A Short Formula for the Area of a Quadrilateral with Two Equal Opposite Sides and it’s Application
Yordan Tabov 1), Stanislav Stefanov 2), Haim Haimov 3), Dobrinka Milusheva-Boykina 4)
1) Institute of Mathematics and Informatics, BAS – Sofia, Bulgaria
2) Todor Kableshkov University of Transport – Sofia, Bulgaria
4) University of Plovdiv “Paisii Hilendarski” – Plovdiv, Bulgaria
https://doi.org/10.53656/math2025-2-1-asf
Abstract. The article develops methods for solving complex problems at the level of mathematics olympiads, which are necessary for both high school students and their teachers. At the heart of these methods lies a formula for the area of a quadrilateral with a pair of equal but non-parallel opposite sides. It obtains as a special case of a more general formula for the area of an arbitrary convex quadrilateral, discussed in (Stefanov et al. 2024). Here we will prove the last formula in a more direct way and will consider its application to efficiently solving geometric problems.
Keywords: a special kind of quadrilaterals; area formulas; area of a parallelogram; problems
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
Application of Systems of Linear Equations in balancing Chemical Equations
Tsvetelina Peeva
High School of Mathematics “Dr. Petar Beron” – Varna, Bulgaria
https://doi.org/10.53656/math2025-2-3-aos
Abstract. The article describes an integrated STEM lesson conducted with students of the ninth grade of High school of Mathematics “Dr. Peter Beron”. It examines the author's attempt to upgrade the students' knowledge of solving systems of linear equations (SLE) by introducing the Gauss method to them. For this purpose, the concepts of matrix, matrix rank and elementary transformations in a matrix are introduced. One application of SLE as a universal method for balancing chemical equations is shown. A conclusion is made about the students' interest in the topic and the degree of assimilation of the presented learning material, which is beyond the scope of the curricula for compulsory and optional training optional training in Grade 9.
Keywords: method of Gauss; matrix; stoichiometric coefficients; STEM education
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
