The purpose of this article is to invite all those who are interested in the subject, experts and the general public at large, to answer the question: ``is a modernization of the school mathematics curriculum necessary, and if it is, what kind of modernization is to be implemented". We shall deal with subjects like the problems of curriculum's content, the bifurcation of the process during the change of its parameter, fractals and chaos, the position of computational mathematics in the curriculum, and particularly the possibility of laboratory work in mathematics. We think that the most important science, which investigates and establishes principles and methods of teaching mathematics, needs a serious conceptual reconstruction. Particular attention will be paid to a new ``area" of mathematics, the so-called ``mathematics of entrance exams".