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数学教学实践告诉我们,当要做的问题获得解决以后,教师进一步因势利导,启发学生去分析思考所研究的问题的逆命题是否成立,显得很重要。经常进行这种训练,不但能激发学生的兴趣,使学生对所论命题的认识更加深化,而且是开拓思路,培养逆向思维能力的有效途径。作为一种极为必要的训练,应当予以重视和加强。下面试举几例说明如何引导学生探索逆命题。例1 设抛物线y~2=2px的任意弦P_1P_2交轴于P_3 求证:x_1、x_3、x_2成等比数列。此题证明从略,解过之后追问学生:例一的逆命题是否成立,即任意弦具有这种特性的曲线是否一定为抛物线?引导学生得到下面的逆命题:
The teaching practice of mathematics tells us that when the problem to be solved is solved, the teacher further takes advantage of the situation and inspires students to analyze and consider whether the contradictory issue of the researched problem is established. It is very important. Frequently carrying out such training not only stimulates students’ interest, but also deepens students’ understanding of the propositions they have discussed. It is also an effective way to develop ideas and cultivate counter-thinking abilities. As an extremely necessary training, it should be emphasized and strengthened. Here are a few examples of how to guide students to explore the reverse proposition. Example 1 Let any parabola P_1P_2 of the parabola y~2=2px be quadrature on P_3. Prove that x_1, x_3, and x_2 are equal to each other. The proof of this question is omitted. After solving the question, ask the student whether the contradictory proposition of Example 1 is true, ie whether the curve of any string with this characteristic must be a parabola? Instruct the student to obtain the following contradictory proposition: