脑算的概念与发展史
脑算是不用任何计算工具对任意数进行计算的方法，俗称心算，由于它是直接用口报出答案的，所以在小学数学里又称口算。
一种好的脑算方法的发明都是建立在相应的计算工具基础上的。我国人民发明的有记载的计算工具方法有十四种。它们分别是积（筹）算，太乙算，两仪算，三才算，五行算，八卦算、九宫算，运筹算，了知算，乘数算，把头算，龟算，珠算，计数算（心算）。除心算不用工具外，其他１３中算具都有相应的算具。
一种好的计算工具就会导致一种相应的脑算方法的问世。其脑算方法主要有：直接心算、筹算式心算（模拟算筹）、珠算式心算（模拟算盘）、笔算式心算（模拟笔算）。
古法“一掌金”概述手算
古法“一掌金”又称“一掌金河图”或称“袖里吞金”，据文献记载，明朝程大位<<算法统宗>>明文记载了一掌金。
一掌金表示数的方法是以左手五指设点作为数码盘，每个手指表示一位数，五个手指可表示个、十、百、千、万五位数字。每个手指的上、中、下三节分别表示１－９个数。
每节上布置着三个数码，排列的规则是分左、中、右三列，手指左边逆上（从下到上）排列１、２、３：手指中间顺下(从上到下)排列４、５、６：手指右边逆上排列７、８、９。
一掌金的计算方法是采用心算办法利用大脑形象再现指算计算过程而求出结果的方法。它把左手当作一架五档的虚算盘，用右手五指点按这个虚算盘来进行计算。记数时要用右手的手指点左手相对应的手指。其明确分工是：右手拇指专点左手拇指，右手食指专点左手食指，右手中指专点左手中指，右手无名指专点左手无名指，右手小指专点左手小指。对应专业分工各不相扰。哪个手指点按数，哪个手指就伸开，手指不点按数时弯屈，表示０。
一掌金的计算是采用珠算的计算方法。而一掌金起到算盘作用。其计算后与珠算类同。由一掌金这里就不赘述了。该法实质上是用手级数，用大脑运算。清朝末年有人用过。但训练难度大，现在已经无人采用。
见仁见智说数学
Mathematics is the Queen of the Sciences，and
Arithmetic the Queen of Mathematics．
（数学是科学的女王，而数论是数学的女王。）
——高斯
The science of mathematics presents the most brilliant example of how
pure reason may successfully enlarge its domain without the aid of
experience.
（数学科学呈现出一个最辉煌的例子，表明不用借助实验，纯粹的推理能成功地扩大人们的认知领域。)
——康德
Pure mathematics is， in its way，the poetry of logical ideas．
（纯粹数学，就其本质而言，是逻辑思想的诗篇。）
——爱因斯坦
But there is another reason for the hign repute of mathematics：
it is mathematics that offers the exact natural sciences a certain
measure of security which，without mathematics，they could not attain．
（数学受到高度尊崇的另一个原因在于：恰恰是数学，给精密的自然科学提供了无可置疑的可靠保证，没有数学，它们无法达到这样的可靠程度。）
——爱因斯坦
Mathematics is the tool specially suited for dealing with abstract
concepts of any kind and there is no limit to its power in this field．For
this reason a book on the new physics，if not purely descriptive of
experimental work，must be essentially mathematical.
（数学是一项工具，特别适合于处理任何一类抽象概念，而且，它在这方面的作用是无止境的。因此，一本论述新物理学的书，如果不是单纯地描述实验工作的，其本质上，必定是一本数学书。）
——狄拉克
Nature’s great book is written in mathematical symbol.
(自然这一巨著是用数学符号写成的)
——伽里略
Mathematics is a creative art because mathematicians create beautiful
new concepts；it is a creative art because mathematicians live，act，and
think live artists； and it is a creative art because mathematicians
regard it so．
（数学是创造性的艺术，因为数学家创造了美好的新概念；数学是创造性的艺术，因为数学家的生活、言行如同艺术家一样；数学是创造性的艺术，因为数学家就是这样认为的。）
——哈尔莫斯
The essence of mathematics lies in its freedom．
（数学的本质在于它的自由。）
——康托尔
知行糅合堪讲究
The instruction of children should aim gradually
to combine knowing and doing . Among all sciences mathematics seems to
be the only one of a kind to satisfy this aim most completely.
（儿童教育的目的应该是逐渐地把知与行结合起来。在所有的学科中，数学似乎是能最完全地满足这一要求的惟一的一类学科。）
——康德
The only way to learn mathematics is to do mathematics.
(学习数学的惟一方法是做数学。)
——哈尔莫斯
If the educational potential of mathematics is to be realized，then the
attention that has to be paid to its technical aspects must be balanced
by attention paid to its structural ，historical，genetic and
philosophical aspects.
（如果要把数学的教育潜能付诸实际，那么，既要注意到数学的技术方面，也必须注意到它的结构、历史、创生与哲学的方面，而且，各方面的注意须适当平衡。）
——谢尼泽
We must teach these people good，useful mathematics in a low-key，relaxed
style，free from authoritarianism，so that they come to feel not
threatened by it，not daunted by its unfamiliar symbols，but
comfortable with it and convinced that they are now better able to reach
rational decisions and cope with our complicated world. Mathematics will
then become not a barrier，but the key to unlock the door to a fuller
life.
(我们必须以低调的轻松的风格,
而不是满堂灌的独裁风格把好的、有用的数学教给这些人，这样，他们才不会感到在受数学的威胁，不会因不熟悉的符号而畏缩，他们才会喜欢上数学，并且相信他们能够很好地作出合理的判断，能够很好地对付复杂的现实问题。那么，数学将不再成为一种障碍，而将成为打开通向更充实生活的大门的钥匙。)
——希尔顿
Some teachers demand that students solve mathematics problems only by
the method taught in class，This process can thwart initiative，compound
frustration，and promote avoidance.
(有的教师要求学生只用课堂上教的方法解数学题。这种做法会阻碍独创能力的发展，导致失败，并造成回避困难的心理。)
——波雅妮
When students are forced to develop their own approaches to unsolved
problems，they benefit much more than from exposure to the mathematics
alone.
（当一个学生被迫大展身手去对付未解决的问题时，比仅仅学习数学知识，其受益要多得多。）
——斯潘尼尔
At any rate，in my opinion，one of the principal aims of the high
school mathematics curriculum is to develop the students’ ability to
solve problems.
（以我看，无论如何，中学数学教育的主要目的之一是发展学生的解题能力。）
——波利亚
When introduced at the wrong time of place，good logic may be the worst
enemy of good teaching .
（在错误的时间或错误的场合讲授，好的逻辑也可能成为好教学的最坏的敌人。）
——波利亚
If，however，the teacher has had no experience in creative work of
some sort，how will he be able to inspire，to lead，to help，or even
to recognize the creative activity of his students？A teacher who
acquired whatever he knows in mathematics purely receptively can hardly
promote the active learning of his students. A teacher who never had a
bright idea in his life will probably reprimand a student who has one
instead of encouraging him.
(但是,如果教师没有某种创造性工作的经历,他怎么能够去唤醒、引导、帮助、甚而赏识他的学生们的创造性活动呢？如果一个教师在数学方面的知识完全是被动地接受下来的
，那么，他基本上无法激励他的学生们去主动地学习。如果一个教师一生从来未有过奇思妙想，那么，他很可能不会鼓励他的学生们这样做，反而可能会责备他。)
——波利亚
越俎珍羞胜老庖
The mathematical method is the essence of
mathematics. He who fully comprehends the method is a mathematician．Music
has much resemblance to algebra. Pure mathematics is real magician’s
wand. The real mathematician is an enthusiast person；without
enthusiasm，no mathematics. One may be a mathematician of the first
rank without being able to compute，it is possible to be a great
computer without having the slightest idea of mathematics. What
logarithms are to mathematics,that mathematics are to the other
sciences.
（数学方法是数学的本质。数学家是能完全领悟数学方法的人。音乐与代数很类似。
纯数学是真正的魔术师的魔杖。真正的数学家，其本质是一个热情洋溢的人，没有热情就没有数学。一个不擅于计算的人，有可能成为一个第一流的数学家，而一个没有丝毫数学观念的人，充其量只能成为一个大计算家。对数之于数学，恰如数学之于其它科学。）
——哈登伯格
It is the man not the method that solves the problem.
(解决问题的是人,而不是方法。)
——马斯科
Mathematics possesses an aesthetic value as clearly defined as that of
music or poetry.
(数学如同音乐或诗一样显然地确实具有美学价值。)
——Jerry P. King
There is a myth that a mathematician must be “very fast with
figures”. This is false——mathematics mainly involves intuition and
the ability to grasp theoretical ideas．If a mathematician has these
talents，he need have very little else.
（有一个神话说数学家必须是“计算神速”的。这是不对的——数学主要靠直觉和掌握理论概念的能力。如果一个数学家有了这些才能，他就别无它求了。）
——理查德
善学贵在乐勤高
Do not imagine that mathematics is
hard and crabbed and repulsive to common sense， it is merely the
etherealization of common sense.
（别把数学想象为硬梆梆的、死绞蛮缠的、令人讨厌的、有悖于常识的东西，它只不过是赋予常识以灵性的东西。）
——开尔文
The attraction of mathematics is that it’s great fun.
（数学的魅力在于它是很有趣的学科。）
——帕克特
If you wish to learn swimming，you have to go into the water，and if
you wish to become a problem solver，you have to solve problems.
（如果你想学会游泳，你必须下水；如果想成为解题能手，你必须解题。）
——波利亚
It appears to me that if one wants to make progress in mathematics，one
should study the masters and not the pupils.
（我似乎觉得，如果一个人想在数学方面有所进步，他必须向大师们学习，而不是向徒弟们学习。）
——阿贝尔
Conceptual thinking is the salt of mathematics.
（ 概念的思考是数学的特色。）
——波士顿
Facts are not much use，considered as facts. They bewilder by their
number and their apparent incoherency. Let them be digested into theory,
however, and brought into mutual harmony, and it is another matter.
Theory is of the essence of facts. Without theory scientific knowledge
would be only worthy of the mad house.
（事实，只认为是事实，没有太大的用处。因其数量之多与显而易见的支离破碎，使人手足无措。但是，把他们融合成理论，并使之谐调，那么，情况就不同了。理论是事实的本质。没有理论，科学知识将只是一处疯人院。）
——希维赛德
What, in fact , is mathematical discovery? It does not consist in making
new combinations with mathematical entities that are already know. That
can be done by anyone , and the combinations that could be so formed
would be infinite in number, and the greater part of them would be
absolutely devoid of interest. Discovery consists precisely in not
constructing useless combinations, but in constructing those which are
useful, which are an infinitely small minority. Discovery is
discernment, selection.
Mathematical facts worthy of being studied are those which,by their
analogy with other facts, are capable of leading us to the knowledge of
a mathematical law, in the same way that experimental facts lead us to
the knowledge of a physical law. They are those which reveal unsuspected
relations between other facts, long since known, but wrongly believed to
be unrelated to each other.
（实际上，什么是数学发现呢？它不在于把早已经知道的数学实体做新的组合。这是任何人都能做的，而能这样形成的组合也将有无限多种，其大多数肯定是没有任何意义的。发现恰恰在于不去构筑无用的组合，而在于构造有用的组合。这样的组合是极其稀少的。发现是辨别，是选择。
值得研究的数学事实是因它们与其它的事实相类似，能够引导我们去认识数学定律，正如实验事实引导我们去认识物理定律一样。值得研究的数学事实是能揭示与其它早已知道的事实有无可怀疑的联系，而被错误地认为是相互无关的那些。）
——庞加莱
宇宙名誉建筑师
God ever arithmetizes.
（上帝是一位数论学家。）
——雅可比
The great Architect of the Universe now begins to appear as a pure mathematician.
（宇宙的伟大建筑师现在开始以纯粹数学家的身份出现。）
——吉恩斯
The greatest mathematicians， as Archimedes，Newton and Gauss，always united theory and application in equal measure.
(如阿基米德、牛顿与高斯这样的最伟大的数学家，总是不偏不倚地把理论与应用结合起来。)
——克莱因
A mathematician， who is not also some thing of a poet，will never be a complete mathematician.
（没有一些诗人气质的数学家绝不是一位完全的数学家。）
——魏尔斯特拉斯
The essence of mathematics is the perception of logical relationships between seemingly disparate ideas；the most successful mathematicians are those whose range is widest and whose ability to see analogies and connect ideas is greatest.
（数学的本质是对表面上看来完全不同的概念认识其内在的逻辑关系。最成功的数学家是知识面最宽、概念的类比、想象能力最强的人）
——爱德华
Lagrange is the lofty pyramid of mathematical sciences.
（拉格朗日是数学科学中高耸的金字塔。）
——拿破仑
What we know is very slight，what we don’t know is immense.Man follows only phantoms.
（我们所知道的是微乎其微的，我们所不知道的还是无穷无尽的。人们只是在追逐幻影。）
——拉普拉斯
庐山秀出南斗旁
Those who assert that the
mathematical sciences say nothing of the beautiful are in error. The
chief forms of beauty are order, commensurability and precision.
（硬说数学科学无美可言的人是错误的。美的主要形式是秩序、匀称与明确。）
——亚里斯多德
The feeling of mathematical beauty , of the harmony of numbers and
forms, of geometric elegance. This is a true aesthetic feeling that all
real mathematicians know.
（感觉到数学的美，感觉到数与形的协调，感觉到几何的优雅，这是所有真正的数学家都清楚的真实的美的感觉。）
——庞加莱
The beauty of mathematics speaks for itself.
（数学之美是很自然明白地摆着的。）
——哈尔莫斯
I think it is correct to say that (the mathematician’s) criteria of
selection, and also those of success, are mainly aesthetical.
（我认为，说数学家选择课题的准则以及判断他是否成功的准则，主要的是美学准则，这是正确的。）
——冯.诺伊曼
My work always tried to unite the truth with the beautiful, but when I
had to choose one or the other, I usually chose the beautiful.
(我的工作总是力图把真与美结合起来，但是，当我不得不选择其中的一种时，我通常选择美。)
——韦尔
Aesthetic judgments transcend both logic and applicability in the
ranking of mathematical theorems: beauty and elegance have more to do
with the value of a mathematical idea than does either strict truth or
possible utility.
（在数学定理的评价中，审美标准既重于逻辑的标准，也重于实用的标准：在对数学思想的评价时，美与优雅比是否严密、正确，比是否有用都重要得多。）
——斯蒂恩
Pure mathematics can be practically useful and applied mathematics can
be artistically elegant.
（纯粹数学可以是实际有用的，而应用数学也可以是优美高雅的。）
——哈尔莫斯
Much research for new proofs of theorems already correctly established
is undertaken simply because the existing proof have no aesthetic
appeal.
（对早已正确认定的定理做进一步的研究，探索它的新证法，只不过是因为现有的证明欠缺美的魅力。）
——克莱因
未能盖棺可论定
Histories make men wise；poems
nitty；the mathematics subtile；natural philosophy deep；moral grave；logic
and rhetoric able to contend．
（历史使人明智，诗歌使人聪慧，数学使人精密，哲理使人深刻，伦理学使人有修养，逻辑与修辞使人善辩。）
——培根
The advancement and perfection of mathematics are intimately connected
with the prosperity of the State．
（数学的发展与完善和国家的繁荣昌盛息息相关。）
——拿破仑
Mathematics is the most powerful technique for the understanding of
pattern，and for the analysis of the relation of patterns．．．If
civilization continues to advance in the next two thousant years ， the
overwhelming novelty in human thought will be the dominance of
mathematical understanding．
（数学是理解模式与分析模式关系的最有威力的工具．．．．．．只要在今后的两千年里，文明继续进步，人类思想中压倒一切的新事物，将是数学理智的统治。）
——怀德海
Mathematics is a crucial component of our culture. It can and should
make a signal educational contribution.
（数学是我们文化中极为重要的一个组成部分。它能够也必将作出显著的教育上的贡献。）
——谢尼泽
For me the attraction of turning to mathematics is that in mathematics
it is possible to actually make progress in a train of thought.
（吸引我转向数学的原因在于数学能够在思想的训练方面实际地作出贡献。）
——苏利文
Algebra is generous，she often gives more than is asked of her.
（代数学是慷慨大方的，它给予人的往往比人们对她所要求的还要多。）
——达朗贝尔
Like it or not, mathematics opens career doors,so it’s downright
practical to be prepared.
（不管你喜欢与否，数学为你打开求职的大门，因此，它是需要加以准备的真正实用的课程。）
——波雅妮
Yet the part which mathematics plays in meteorology increases every year，and
seems defined to increase.
（此外，数学在气象学方面所起的作用在逐年增大，而且，似乎还在不断地继续增大。）
——史密斯
More recent advances in the mathematical sciences have helped improve
our ability to predict the weather ，to measure the effects of
environmental hazards，to study the origin of the the universe，and to
project the outcomes of elections，Mathematical methods have become
indispensable to the proper functioning of our technological society.
(数学科学最近的进步帮助我们提高预测气象的能力,估计环境危险的影响的能力,
研究宇宙起源的能力，以及筹划选举结果的能力。数学方法对于我们这个技术社会真正发生效能已经变得不可缺少了。)
——哈尔莫斯
Number rules the universe.
(数统治着宇宙)
——毕达哥拉斯
数学艺术一家春
A mathematician , like a painter
or a poet, is a maker of patterns ……The mathematician ’s patterns,
like the painter’s or poet’s ,must be beautiful……Beauty is the
first test:there is no permanent place in the world for ugly
mathematics.
（数学家如画家或诗人一样，是款式的制造者．．．．．．数学家的款式，如同画家或诗人的款式，必须是美的……世上没有丑陋数学的永久立身之地。）
——哈代
A peculiar beauty reigns in the realm of mathematics, a beauty which
resembles not so much the beauty of art as the beauty of nature and
which affects the reflective mind , which has acquired an appreciation
of it , very much like the later.
（一种奇特的美统治着数学王国，这种美不像艺术之美与自然之美那么相类似，但她深深地感染着人们的心灵，激起人们对她的欣赏，与艺术之美是十分相象的。）
——库默
May not Music be described as Mathematics of sence, Mathematics as Music
of the reason? The soul of each the same! Thus the musician feels
Mathematics, the mathematician thinks Music—Music the dream,
Mathematics—the working life—each to receive its consummation from
the other when the human intelligence, elevated to its perfect type,
shall shine forth glorified in some future Mozart—Dirichlet or
Beethoven—Gauss—a union already not indistinctly foreshadowed in the
genius and lobours of a Helmholtz.
(难道不可以把音乐描绘成感觉的数学，而把数学描绘成理性的音乐吗？这样，音乐家感觉到数学，数学家想到音乐——音乐是梦想，数学是工作的一生——每一方都经由对方达到尽善尽美的境地，那时，人类的智慧达到完美的典型，将在某个未来的莫扎特——狄利克雷或贝多芬——高斯的歌颂下而光彩夺目。这种联合已经在一个赫姆霍尔兹的天才和工作中清楚地预示出来了。）
——西尔弗斯特
I like to look at mathematics almost more as an art than as a science,
for the activity of the mathematician, constantly creating as he is ,
guided though not controlled by the external world of the senses, bears
a resemblance, not fanciful I believe but real, to the activity of an
artist, of a painter let us say. Rigorous deductive reasoning on the
part of the mathematician may be likened here to technical skill in
drawing on the part of the painter. Just as no one can become a good
painter without a certain amount of skill, so no one can become a
mathematician without the power to reason accurately up to a certain
point. Yet these qualities, fundamental though they are , do not make a
painter or mathematician worthy of the name, no indeed are they the most
important factors in the case. Other qualities of a far more subtle
sort, chief among which in both cases is imagination, go to the making
of a good artist of good mathematician.
（一般地说，我更想把数学视为是艺术，而不是科学。因为我们可以说，数学家的活动，当他受外部的理性世界所引导，而不是被控制时，不断地进行创造性的活动，与一个艺术家、一个画家的活动相类似，有着实在的，不是虚幻的相似点。数学家这一方面的严密演绎推理可以比喻为画家那一方面的绘画技巧。恰如没有一定技巧的人不能成为一位好画家一样，没有一定的精密推理能力的人不能成为一位好的数学家。但是，这些尽管是他们的基本特质，还不足以使一个画家或数学家名副其实，画图技巧与推理能力，说实在的，终究不是最重要的因素。远为敏感的，为二者都是主要的一类特质是想象力，它才能造就一名杰出的艺术家或杰出的数学家。）
——博歇
We can hope that as education and leisure develop more people will enjoy
music and painting. But the number of people who really enjoy
mathematics is very small.
(我们能够期待，随着教育与娱乐的发展，将有更多的人欣赏音乐与绘画。但是，能够真正欣赏数学的人数是很少的。)
——贝尔斯
冥王星软件工作室
相关页面：行业管理 图像处理 媒体工具 编程开发 应用软件 系统工具 游戏娱乐 网络工具 网站源码 教育教学
相关软件：少儿速算练习机
秘密日记本 电子琴跳舞机
百变珠宝俄罗斯
人物分析机 微型个人理财
