Abstract
In the paper "A Study of Number System in School Mathematics (III)", I defineed the framework of systematic understanding of numbers. The purpose of this research is to reconsider this framework, and to investigate systematic understanding of numbers under the framework. I considered following framework of the systematic understanding of numbers. 1. Understanding the structure of a set of numbers ・Understanding a calculation rule ・Understanding a definition of numbers, and obtaining a image of numbers ・Understanding the operation in a set of numbers ・Understanding the linear order relation in a set of numbers, and understanding the density of a set of numbers 2. Understanding the density of a set of numbers 1 and 2 are complementary rather than independent. Under this framework, I investigated the degree of achievement of systematic understanding of numbers. And I analyzed how the understanding of the structure of set of numbers influenced the Understanding of a relation between sets of numbers by discriminant analysis. As a result, the following was clarified. Understanding the closure properties of set if numbers and the density of a set of numbers are effective to the understanding of the density of a set of numbers. In response to this result, I introduced Concrete teching methods that focus on the closure properties and the density of a set of numbers, and considered these teaching methods. A future subject is to consider systematic instruction of the number in school mathematics on the basis of these results of an investigation.
