Semiotic and cognitive research on social interaction in the process of mathematical generalization

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Regarding to the first research, we introduced "synectics" by Gordon, W. to clarify the substantial difference between extensional and intensional generalization from the angle of cognition although DOrfler distinguished them logically. The definition of those terms has the following predicates. The former is to make the familiar strange in addition to its logical sense of enlargement of the extent of a set. It gives us the new point for integration of various pieces of old knowledge by making the familiar strange as a consequence. On the other hand, the latter is to make the strange familiar in addition to its logical sense of elaboration of the properties of objects. We proposed "separation model" which has both extensional generalization and intensional generalization as the alternative of two cognitive processes just after "symbols as object" in Dorfler's generalization model.Regarding to the second research, we focused on difficulties of division with fractions. The cause of this difference could be explained as qualitative difference of generalization after "symbols as objects" in terms of separation model. We designed an alternative teaching of division with fractions for extensional generalization, which could induce the transition from arithmetic to mathematics. The teaching experiment based on it was practiced for the sixth graders in this research. It showed us not only the effectiveness of an alternative teaching but also the cognitive process in the separation model.The third research was to elaborate a theory of Wittmann's "Teaching Units (TU)" in terms of our separation model. We complemented TU diachronically by incorporating his theory into separation model. The fourth research was to consider current teaching way of "subtraction with negative numbers" critically in terms of our separation model to develop a teaching material for transition from arithmetic to algebra.