Една кратка формула за лице на четириъгълник с две равни срещуположни страни и приложението ѝ
Йордан Табов 1), Станислав Стефанов 2), Хаим Хаимов 3), Добринка Милушева-Бойкина 4)
1) Институт по математика и информатика, БАН, София (България)
2) Висше транспортно училище „Тодор Каблешков“, София (България)
4) Пловдивски университет „Паисий Хилендарски“, Пловдив (България)
https://doi.org/10.53656/math2025-2-1-asf
Absract. В статията са развити методи, с които се решават сложни задачи от нивото на олимпиади по математика, които са необходими както на ученици от гимназиалния курс, така и на техните учители. В основата на тези методи лежи една формула за лице на четириъгълник с двойка равни, но неуспоредни срещуположни страни. Тя бе получена като частен случай на по-обща формула за лице на произволен изпъкнал четириъгълник, разгледана в (Stefanov et al. 2024). Тук ще докажем тази формула по по-пряк начин и ще разгледаме приложението ѝ за ефективно решаване на геометрични задачи.
Keywords: специален вид четириъгълници; формули за лице; лице на успоредник; задачи
Artificial Intelligence – Assisted Learning in Moodle
Maria Gorgorova, Silvia Gaftandzhieva, Stanka Hadzhikoleva
University of Plovdiv “Paisii Hilendarski”, Plovdiv (Bulgaria)
https://doi.org/10.53656/math2025-2-2-aia
Abstract. Generative Artificial Intelligence (AI) has unveiled numerous opportunities in education. All participants in the educational field have found benefits for themselves. Educators use it to prepare, organize, and conduct the educational process, including developing educational resources and exam materials, grading tests and assignments, preparing educational documentation, and more. Learners find its application useful for acquiring new knowledge, self-study, and completing coursework and homework, among other things. Administrative staff have discovered that AI can assist in preparing administrative documents, process and analyze data. The exploration of AI’s potential in education is increasingly capturing the attention of researchers. This paper outlines several possibilities for integrating AI tools into the Moodle e-learning management system through external service APIs. It discusses the AI Connector, AI Text to Questions Generator, AI Text to Image, OpenAI Chat Block, and the benefits they offer to educators and learners. Practical examples of their usage and the results of an experiment with students are presented. Some risks and limitations of using AI in education are also discussed.
Keywords: artificial intelligence; education; innovations in education; Moodle
Приложение на системи линейни уравнения в изравняване на химични уравнения
Цветелина Пеева
Математическа гимназия „Д-р Петър Берон“, Варна (България)
https://doi.org/10.53656/math2025-2-3-aos
Abstract.. Статията описва СТЕМ урок, проведен с ученици от девети клас на МГ „Д-р Петър Берон“. В нея се разглеждат опитът и целта на авторката да надгради знанията на учениците за решаване на системи линейни уравнения (СЛУ) чрез запознаване с метода на Гаус. За целта са въведени понятията „матрица“, „ранг на матрица“ и „елементарни преобразувания на матрица“. Показано е едно приложение на СЛУ като универсален метод за изравняване на химични уравнения. Направен е извод за интереса от страна на учениците към темата и степента на усвояване на поднесения учебен материал, който е извън предвидения в учебните програми за задължителна и избираема подготовка в девети клас.
Keywords: метод на Гаус; матрица; стехиометрични коефициенти; STEM обучение
Enchancing Visual Reasoning in Calculus: introducing a New Typology of Graphical Tasks
Valentina Kostic1), Tanja Sekulic2)
1)Academy of Applied Technical and Preschool Studies Nis – Department Pirot (Serbia)
2)Technical College of Applied Sciences in Zrenjanin (Serbia)
https://doi.org/10.53656/math2025-2-4-evr
Abstract. This paper emphasizes the introduction of the innovative typology of tasks in calculus education to enhance students’ visual reasoning skills. The proposed typology includes graphical tasks that require students to engage with images to derive meanings, justify solutions, and foster a deeper conceptual understanding of function derivatives. By integrating these tasks, we aim to address the imbalances in mathematics education regarding traditional approach, which often favors algebraic over graphical representation. This new typology will help students develop a well-rounded understanding, bridging the gap between symbolic manipulation and graphical interpretation.
Keywords: visualization; graphical tasks; calculus; derivative
Interdisciplinary Connection between Mathematics and Physics in a Grammar School for Gifted Mathematics Students
Aleksandar Milenković1), Nemanja Momčilović2)
1) University of Kragujevac, Faculty of Science, Kragujevac (Serbia)
2) First Grammar School Kragujevac, Kragujevac (Serbia)
https://doi.org/10.53656/math2025-2-5-icb
Abstract. Interdisciplinary connection of teaching content, as an approach that enables the deepening of knowledge in different subjects while simultaneously fostering various student competencies, is not sufficiently present in schools in the Republic of Serbia. Moreover, it is almost neglected in working with students gifted in mathematics. In this paper, we describe a two-lesson session on the topic Applications of Differential Calculus in Physics, in which we connected teaching content from Mathematical Analysis with Algebra and Physics in working with fourth-year high school students in Kragujevac, specifically those enrolled in the specialized mathematics program. Students had the opportunity to revisit and deepen previously acquired knowledge in Physics, and simultaneously recognize the application of differential calculus, which they had recently learned. The paper presents the problems solved by the students during the session, as well as the results of a survey, in which students evaluated this approach as useful, engaging, and they expressed that it increased their interest in studying mathematics and physics in greater depth.
Keywords: interdisciplinary connection; calculus; mathematics; physics; students gifted in mathematics
Teaching Mathematics under Extreme Conditions: Ukrainian Realities and Experience
Olha Matiash1), Oleksandr Shkolnyi2)
1)Mykhailo Kotsiubynskyi State Pedagogical University, Vinnytsia (Ukraine)
2)Dragomanov Ukrainian State University, Kyiv (Ukraine)
https://doi.org/10.53656/math2025-2-6-tmu
Abstract. Since February 2022, Ukrainian education has been in extreme conditions of martial law. In this article, we analyze Ukrainian realities and experience in ensuring mathematical training of students in extreme conditions of martial law in the country. How do Ukrainian schools, teachers, and students work in wartime conditions? How is school mathematical education being restored and is it being restored in communities affected by the war? The acquired Ukrainian experience in organizing educational processes in such conditions is relevant against the background of increasing international interest in new extreme challenges.
Research methods: observations, conversations, interviews with teachers, students, parents of students, analysis of documents of the Ministry of Education and Science of Ukraine, analysis of current publications in the Ukrainian information space.
The article identifies and characterizes three special periods of the organization of educational processes: from February 24, 2022 – until the end of the 2021 – 2022 academic year; 2022 – 2023 academic year; 2023 – 2024 academic year.
Each of the specified periods has its own characteristics of education. For each of the specified periods, the conditions of martial law and the corresponding actions of the Ministry of Education and Science of Ukraine, the activities of mathematics teachers and the activities of students are characterized. The main challenges (power outages, limited access to distance learning tools, air raids, lack of shelters) faced by Ukrainian students and teachers are considered. The state of organization of distance and blended learning in mathematics is described, and the problems that arise during its organization are outlined. Specific examples of organizing mathematics teaching in wartime and using interactive techniques that promote students’ emotional resilience and support their motivation to learn are provided.
Our general conclusions, which may be useful for other countries in crisis situations: the activities of all those involved in education in new extreme conditions must be clearly adjusted; the efficiency and balance of management documents are important; it is important to treat the difficulties of students in extreme learning conditions with increased attention and understanding; it is necessary to establish interaction and mutual support among teachers to find and implement new forms and means of learning in extreme conditions; methodologists, through various seminars and webinars, can help increase the ability of mathematics teachers to work effectively in extreme conditions.
Keywords: distance and blended learning; mathematics education; extreme learning conditions; wartime learning strategies
