In Chinese Mathematics, Jia Xian(賈憲) introduced zengcheng kaifangfa(增乘開方法) to get approximations of solutions of polynomial equations which is a generalization of square roots and cube roots in Jiuzhang Suanshu. The synthetic divisions in zengcheng kaifangfa give rise to two concepts of Fan ji(fanji) and Yi ji(yiji) which were extensively used in Chosun Dynasty Mathematics. We first study their history in China and Chosun Dynasty and then investigate the historical fact that Chosun mathematicians Nam Byung Gil(Nam Byeong-gil) and Lee Sang Hyuk(Lee Sanghyeog) obtained the sufficient conditions for Fan ji and Yi ji for quadratic equations and proved them in the middle of 19th century.

This paper examines the gap between procedural fluency and conceptual understanding in mathematics learning through the lens of "hidden variables." In school mathematics, problem solving often prioritizes procedural execution under idealized assumptions with key variables suppressed; yet this practice can impede substantive understanding, especially in real-world and geometric contexts. Using a set of illustrative tasks-including origami puzzles, traffic-flow calculations, and counting images in mirrors-we investigate how implicitly embedded variables reshape learners' interpretations of a problem and the depth of understanding they attain. We argue that, rather than eliminating variables, introducing additional variables or increasing dimensionality can make the problem structure explicit and facilitate concept formation. Extending beyond notions of maxima and minima, we further analyze geometric ideas naturally characterized by saddle points, thereby probing mathematical mechanisms underlying concept development. These findings suggest pedagogical directions for fostering conceptual understanding beyond procedural competence and highlight the growing importance of creative problem comprehension in AI-rich learning contexts.