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

元认知视角下的泛化符号学分析——分数除法和小数除法的泛化过程研究

基本信息

批准号：
14580196
负责人：
IWASAKI Hideki
金额：
$ 2.18万
依托单位：
HIROSHIMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

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

本研究的目的有以下两点。第一，从数学概括的观点出发，建立了教学轨迹的理论框架。第二个目标是分析分数除法的困难，针对第一个目标，我们在Dorfler的综合模型的基础上，建立了描述性和规范性的综合过程模型。尽管多夫勒在逻辑上对外延概括和内涵概括进行了区分，但本研究还是批判性地思考了他的理论，并从认知的角度阐明了外延概括和内涵概括之间的实质区别。在此基础上，我们提出了“分离模型”，将这两种概括放在Dorfler模型中的“符号作为对象”之后，并以“扩展元认知”作为补充。对于第二个目标，我们重点讨论了全国成绩中出现的分数除法的困难，并提出了一个新的目标，即“扩展元认知” ...更多信息教育部进行的调查。结果表明，虽然分数除法与小数除法具有相同的数学结构，但两者在比例推理上的成就有很大的差异。我们指出这种差异的原因可以用分离模型解释为概括的质的差异，即内涵概括和外延概括，在此基础上，我们设计了一种分数除法的替代教学。我们建议的第一点，是以“比较”的模式，而不是以比例的模式来表达。二是根据分数的性质或除法的规则，逻辑地推导出“倒乘”算法。本研究在此基础上对六年级学生进行了教学实验。它不仅向我们展示了替代教学的有效性，而且还展示了分离模式下的认知过程。少