A Study on Teaching“Transformation on Plane” in Study of“Complex Plane”: From the View Point of APOS Theory

Abstract

　　At present, “matrix, linear transformation” is virtually not taught in Japanese upper secondary schools, and “complex plane” is the only unit where the concept of transformation can be involved, while it is pointed out by not a few mathematicians that the concept of transformation is important.  In fact, transformation (on plane) is also taught in junior secondary schools through the study of “motion of figures”, and is, we consider, a fundamental object in mathematics as well as function.  If so, is it possible to teach the concept of transformation sufficiently in study of “complex plane”? And what conception of transformation should be obtained in the study of “complex plane”?  

　　We considered the concept and the conception of transformation from the view point of APOS theory, and identified its conception which should be studied through the study of “complex plane” as a framework.  In APOS theory, mental structures, which are necessary to learn a specific mathematical concept, are studied. They consist of Action, Process, Object and Schema structures.  Referring to this theory, we identified in detail the proper conceptions of each stage of APOS in the study of transformation, and in particular we claimed that conceptions up to the Object conceptions of transformation should be acquired.  Also we surveyed the textbooks from the previous framework and pointed out that there exist some problems in constructing Object conceptions of transformation in the current teaching.  Then, we proposed a genetic decomposition, in which students could construct the required concept of transformation.  A genetic decomposition　is a hypothetical model in APOS theory that describes the mental structures and mechanisms that a student might need to construct in order to learn a specific mathematical concept.  Finally, we proposed teaching materials of mathematical activities which prompt students to encapsulate the Process conception of transformation into the Object conception of transformation.