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We will clarify the range and judgment of the number of children who can process the Estimated the number.We conducted a survey on children of grade 1st to 3rd by Elementary school.Result of investigation, Grade 1st and 2nd children are "Overall view". But, Grade 3rd children are "Decomposition view".We must consider the relationship between guidance of Numerical calculation and guidance of Number calculation.

In the last paper, we considered the lesson of the school mathematics about the improvent of the lesson by using "Model Plate". Until now, We were considering the teachig learning processes to validate the "Model Plate". On this paper, we present a case study that the "Model Plate" has the function of improving of the lesson. In this case study, the experienced teacher revised lesson plan referring to the "Model Plate", After all, in the revised lesson of a case study, students think in a Developmental Way and present many developmental ideas.

This research is examining the meaning of 'Developing Way'. Especally, in this paper, we consider it from the purpose of structuring the math. In conclusion, this meaning of mathematics to make students consider in 'Developing Way' is named to be 'Structured Development' which was classified into "Integration" "Generalization" and "Concise and Clear". Finally, we considered the teaching and learning processes of the topic of Area of a Sector from the structured development.

The viewpoint of this research is the teaching of calculation that raises the awareness of relative to the number.We have developed teaching materials to spread the idea of Complement Number by the ways of addition of two one-digit numbers and subtraction as the inverse operation.We verified the effect from the actual lessons. The idea of Complement Number improves the the efficiency and certainty.The idea of Complement Number has a cultivated value, a practical value and a systematic value.To be specific, We have decided to teach the Unit Number of 20 and the Unit Number of 100.We was the development of the number models. When carrying out the verification lessons, more than 90% of all children had been able to understand. However, it was found to be difficult to spread the idea of Complement Number. In addition, it was found that the ways of the Unit Number of 10 and the ways of the Unit Number of 100 were needed teaching same as seen.

This study will consider the significance to develop teaching materials Complement. In addition to clarify the perspective to develop teaching materials based on the problems of the past. I have considered previous research on Number-sense and Mental-arithmetic. I was able to find 4 points of Number-senses problem ‘‘The difficulty of defining’’ ‘‘Consider the system of teaching contents’’ ‘‘Teaching associate Number-sense and calculation’’ ‘‘Validation of teaching materials the theoretical and practical’’. In addition, mental arithmetic was not appreciated enough in research and teaching case in the past. However, foresight about the teaching calculation is valid the presence of number is growing significantly now. I have found the need for develop teaching materials ‘‘Complement’’ that takes into account the lessons learned from the ‘‘Anzan-ronso’’ era. It is important that we consider 4 points ‘‘Degree of short-term memory loads’’ ‘‘Development of mathematical model for teaching Complement’’ ‘‘The teaching system of Number and Calculation’’ ‘‘Method of teaching’’.