Faculty of Education , Fukui University, Fukui − shi , 910 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 mathemat 量 cal situation , they can make mathematica 正 propositions consecu . tively by their own effort . b)Under such circumstances , all what the teacher have to do is just to give information about how to extend

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... the thinking of the students . His proposal moved H . Freudenthal to propound the the following problem . " Make scientific research of a teaching − 1earni 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 construct − ing 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 .