EXPLORING DIVERSE PROBLEM-SOLVING APPROACHES AS A CATALYST FOR STUDENTS' PERSONAL GROWTH

Abstract

Our rapidly changing world demands that we adapt quickly to new circumstances and effectively solve problems. Contemporary realities present us with challenges that require flexible thinking and the ability to view problems from multiple perspectives. From an early age, it is crucial to cultivate in students the skills of seeking alternative solutions, formulating hypotheses, and constructing logical arguments. Solving geometric problems using various methods is an effective way to achieve this goal. In this process, students learn to analyze conditions, propose hypotheses, and justify their conclusions, which are essential skills for any field of activity. Each new approach not only reveals new facets of the problem but also contributes to a deep understanding of mathematical concepts and develops critical thinking. In this article, the author shares their experience working with mathematics-oriented classes, specifically highlighting the methodological aspects of working with students using the example of finding a solution to a geometric problem. The article examines one stereometric problem and presents five different ways to solve it. Each new method is based on the use of knowledge from various sections of geometry, including finding the angle between planes by definition, using the theorem of three perpendiculars, applying the method of orthogonal projections, using the cosine theorem for a trihedral angle and the coordinate method; while using trigonometric ratios, elements of vector algebra, the equation of a plane in segments, and specific cases of the plane's position, in particular, the conditions of parallelism of the plane to the coordinate axis, the scalar product of vectors; the method of areas. The article notes which basic knowledge underlies each solution method, including the Pythagorean theorem, the cosine theorem, the similarity of triangles, and the conditions of perpendicularity of a straight line and a plane. Particular attention is paid to studying the mutual arrangement of points on a segment; an initial condition is selected to obtain a tabular value of the angle between planes; all solution methods contain a complete theoretical justification and necessary calculations. The article emphasizes the importance of teaching students to seek different approaches to solving geometric problems. Such an approach not only deepens the understanding of geometric concepts but also contributes to the all-round development of their logical and critical abilities, preparing them for the successful solution of complex mathematical problems. This skill is key to successfully applying mathematical knowledge in future life and professional activities.