Competency-based approach in teaching mathematical disciplines: modern methods of assessing learning outcomes using the example of tasks in analytical geometry

Abstract

Abstract: The purpose of the article is to comprehensively substantiate approaches to the implementation of the competency paradigm in teaching mathematical disciplines in pedagogical institutions of higher education and to determine the potential of second-order curves and surfaces as a key means of forming professional competencies of future teachers. The research methods include the analysis of current scientific publications, comparison of modern models of assessing learning outcomes, pedagogical modeling and development of competency-oriented tasks integrated with analytical geometry, mathematical modeling and interdisciplinary connections. The results outline a systematic approach to designing educational tasks that contribute to the development of spatial imagination, logical thinking, research skills and the ability to apply mathematical methods in practical situations. Examples of tasks based on the properties of ellipses, hyperbolas and paraboloids are given, which reflect real professional contexts: trajectory analysis, modeling of optical systems, research of technical structures. It is shown that such tasks enhance the applied nature of mathematical training and ensure a high level of student involvement. The conclusions emphasize the importance of the transition from reproductive methods to activity-based methods focused on analysis, research and interpretation of models. The need to create a flexible assessment system that reflects the ability of students to solve complex problems, justify the choice of methods and critically analyze the results obtained is substantiated. It is concluded that the use of second-order curves and surfaces is an effective means of forming integral, general and professional competencies of future teachers and meets the requirements of modern educational standards.