知识与智慧

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近日来接触到了好几个这样的学生:学习不可谓不努力,每天熬到深夜;概念和公式不能说没有掌握,一问都能答出来;练习不能说做得不够,因为老师布置的作业都做了。可是,考试却永远得不了高分,竞赛是永远榜上无名;为什么?

有人说是学习方法不对,可是预习、听课、复习、总结都做了;有人说是不能举一反三,触类旁通,这有一定道理,现在的学生是没有人愿意花功夫去把类似的东西归纳在一起的。但是,最关键的是,不知道把学过的知识联系在一起!说白了,就是脑子里少根筋,也就是笨。

昨天一个12年级学生给我看了他最近的一次向量考试。最后的TIPS问题是证明向量的点积和叉积之间的一个关系式。明眼人一看就知道怎么做:只要把点积和叉积的定义写出来,比较一下,用上最基本的三角恒等式,答案就在眼前。可是该生却在那里写了一大堆毫不相干的东西!即使把点积和叉积的公式摆出来,等式也就出来了,可为什么有人就是想不到?

这让我想起代数学上的一件惊奇之事。有史以来,数学家们一直认为保距变换只能是正交变换,可一直没有人能够证明。十多年前,MIT的一个女学生,竟然用很简单的、向量的点积给证明了!看到证明的当时,我很长时间说不出话来。

这就是独立思考的意义所在。上天赋予人类大脑,就是用来思考的,不是用来感官享受的。老师只能教我们知识,或许一点点思考的方法,但是,真正的思维是无法言传的,要不然,智能机器人早就横行于世了,人类也就没有存在的必要了。

智慧只属于善于思考的人。

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