and methodology of professional education
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creator: Yaroslav Chkana
rights: Copyright (c) 2026 Ярослав Чкана https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 86
ending page: 93
doi: 10.31110/fmo2026.v41i2-08

EXPERIMENTAL VISUALIZATION AND NUMERICAL SIMULATION OF FRAUNHOFER DIFFRACTION FROM ORDERED AND DISORDERED OPAQUE MICROSTRUCTURES FOR EDUCATION PRACTICE
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description: 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 …
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subject: Actual problems of physics and methods of teaching physics
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creator: Oleksii Voronkin
creator: Sergiy Lushchin
rights: Copyright (c) 2026 Oleksii Voronkin, Sergiy Lushchin https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 28
ending page: 45
doi: 10.31110/fmo2026.v41i2-03

ALGORITHMIC SUPPORT FOR REVERSE GENERATION OF TRIGONOMETRIC TASKS USING OBJECT-ORIENTED MODELING
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description: 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…
subject: Articles
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subject: Actual problems of mathematics and methods of teaching mathematics
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subject: Computer sciences and methods of teaching them
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creator: Svitlana Bodnaruk
creator: Nazarii Rybchynskyi
rights: Copyright (c) 2026 Світлана Боднарук, Назарій Рибчинський https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 18
ending page: 27
doi: 10.31110/fmo2026.v41i2-02

CLASSIFICATION OF MOBILE LEARNING SUPPORT TOOLS IN GENERAL SECONDARY EDUCATION
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description: 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…
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subject: Information technology in education
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creator: Yuliia Nosenko
rights: Copyright (c) 2026 Юлія Носенко https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 64
ending page: 70
doi: 10.31110/fmo2026.v41i2-06

USE OF GAMIFICATION ELEMENTS IN TEACHING MATHEMATICAL ANALYSIS AT HIGHER EDUCATION INSTITUTIONS
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description: 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 communicati…
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subject: Actual problems of mathematics and methods of teaching mathematics
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creator: Hanna Linnik
creator: Iryna Morachkovska
rights: Copyright (c) 2026 Ганна Лінник, Ірина Морачковська https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 56
ending page: 63
doi: 10.31110/fmo2026.v41i2-05

FOUR PROMPTING TECHNIQUES FOR ANALYTICAL WORK WITH TEACHING AND LEARNING RESOURCES
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description: 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 …
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subject: Information technology in education
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creator: Taiana Dieorditsa
creator: Maryna Voronina
creator: Raisa Hladushyna
creator: Olga Yepifanova
creator: Olena Kozmenko
creator: Volodymyr Tolmachov
rights: Copyright (c) 2026 Таяна Дєордіца, Марина Вороніна, Раїса Гладушина, Ольга Єпіфанова, Олена Козьменко, Володимир Толмачов https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 46
ending page: 55
doi: 10.31110/fmo2026.v41i2-04

SOFTWARE IMPLEMENTATION OF ALGORITHMS FOR CALCULATING PEARSON, SPEARMAN AND KENDALL CORRELATIONS IN C#
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description: 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 implemen…
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subject: Computer sciences and methods of teaching them
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creator: Vitalii Bazurin
creator: Sofia Bazurina
creator: Svitlana Kolesnyk
creator: Vyacheslav Kolesnyk
creator: Nina Dulenko
rights: Copyright (c) 2026 Vitalii Bazurin, Sofia Bazurina, Svitlana Kolesnyk, Valerii Kolesnyk, Nina Dulenko https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-04-30
publication date: 2026-04-30
starting page: 6
ending page: 17
doi: 10.31110/fmo2026.v41i2-01

SYSTEM OF APPLIED MATHEMATICAL PROBLEMS: CRITERIA FOR DEVELOPMENT AND FEATURES OF SOLUTION
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description: 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, …
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subject: Actual problems of mathematics and methods of teaching mathematics
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creator: Vasyl Shvets
creator: Alla Prus
rights: Copyright (c) 2026 Василь Швець, Алла Прус https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-03-02
publication date: 2026-03-02
starting page: 37
ending page: 47
doi: 10.31110/fmo2026.v41i1-06

OFFLINE-FIRST PWA WITH CONTROLLED GENERATIVE AI SUPPORT FOR TEACHING INFORMATICS IN A NEAR-FRONTLINE SCHOOL
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description: 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 minimizati…
subject: Articles
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subject: Theory and methodology of professional education
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subject: Information technology in education
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subject: Secondary education
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creator: Dmytro Turchyn
rights: Copyright (c) 2026 Dmytro Turchyn https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-03-02
publication date: 2026-03-02
starting page: 32
ending page: 36
doi: 10.31110/fmo2026.v41i1-05

THREE STEPS FOR SOLVING PROBLEMS USING THE DEFINITION OF THE LIMIT OF A SEQUENCE
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description: 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 …
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creator: Mykola Bilotsky
rights: Copyright (c) 2026 Микола Білоцький https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-03-02
publication date: 2026-03-02
starting page: 6
ending page: 14
doi: 10.31110/fmo2026.v41i1-01

ASSESSMENT OF THE LEVEL OF STUDENTS' AWARENESS REGARDING THE SAFE USE OF DIGITAL TECHNOLOGIES IN THE CONDITIONS OF TRANSFORMATION OF THE EDUCATIONAL SPACE
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description: 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 ed…
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creator: Nataliia Priadko
creator: Dariia Liubchykova
rights: Copyright (c) 2026 Наталія Прядко, Дарія Любчикова https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-03-02
publication date: 2026-03-02
starting page: 15
ending page: 20
doi: 10.31110/fmo2026.v41i1-02

MS EXCEL TEMPLATES FOR MODELING PROBLEMS ON INTEGRATION OF RATIONAL FUNCTIONS
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description: 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…
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subject: Actual problems of mathematics and methods of teaching mathematics
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subject: Information technology in education
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creator: Serhiy Radchenko
creator: Mariia Astafieva
creator: Anatoly Mazur
rights: Copyright (c) 2026 Сергій Радченко, Марія Астаф'єва, Анатолій Мазур https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-03-02
publication date: 2026-03-02
starting page: 21
ending page: 25
doi: 10.31110/fmo2026.v41i1-03

VISUALIZATION OF KNOWLEDGE BY WEB TECHNOLOGY IN THE FORMATION OF PRE-SERVICE COMPUTER SCIENCE TEACHERS’ CRITICAL THINKING
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description: 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 throug…
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subject: Theory and methodology of professional education
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subject: Information technology in education
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creator: Artem Yurchenko
creator: Roman Momot
creator: Mariia Ostroha
creator: Olena Semenikhina
rights: Copyright (c) 2025 Артем Юрченко, Роман Момот, Марія Острога, Олена Семеніхіна https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2025-11-28
publication date: 2025-11-28
starting page: 80
ending page: 88
doi: 10.31110/fmo2025.v40i5-11

OPTIMIZATION OF FACTOR VARIABLE SELECTION IN ECONOMETRIC MODELING: PRACTICAL RECOMMENDATIONS
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description: 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 rig…
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subject: Actual problems of mathematics and methods of teaching mathematics
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creator: Oksana Hlushak
creator: Svitlana Semeniaka
creator: Nadiia Zinchenko
rights: Copyright (c) 2025 Оксана Глушак, Світлана Семеняка, Надія Зінченко https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2025-11-28
publication date: 2025-11-28
starting page: 29
ending page: 35
doi: 10.31110/fmo2025.v40i5-04

SOLVING MATHEMATICAL PROBLEMS BY HIGH SCHOOL STUDENTS AS AN EFFECTIVE TOOL FOR DEVELOPING THE COMPETENCE OF «ENTREPRENEURSHIP AND FINANCIAL LITERACY»
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description: 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 mathematic…
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subject: Secondary education
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creator: Olena Tkach
rights: Copyright (c) 2025 Олена Ткач https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2025-11-28
publication date: 2025-11-28
starting page: 68
ending page: 73
doi: 10.31110/fmo2025.v40i5-09

CONSTRUCTION OF MATHEMATICAL MODELS WHEN SOLVING APPLIED PROBLEMS OF ECONOMIC CONTENT
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description: 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…
subject: Articles
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subject: Actual problems of mathematics and methods of teaching mathematics
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creator: Tetyana Hrytsik
rights: Copyright (c) 2025 Тетяна Грицик https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2025-11-28
publication date: 2025-11-28
starting page: 36
ending page: 43
doi: 10.31110/fmo2025.v40i5-05

FEATURES OF TESTS AND TEST CONTROL OF ACADEMIC ACHIEVEMENTS IN PHYSICS
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description: 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…
subject: Articles
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subject: Actual problems of physics and methods of teaching physics
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creator: Mark Bosin
creator: Larysa Rykova
creator: Halyna Broslavska
rights: Copyright (c) 2025 Марк Босін, Лариса Рикова, Галина Брославська https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2025-11-28
publication date: 2025-11-28
starting page: 14
ending page: 20
doi: 10.31110/fmo2025.v40i5-02

AUGMENTED REALITY IN PHYSICS LESSONS: EXPERIENCE USING PHYSICS LAB AR IN BLENDED LEARNING
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description: 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 pathw…
subject: Articles
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subject: Actual problems of physics and methods of teaching physics
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creator: Olha Slobodianyk
rights: Copyright (c) 2025 Ольга Слободяник https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2025-11-28
publication date: 2025-11-28
starting page: 61
ending page: 67
doi: 10.31110/fmo2025.v40i5-08

STUDYING COORDINATES AND VECTORS IN 8TH GRADE OF NUS ACCORDING TO THE AUTHOR'S TEXTBOOK OF THE INTEGRATED COURSE OF MATHEMATICS
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description: 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: “Coo…
subject: Articles
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subject: Actual problems of mathematics and methods of teaching mathematics
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creator: Oleksandr Shkolnyi
rights: Copyright (c) 2026 Oleksandr Shkolnyi https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2026-03-02
publication date: 2026-03-02
starting page: 48
ending page: 54
doi: 10.31110/fmo2026.v41i1-07

CONCEPTUAL MODEL OF GENERATIVE ARTIFICIAL INTELLIGENCE INTEGRATION INTO THE PROCESS OF DEVELOPING INFORMATION AND DIGITAL COMPETENCE IN STUDENTS AT THE BASIC SECONDARY EDUCATION LEVEL
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description: 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. Conclu…
subject: Articles
  topic: section
subject: Information technology in education
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subject: Secondary education
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subject: Educational, pedagogical sciences
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creator: Svitlana Lytvynova
creator: Yulia Nosenko
creator: Kateryna Osadcha
creator: Olha Pinchuk
creator: Natalia Rashevska
creator: Alisa Sukhikh
rights: Copyright (c) 2025 Світлана Литвинова, Юлія Носенко, Катерина Осадча, Ольга Пінчук, Наталя Рашевська, Аліса Сухіх https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2025-11-28
publication date: 2025-11-28
starting page: 44
ending page: 52
doi: 10.31110/fmo2025.v40i5-06

FORMATION OF CIVIC IDENTITY IN SCHOOLCHILDREN DURING MATHEMATICS EDUCATION IN GRADES 5-6
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description: 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.
subject: Articles
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subject: Actual problems of mathematics and methods of teaching mathematics
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creator: Vasyl Shvets
creator: Kateryna Martyniuk
rights: Copyright (c) 2025 Василь Швець, Катерина Мартинюк https://creativecommons.org/licenses/by-nc-sa/4.0
date: 2025-09-29
publication date: 2025-09-29
starting page: 65
ending page: 72
doi: 10.31110/fmo2025.v40i4-10