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§ 1 在黑板上画一个半径为15cm的圆,过离圆心10cm的一点作一条直綫。显然,这条直綫交圓于两个点。这一知識是由实驗方法得到的。 設想在平面上画一个半径为15,000,000公里的圆,过离圓心10,000,000公里的一点作一条直綫。当然,你們会說,这条直綫也是与圆相交于两个点。你們并沒有看見这个圆,这条直綫也沒有真正作出来,你們这个信念的根据是什么呢?不可能实际地証实这样的直綫与圆相交。即使要証明这个定理也并非易事,而在中学阶段是不可能証明的。你們这个信念是一定的經驗与直觉所给与的。中学数学課不是、也不可能是具有严密邏輯系统的課程。許多数学事实是由实驗方法得到的,而只有一部分才經过了邏輯证明。为了改进数学教学,必須正确地理解在学生获得知識的过程中,实驗、直觉与邏輯的相互关系的意义。
§ 1 Draw a circle with a radius of 15cm on the blackboard and make a straight line from a point 10cm away from the center. Obviously, this line crosses two points. This knowledge is obtained experimentally. It is envisaged to draw a circle with a radius of 15,000,000 kilometers in the plane and make a straight line from the point at the center of 10,000,000 kilometers. Of course, you would say that this line is also intersecting the circle at two points. You did not see the circle. This line was not really made. What is the basis of your belief? It is impossible to actually prove that such a line intersects a circle. Even if you want to prove this theorem is not easy, it is impossible to prove in the middle school. Your belief is given by certain experience and intuition. High school mathematics classes are not and cannot be courses with a tightly logical system. Many mathematical facts are obtained experimentally, and only a few are logically justified. In order to improve mathematics teaching, it is necessary to correctly understand the meaning of the interrelationship of experiment, intuition and logic in the process of students’ acquisition of knowledge.