高校数学における複素関数の考え方を引き出す授業の開発 ―代数学の基本定理の原理的理解を題材に―

抄録

　　The theory of complex functions is equipped with many beautiful aspects, which cannot be seen in the theory of real functions, and we have been interested in introducing such aspects of complex functions. However, such an approach has many difficulties.  In fact, though the current senior secondary mathematics includes study of complex number and complex plane, the focus of their study tends to be in algebraic treatment, rather than dynamic aspects of complex number and their operations, which deeply relates with complex functions.  There exist previous studies about this issue, which focuses on dynamic aspects by recognizing complex number operations as transformations in complex plane, which is virtually complex functions.  First, in this paper, we identify and clarify the view of complex functions that is possible and realistic to treat in senior secondary mathematics.  Complex functions involve transformations of 2-dimensional regions, which is difficult to seize for students, while it is not so difficult to consider images of finite 1-dimensional figures by a complex function.  Then we focus on considering how the image of 1-dimensional figure changes, when the original 1-dimensional figure changes from one considered figure to another: we think these considerations are essential in considering complex functions in secondary mathematics.  Secondary, it is important to show the rationale of this new view.  For this purpose, we developed the teaching material and teaching practices relating the proving principle of the fundamental algebraic theorem, which cannot be achieved by usual view but can be achieved by the new view of complex functions.  As the result, we could see the possibility of this practice to a certain degree and obtained some suggestions for further improvement.