# 身体运动蕴含着不可见的思维，数学思维启示着内在性的运动

[2011-03-14 10:57:39]
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ody Movements Are Invisible Thinking, Mathematical Thinking Is Inner Movements
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身体运动蕴含着不可见的思维，数学思维启示着内在性的运动
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On one hand mathematics is thinking; on the other it is life forces. The path from childhood to adulthood starts with the body and then moves to the head, in other words from movement to thinking. This is how the small child demonstrates its method to the rest of the world in the first three years: Walk, talk and think! Mathematics is under a lot of pressure to change. The teacher-controlled pedagogy, where the teacher presents the problem and then answers it, is a method that effectively develops the ability to understand abstractions, something that many consider to be old-fashioned. In which direction shall we head? Professor Befring at the University of Oslo wrote in a news*** article the Fall of 1998 that the best thing to do would be to stop the subject of mathematics all together and instead incorporate problem solving in other school subjects. A direction that has won more and more favor in the past years is to allow pupils to learn mathematics by doing. The teacher can arrange a store in the classroom where the pupils count their buying and selling transactions. If we give them *** and scissors and let them make volume forms, they will better understand volume calculations than they could have imagining the forms. The idea is to let und hb erstanding spring from practical work. Then each situation becomes experiential and connected to the child’s joy of discovery, while providing exercises in being independent. A child must be allowed to discover the world, not just learn about it. Later they develop the formal and abstract language of mathematics. This summarizes the public debate as of now.
Within a larger perspective we can follow a similar polarization in the historical development of mathematics.
In classic times geometry, which is the perceivable part of mathematics, was considered primary. Numbers and counting were secondary. First in the Middle Ages and in the Renaissance did mathematics become algebra-oriented and not until the 19
In the 1960s there was a renewal of algebra’s dominating position. An international research group was named to “create” a new mathematics. All mathematical concepts were to be developed algebraically from the most *** to the most difficult. Mathematics should become mere thought, and geometry was considered a secondary discipline. The project was called Bourbaki and is written down in multiple bands of a large work. In connection with this thinking the so- called New Math was created in the 1970s. Pupils should jump over the heavy and time-consuming learning of elementary counting and arithmetic. Lessons should lead directly into abstract thinking at an early stage, actually in the first grade. Measuring quantities replaced elementary arithmetic. The experiment speaks for itimmolation. In the last decade educational circles have clearly seen that pupils need practical arithmetic and geometry. With concrete problems practically everyone can work and the ability to abstract results as a product of their work. Within mathematical circles today many are searching for games, movement and invention that awaken the child to independent thinking. Yet do they understand why that children experience pedagogy positively? What is the relationship between physical body movements and mathematics, which are after all, pure thinking processes? Before we reach a degree of understanding of such questions, the movement back and forth between which method is correct will continue both in academic circles and in schools. Can we find ideas that unite these two directions?
The Waldorf Schools work from the conviction that both directions are expressions of the same thing. Especially during the classical school years between the seventh and the fourteenth years the child’s inner world of thoughts, feelings and willpower are connected with each other. To think entails bringing thoughts together willfully, to move is to carry out the forms you thought of. We influence our children’s life of thought by letting them move in forms and we influence their forces of will by thinking in movement. （ Remark by Translator: This idea is in the same line of Chinese Inner KungFu, which emphasis on “Improve Soul and Mind by Body Forms”, instead of “Improve Soul and Mind by Soul and Mind themselves”. The author leaves the details of such Body Forms in the subsequent articles. For easy comprehension, the ***st example can be that, the children walk in a circle, or walk in a “8” loop. ） In this way we can consider movement to be an outer life of thoughts and thinking to be an inner force of will. One of the clearest signs of this is found in the child’s development during the first three years. At approximately the age of one a child stands and walks; at the age of two they begin to speak and at the age of three thinking first appears. The path starts with movement of the body and continues with movement of thoughts. If you observe this connection you can also see that our entire childhood is a transition. The process takes place over many 2011大学生英语竞赛>years and there are clear stations along the way. The ability to do mathematics increasingly displays that the child’s willpower is not bundled into movements of the body but rather to movements of thought. To do mathematics, in the sense of developing abstract thoughts, can therefore be understood like a thermometer that measures how long a process has evolved. When you go into details there is a gradual liberation of willpower that is not directly easy to describe. Despite this, let us try to formulate some of the stations.
During the first three years children learn to walk, to speak and to think, but they do not think the way adults think. Very few of us can remember anything from this time and we cannot explain how we understood or learned anything. Memory and consciousness were present but connected so thoroughly to the processes in the physical body that we could not consciously and freely relate to them. Our thinking at this age was connected to the physical body and its movements. Yet during these years human beings begin a lifelong theme for our development of consciousness: Walk, speak and think. Can children truly count and calculate at this age? Normally not. But the basis is developed. The first experiences with the concepts of identity, difference and equilibrium are made. All progress is inspired by closeness and love between parents and children.
Children spend their fourth to seventh years in kindergarten, in play and in fantasy. As children we sensed so much but could not speak about it C it was not understood until we became adults. We could play, move and find things. Most of our activities that took place within mental images and concepts were enveloped by fascinating situations in our lives. During these years mathematics can be found in the childd body, in its movements. Everything from walking normally to running, hopping and climbing are developed in an emerging understanding of balance and imbalance. Children use their bodies to create experiences that can awaken reflections, that is thoughts. In this stage thoughts are pictures from life and concepts lie hidden as secrets within those pictures.
In the first three or four school years a new level is reached. For the first time children can speak about the world. What does life show us then? It displays forms, quantities and situations in life. Stories – from life C and arithmetic belong together. Reflections appear when the world is described. Counting is freed from the objects and arithmetic is freed from the situations in life. For example, after you move in an octagon the form can be drawn. Movements are the basis for the beginning of geometry, namely form drawing. Multiplying, running and hopping provide the basis for number rhythms. The movements of our bodies provide the basis for the movements of our thoughts.
At the age of nine another leap in childrenn development takes place. The distance between objects on the outside and the objects inside, i.e. mental images become markedly greater. The development of thoughts and concepts is more removed from the situations in life. The realities that directly appear to us have a symbolic meaning. The objects in the world become aw materiall?for analytical reflection. This process intensifies during the years between twelve and thirteen. Pure mathematics provides the basis for conscious work with parts of the whole, i.e. fractions. This is a major step for children. They cannot only divide the whole into parts but the parts can be put back into the whole. Consciousness works within comparisons of the whole, for example how large a part of the whole a thing may be. In my consciousness I move in the parts, measure them, weigh them and compare them. From my inner feeling for symmetry I judge the fraction.
From the twelfth to fifteenth year the liberation of thoughts and concepts from life situations reaches a level where the sensory qualities are less apparent in the mental images. The gradual liberation of the forces of thinking results in the world becoming more and more lifeless. This becomes the basis for abstract thoughts. Now thoughts have their own world and youth stand with a keletonn?in their thoughts. Thoughts are no longer a part of life processes in the body. The abstract thoughts are indeed threatened by their own stiffness and can only be positive if they are placed in a meaningful context. A superior process based on human judgment must be put in place. During these years it is important to use problems from daily life as the entrance to mathematics. Projects and testing the pupill independence are used consciously as part of mathematics. Now the time has come for algebra, equations and geometry. This work is built upon the movement, rhythm, balance, equilibrium and symmetry that have become inner qualities in the childd thinking. What is an equation other than an inner equilibrium? How important is symmetry for understanding geometry! Movements and transformations in normal arithmetic can now be repeated algebraically.
In the first years after puberty, roughly from fifteen to seventeen, there are big challenges to address. How can we prevent that a personn concepts of the world become a collection of abstractions void of any context? The goal must be to rediscover the same motives and truths in both worlds. The life forces that were important experiences in childhood, - movement, equilibrium, symmetry and rhythm may now be found in nature by using the abstract thinking. At its best thinking can be an imaginative game filled with life and also a path to objective truths. If we develop the principles behind a parable mathematically and then recreate it experimentally in a fling we have an example of how to meet teenagers pedagogically. Simple probability calculations, combinations and exponential growth point in the same direction. The (gyldne snitt) olden rulee?is a third theme that is very appropriate. The math is easy and you can easily find the physical proportions of the bodies.
Does the world make sense when thinking about it, and can I create a meaningful picture of it? Or is the world fragmented and random? Are there limits to our knowledge? These and many more existential questions live in our youth between the ages of seventeen and twenty-one. It is important to challenge our thinking with paradoxes where the answer is not just a ess?or oo?but often oth..?Thinking does not give me an absolute picture of the world, but a basis with which I relate to the world. I am not my thinking, but free thinking can give me an objective insight into the world. We remain conscious of the world in many ways and mathematics is just one of them. At the same time we can say that everything in the world can be analyzed and researched. Within every context there is mathematics. Life itimmolation and nature can be judged as thinking placed behind a veil. During these time we work with themes such as: derivation, integration, and projective geometry.
As adults we have hopefully achieved a quality of thinking that is alive and in motion. Of course we also meet the opposite qualities of thinking. If you are able to turn your forces inwardly in a positive way rich, flexible thinking appears. That thinking is not necessarily oriented towards mathematics but you will develop a rich world of flexible, pliable concepts. Movements of the body are hidden thinking. And mathematical thinking is inner movement. Therefore mathematic lessons must contain both flexible and steady concepts. |
>数学一方面是一种思维，另一方面更是生活中的气力。人生之路，启?strong>>身?/strong>>，到达于头脑，换言之，?strong>>身?/strong>>运动到心智思维。这正?/span> 0 >?/span> 3 >岁的儿童向外部世界展示自己的方式：走路，说话，思考?/span>
>数学正面临着相当多的教育改革压力。教师为主导的教学?/span> --- >老师提出题目、回答题目，固然在发展学生对抽象事物的接受能力时，是有效的，但是仍然被越来越多的人以为过期了?/span>
>我们该朝哪个方向走呢?/span> 1998 >年秋天?/span> Oslo >大学?/span> Befring >教授在报纸上发表文章，以为最好完全停止数学课，取而代之的是将数学题目融进到学校的其他课程之中。让学生在“做”的过程中学习数学，这种尝试，在过往几年中得到越来越多的赞同?/span>
>老师可以在教室里布置一个模拟商店，让学生进行买卖交易。假如我们给他们纸和剪刀让他们制作各种立体外形，比起凭空想象，他们将更好地理解体积的计算。这种想法就似乎是通过实际的生活来了解春天一样?/span>
>让孩子独立地练习，就会使得每节课堂都成为他们的经验，引发孩子探索的乐趣?/span>
>孩子必须被准许往观察世界，而不止是从书本中学习这个世界。然后，他们才能发展出规范和抽象的数学语言。我们可从迄今为止的公众讨论中，总结出这一点?/span>
>沿着数学发展史，在更长的标准上，我们能看到一个类似的两极化现象?/span> >在古典时期，直观的几何学，被以为是数学中最重要的部分，数字和计算居于第二位。在中世纪和文艺复兴时期，数学第一次转变为以代数为主导。直?/span> 19 >世纪，这两个学科中的原理才得到综合，并获得了富有成就的结果?/span>
>?/span> 1960 >年代，代数的主导位置重新得以突出。一个国际研究组织试图构建新数学，即，仅用代数思想，往构建由最简单到最难的所有数学概念，数学应该只是思考（代数推理），几何从而被以为是第二位的?/span>
>这个计划被称?/span> >布尔巴基计划?/span> >囊括了很多部著作。在这种思潮的影响下，取名为新数学的教育改革实验?/span> 1970 >年代产生了。其含义是，学生跳过繁重的、耗费时间的初等计算和算术的学习。学生在很早的阶段，确切说是在一年级，就直接进行抽象思考。丈量识数代替初等算术。从这些内容上看，其结果不言自明?/span>
>在过往几十年里，教育界清楚地熟悉到，学生需要从实践中学习算术和几何。通过具体的实际题目，引导大家求解，抽象能力就会在这个过程中自然产生?/span>
>在今天这个数学环境下，很多人在寻找游戏，运动和探索来唤醒孩子的独立思考。然而他们真正理解为什么孩子需要体验性教育吗？身体运动和数学思维是什么关系呢？究竟哪个才是纯粹的思考过程呢?/span>
>我们若对这些题目不具备相当程度的理解，面对改来改往的教育运动，学术界和教育界就会关于其正确性继续争论不休?/span>
>我们能发现出一些观念，用以综合这两个方向吗?/span>
>华德福学校的工作宗旨在于将这两种教学途径兼收并蓄，看作是一种事物的两种不同表达方式。特别是对传统的学龄期在七到十四岁的孩子，他们内心世界的思维，情感和意志是密不可分的?/span>
>思考题目，需要有意识地整合各种想法；运动变化，展现的是内心潜伏的范式。我们影响孩子们终生思维习惯的重要途径是，让他们运动起来，进进范式，然后再通过这种身心合一，来培养孩子们的意志力。（译者注：这跟中国古代内家拳思想是一致的，夸大“以武治心”，而不是“以心治心”，内家拳武术，即是一种人类上千年来积累下的身体范式。本文作者尚未谈及这些范式的具体教法。简单的例子有，让孩子们围着圆圈这种最简单的外形运动，或者再复杂一些，让孩子们围?/span> 8 >字形运动。?/span>
>以这种方式，我们可以以为运动是思维的外部展现，而思维相当于内在的意志力?/span> >孩?/span> 0-3 >岁时，最能清楚地说明这一点。在一岁左右的时候孩子开始站立走路；两岁的时候他们开始说话，三岁的时候开始思考。儿童长大之路，先启于身体运动，后继以心智运动?/span>
>假如你留意到这种联系，你也会熟悉到实在我们的整个童年就是一个转变的过?/span> >。这个过程要经历几年，并且这条路上有明确的站点?/span>
>在发展数学能力的过程中，孩子们的意志力，从跟身体运动有关，越来越多地转变为跟心智运动有关。学习数学，在发展抽象思维能力的角度看，正像是用温度计监测一个长期发展演变的过程，（意思是，数学是抽象思维能力的检测标志）?/span>
>当你进进细节时，意志力将会渐渐解放，固然难以描述，但还是让我们试着规定出一些站点?/span>
>孩?/span> 0-3 >岁学习走路，说话和思考，但是他们不会像成年人那样思考。我们中很少有人记起这段时间发生的事情，而且我们不能解释我们是怎么理解或学习的。记忆和意识会出现，但是很难跟身体这个过程联系起来，实际上在这个年龄我们的想法是和身体、身体运动联系起来的?/span>
>于是，在这初始的几年里，我们就开始了一项人类意识发展的终身主题：走路，说话和思考。孩子真地能在这个年龄数数和计算吗？一般不行，但是其内在基础已经形成。第一次开始体验“等同”，“差异”，“平衡”这些概念，所有这些进步都被父母与孩子间的亲密和爱激励着?/span>
>孩子们在幼儿园、在游戏中、在幻想中度过他们?/span> 4-7 >岁。儿时的我们，难以言说自身的感受——直到我们成年才能理解。我们会玩、会动、会探索。我们大部分的心智活动，概念理解，是用生活中的出色事例作为标签，存于记忆的?/span>
>数学能在孩子年复一年的身体运动中被发现。从会走路到会跑、会跳、会攀爬，这些过程中的每一点，都在启发孩子们往理解平衡和不平衡。孩子们运用身体天生经验，才能唤醒思考，产生思想?/span> >在这个阶段，脑海中积累的是生活中的图片，而概念和道理，是像秘密一样躲于那些图片里的?/span>
>在学校里开始的三到四年，孩子进进一个新阶段。孩子第一次可以描述这个世界。然后生活展现给我们什么了呢？它展现了生活中的形式、数目和场景。源于生活的故事与算术息息相关。当世界被描述的时候，思考随之出现。数数从实物中脱离出来，算术从生活的场景中脱离出来。例如，你进进一个八边形之后，你就能画出它。运动是几何进门的基础，后者即为形线画课程。将乘法结合于跑和跳，就为体会数字节奏提供了基础。我们身体运动提供了我们思维运动的基础?/span>
>?/span> 9 >岁的时候，在孩子的长大中又发生了另一个奔腾。外在事物和内在事物（即心理印象）之间的间隔明显地变大。思维和观念的发展更加能抽离出现实生活中的情境。直接展现在我们眼前的事实具有象征含义，那些世界中的物体提供了分析思考的原料，这个过程在孩子?/span> 12-13 >岁时得到强化。我们的意识离不开“整体到局部，局部到整体”的思考，纯数学为此提供了基础，比如分数的概念。这是孩子们长大的一个主要台阶。他们不能只会将整体分成部分，还要会将部分还原成整体。意识在整体之中通过比较而工作，例如，一个物体中的一部分可能有多大，我的意识会进进到细节和局部，丈量它们，称量它们，比较它们，从我内心具有的对称感判定它们?/span>
12 >岁?/span> 15 >岁之时，思维和观念进一步从生活中解放出来，感性的元素在内心图像中可以更加淡薄。思维逐渐解放，往生活化，这成为抽象思维的基础。现在思维有它自己的世界，青年人的思维开始定型。思想不再囿于身体中的生命过程。抽象思维确实易受僵化教条的危害，只有置身于有意义的内容，才会是积极的、正确的。人类判定性思维是一个高级的过程，必须恰如其分?/span>
>这段时间里，运用日常生活中的题目作为数学的引进点是重要的。有意识地往考察学生的独立性已经成为数学教育的一部分。现在学习代数、几何和方程的时间到了。这个工作建立在运动、节奏、平衡、等价和对称之上，形成孩子们的内在思维特质?/span> >什么是方程而不是内心的平衡？对于理解几何来说对称是多么的重要！常规算术里的活动和变换现在能够得以代数化?/span>
>在青春期后的头几年，大该?/span> 15-17 >岁，我们碰到很大的挑战需要解决。我们怎么防止一个人的世界观成为空虚的抽象教条的集合。为此必须在这两个世界里重新发现相同的动机和真理。生命的气力在童年经验中非常重要，运动，平衡，对称和节奏，重新在抽象思维中得到自然的发现。最好的思维是布满生活想象的游戏，而且也是一条通向真理的道路?/span>
>假如我们以数学的观点认知寓言中的原理，寓教于乐地在实验中重现它们，从而得到范例说明怎样往实现青少年的教学法。简单的概率计算，排列组合及同向指数增长，然后将“黄金分割”作为第三大主题也非常合适。数学是轻易的，恰如你能轻易地发现身体的比例?/span>
>当思考世界的时候，它是有意义的吗，我能创造一张关于世界的有意义的图片吗？或者这个世界是零散的和随机的？我们拥有的知识有极限吗？这些题目和很多更实质的题目在我们年轻的时候被提出来，大概?/span> 17-21 >岁的时候。用一些自相矛盾的题目往挑战我们的思维是很重要的，这些题目不单单能说它是对或是错，而是两者都可以。思维没有给我这个世界的一张完整图片，但却是与这个世界联系的一个基础。我不是我的思维，但自由的思考能给我一个客观的视野往看待这个世界?/span>
>我们有很多感知这个世界的方法，数学只是其中一种。同时我们敢说世界上任何事情都能被分析和研究。任何事物中都有数学。生活本身和自然是能够被判定的，只是暗躲在一层面纱之后。在这段时间，我们研究的是像求导、积分和射影几何这样的题目?/span>
>作为成年人我们有希看达到活跃和富有感情的思维。当然我们也见到相反特点的思维。假如我们能把我们的气力用到正确的方向上，那么我们就会拥有丰富灵活的思维。这种思维不是特指数学思维，但是你将拥有一个活跃弹性的丰富世界?/span>
>身体的运动暗躲着思维。而数学思维是内在的运动。因此数学课必须兼顾灵活和稳定的观念?/span> |

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