理数探究における数学的探究の可能性 ―「ラングレーの問題の一般化」の数学的探究の事例分析を通して ―

抄録

　　The aim of this paper is to clarify potential of a mathematical inquiry in “Risutankyu” and which elements of the inquiry work well in order to progress forward.  To achieve the purpose, the mathematical intuiry into “generalization of Langley’s problem” which was conducted in “Kadaikenkyu”, is analyzed by “Herbartian schema” which is an analytical tool within “anthropological theory of didactic” and can clarify resources such as established answers (A^♢_i), derived questions(Q_k), works(W_j), and data(D_l), used in inquiry.  Three categories (topogenesis, chronogenesis, and mesogenesis) is also used as a analytical tool for the inquiry.

　　“Kadaikenkyu” is the inquiry for 2^nd grade students who are in a science course and can choose their fields (physics, chemistry, biology, geology and mathematics).  The mathematics group is composed of 5 students (all of them chose mathematics as a first choice), who decide “generalization of Langley’s problem” as a theme of inquiry. “Langley’s problem” is an elementary geometry problem which was presented by Edward Mann Langley in 1992.  

　　Topogenesis of the inquiry is analyzed thtough the topos of each student.  The elements regarding to topogenesis are dialectic of persons and institutions and the didactics contract for inquiry.  

　　Chronogenesis of the inquiry is analyzed through the transitions of phases.  The elements are the condition related to the initial question and dialectic of dissemination and reception.   

　　Mesogenesis is analyzed through the evolution of the didactic milieu.  The element is the condition about persistence of the area where the mathematics group consider the own answer.  

　　The contributions of this study are as follows: (i) analyzing the mathematical inquiry conducted in “Risutankyu” using “Herbartian schema”, (ii) characterizing the mathematical inquiry by three categories (topogenesis, chronogenesis, and mesogenesis).