The learning and understanding of mathematical concepts can be facilitated through detailed textual or oral explanation. Here explanation has two different meanings: Explanation as didactical explication of mathematical ideas or explanation as explication of conditions under which some people develope mathematical ideas. The main goal of mathematics instruction is two delineate explanations in such a way that students dont have to make additional inferences in producing a coherent text base. To construct such explanations the application of the theory of text comprehension of Kintsch is being applied to the verbalization of mathematical concepts. The core of real-science explanations are hypotheses. Empirical science has to test such hypotheses in prospective studies in a way that eliminates alternative explanations. After that the contribution of different research methods to scientific progress is discussed. Four different research methods were differentiated: (1) Diagnostic / descriptiv research of learning behavoirs, attitudes and strategies, (2) research of mathematics instruction (teaching and student behavior) (3) experimental research as prospective research and (4) developmental research. It is concluded that all four types of research have their own goals and it seems unfortunate to prefer one type at expense of others. © 1997 GDM - Gesellschaft für Didaktik der Mathematik.