对数学分析教材中一道错题的剖析

本文给出在函数级数方面一道错题的分析,并给出反例说明所给命题不成立.     

例谈反例在初中数学教学中的妙用

数学中的反例,是指符合某个命题的条件,而又不符合该命题结论的例子．说得更简洁一点,反例就是一种指出某命题不成立的例子．当然,从某种意义上来说,所有例子都可以称为反例,因为它总可以指出某命题（甚至是非常荒谬的命题）不成立．但这里,我们讨论的反例,是建立在数学上已证实的理论与逻辑推理基础上的,并且具有一定作用的反例．举反例也是一种证明的特殊方法,     

从“Lagrange中值定理的逆命题”谈起

本文指出“Lagrange 中值定理的逆命题”一文的错误,给出反例说明“Lagrange 中值定理的逆定理”不成立,然后给出一个新的命题,建立了可微函数与其导数之间的一种联系。     

注意运用反例 准确把握概念

在数学教学中常用给出反例的方法,来强调命题的前提条件或证明某一个命题不成立。初学者阅读教材时往往不重视定义定理后面的说明叙述,把给出的反例等同于一般的例题,而它们对帮助我们准确把握概念恰恰有着重要作用。例如在§3.2中,关于可导和连续的关系,给出了“函数在某点连续是函数在该点可导的必要条件.但不是充分条件”的结论。结论的前一半:“可导必然连续”,由定理3.1叙述,并加以证明。而结论的后一半:“连续不一定可异”,是用一个反例说明的,即,给了一个具体的函数y=f(x)=|x|,它在X=0处是连续的,但在该点导数不存在。给出此例的目的,并不是为了示范如何用导数定义求分段函数在分段点处的导数(分段函     

举反例与反证法

举反例和反证法是判断命题真假的两种方法,但本质不同. 所谓反例,通常是指用来说明某个例题不成立的例子.举反例就是证明某     

浅谈数学教学中的反例

众所周知,要证明一个命题正确,必须经过严密的逻辑推理。而要证明一个命题是错误的,十分简明而又有说服力的是举出一个反例。例如,“自然数不是质数,就是合数”这一命题,只要举出1是自然数,但它既不是质数,也不是合数,即可说明这个命题是错误的。又如,要想说明“两个质数的乘积一定是奇数”的结论不成立,也只要举出一个反例就行了。例如,2是质数,那么它和任何质数的乘积都是偶数,而不是奇数,这就说明这一结论不成立。这种与命题相矛盾的例子,数学上叫反例。     

浅析“反例”在几何教学过程中的重要作用

命题有真有假,要说明一个命题是真命题,并不是一件容易的事,有些命题的正确性只能靠实践来检验,并总结出来,有些命题的正确性可以靠逻辑推理来证明。而要说明一个命题是假命题只需要举一个反例足矣!所谓反例,就是它符合命题的题设,但不满足命题的结论的例子。可以这样说:数学由两个大类——证明和反例组成,而数学发现也朝着两个主要目标——提出证明和构造反例来进行。举“反例”占了数学的另一半!就初中几何而言,如何证明几何题,教材、教师都予以了足够的重视,而利用构造反例来说明一个命题是假命题,就略显薄弱些。下面就来看看这几个反例…     

反例在中学数学教学中的作用

反例通常是指符合某个数学命题题设条件,但不符合该命题结论的例子.举出反例即指出某命题不成立的例子.美国数学家盖尔鲍姆指出:“数学由两大类——证明和反例组成.而数学发现也是朝着两个主要目标——提出证明和构造反例”.数学问题的探索中猜想的结论未必正确,正确的需要证明     

数学分析中反例构造的探讨

数学本身可以说就是证明和反例所组成。因此数学的发展也大致朝着与此相应的两个方向,即预料结论正确的,就应想方设法给出证明;怀疑猜想不真的,则需设法找出其反例。举反例的目的就是为了说明被考察的命题不真;这无异于断定它的反面成立。因此举反例实质上也是反证法,只不过是应用范围更加特殊而已。     

适当运用反例提高数学课堂教学效率

所谓数学中的反例,是指符合某个命题的条件而又不符合该命题结论的例子。简单地说,反例是一种指出某命题不成立的例子。在数学的发展历史中,反例和证明同样重要。一个数学真命题往往需要严密的证明,而假命题则靠反例加以鉴别。数学家B·R·盖尔鲍姆和J·M·H·奥姆斯特得曾指出,数学有两大类———证明和反例组成,而数学发现也是朝着两个主要的目标———提出证明和构造反例。一个数学问题,用一个反例予以解决,给人的刺激犹如一出好的戏剧,所以在数学教学中有意识地构造反例来解决实际问题,让学生从中领会神奇功效,从而使学生切实有效地掌…     

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Many college freshmen arrive woefully unprepared to do college work, and as disadvantaged populations continue to grow, the share of the American work force that has made it through college is expected to plummet. Many experts blame that educational failure not just on high schools but also on colleges. School & College, a special report by The Chronicle, looks at efforts to fix the system. What reforms would better prepare students for college? What should schools and colleges be doing differently? How should state and federal officials help?     

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