Keyword: «teaching mathematics in secondary school»

Pedagogical science has always faced an issue of finding effective means to achieve the educational results of students. This problem is especially urgent today, when the world is changing rapidly, and those tools, that yesterday could keep pupils interested in studying mathematics and provided an opportunity to develop their mathematical abilities, quickly become out-of-date. Today, the search for a new toolkit that contributes to the development of students by means of mathematics, as well as mechanisms for incorporating it into the educational process, is highly relevant. Thus, the purpose of the article is to study the possibilities of using a new learning tool - puzzles in all their diversity in the process of teaching mathematics to children and teenagers, both in class and in extracurricular activities. The leading method here is the modeling of the methodological system of training in basic and additional mathematical education of children and teenagers, with the inclusion of a new didactic toolkit in it, which will contribute to the increase of students' interest in the subject, as well as to the development of certain mathematical abilities: logical thinking, abstraction, combining, operating with spatial images, critical thinking, mathematical memory. As a result of the research, the author of the article determined the place, opportunities and methodological aspects of including puzzles in the process of learning mathematics, both in the system of classical and nontraditional (creative) lessons of mathematics, and in the structure of extracurricular activities of students: mathematical circle, system of mathematical competitions, mathematical summer camp, etc. Practical use of this system makes it possible to reduce the disadvantage of tools deficiency for the development of mathematical abilities of students in the pedagogical practice, which gives us an opportunity to speak of high academic results in mathematical activity of children and teenagers.

In the modern world, the role of mathematics as the universal language of science and communication is increasing. These trends are reflected in the system of school mathematical education: the priority goals of teaching mathematics lie in the field of the subject knowledge formation, meta-subject skills and creativity of students. This actualizes the problem of designing an appropriate educational environment as a set of conditions for student's self-development and self-learning. The openness of the educational environment presupposes the involvement of students in the process of its design; however, the methodological aspects of such joint activities have not been sufficiently studied. The aim of the article is to present the characteristics of the subject educational environment (SEE) as a factor in achieving the priority goals of teaching mathematics in secondary schools using the example of joint design of visual technology with students. To achieve this goal, theoretical methods of scientific research were applied: analysis and generalization of scientific, pedagogical and methodological literature; systematization of the information received; modeling. The following empirical methods were chosen: questionnaires; analysis of diagnostic work of students; presentation of data in graphical form. As a result, the basic principles of SEE design were formulated, its model was constructed, and the author's original approach was tested. The theoretical significance of the research is represented by the formulated principles of designing a subject-based educational environment: student-centered approach; openness of the educational environment; integration of the digital environment, where artificial intelligence is its important component; three-part structure of the educational environment; dosed gamification of the educational environment. The practical significance of the research lies in the development and testing of a model for the joint design of a visual novel game based on probability theory for ninth grade students. The experience of implementing the model is shown by describing each of its stages: identifying emotional triggers for studying mathematics; defining the subject content of a novel; creating a storyline based on storytelling; developing character images using neural networks; choosing a platform and creating a novel; testing the novel and analyzing its results. Checking the effectiveness of the author's approach allowed us to conclude that the novel game has a positive effect on overcoming mathematical anxiety and developing students' subject and meta-subject skills.

Modern school education is focused not only on acquiring subject knowledge, but also meta-subject skills (universal learning actions – ULA). In the context of digitalization of education, knowledge management is complemented by skills in working with digital tools, and mathematics is an effective tool for their development. An analysis of pedagogical research on the problems of developing meta-subject skills using the potential of the digital environment has shown that meta-subject tasks can effectively develop all types of ULA, and the digital environment improves this process and enriches the content of meta-subject skills with additional actions. However, there is an insufficient number of such tasks in the practice of teaching mathematics. The aim of the study is to identify techniques for transforming mathematical tasks into a meta-subject task using a digital environment to enrich the content of ULA and optimize the process of their development. To achieve this goal, we applied a comparative analysis of psychological, pedagogical and methodological literature, systematization of the information received, pedagogical modeling, student survey and graphical representation of data. The transformation techniques are based on problem-solving, personality-oriented approaches, and the theory of generations, as well as the concept of a mathematical problem as a four-element system. The author's approach is illustrated by the example of the well-known problem of the area of regular nine-sided rectangle parts, which is transformed into a meta-subject task consisting of a series of tasks united by a common plot and a solution idea. Based on the survey results, it was concluded that there was an increase in motivation, a predominance of positive emotions, and a positive self-assessment of the development of ULA and digital skills among students when completing meta-subject tasks. The theoretical significance of the study is represented by a model of transformation of a mathematical problem into a meta-subject task, including the name of transformation techniques, an indication of the transformed element of the task, a description of the essence of each technique and ULA in a digital environment. The practical significance of the research lies in the description of methodological aspects of the introduction of meta-subject tasks into the educational process based on the identified techniques: "real context", "elements of gamification", "open final", "competition with a neural network", "research in a dynamic mathematics program", "infographics".