How to teach mathematics so that students truly understand it

Creating favorable conditions for students to master mathematical content is one of the primary tasks of teaching mathematics in secondary school. Psychological studies have shown that comprehension becomes possible when new information becomes part of the student's subjective experience and when connections can be made between aspects of the new information and various ways of representing it. The act of understanding includes the ability to construct a functional situational model, which therefore becomes an important means for solving mathematical problems. This article describes a holistic approach to teaching mathematics aimed at solving the basic tasks of such instruction—particularly, the problem of students’ comprehension. Essentially, this approach consists of: (1) establishing connections between the properties of a concept—meaning, value, and term; (2) taking into account the components of students’ subjective experience (value-emotional, contextual, operational, and communicative) in the educational process; (3) integrating various forms of representing mathematical information; and (4) identifying and using the interactions between real, perceptual, and conceptual (scientific) mathematical spaces through the use of modeling real-life situations. Practical approaches to using these foundational principles are described. Identifying how the proposed techniques and methods influence the understanding of material used during the teaching of mathematics was the principal goal of the research. The study included an experiment involving 256 general secondary school students: 142 in the experimental group and 114 in the control group. All students had the same level of mathematical achievement and were following the same curriculum. During the experiment, comprehension was assessed for two topics—“Derivative” and “Trigonometric Functions.” Students in the control group were taught using traditional methods, whereas students in the experimental group were taught using the holistic approach: under the teacher’s guidance, they completed tasks aimed at establishing connections between characteristics of concepts and translating information from one form of representation to another.