DIFFERENT WAYS OF SOLVING PROBLEMS AS A TOOL FOR FORMING FLEXIBILITY OF THINKING (ON THE EXAMPLE OF A GEOMETRIC PROBLEM)

Abstract

The modern world is full of many problems that require quick action and several ways to successfully solve them. It is important to develop in students the skills of searching for alternative options for finding the correct answer, making and validating certain assumptions, and building fragmentary theoretical generalisations. The development of these skills is especially facilitated by solving geometric problems in different ways, because each new way reveals the problem from a different angle, shows its deep properties, encourages students to analyse the condition of the problem, find possible ways to solve it, while applying knowledge from different areas of mathematics. In addition, solving a problem in several ways increases the guarantee of the correctness of the answer. In the article, the authors share their practice of working in mathematics classes, in particular, they reveal the methodological aspects of working with students on the example of finding a solution to a geometric problem - the paper considers one geometric problem and provides ten different ways to solve it. Each new method is based on the use of knowledge from different areas of mathematics, including: trigonometric relations (special attention is paid to the use of the basic trigonometric identity, the formulas of cosine and sine of the sum), geometric transformations (shown as the completion of similar triangles and axial symmetry); different types of equations and inequalities; applied the method of areas, the method of coordinates and the method of estimation; described the features of using each of them and the feasibility of such use; noted what basic knowledge underlies each of the methods of solving, including the theorems of cosines, sines, Pythagoras, the bisector formula, its property and others; special attention is paid to the study of the type of a given triangle, all the steps of the study are described in detail and it is substantiated which triangles are possible under the conditions of a given problem and which do not satisfy it. A generalised version of the original problem is also considered, its features are commented on, and it is indicated which methods among those given in the first problem can be used to solve the second. The article emphasises the importance and feasibility of finding different ways of solving geometric problems in the course of studying geometry in mathematics classes and notes the impact of such a practice of organising the educational process on the development of logical and critical thinking and improving the mathematical education of students.