• School Mathematics
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• v.18 no.1
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• pp.215-229
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• 2016

This study aims to identify Archimedes' idea used while proving proposition 23 in 'Quadrature of the Parabola' and to provide an alternative way for finding the sum of geometric series without applying the concept of limit by extending the idea though metaphor. This metaphorical model is characterized as static and thus can be complimentary to the dynamic aspect of limit concept adopted in Korean high school mathematics textbooks. In addition, middle school students can understand $0.999{\cdots}=1$ with this model in a structural way differently from the operative one suggested in Korean middle school mathematics textbooks. In this respect, I argue that the metaphorical model can be an useful educational tool for Korean secondary students to overcome epistemological obstacles inherent in the concepts of infinity and limit by making it possible to transfer from geometrical context to algebraic context.

• Journal for History of Mathematics
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• v.21 no.2
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• pp.19-38
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• 2008

This study aims to investigate the evolution of calculating the volume of a sphere in eastern and western mathematical history. In western case, Archimedes', Cavalieri's and Kepler's approaches, and in eastern case, Nine Chapters';, Liu Hui's and Zus' approaches are worthy of noting. The common idea of most of these approaches is the infinitesimal concept corresponding to Cavalieri's or Liu-Zu's principle which would developed to the basic idea of Calculus. So this study proposes an alternative to organization of math-textbooks or instructional procedures for teaching the volume of a sphere based on the principle.