Igor V. Kiselnikov

The article examines a topical problem of teaching methods in mathematics in basic school – identification and analysis of persistent students’ errors in solving plot-based mathematical problems. The aim of the article is to study the errors of students in solving plot-based tasks of the main state examination in mathematics in basic school and, on this basis, to identify persistent errors of basic school students and search for ways to prevent and correct them in further education. Theoretical and empirical methods are used: analysis, synthesis, generalization, expert assessments. The author notes the influence of successful solution of plot-based tasks on the development of students' analytical thinking. Modern scientific and pedagogical literature devoted to the problem of teaching to solve plot-based mathematical tasks in basic school is analyzed. A review of domestic and foreign sources is provided. Plot-based tasks presented in the control and measuring materials of the main state examination are characterized. The article describes a statistical tool that allows analyzing the distribution of exam participants’ answers to various types of tasks: a “fan” (range) of exam participants’ answers, its functions, specifics of its construction, and an example of its application. The article specifies the main persistent errors in solving plot-based tasks in basic school: incorrect understanding of the problem statement, incorrect choice of solution method, calculation errors, and inability to check the correctness of the solution. The article specifies methodological features of implementing an estimate when solving plot-based tasks. The article suggests effective ways to overcome errors in understanding the conditions of a plot-based task: analyzing and structuring information, reading the task conditions aloud, writing down key data, using examples and analogies, a step-by-step solution plan, practice and repetition, and feedback from the teacher. Calculation errors are indicated as one of the most common problems in solving plot-based mathematical tasks. The article reflects strategies for minimizing calculation errors: correct organization of calculations, checking intermediate results, double checking the final result, using inverse operations, practice and training, concentration and attention, using technology, training, and consultations. The paper also specifies the approaches to finding optimal methods for solving tasks. The theoretical significance of the study consists in the development of the concept of "persistent errors in solving tasks". The novelty of the study is expressed in the identification of specific persistent errors in solving plot-based tasks and methodological approaches to eliminating errors.

The article is devoted to topical issues related to the development of a methodology for teaching mathematics, aimed at stimulating the educational and cognitive activity of students. There are the contradictions, the resolution of which can contribute to an increase in the activity of students in the process of learning mathematics. The purpose of the article is to develop the main provisions of the methodology for teaching irrational equations, aimed at stimulating the cognitive activity of students. The author uses modern works on the problem of stimulating the cognitive activity of students in the following directions: organization of independent cognitive activity; development and organization of creative research cognitive activity; stimulating of schoolchildren in learning; formation of the cognitive interests of schoolchildren in the general educational aspect; search for rational methods and individual techniques of cognitive activity and teaching students to use them; particular methodological procedures contributing to the stimulation of cognitive activity. The leading approach is the process approach to learning, ensuring that students understand the mathematical content. The causes of reduced attention and understanding of the studied material are revealed, and the methodology for teaching irrational equations at school is built on this basis. The developed teaching methodology meets the requirements of modern federal state educational standards. The main result is the selection and substantiation of the conditions for the stimulation of students to study irrational equations in the secondary school. The theoretical significance of the article lies in the development of methodological approaches to the study of irrational equations in secondary school. The results of the study can serve as a basis for writing other scientific papers on a given topic. The practical significance is due to the fact that the results of the study can be used for educational purposes, as well as the possibility of applying and implementing the program of extracurricular activities in mathematics "Mathematical Workshop on Solving Irrational Equations".