Mathematics education as theoretical knowledge

Didactics of mathematics (DM) is the theoretical part of our knowledge in mathe- matics education. Its connections with the mathematical sciences and with school mathematics determine the independent character of DM as a scientific discipline. Features of school mathematics such as its unique aims, the highly abstract nature and hierarchical construction of the material to be studied, and the varied kinds of educational activities lead to its specific character, situating it among the school subjects and making accepted theoretical conclusions exclusively applicable to mathematics education. The social character of DM generates the approaches to constructing categories within the discipline (inexact and "inaccurate" conceptions, exclusion principles in formulating absolute statements, plausible reasoning, diversity of proofs) and promotes the selection of adequate methods of research that are not typical of positivistic science (expert review, discussion, pedagogical experiment). The applied character of DM determines an appropriate methodology of research and efficient ways of overcoming contradictions. The concrete practice of education (partly in the form of experimental tests) gives teachers an opportunity to use "inaccurate" empirical methods, reasonable considerations, intuitive choice, and so forth. Some practical advice is given for discussing and conducting research projects.