反例在数学课堂教学中的作用

On the Functions of Counterexamples in Mathematics Classroom Teaching

证明命题是否成立,必须是在一个给定的条件下,通过严格的逻辑推理得出结论。其中简洁而极具说服力证明一个命题是错误的一种方法就是举出一个反例,这就能证明它是错误的。反例其实就是与命题相矛盾的特例,而正是恰当的反例推动了数学的发展,但是在当下,教师对反例教学的重视程度还不够,而教材中的反例案例也很少,从而需要重视此方面的教学。证明固然很重要,但是从哲学角度出发,反例同样也很重要。而在实际生活中,反例同样是很有意义的。

Whether a proposition is true or not must be concluded through rigorous logical reasoning under a given condition. A simple and convincing way to prove that a proposition is wrong is to illustrate a counterexample. Countermeasures are virtually special examples that contradict the given propositions, while it is proper countermeasures that promote the development of mathematics. But currently, teachers attach less importance to counterexamples and there are few cases of counterexamples in the textbook, so teachers must pay more attention to the issue. It is true that proving is important, but from the philosophical perspective, counterexamples are equaUy important. Likewise, counterexamples are of great significance in practical life.