2004 Fiscal Year Final Research Report Summary
Semiotic Analysis of Generalization from the Viewpoint of Metacognition : Research on Generalization Process of Division with Fractions and Division with Decimals
| Project/Area Number |
14580196
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Science education
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| Research Institution | HIROSHIMA UNIVERSITY |
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Principal Investigator |
IWASAKI Hideki Hiroshima University, Graduate School of Education, Professor, 大学院・教育学研究科, 教授 (50116539)
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| Co-Investigator(Kenkyū-buntansha) |
YAMAGUCHI Takeshi Fukuoka University of Education, Faculty of Education, Associate Professor, 教育学部, 助教授 (60239895)
UEDA Atsumi Hiroshima University, Graduate School of Education, Associate Professor, 大学院・教育学研究科, 助教授 (50168621)
BABA Takuya Hiroshima University, Graduate School for International Development and Cooperation, Associate Professor, 大学院・国際協力研究科, 助教授 (00335720)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Dorfler's Generalization Model / Separation Model / Expanded Metacognitive Model / Intensional Generalization / Extensional Generalization / Division with Decimals / Division with Fractions |
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| Research Abstract |
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
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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
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