An Epistemological Analysis on the Definite Integrals Using History of Mathematics: Toward the Design and Implementation of Inquiry-Based Learning

Abstract

　　The aim of this paper is to analyze the nature of the definite integrals using the history of mathematics. We investigated, especially, the raison d’être of the definite integrals with the concept of praxeology within the Anthropological Theory of the Didactic.  In this paper, we compared the types of tasks calculating the area or the integrals in the history of mathematics, and identified types of tasks which can only be applied by techniques or technologies in the praxeology as the raison d’être to calculate the area and the integrals.  This analysis reveals that one of the raison d’être of Fundamental Theory of calculus is to find the area bounded by transcendental curves, and that of trapezoidal rule and Taylor expansion is to calculate elliptic integrals and Fresnel integrals.  Through this analysis, we obtained a suggestion toward the design and implementation of inquiry-based learning.  We confirmed, in addition, that comparing the range of task types to which it can be applied technologies in the praxeology allows to analyze the raison d’être of mathematical knowledge.