Fundamental study on lesson design on the basis of the roles of representation in mathematical problem solving

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The aims of this study was to clarify the patterns of construction/transformation of (problem) representation in mathematical problem solving, to investigate its roles in progress of problem-solving process, and to consider the methodology of lesson design on the basis of the roles. The main results were to identify five types of wide-purpose representations for mathematical problem solving, to investigate three constructional/transformational patterns of problem representation in mathematical problem solving (i.e.; construction/reconstruction, abstraction/concretization, shift), and to analyze the concrete roles of “abstraction/concretization” process of problem representation.Furthermore, a methodology of lesson design on the basis of the roles of abstraction process was considered.