Research on Development and Utilization of Instrument of Potential Ability for Learning Mathematics

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First, we developed the final version of the reliable instruments of potential ability for learning mathematics (PALM) for 4th graders and 8th graders. Second, in order to utilize the instruments, we developed the method to analyze pupils' mathematical learning ability in terms of "the divided attainment (attainment÷potential ability)" and "the subtracted attainment (attainment-potential ability)". Third, as a result of analysis of the relationship between potential ability and mathematical attainment, it was found out that the PALM components of "pattern recognition", "manipulation", and"flexible thinking" correlate relative-strongly with potential ability while"logical reasoning" correlates relative-weakly with potential ability. Fourth, we showed that pupils' potential ability can be improved through teaching some materials for potential ability in schools, and that the improvement of pupils' potential ability has a positive effect on improving their mathematical thinking ability.