Abstract: In the paper, the results of a study into a problem of cognitive barriers (difficulties) related to the third component of a pedagogical triad “how to teach what to teach to how to be learned study” are reported. At the first stage, methodical approaches to academic performance rating for students learning mathematical and natural-science disciplines were proposed. At the following stage, the cognitive barriers of students arising in the course of studying the specified disciplines were established. The results, obtained during realization of the above two stages were used as a basis for the development of methodology for teaching the discipline “Concept of modern natural sciences”, which is intended for the students of humanitarian and economic profiles. The core of the methodology for teaching the mentioned discipline is a theoretical course, guided by the fundamental ideas of physics, which have been formulated conceptually. Also, the content of a special course entitled “Physical foundations of new composite materials” is disclosed. The special course is intended for senior students of the technical universities studying in the directions training technical officers of industrial productions. The purpose of this course is to give to students the basic knowledge on the physical phenomena of material world of nano – and microscales, which are used for design of the advanced materials.