The article deals with the problems of higher education in Russia by the example of mathematics. Special attention is given to the fact that university applicants are unprepared and unmotivated to higher education, which leads to impossibility to implement the main educational task - teach them to study. The author arrives at a conclusion, that universities should start teaching maths with elementary mathematical operations, as some educators said: “In order to learn something, you have to know something already”. The serious teaching of students should be carried out at university from the very first courses. It seems to be impossible to give solid knowledge to students in a short time. “To learn something, you have to work hard”. A student is not able to do long calculating operations without any mistakes, so he needs “helpers” which might help him to solve the problem. Such “helpers” can include computer tutorials which are supposed to implement definite tasks. Distance education could be also very supportive. The study investigates working conditions for university teaching, analyses of such requirements, as working with students who pay fees, academic teachers’ activity and the process of teaching staff renewal.