Author: John holt
Source: What Do I Do Monday
Translator: Esther
Let me sum up what I have been saying about learning. I believe that we learn best when we, not others, are deciding what we are going to try to learn, and when, and how, and for what reasons or purposes; when we, not others, are in the end choosing the people, materials, and experiences from which and with which we will be learning; when we, not others, are judging how easily or quickly or well we are learning, and when we have learned enough; and above all when we feel the wholeness and openness of the world around us, and our own freedom and power and competence in it. What then do we do about it? How can we create or help create these conditions for learning?
让我总结一下我所讲的关于学习的内容。我相信当我们——而不是其他人——决定我们要学什么、何时学、如何学、出于什么原因或目的而学，当我们——而不是其他人——最终选择跟谁学，用什么资料和经验学；当我们——而不是其他人——判断我们学得多容易、多快、多好，何时算是学到位；最重要的，当我们感受到周围世界的整体性与开放性，感受到我们身在其中的自由、权利和能力时，我们学得最好。那么，我们该怎么做？我们如何创造或协助创造这些学习条件？
Perhaps I can make more clear what I mean by the wholeness of learning or experience by talking about my own discovery of mathematics. At school, I was always a fairly good math student. It bored me, but it didn’t scare me. With any work at all, I could get my B. But after many years I knew that although I could do most of the problems and proofs and remember the theorems and for-mulas, I really didn’t have the slightest idea what it was all about. That is, I didn’t see how it related to anything—where it had come from, what it was for, what one might ever do with it.
或者我可以通过讲述我自己的数学发现将我所谓的学习或经验整体性讲述得更清楚。上学时，我数学一直都不错。我也觉得枯燥，但并不怵它。稍微做点功课，我就能得个B。可多年之后，我知道虽然我能够把大部分题目解答证明出来，也记得定理和公式，可压根不知道数学到底是什么。即，我不明白数学和任何东西有什么联系——数学起源自何处，存在的意义如何，有什么用。
Some years after I left the Navy I came across a series of books, written to help people with little or no math training under-stand some of the new and large ideas in mathematics. They were written by a Mr. and Mrs. Lieber. The first of them was The Educa-tion of T. C. Mits. There was a character called SAM, whose initials stood for Science, Art, and Mathematics. The point of the books was that people should not be afraid of new ideas in these fields, and that if they took the plunge, exposed themselves to them, they would find them not so terrifying or difficult.
我离开海军几年之后，看到了一套书——帮助那些有一点或者根本没有数学基础的人理解数学中的重要概念。该系列由Lieber夫妇所著。第一本是《T.C.Mits的教育》，里面有个人物叫萨姆（SAM），是科学、艺术和数学的首字缩写。该套丛书的主旨是人们不应害怕这些领域里的新概念，如果他们勇敢尝试，勇敢面对，就会发现数学既不可怕也不困难。
The books themselves were very well done. Mr. and Mrs. Lieber, in one sense at least, were excellent teachers. They would have been very good at writing out programs. They understood how easily and quickly a learner, moving into new territory, is frightened by uncertainty, contradiction, or logical steps that cover too much ground. So they were very careful to define their terms in words the learner would understand, to move ahead slowly and patiently, taking time to illustrate their points and to reassure the reader. Anyone who didn’t panic could follow them through their argument.
书本身写得非常好。Lieber夫妇至少在某种意义上是非常出色的老师。他们要是写教学大纲，肯定写得不错。他们了解学习者在踏进新领域时，很容易立即就被遍地皆是的不确定性、矛盾或逻辑步骤吓到。所以他们小心谨慎地用学习者理解的词语定义名词术语，缓慢而耐心地前进，花时间图解自己的观点，花时间给读者打气。任何耐住性子的人都跟得上他们的推理。
But at the end of each of their books, though I had enjoyed being able to follow them on their journey, and liked the feeling of knowing something I hadn’t known before, I was still uneasy, dis-satisfied. I was not sure why. It seemed that there must be more to this new idea than I had been told. I was not able to bring my un-ease into focus, to get hold of it, find words for it, until I had fin-ished their book on Galois and the Theory of Groups. I had been able to follow them, step by step, to the end of the book. But at the end I felt as if I had been blindfolded and then led along a care-fully prepared path. “Now put your foot here, easy now, that foot there …” I didn’t stumble, but I wanted to take the blindfold off and say, “Where are we, anyway? How did we get here? Where are we going?” What had led Galois to invent this theory? What had made it seem worth inventing? Had he been working on a problem that he and others had not been able to solve? What was the prob-lem, what had he and the others been doing to try to solve it, what had started him in this direction? As it was presented to me, the Theory of Groups seemed disconnected from everything, or at least anything I could imagine. And once Galois had started to work on it, had he made any false starts, gone down any dead ends? Or did he go straight along, like the Liebers? And then, when he got the theory worked out, came to where I was at the end of the book, what did he do with it, how did he use it, where did he go next? Did it help him with the problem he had been trying to solve, and how?
可是在每本书读完的时候，虽然我很享受能够跟上他们的旅途，也喜欢那种了解原本不知道的东西的感觉，我还是觉得不满意。我不清楚这是为什么。看上去，关于这个新观点，肯定还有很多我不知道的东西。在读他们关于伽罗瓦及阶群理论之前，我无法理解自己的这种不满足，既无法掌握也无以言表。我能够一步一步地跟着他们把书读完。可最后，我觉得自己好像被蒙上了眼睛，让人领着我沿一条精心准备的路径前进。“现在，把脚放在这里，慢点，那只脚踩那里…..”我倒没有摔跟头，可是我想把眼罩取掉，问问，“可我们到底在哪？我们怎么来的？我们要去哪里？”是什么导致伽罗瓦发明这条理论？是什么让这理论有价值？他是不是在解决他和其他人未能解决的问题？那又是什么问题，他和别人做了哪些事情来解决这个问题？是什么促使他朝着这个方向努力？因为就我所见，阶群理论和任何事情都毫不相关，至少是我能够想象到的任何事！一旦伽罗瓦开始研究，他有没有犯过错误，最终走进了死胡同？还是他长驱直入，像Lieber夫妇一样？然后，当他研究出这一理论，到了我在书末所处的境况，他用它做什么了呢？他怎么用？后来他又研究了什么？这对他一直在解决的问题有没有帮助，又是如何帮助的？
In short, I felt like saying to my patient and hard-working guides, the Liebers, “Thanks for your help, but you haven’t told me anything important, you’ve left out the best part.”
简言之，我想对极具耐心且努力的向导Lieber夫妇说，“谢谢你们的帮助，可你们没有告诉我任何重要的东西。你们落下了最好的东西。”
Some years later, a former pupil and good friend of mine, then at college, was meeting calculus for the first time. Like many people, he was having trouble. He had the feeling I had had years before of being able to go through the motions, writing formulas and doing problems, but without any idea of what they were all about, seeing them only as a kind of mumbo-jumbo, meaningless recipes for getting meaningless answers to meaningless questions. He asked me one day if I would try to make some sense of it for him. I said I would. I began by trying to give him a very rough idea of the problem, philosophical as much as mathematical, that had started man on his search for the calculus. (What little I knew about all this I had picked up after I left school.) So I talked about the Greeks trying to think about instantaneous motion, described some of the Paradoxes of Zeno—the arrow, Achilles and the tor-toise, etc. At any instant the arrow is not moving, since motion is distance covered in time; but then, since time is made up of a sum of instants, how can motion be possible? It is easy to say, if a car traveled five miles in ten minutes, its average speed in that time was thirty miles per hour. But what does it mean to ask how fast it is going at any instant, and how can we find out?
几年之后，我从前的一位学生，也是朋友，当时在大学初次接触微积分。和很多人一样，他遇到了麻烦。他跟我多年前的感觉完全相同，觉得自己能够跟上进度，能写公式，也能解题，可对于这些到底是什么，却摸不着头脑，只能当天书，当做是解决毫无意义的问题的毫无意义的答案的毫无意义的秘诀，仅此而已。有天，他问我，能不能帮他理清这些东西。我说可以。开始时，我尝试让他从哲学和数学两方面对促使人类开始研究微积的问题有个大概的了解。（毕业后，我又重新拾起了对这个问题微乎其微的了解。）所以，我谈到了希腊人想了解瞬时运动，讲了讲齐诺悖论——箭、阿基里斯和乌龟等等。在任一瞬间箭是不动的，因为运动是一定时间内走过的距离；可由于时间等于瞬时之和，怎么可能有运动呢？如果一辆车10分钟内开了5英里，其平均速度为每小时30英里。可问在任何瞬时车速如何的意义是什么，我们又如何解决？
My friend saw the sharpness of the dilemma. I then showed how Cartesian or coordinate geometry made it easier to think about the problem, and thus prepared the way for men to solve it, by giving us a way to make a picture or map of something moving at various rates in space and time. We simply plot a graph of dis-tance traveled against time. It could then be seen that the average speed between two points could be seen as the slope of the line joining them on the graph. From there we could see that the ques-tion: How fast is this object going at a particular instant?, could be asked as: What is the slope of the curve, or the tangent to the curve, at that particular point? We had then to find out what hap-pened to that slope as the interval of time became smaller and smaller, and indeed what it meant to have something approach zero as a limit. My friend and I did some arithmetic, some algebra, derived the general formula for the differential at a point—all stuff he had had in the course. But now he said, “So that’s it. Why didn’t anybody tell me that? It’s so simple when you see what it’s about.”
我的朋友看到了这一难题的尖锐性。接着，我展示了笛卡尔的坐标几何如何通过绘制一幅某个东西在时空以不同速度运动的图像，将这个问题的思考变得容易，从而为人们解决这个问题准备了道路。我们画了一幅图，表示在某段时间内走过的距离。可以这样理解，两点之间的平均速度可以看做图上将其连接的线段坡度。从这，我们可以明白，“在某个具体瞬间，这个物体移动得多快？”这一问题也可以这样问，在某一点曲线弧度是多少，或者曲线正切是多少？我们必须得理解随着时间间隔越来越小，坡度变化如何。还有，当某样东西接近零极限时，到底意味着什么。我和我的朋友做了些算术和代数，得到了某一点的微分公式——这些都是他上课学过的。但现在他说，“原来如此。怎么以前没人告诉我呢？你要是明白了原理，就变得易如反掌。”
Exactly. What I had done, clumsily enough, was not to try to hand him a lump of knowledge, which people had already handed him and which he could not take hold of, but to take him on a kind of human journey with the people who had first thought about and discovered these things.
没错。我所做的，并非传授给他海量知识——人们已经向他传授了他无法掌握的东西——而是带着他与那些最初思考发现这些问题的人们一起开始人类之旅。
