Abstract
The purpose of this research is to develop instruments to measure pupils' potential ability for learning mathematics and to study theoretically and empirically the ways to utilize the instruments for improving mathematics education at elementary and secondary school levels. In the series of our study (Nakahara, 2002; Nakahara et al., 2008), we has elaborated the concept of "potential ability" and we constructed the theoretical framework for potential ability which is constituted of two aspects, "Mathematical Thinking" and "Mathematical Content". "Mathematical Thinking" is categorized into four components: logical reasoning, pattern recognition, manipulation, and flexible thinking and "Mathematical Content" is categorized into three components: number and quantity, shape and space, and function and relation. According to the theoretical framework, we developed the instruments to measure pupils' potential ability for learning mathematics (Nakahara et al., 2008). These instruments for 4th graders and for 8th graders have 20 items each, and the reliability of the instruments were validated. The purpose of this article is to investigate and consider the relationships between potential ability and mathematical attainment of 4th grade pupils. 4th graders (N=363) from three public elementary schools were administered the instrument of potential ability and mathematical attainment test. As a result, the following findings were mainly found out. (1) There were relatively strong correlation between potential ability and mathematical attainment. From the viewpoint of the "Mathematical Thinking" aspect, pattern recognition, manipulation, and flexible thinking correlate relatively strongly with potential ability, but logical reasoning correlate relatively weakly with potential ability. (2) By introducing "the subtracted attainment (attainment-potential ability)" and "the divided attainment (attainment÷potential ability)", we could characterize relatively the difference of the effect of teaching mathematics conducted in each school. (3) It was found that manipulation and logical reasoning components have relatively strong effect toward mathematical attainment and have a certain effect toward pattern recognition component.
