The purpose of this study is to develop the instruments of measuring pupils' potential ability for mathematics learning. Through the pilot tests and discussion, we elaborated an instrument composed of 20 problems which can be categorized into five components: logical reasoning, pattern recognition, manipulation, flexible thinking and application. By using this instrument, we investigated the potential ability and achievement of lower secondary school students: 73 1^<st> graders (A school), 62 1^<st> graders (B school), 73 2^<nd> graders (A school) and 59 2^<nd> graders (B school). As a result, we could find the following points. 1) The average of potential ability of different school students were almost same. This point suggests the validity of this instrument. 2) The 2^<nd> graders' potential ability was remarkably higher than that of 1^<st> graders of the same schools. This point suggests that potential ability can be advanced by their learning. The advance about the components of logical reasoning and application was more remarkable than that of flexible thinking. 3) The average of achievement of different school students were very different. Because the difference of potential ability was not so remarkable, we could grasp the difference of the effect of mathematics teaching. 4) The difference (achievement-otential ability) and the proportion (achievement÷potential ability) can be introduced as an index of the effect of mathematics reaching.
