The Problems of Mathematical Reasoning on Teaching Geometry

Proof has two aspects in the from of thinking. Plato said that one is synthesis which goes asendently in cognition, and the other is analysis which goes descently. Methematical reasoning is the key concept of new teaching gemetry of grade 7. It is clarified when we compare a new text with old one and research into the mathematical course of study. Mathematical reasoning has two educational problems. One is the Necessity of deduction and the other is the strictness of deducion. Philosophical ambiguity of the proof corresponds with two problems of mathematical reasoning. We can conprehend the problem of the necessity of deduction in the light of synthesis and the strictness of deduction in light of analysis. Necessity and Strictnss of deduction are logically interrelated and dependent upon each other in the content of teaching geometry. Locally axiomatic, which is proposed by Freudenthal, H. and Schuster, S. independently, is considered as the point of view which integrates the necessity and strictness of deduction. The problems of mathematical reasoning are oriented to solutions in the terms of locally axiomatic.