The paper will consist of three parts. In part I we shall present some background
considerations which are necessary as a basis for what follows. We
shall try to clarify some basic concepts and notions, and we shall collect
the most important arguments (and related goals) in favour of problem solving,
modelling and applications to other subjects in mathematics instruction.
In the main part II we shall review the present state, recent trends, and
prospective lines of development, both in empirical or theoretical research
and in the practice of mathematics instruction and mathematics education,
concerning problem solving, modelling, applications and relations to other
subjects. In particular, we shall identify and discuss four major trends: a
widened spectrum of arguments, an increased globality, an increased unification,
and an extended use of computers. In the final part III we shall comment
upon some important issues and problems related to our topic.