Mathematics at school

ON ELEMENTARY METHODS OF SOLVING EXTREMAL TASKS

2026-05-16T20:57:54+00:00

The article is devoted to the consideration of extremal tasks studied in the school mathematics course by various elementary methods (algebraic, geometric, analytical). The concept of an extremal tasks ( or task of extrema) is briefly given, it is noted that even in ancient times great scientists of the past were engaged in such tasks. It is interesting that in the school mathematics course comparatively less attention is paid to extremum tasks, and the existing tasks are mainly solved by the method of mathematical analysis based on the study of the function. In this regard, it is noted that there are extremum tasks of a wide range, the solution of which is most appropriately considered not by the method of mathematical analysis, but by the so-called elementary methods, which use a number of properties of standard ratios, and also apply various, sometimes non-standard techniques and methods. It is stated that these techniques and methods have not found their full reflection in the course of elementary mathematics, since there are no clearly developed methodological systems for their use. The article considers a number of specific methods for solving extremal tasks without using a derivative, including the method of standard inequalities, the method of importing auxiliary parameters, the method of conditional assumptions, the coordinate method, the method of auxiliary constructions, the linkage method and iteration method.

ON SOME CONTENT-RELATED FEATURES OF TEACHING RATIONAL FRACTIONS

2026-05-27T20:58:30+00:00

This article examines several important content-relevant aspects of teaching rational fractions in middle school algebra. First, the complex role of the concept of a rational fraction is demonstrated depending on the grade level of middle school. In seventh grade, the concept of a rational fraction as an algebraic expression is a fundamental component of algebraic language, including the theory of polynomials with many variables. In eighth grade, the situation is simplified by introducing the concept of a rational fraction, and in ninth grade, the approach is refined for polynomials with one variable. Another important feature is the role of rational fractions in defining the concepts of equality and identity. The role of rational fractions in modeling and solving problems arising in applied settings is also discussed. Various situations related to rectilinear and circular uniform motion, filling and emptying of a pool, joint work, mixtures and alloys are considered as situations of the applied environment, and a mechanism for formulating the corresponding problems is presented.

ARMENIAN PRINTED GEOMETRY TEXTBOOKS OF THE 19TH CENTURY

2026-05-27T21:37:32+00:00

This article examines a brief history of the development of methodological thought in teaching geometry and its influence on the development of Armenian educational literature on geometry. Geometry, as an academic subject, has been taught in Armenian schools since ancient times. Until the end of the 18th century, it was taught from manuscript translations of Euclid's Elements (by G. Magister and G. Kesaretsi). General comments are given on the nature of Euclid's presentation of the Elements. In the second half of the 18th and early 19th centuries, influenced by the advanced French methodological ideas of the time and based on the textbooks of Bezout, Legendre, and Lacroix, Armenian authors (A. Taghiyan, S. Pronyan, Papazyan, A. Charyan, G. Terteryan) compiled valuable textbooks.In the second half of the 19th century, Armenian methodological thought in the field of teaching geometry embarked on new research. Armenian methodologists and leading teachers studied the textbooks and practices of advanced European and Russian schools. The best French, German, and Russian geometry textbooks of the time, by Karbantiz, Moshnik, L. Thilo, A. Diesterweg, and A. Davidov, were translated into Armenian. Finally, creatively reworking both previous Armenian and the best European and Russian textbooks, Armenian authors (A. Papikyan, M. Sahakyan, V. Azarapetyan, A. Falagashyan, M. Ter-Sarkisyan, and others) have compiled original geometry textbooks. The article provides brief descriptions of the structure and scientific and methodological frameworks of the textbooks by these authors.

ARMENIAN GEOGRAPHY TEXTBOOKS AND THE TEACHING OF GEOGRAPHY IN ARMENIAN SCHOOLS IN THE SECOND HALF OF THE 19TH CENTURY

2026-05-27T22:20:11+00:00

This paper examines the creation of Armenian geometry textbooks in the second half of the 19th century. Research conducted by Armenian specialists in compiling these textbooks is presented. The methods underlying their compilation and the innovations introduced during this period are presented, with an emphasis on the use of illustrations.

Two trends in the history of geometry teaching methods in Armenian schools, observed in Armenian textbooks published until the end of the 19th century, are presented.

It shows how Armenian methodologists and progressive teachers studied the textbooks and practices of progressive European and Russian schools. They translated the best French, German, and Russian textbooks of the time into Armenian. Finally, by creatively reworking both previous Armenian textbooks and the best European and Russian textbooks, and drawing on scientific advances in mathematics, Armenian authors have compiled several original textbooks.

SHIRAKATSI AND HIS MATHEMATICAL LEGACY

2026-05-27T22:39:58+00:00

The article examines the didactic potential of the mathematical heritage of the prominent 7th-century Armenian scholar Anania Shirakatsi and its effectiveness in modern general education. The aim of the study is to identify the pedagogical possibilities of Shirakatsi’s arithmetic problems and their impact on students’ logical thinking, analytical skills, and learning motivation. The research is based on a qualitative pedagogical methodology and includes an experimental study conducted in the 6th grade, with a comparative analysis of experimental and control groups. The results demonstrate that the purposeful integration of historical-mathematical material significantly enhances students’ cognitive engagement, independent thinking, and ability to provide reasoned solutions. At the same time, the value-based component of education is strengthened, as Shirakatsi’s heritage fosters respect for national scientific traditions. The article also presents methodological approaches, a SWOT analysis, and an interpretation within Bloom’s taxonomy framework. The findings confirm that Shirakatsi’s mathematical heritage can be effectively applied as a modern didactic tool to improve the quality of mathematics education.

THE ROLE OF INTRA-SUBJECT CONNECTIONS IN THE DEVELOPMENT OF MATHEMATICAL THINKING

2026-05-27T23:10:39+00:00

This research is devoted to the study of the role of the use of interdisciplinary connections in the process of forming mathematical thinking. Modern approaches to teaching mathematics show that the isolated transfer of individual knowledge does not ensure the full intellectual development of students. Therefore, it is important to ensure the establishment of substantive and methodological connections between different subjects in the learning process, which contribute to the systematic perception of knowledge and their effective application.

The results of the research show that the purposeful and systematic use of interdisciplinary connections significantly contributes to the development of logical, analytical and creative thinking of students. The connection of mathematics with natural sciences, informatics, technology, as well as art and everyday life allows students to perceive the applied significance of mathematical ideas, see their role in the process of explaining various phenomena and solving problems. Such an approach not only deepens the assimilation of knowledge, but also stimulates the cognitive interest of students, active participation in the educational process and the ability to think independently.

The study also revealed that the effective use of interdisciplinary connections requires careful planning of the learning process, appropriate methodological support, and the use of innovative pedagogical approaches by the teacher. At the same time, some difficulties may arise due to the limited number of lessons, the content load of the curriculum, and the lack of methodological materials. However, with proper organization and targeted methodological work, interdisciplinary connections can become an important factor in increasing the effectiveness of mathematics teaching. Thus, the use of interdisciplinary approaches in the teaching of mathematics contributes not only to the integration of knowledge, but also to the development of mathematical thinking of students, which is one of the most important goals of modern education.

PEDAL TETRAHEDRONS

2026-05-28T12:05:35+00:00

The concept of the pedal triangle of the given triangle with respect of the given point located in the plane of this triangle was introduced in the middle of the XX century. It was established that the third pedal triangle of the given triangle with respect of a given point in the plane of this triangle was similar to the initial. This are staid without study for a long time. The author of the present work identified some problems about pedal triangles. In particular, he proved the existence of some points in the plane of triangle such that their second pedal triangles with respect of these points were similar to initial. Then this problem was diffused onto the class of convex quadrangles and it was proved that fourth pedal quadrangle of the given quadrangle with respect of a given point in the plane of this quadrangle was similar to the initial. Besides, were studied the cases when second (third) pedal quadrangle of the given quadrangle with respect of a given point in the plane of this quadrangle was similar to the initial. The special case of the semicanonical trapezium was studied, some interesting geometric results were discovered here.

In the present work the next natural step of research is made, the concept of the pedal tetrahedron of the given tetrahedron with respect of a given point is introduced. The main result states that for any point of the space the fourth pedal tetrahedron of the given tetrahedron with respect of this point is similar to the initial. As an example the case of a right tetrahedron with canonical triangle in the base and pairwise congruent other three sides. It is proved that the altitude of this tetrahedron contains a point such that the first pedal tetrahedron with respect of this point is similar to initial.