The proposed article substantiates an effective way to use the combined method of algorithmization and reasoned learning for reliable consolidation of theoretical knowledge and skills in mathematics and provides a specific example to demonstrate its effectiveness. The proposed methodological approach allows the student to organize and study independently more effectively, successfully engage in search, and thoroughly study a specific question of mathematics or a specific topic at their own discretion. In this regard, a modern mathematics teacher should know and possess modern technologies and methods of teaching mathematics, be able to effectively apply them in practice. The main methodological ideas proposed by the authors in this article can be successfully used to improve the General professional level of a student of any specialty.