小学数学理解与教学外文翻译资料

Understanding and Teaching Elementary School Mathematics

原文作者 David Rappaport 单位 Professor of Education Ilinons Teachers College Chicago (North)

摘要：随着数学课程的不断改革，小学、初中、高中数学课程的设置都受到了一定的影响。数学课程内容的改革不仅引起了人们的关注，更要求教师妥善处理新旧内容之间的关系。此书对新旧内容进行了对比，同时对教师在课堂中如何展开新课程教学进行了探索研究。

关键词：数学课程； 教材内容； 教学模式；新旧知识

1.1数学在当今的学校

人类在很早的时候就已经开始使用某些形式的数学了。人类会计算和测量。但人类还缺一种语言去和别人沟通这种数学。随着文明的发展，数学对于人类而言愈发的重要了。这是不言而喻的，没有数学的发展，科学和技术将不可能进步。数学不仅在文明的发展中起了到了重要作用,它也成为了不可或缺的工具。

学校教授数学是为了满足社会的需求。但这样做经常，会让人们对数学造成不好的印象。来看一下数学教科书上的标题，《一到六年级的算术》,《8年级数学》,《看透算术》，《看透数学》这就是最近的教科书常使用的标题。 这种明显的区别给人一种算术是给小孩子的玩意儿，而数学才是成年人的印象。 大多数人会说他们在小学学习算术，在高中学习数学。这种说法是不科学的。一个人在小学学习基础的科学知识，在中学以及大学进一步研究、学习、拓展科学知识，但我们在这些初等阶段和高等阶段学习的内容都有一个统一的名字——科学。不过数学家们不这么认为。在数学家眼里，算术不仅仅是数学的一部分，更是非常重要的一部分。著名数学家K. F.高斯（1777-1855）给予算术非常高的评价，他说：数学是科学的皇后，而算术是数学的皇后。她经常谦卑地为天文学和其他自然科学提供服务。

许多教师和学校管理人员认为算术一个工具。在他们看来学校教授算术的主要原因是社会需要会算术的人才。他们认为算术对于实际问题和日常生活问题的解决是一个重要科学工具和技术，因此才会对算术有这样的印象,即数学是用来处理理论的，而算术是用来处理实际应用的。在这种功利主义下解释的算术过分强调学习的技能和步骤，却忽视了算术意义。

一个相反的观点认为学习算法就是研究数论。根据这一解释，算法是数学的一个分支，这已经算得上是一种是高度发达和成熟的理念了。虽然对于孩子们而言第一次接触数学就是通过算术的方式，他们在小学学习算术的概念，当然他也可能在他日后的学习生涯中继续接受算术的教育。

L4教育原则

有先进理念的教育者强调数学的教学是侧重理解的教学而不是计算的教学。虽然在练习的情况下，我们要求孩子们能够进行计算并且得出正确的答案。孩子们学习数学还是应该用数学的方法来学习比较好，也就是说，基本的数学含义应该理解，同时又能知道数学的基本原理。

只有当孩子自己有所发现时，他们才会对知识有着最好的理解。而直接告诉孩子的“什么”和“如何”的教学方法是不好的。教师应该以这样的方式进行数学课，从而培养儿童发展的洞察力，促使他们看基本数学概念内在的关系。我们发现孩子在有所发现的时候通常也能同时理解“为什么”。

为了帮助孩子发现重要的小结结论，我们应该有足够的工具，保证每个孩子都可以使用到。这些混辅助工具包括：算盘，分数，计算框架，占位符图，数线。上述工具都可以让孩子们在发现的第一时间使用上。

孩子应该体会从具体到抽象的过程。一些孩子可能会用图来代替具体的对象。当孩子学习到某种程度时，他会放弃使用具体的对象，而是用半抽象半具体的方式来学习，比如说使用线或圆等。比如，儿童在学习5加3时，可能用5条杠(/ / / / /)来表示5，然后用3条杠(/ / /)来表示3，数一数得到和为8的结论。在孩子学习水平到了最高程度以后，他们就可以不再使用一点具体的对象，完全依靠抽象的符号来进行数学学习。

虽然心理学不赞成使用半具体半抽象的方式来学习数学，因为它们只是学习的教具辅助，现代教育家支持使用教具辅助的原因是因为它们帮助孩子看到了内在的关系。前者遵循桑代克的名言，认为学习是通过联结-试误的方式。通过不断尝试，改变原来的认知结构，习得新知识。后者则认为认为，内在洞察力的习得是由于孩子对现实具体事物的依赖性，但是当孩子学会（思维）从具体到抽象的转变时，他们会抛弃对实物的依赖。如果他们又再度需要具体实物的辅助时，这种方法同样又是适用的。一味的要求孩子不能通过具体实物（学习）的方式，有时候会导致孩子没有安全感、产生挫败感。儿童需要被挑战、被鼓舞，以及最大可能的去激发他们的才能。数学应该是一门在符合学生认知水平的基础上，对儿童具有挑战性的一门学科。孩子们需要被鼓励去发现更多广泛的事物。这就意味着孩子们需要接受和提供更多符合他们学习认知能力发展的机会。

1.5心理原则

孩子们的学习能力不同。 学生在教育教学时总是被提醒他们必须展示自己的个性。 这个建议说起来容易的,但是要在教学中真正的实现对教师而言是一个极大的挑战。 如果正如所说的，孩子们的个人能力具有差异性，那么就不应该分配同样的任务给不同的学生。有些人学的快，有些人学的的。教学原则应该根据学生自身的学习速率制定。强求用高于他们自身能力的学习方式学习会让孩子产生挫败感，如果低估孩子的学习方式，则会对他们的学习帮倒忙。学的慢的学生应该比学的快给予更多的时间，让他们去做一些可以获得成功的任务（增强自信心）。就是说，给他们尝试一些相对简单的练习。在面对不同的情境时，需要为能力差的孩子提供一定的操作辅助，但是当他们一旦学会处理时，就可以让他们尝试更难的一点任务。学习能力强的孩子，教师要根据他的学习水平制定任务，促使他去挑战一些比较难的学习任务。这样的孩子很少需要实物操作的帮助，因为他们的抽象思维能力已经发展较好了。如此便意味着，数学课程应该根据不同孩子的学习能力，提供不一样的知识。同样也说明，每一个孩子都已应该从事对他们有意义的任务。

1.6水平的学习

教师必须的理解和认识学生不同层次的学习水平，这样才有助于在教育教学过程让学生更好的学得算术的概念与意义，以及表现他们的个体差异。学生应该具备解决问题的能力、数学学习模式以及数学概念的学习、能够表述自己数学思考的过程。 学习水平具有三个层面分别是为操作、概括和验证。

操作层面，这是学习的第一或最简单的水平。在这一水平的孩子采取具体实物辅助的方法。例如孩子在数数时，会掰手指；或者在纸上画圈圈的方式。当学习分数时，孩子会采用画圆圈等分的办法等。

当孩子在学过分数的基础上，即已经知道如何表达分数。当他们被要求比较不同形式表达分数时，他们就会知道一半就等于二分一、四分之二，六分之三，八分之四，或十二分之六。他们也会学习，三分之一等于六分之二，四分之三等于十二分之九，六分之五等于十二分之十。因为他们的学习是通过具体模具的运用，仍出于实物操作层面。例2

孩子们被要求找出四分之三乘以六分之五。他们可能执行以下步骤。他们会先将一个矩形六相等的部分，保留其中的五个部分。然后他们会把这五部分成四个相等的部分，保留其中的三。他们将结束对这个问题的答案二十四分之十五。所有的乘除方法都采用了实物辅助的分式。这再一次学习依然处在操作水平。

示例3

要求学生求两个奇数的和。 他们会利用5和3 、7和5、9和3、15和7等，用特殊数值带入法解决问题。

概括水平。 当孩子们对以上三个例题做了足够多的次数后,他们可能会产生一些概念性的理解。 在示例1,他们可能会发现,如果他们的把分子分母约分那么的答案是相等的。 在示例2中,他们可能会发现计算二个分数相乘，可以将它们的分子分母相乘。 一旦这些概念被发现孩子不再依赖于实物模型了。他找到了一种方式可以被理解为是概括化的学习方式。 在例3中,他可能会发现或推广,两个奇数之和总是偶数。

验证水平。这是学习的最高水平。现在老师问孩子们证明概念或解释为什么上述概念是真的。这通常是相当困难的。它可能不是很容易证明。 试图找到解释每个描述的三个概念需要更高层次的推动。 一般知道二个小数相乘就是先相当于二个小数点后面的实数相乘在增加小数点位数。但是这个规律又是为什么呢？

概括就有规则性。 验证就是最好的解释。 一个孩子可能会发现二个奇数相加总是偶数。 因为7 3 = 10,11 15 = 9 5 = 14日26日。 他概括了任何两个奇数的和总是一个偶数。 这是为什么概括是真的? 他可能学会用2 n 1代表一个奇数或2 m 1代表另一个奇数。 然后两个奇数的和可以表示为(2 n 1) (2m 1)或2 n 2 m 2或2(n m 1)。尽管n m 1可能是奇数也可能是偶数,但是2(n m 1)总是一个偶数。 这是一个证明,两个奇数的和总是偶数。

孩子可能会问如果他将一个奇数减去另一个奇数。 他发现9 - 3 = 6,17-5=12，23-15 = = 8概括了两个奇数的差总是偶数。为什么这是真的吗？他可以使用相同的方法用2 n 1代表一个奇数或2 m 1代表另一个奇数。 然后两个奇数的差可以表示为(2 n 1)-(2m 1)或2 n -2 m 或2(n - m)。同样，n-m不能确定奇偶性，但是2(n - m)一定是偶数。

外文文献出处：Understanding and Teaching Elementary School Mathematics ，Copyright by 1966 Jhon Wileyamp;Sons.Inc ,Library of Congress Catlog Card Number.66-10587 .Printed in the United States of America

附外文文献原文

1.1 MATHEMATICS IN TODAYS SCHOOLS

Some form of mathematics has been used by man from the very

beginning of his existence. He learned to count and to measure. But

he also had to invent a language by which to communicate his ideas of

quantity to other men. As civilization became more complex, mathe-

matics became more important to mans existence. It is self-evident

that advances in science and technology would not have been possible

without the development of mathematics. Mathematics has not only

achieved a prominent position in the development of civilization but it

has also been an indispensable instrument in its development.

The schools have responded to the mathematical needs of society

by giving a prominent place to arithmetic in the elementary school

curriculum. But in doing so the impression has often been given that

arithmetic is not really mathematics, or that it is lowly mathematics.

Examine the titles of textbooks. Arithmetic Grades I to 6, Mathematics

Grade 8, Seeing through Arithmetic, Seeing through Mathematics are

a few titles of recent textbooks. This distinction between arithmetic

and mathematics gives the impression that arithmetic is for children

whereas mathematics is for adults. Most people will say that they

learned arithmetic in elementary school and mathematics in high school.

This kind of distinction is generally not made for science. One studies

elementary science in elementary school and advanced

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1.1 MATHEMATICS IN TODAYS SCHOOLS

Some form of mathematics has been used by man from the very
beginning of his existence. He learned to count and to measure. But
he also had to invent a language by which to communicate his ideas of
quantity to other men. As civilization became more complex, mathe-
matics became more important to mans existence. It is self-evident
that advances in science and technology would not have been possible
without the development of mathematics. Mathematics has not only
achieved a prominent position in the development of civilization but it
has also been an indispensable instrument in its development.

The schools have responded to the mathematical needs of society
by giving a prominent place to arithmetic in the elementary school
curriculum. But in doing so the impression has often been given that
arithmetic is not really mathematics, or that it is lowly mathematics.
Examine the titles of textbooks. Arithmetic Grades I to 6, Mathematics
Grade 8, Seeing through Arithmetic, Seeing through Mathematics are
a few titles of recent textbooks. This distinction between arithmetic
and mathematics gives the impression that arithmetic is for children
whereas mathematics is for adults. Most people will say that they
learned arithmetic in elementary school and mathematics in high school.
This kind of distinction is generally not made for science. One studies
elementary science in elementary school and advanced science in high
school or in college, but at all levels it is called science. This attitude
toward arithmetic is not shared by the mathematicians. In their opinion
arithmetic is not only mathematics; it is also important mathematics.
The renowned mathematician K. F. Gauss (1777-1855) assigned the
greatest importance to arithmetic when he said:

Mathematics is Queen of the Sciences and Arithmetic the Queen of Mathe-
matics. She often condescends to render service to astronomy and other natural
sciences, but under all circumstances the first place is her due.*

Many teachers and school administrators consider arithmetic to be
a tool subject. In their view the main reason for teaching arithmetic is
social utility. It is necessary for the solution of practical problems of
everyday life or as an important tool for the solution of problems in
science and technology. Thus there is the impression that mathematics
deals with theory whereas arithmetic deals with practical applications.
This utilitarian interpretation of arithmetic has resulted in an over-
emphasis on the learning of skills and processes and a neglect of the
meanings in arithmetic.

An opposing view considers the study of arithmetic as a study of
number theory. According to this interpretation arithmetic is a branch
of mathematics that is highly advanced and a challenge to the most
mature. Although the childs first experience with mathematics is
through arithmetic, he learns only the elementary concepts of arith-
metic in elementary school. He may continue to study arithmetic
throughout his educational career.

L4 EDUCATIONAL PRINCIPLES

Leading educators emphasize the teaching of mathematics for under-
standing rather than for the sole purpose of learning computational
skills. Although it is expected that children should be able to compute
and to derive the correct answers when required in practical situations,
it is important for children to learn mathematics as mathematics.
There are basic meanings in mathematics which should be understood.
There is a rationale to mathematics.

The child understands the meanings best when he makes his own
discoveries. Instead of showing children the 'what' and the 'how,'
teachers should conduct the mathematics classes in such a way that
children will develop insight and will see the relationships that con-
stitute the basic concepts in mathematics. The child who discovers
the relationships will also be able to explain the 'why.'

In order to help the child discover the important generalizations
there should be sufficient manipulative devices that each child can use.
These are concrete aids such as number blocks, abaci, fraction kits,
counting frames, placeholder charts, and number lines that the child
can use when he finds it necessary.

The child should proceed from the concrete to the abstract. Some
children may substitute pictures for concrete objects. At a certain
period in his development the child will discard concrete devices and
substitute semiconcrete devices such as lines or circles. He may learn
to add 5 and 3 by making 5 marks (/////) and then 3 marks (///) and then
count to find the sum 8. At the highest level the child discards all

manipulative devices and achieves the required results by operating
with the abstract symbols.

Although the advocates of bond psychology frow Although the advocates of bond psychology frowned upon the use of
semiconcrete devices because they were crutches, modern educators
support the use of crutches because they help the child see relation-
ships. The former follow Thorndikes dictum that no bond should be
established that will later have to be broken. The latter support the
view that insight often results from the use of crutches and that
crutches will be discarded by most children themselves because
understanding will make the use of a crutch unnecessary. As the child
proceeds from the concrete to the abstract he discards manipulative
devices, but these aids should be available to the child when he finds
it necessary to use them. Forcing children to avoid all use of concrete
aids

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