The purpose of this paper is to sequentialize Mathematics-learning contents, and to explore teaching-learning model for mathematics, with on the basis of the theory of cognitive development and the period of conservation formation for children. The Specific topics are as follows: (1) Systemizing those theories of cognitive development which are related to Mathematics - learning for children. (2) Organizing a sequence of Mathematics-learning, on the basis of experimental research for the period of conservation formation for children. (3) Comparing the effects of 4 types of teaching-learning model, on the basis of inference activity and operational learning principle. ① Induction-operation(IO) ② Induction-explanation(IE) ③ Deduction-operation(DO) ④ Deduction-explanation(DE) The results of the subjects are as follows: (1) Cognitive development theory and Mathematics education. ① Cognitive development can be achieved by constant space and Mathematics knowledge is obtained by the interaction of experience and reason. ② The stages of cognitive development for children form a hierarchical system, its function has a continuity and acts orderly. Therefore we need to apply cognitive development for children to teach mathematics systematically and orderly. (2) Sequence of mathematical concepts. ① The learning effect of mathematical concepts occurs when this coincides with the period of conservation formation for children. ② Mathematics Curriculum of Elementary Schools in Korea matches with the experimental research about the period of Piaget`s conservation formation. (3) Exploration of a teaching-learning model for mathematics. ① Mathematics learning is to be centered on learning by experience such as observation, operation, experiment and actual measurement. ② Mathematical learning has better results in from inductional inference rather than deductional inference, and from operational inference rather than explanatory inference.