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A Study on the Pedagogical Application of Omar Khayyam's Geometric Approaches to Cubic Equations  

Ban, Eun Seob (Heungdeok High School)
Shin, Jaehong (Korea National University of Education)
Lew, Hee Chan (Korea National University of Education)
Publication Information
School Mathematics / v.18, no.3, 2016 , pp. 589-609 More about this Journal
Abstract
In this study, researchers have modernly reinterpreted geometric solving of cubic equations presented by an arabic mathematician, Omar Khayyam in medieval age, and have considered the pedagogical significance of geometric solving of the cubic equations using two conic sections in terms of analytic geometry. These efforts allow to analyze educational application of mathematics instruction and provide useful pedagogical implications in school mathematics such as 'connecting algebra-geometry', 'induction-generalization' and 'connecting analogous problems via analogy' for the geometric approaches of cubic equations: $x^3+4x=32$, $x^3+ax=b$, $x^3=4x+32$ and $x^3=ax+b$. It could be possible to reciprocally convert between algebraic representations of cubic equations and geometric representations of conic sections, while geometrically approaching the cubic equations from a perspective of connecting algebra and geometry. Also, it could be treated how to generalize solution of cubic equation containing variables from geometric solution in which coefficients and constant terms are given under a perspective of induction-generalization. Finally, it could enable to provide students with some opportunities to adapt similar solving procedures or methods into the newly-given cubic equation with a perspective of connecting analogous problems via analogy.
Keywords
cubic equations; geometric solving of equation; mathematical connection; induction; generalization; analogy;
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