In a programmatic discussion to establish mathematical education as a scientific discipline in the early 1970s in Europe, we could point out the academical construction rationale of Object-Method-Problem in mathematical education and the dialectical innovation perspective of Practice vs. Theory in it. It also follows that both are complementary to each other in research of mathematical education.
Based on the dialectical innovation perspective, it is reported that the research of mathematical education in Japan had begun with the reform movement of mathematical education and it has been institutionalized gradually for a scientific discipline after the modernization movement of it.
From the dialectical innovation perspective, Wittmann, E. introduces "Design Science" into Didactics of Mathematics. He succeeds in clarifying both structure and core of it. If, however, each component of the core is peculiar to Didactics of Mathematics, their peculiarity must be explained by "Design Science". But it has not had such capability because originally it is not a science for education. There must be educational idea or an awareness of issues which integrates each component of the core. We seek it in the transition between elementary education and secondary education. Based on this idea, each of them could be an issue in Mathematics Didactics in the proper meaning of these words.