With elementary school students who have learned the primary concepts of the four arithmetic operations as its subjects, this study has investigated in depth how schema and transformed schema are composed by recognition of the correct concepts and connection of concepts, that is to say, what schema learners form along with transformed schema with the primary concepts of the four arithmetic operations to understand the secondary concepts when power and mixed calculations are taken into contents. It has also investigated how the subjects use the schema they have formed for themselves and the transformed schema to approach problem solving, and how their composition of concepts and schema in problem solving ability achieve transformations. As a result, we can tell that the recognition of precise primary concepts and transformed schema work as important factors in the development from the primary to the secondary concepts. Here, we can also see learn that the formation of the schema created due to the connection among the primary concepts and the recognition of them and of the transformed schema play more important roles in the development toward the secondary concepts and the solution of arithmetic problems than any other factors.