A Study on the Potential of “The Problem of Grid-points on Circle” as an Object of Mathematical Inquiry

Abstract

　　The aim of this paper is to investigate the potential of the mathematical question we call “the problem of grid-point on a circle” (specifically “How many integer solutions does the equation x² + y² = r² has?) as a mathematical inquiry theme. We investigated in detail the mathematical background of this question with an analytical tool, called Question and Answer map, which has been developed within the research community of the Anthropological Theory of the Didactic (ATD). The results of analysis show that inquiring this question leads to different mathematical ideas, such as Pythagorean numbers, Fermat’s theorem on sums of two squares, Jacobi’s two square theorem, and so on. In addition, through this analysis, we confirmed the power of Question and Answer map in the epistemological analysis of a given mathematical theme as well as the didactic analysis that implies the clues for teachers to carry our inquiry-based lessons.