Aspects of the Nature and State of Research in Mathematics Education

This paper offers an outline and a characterisation of the didactics of mathematics, alias the science of mathematics education, as a scientific and scholarly discipline, and discusses why its endeavours should be of interest to research mathematicians (and other mathematics professionals). It further presents and discusses a number of major, rather aggregate findings in the discipline, including the astonishing complexity of mathematical learning, the key role of domain specificity, obstacles produced by the process-object duality, students' alienation from proof and proving, and the marvels and pitfalls of information technology in mathematics education.This revised version was published online in September 2005 with corrections to the Cover Date.     

Applied mathematical problem solving,modelling, applications,and links to other subjects — State,trends and issues in mathematics instruction

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 (applied) 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.Written version of a Survey Lecture given jointly at the Sixth International Congress on Mathematical Education (ICME-6, Budapest 1988). A condensation was published in a volume with contributions from the ICME-6 Theme Groups on Problem Solving, Modelling and Applications and on Mathematics and Other Subjects (Blum/Niss/Huntley 1989, pp. 1–21).