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

对于第一个研究，我们引入了Gordon，W.的联结理论，从认知的角度阐明了外延泛化和内涵泛化之间的本质区别，尽管Dorfler在逻辑上区分了它们。这些术语的定义有以下谓词。前者是为了把熟悉的东西变得陌生，除了它的逻辑感放大了一个集合的范围。它通过使熟悉的事物变得陌生，为我们整合各种旧知识提供了新的切入点。另一方面，后者是为了让陌生的事物变得熟悉，除了它对物体属性的逻辑阐述。在Dorfler的泛化模型中，我们提出了既有外延泛化又有内涵泛化的“分离模型”，它是继“符号为客体”之后的两个认知过程的替代。造成这种差异的原因可以从分离模式的角度解释为“符号为物”后泛化的质的差异。我们设计了一种用于推广的分数除法的替代教学，可以诱导从算术到数学的过渡。本研究以六年级学生为对象，进行了以此为基础的教学实验。它不仅展示了替代性教学的有效性，而且展示了分离模式下的认知过程。研究三是从我们的分离模式出发，阐述了维特曼的教学单元理论。我们通过将他的理论纳入分离模型，在历时上对TU进行了补充。第四项研究是根据我们的分离模式，批判性地反思目前的负数减法教学方式，开发出一套算术向代数过渡的教材。

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.