Pedagogical science has always faced an issue of finding effective means to achieve the educational results of students. This problem is especially urgent today, when the world is changing rapidly, and those tools, that yesterday could keep pupils interested in studying mathematics and provided an opportunity to develop their mathematical abilities, quickly become out-of-date. Today, the search for a new toolkit that contributes to the development of students by means of mathematics, as well as mechanisms for incorporating it into the educational process, is highly relevant. Thus, the purpose of the article is to study the possibilities of using a new learning tool - puzzles in all their diversity in the process of teaching mathematics to children and teenagers, both in class and in extracurricular activities. The leading method here is the modeling of the methodological system of training in basic and additional mathematical education of children and teenagers, with the inclusion of a new didactic toolkit in it, which will contribute to the increase of students' interest in the subject, as well as to the development of certain mathematical abilities: logical thinking, abstraction, combining, operating with spatial images, critical thinking, mathematical memory. As a result of the research, the author of the article determined the place, opportunities and methodological aspects of including puzzles in the process of learning mathematics, both in the system of classical and nontraditional (creative) lessons of mathematics, and in the structure of extracurricular activities of students: mathematical circle, system of mathematical competitions, mathematical summer camp, etc. Practical use of this system makes it possible to reduce the disadvantage of tools deficiency for the development of mathematical abilities of students in the pedagogical practice, which gives us an opportunity to speak of high academic results in mathematical activity of children and teenagers.