The aim of the research is to study the possibilities of the final stage of working with a mathematical problem as a means of forming schoolchildren's creative activity. The leading method of investigating this problem is to establish the correspondence between the components of the final stage of working with the mathematical problem and the procedural features of the student's creative activity. The study resulted in defining the structure of the final stage of working with a mathematical problem, which made it possible to identify a certain set of activities that make up the ability to work with the problem on the final stage of its solution. The article establishes the relationship between actions appropriate to this stage of work with the task and signs of the student's creative activity. It is proved that in the process of working with the problem on the final stage of its solution, students develop procedural features of creative activity. The author's method of forming students' creative activity suggested in the article can be used by the teachers of mathematics in school practice, by the authors of methodological manuals for students and teachers, and also can be used as the basis for a special course for students of pedagogical universities.
The introduction of new standards of mathematical education requires to stop understanding of the learning process as the transfer of ready-made knowledge and experience. Educational activity built on the principle of self-construction of knowledge by schoolchildren is highly demanded in new environment. Tasks with parameters have high learning, development, research and diagnostic potential. It allows to identify and in the process of teaching mathematics to prepare students who possess subject knowledge at the highest level, corresponding to the trends of the time. The urgency of the problem under study is determined by the need for students to form the ability to solve problems with parameters in order to achieve high results in mathematical, intellectual and personal development. The aim of the research is to develop a methodology for teaching students how to solve problems with parameters as an effective means of high-quality mathematical studies. The authors have identified main methods for solving problems with parameters and approaches to their study, and proved the theoretical basis for the application of these methods in the learning process. Therefore, they have shown the role of the propaedeutic stage of teaching graphic methods for solving problems with parameters, its goals, objectives and content. The authors suggest a methodology for designing a system of tasks that contributes to achieving high learning outcomes, which has passed multi-stage approbation. Moreover, they prove the need to use the Live Mathematics software as an effective visualization tool for studying graphic methods for solving problems with parameters. The methodology described in the article can be used by teachers at school and extracurricular mathematics classes, by the authors of textbooks for students and teachers, and it can also be the basis for a special course for students of pedagogical universities.
Ensuring a high quality of teaching mathematics to students is inextricably linked with teaching how to solve creative mathematical problems. These problems traditionally include tasks of high educational, developmental and diagnostic value. Teaching how to solve such problems allows training graduates with the highest level of the subject knowledge. The problem under study is relevant due to the need to form students' ability to solve problems with parameters in order to achieve high results in mathematics, intellectual and personal growth. The problem has a new meaning during temporary distance learning, emphasizing the need to create educational and methodological materials that allow students to organize self-education on complex topics in mathematics. The purpose of this research is to develop and describe a teaching methodology for solving problems with parameters based on the allocation of basic (key) problems. The authors have developed a methodology for constructing a system of tasks based on systematizing the theoretical and task material, highlighting students' basic knowledge and skills, describing the intra-and inter-subject connections of the topic "Equation of a circle in tasks with parameters". This methodology includes a typology of problems with parameters containing the equation of a circle, a substantiated description of the system of basic (key) problems, the role and content of the propaedeutic stage of teaching. The materials have passed multi-stage approbation and shown their consistency in achieving high results in mathematics. They can help teachers to prepare lessons and extracurricular activities, authors to provide teaching aids for students and teachers, and serve as the basis for a special course for students of pedagogical universities.
The issue under study is urgent today because there is a necessity for students to develop skills in working with the mathematical problem at the final stage of its solution in order to get excellent results while learning geometry and when encouraging intellectual and personal development. The aim of the research is to develop the theory and methodology of the final stage of working with planimetric problems as a means of improving the quality of geometry knowledge of schoolchildren. The key research method of the issue is match making between the components of the final stage of working with the mathematical problem and their corresponding operations. The research has resulted in defining the structure of the final stage of working with mathematical problems. It allowed to perform a certain set of operations composing the skill of working with the mathematical problem at the final stage of its solution. The article shows the technique for composing special tasks in order to form operations corresponding to the final stage of working with mathematical problems. It is proved that students' ability to carry out the above described stage of solving a mathematical problem helps them to get excellent results while learning geometry. The author's technique of teaching students to work with the mathematical problem at the final stage of its solution, proposed in the article, can be used by mathematical teachers in their practical work, by the authors of resource books for students and teachers and by students of pedagogical universities while doing their special courses.Keywords: mathematical problem, final stage of working with the mathematical problem, technique, results of teaching geometry, system of tasks Zelenina et al. / Improving Geometry Knowledge in School Students 2 / 13and research skills and creative abilities. Hence, the final stage of solving the mathematical problem is of key importance because its implementation involves the acquisition, revision, systematization and compilation of the information learnt and the discovery of new knowledge by school students (Galushkin, 2018;Kryukova et al., 2017;Kvon et al., 2018;Mutavchi et al., 2018;Potapova et al., 2018). Various aspects of using the final stage of solving the problem in teaching mathematics are widely discussed in the scientific and methodological literature, the works of famous mathematicians, methodologists and teachers. Despite different aspects of the papers analyzed in the present research, they all have a common thesis, namely: the final stage of solving the mathematical problem is a necessary and significant part of the solution and has considerable potential for teaching, developing and educating students and for improving Mathematics teaching. At the same time studying the experience of mathematical teachers shows that the capacities of the final stage of solving the mathematical problem are not fully used in school teaching. Many teachers pay no attention to this stage or think that it is just enough to obtain the answer to the problem (A...
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