An accurate formation of the concept of number related to quantity should precede any other learning in mathematics. Taking the following three steps is essential to the formation of the concept; (1) making various sets based on various properties, (2)eliminating all the properties except number pieces and making supersets based on it, (3)comparing and estimating the size of several supersets. In this research, the authors constructed a learning course that meets (1) and (2) as preconditions for (3). In the construction of the course, we gave careful consideration to the identification of the laws for the formation of the three concepts (set making, set re-making within hierarchy, set re-making between hierarchy) and to making a systematic combination of three factors (term-manipulation, experience of scientist, experience of everyday life). We taught 13 classes to first graders immediately after the entrance into public elementary school. Whereas in the pretest the students were unable to solve almost any of the problems, in the post-test all of them were able to give the correct answers to all the problems within several difficulty levels. Many spontaneous hypothesis-verification activities also occurred, and we confirmed the formation of cardinal number concept in students at quite an early stage.
Magara (1993) showed that many undergraduates had the misconception that in the Edo period the Tokugawas shogunate collected a land tax from daimyos (feudal rulers). This study dealt with such a misconception. In the textbooks used in high school, the following content is described. "The shogunate made daimyo give up a certain amount of rice as tax on their territories during 9 years in the Edo period. This system is called Agemai-no-sei" (We call the content of the first sentence proposition A.) Proposition A implies that the shogunate did not collected a land tax from daimyos for the rest of the 9 years in the Edo period (We call this content proposition B.) In experiment 1 (N=62), it was suggested that subjects who abandoned the misconception could transform proposition A into proposition B. In experiment 2 (N=34), we taught subjects that proposition A could be transformed into proposition B. As a result, most of them could abandon the misconception. Those results were discussed from the standpoint of operational thinking proposed by Kudo (2010).