数学的严密性,一个重要的方面反映在特殊与一般的关系的处理上。数学教学的一个目的,是提高学生的数学素养,养成周密思考的习惯,而这个目的的达到,很重要的是要通过处理好特殊与一般的矛盾来完成。一、“特殊”不能代替“一般”“特殊”不能代替“一般”,是很显然的道理,我们在教学中却往往忽视。例如,在讲等比数列求和公式时,只重视推导出q≠1的情况,它并不能代替等比数列的“一般”情况(这里的“一般”,不应理解为“大多数情况”)。在讲两点问的距离公式时,只讲了两点的联线既不平行于x轴、也不平行于y轴的情况。因而,要求学生解题时有多严密就不可能。例1求证以(x_1,y_1)、(x_2,y_2)为直径端点的 Mathematically tight, an important aspect is reflected in the treatment of special and general relations. One of the purposes of mathematics teaching is to improve the students' mathematical accomplishments and develop the habit of thinking carefully. What is important to achieve this goal is to accomplish the special and general contradictions. First, “special” can not replace “general” and “special” can not replace “general”. It is quite obvious that we often neglect our teaching. For example, in the case of the geometric summation formula, only the case of deriving q ≠ 1 is not emphasized, and it does not replace the “normal” case of a geometric sequence (“normal” here, not to be understood as “most cases” ). Speaking about the distance formula between two points, we only talk about the situation where the two points are not parallel to the x-axis or parallel to the y-axis. Therefore, it is impossible to ask students how strict the problem solving is. Example 1 Verify that (x_1, y_1), (x_2, y_2) are the diameter endpoints