This article is devoted to the study of two main problems – whom and how to teach in a university. Theoretical analysis substantiated the thesis that the inefficiency of teaching students is connected directly with the lack of teachers’ knowledge about psychological peculiarities of modern student community. Accordingly, the research shows the contrariety between this age and forms of its appearance as one of students’ specifics. One of main consequences of those problems is the genesis and development of student infantilism, which has such essential characteristics: ambivalence, motivational stimulating force, behavioral form, protracted nature of infantilism and its extending, possibility of its overcoming and correction. The particular research set effective algorithm and the program of psychological correction of student’s infantilism while obtaining higher education. The second problem is devoted to the necessity of using teaching psycho didactical methods to create conditions for personal cognitive development. Theoretical analysis and empiric research identified the method system to develop cognitive process: for memory – enlarging didactic units as factors for systematizing learning materials and students’ knowledge; attention – improving individual controlled work of students; information representation – text manipulation and multimedia didactics; feelings – emotional and willful control, decision taking ability logic development; imagination – didactical games.