Semiotic Analysis of Generalization from the Viewpoint of Metacognition : Research on Generalization Process of Division with Fractions and Division with Decimals

本研究的目的主要有以下两点。一是从数学泛化的角度构建教学轨迹的理论框架。针对第一个目标，在Dorfler推广模型的基础上，建立了推广过程的描述性和规范性模型。在这项研究中，我们批判性地考虑了他的理论，并从认知的角度阐明了外延概括和内涵概括之间的本质区别，尽管多夫勒在逻辑上区分了它们。由于我们的考虑，我们提出了分离模型，将这两种概括放在多夫勒模型中的符号对象之后，并辅之以扩展的元认知。针对第二个目标，我们重点讨论了国家Achie…中出现的分数除法的困难教育部进行了更多的变化调查。结果表明，分数除法与小数除法虽具有相同的比例推理数学结构，但两者的成绩有很大差异。我们指出，这种差异的原因可以从分离模型的角度解释为泛化的质的差异，即内涵泛化和外延泛化。在此分析的基础上，我们设计了分数除法的替代教学。我们建议的第一点是根据“比较”的图式而不是比例的图式来表达。二是根据分数的性质或除法规则，从逻辑上推导出“倒乘”算法。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。它不仅向我们展示了替代教学的有效性，也向我们展示了分离模式下的认知过程。较少

The objectives of this research are the following two points. The first is to develop a theoretical framework of teaching and learning trajectory from the viewpoint of mathematical generalization. The second is to analyze difficulties of division with fractions based on our framework.Regarding to the first objective, we developed descriptive and normative model of generalization process based on Dorfler's generalization model. In this research, we critically considered his theory and clarify the substantial difference between extensional and intensional generalization from the angle of cognition although Dorfler distinguished them logically. As a result of our consideration, we proposed "Separation model" which are put these two kinds of generalization just after "symbols as objects" in Dorfler's model and are complemented by "Expanded Metacognition".Regarding to the second objective, we focused on difficulties of division with fractions which appeared in a result of the National Achie … More vement Survey conducted by Ministry of Education. It showed that there was a big difference between the attainments on making an expression of division with fractions and that of division with decimals although both problems had the same mathematical structure with proportional reasoning. We pointed out that the cause of this difference could be explained as qualitative difference of generalization, i.e. intensional and extensional generalization, in terms of separation model.In addition to this analysis, we designed an alternative teaching of division with fractions. The first point of our proposal is to make an expression based on the schema of "comparison" instead of that of proportion. The second is to deduce the algorithm "invert and multiply" logically based on properties of fractions or rules of division. 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. Less