Prospects in Development of Teaching Materials as Mathematics Education Research:For Mathematical Proof Instruction throughout Six Years of Secondary Education

Abstract

　　This paper simply aims at development of teaching materials on mathematical proof based upon the theory of substantial learning environment proposed by Wittmann.  We think proof attitude and proving skills will become a core for new literacy in future days after modern times.  Mathematics education has to play a big role to realize it.

　　So far mathematics education research, however, paid much attention to the entrance of school mathematics, i.e. arithmetic teaching in Japan.  She missed the exit of school mathematics, especially high school mathematics.  In other words the issue has not been problematized academically on secondary mathematics throughout six years although almost all of students go up to senior high school after graduation of compulsory junior level.  Oncoming mathematics education research, therefore, should focus on the exit of school mathematics from the various angles as well as the entrance.  

　　On the other hand, senior high school mathematics still seems to be a cluster of teaching materials or collection of new instructional resources.  They are not always examined and surveyed carefully and academically.  They need to be scrutinized another angle besides mathematics because mathematics in senior high school stands at the exit of whole school mathematics.  That is to say institutional angle, cultural angle, and societal angle are inevitable.  Many students more than half launch out society with relevant citizenship after graduation of senior high school.  Moreover they must engage in lifelong learning and highly advanced information society whether they like it or not.  

　　In this paper, “Sylvester’s  partition theorem” is chosen as a throughout proof teaching material.  She will be prepared and considered in terms of substantial learning environments as an authentic teaching material first.  She will be analyzed by means of the core of mathematics education as a scientific discipline proposed by Wittmann second.  As consequences, some methodological techniques for developing instructional materials are identified.  The techniques imply not only developmental methods in mathematics education research, but also new direction of training-system for mathematics teachers in educational practice.