Abstract
C. Gattegno proposed a working field of mathematics education on the basis of a notion which he called a mathematical situation, and which is consisted of the following a) and b). a) If the students are put under a certain mathematical situation, they can make mathematical propositions consecutively by their own effort. b) Under such circumstances, all what the teacher have to do is just to give information about how to extend the thinking of the students. His proposal moved H. Freudenthal to propound the the following problem. "Make scientific research of a teaching-learning process where the students can make mathematical propositions and their proofs by their own effort. "But C. Gattegno did not show any concrete way of constructing the mathematical situation. And H. Freudenthal did not give any kind of the answer to his problem either. Therefore, the principle of designing the above teaching-learning process is not been established yet. The main purpose of this paper is to give an answer to the problem propounded by H. Freudenthal in the geometry learning.
