《怎样解题》书评

为人师，当如是！

这本书影响了我的生活轨迹。我属于那种没事爱瞎琢磨的人，波利亚没事瞎琢磨琢磨出了一套解题的本领，我没事瞎琢磨啥也没琢磨出来。自初中数学开始，我就没有真正理解数学了，尽管也有一套对付数学题的办法，但是再没有小学时的那种澄明的感觉了。因为那些抽象的概念，一直没有消化好。看完波利亚的书，我知道那些老师全教错了。教数学，教解题，应该象波利亚这样！有一些系统的路数来使我们逐渐接近答案，而不是象一般老师象变魔术一样的介绍突然杀出来的解决办法。在《合情推理》和《数学的发现》里，波利亚细致的展开了这些过程。《数学的发现》没有再版，很可惜，只能去图书馆借。要不是小学教师资格考试的那些考题太无聊，补习老师过于功利，可能我已经在某个小学给孩子们上课了呢。但是波利亚说到的还有一个优点我是做不到的，就是在讲那些你早已了解的内容的过程中，你要有演员的本事，要入戏，好像这个过程你也是第一次经历一样。我明白他是对的，我们以前的语文老师就是这样，在两个班上课，充满感情的讲。后来有个同学不小心在另一个班听了一堂课，差点晕倒，连笑话都是一模一样的，可还是那么投入。这本书值得每个没事瞎琢磨的人参考。

数学老师和数学爱好者应该看的一本书

没有见到这本书之前，看到的只是关于解题的流程图。后来买到了这本书，学习过后真是觉得博大精深，今年是第二遍阅读了。如今搞素质教育，新教材换来换去，说到底还是个教育理念、教育素养问题。为师者应该勤于学习呀

翻来覆去的几句话

解题步骤：熟悉题目深入理解题目寻求有用思路执行方案回顾（能在别处应用这个结果吗？）是否用到所有的已知量？如果不能解所提的题目，先去尝试解某道有关的题目。创造一道相关题目并尝试解答。寻找有关的题目或可能有用的定理为了可能应用它，是否应该引入某个辅助元素？条件有可能满足吗？类比 归纳检验

数学的方法

数学不仅是解题的过程，还是一个思考、沉默、寻找、模拟等等多方面的过程，因此解题是一个令人心驰神往的过程。解题的方法是随着题目的问句，掉头和从头开始，从两个方向，同时思索，有时会发现这两种思索方式会结合在一起，相遇，因此这个题目就得到了解决。

好书、值得一读，但是有些要注意

这本书无疑值得任何一个想提升解题能力的人来阅读，我甚至觉得，我们能在这本书上找到一些解决问题的普遍原则，这些问题并不单指数学问题，我觉得还包括怎样学习＆怎样解决生活中的问题。在书的最开头，作者列出了解题步骤的简单表，我觉得这是这本书最有价值的地方。每个读这本书的人都应该对这个解题表烂熟于心，由此来指导自己解题，肯定会事半功倍。但是应该注意的是，解题是一件实践的事，我认为，任何解题能力的提升都是直接来自于个人在解题过程的体验，换而言之，你不要妄想自己读了这本书之后就能立即获得解题能力的提升。这本书适合在进行了大量解题实践之后来阅读。打个比方，波利亚的这本书是一门高深的内功心法，学习它可以提升内力成为高手，而解题技巧则是武功招数，解题实践则是临阵对敌的经验。如果只练了内功心法，就跟《天龙八部》里获得了活死人百年内力的虚竹一样，不懂得如何运用和驾驭这股内力，反而浑身难受。波利亚的这本书，乃是对解题实践的研究，在他的解题表里强调的一点，就是充分联想自己曾经解决过的问题，然后类比性的解决新题，并在回顾中获得新的解题经验，所以，如果你以为看了这本书就如同踏上了解题的捷径，那就大错特错了，波利亚只是告诉你：有这么一条捷径，但是你自己得找到它。这本书是对解题方法的梳理和抽象归纳，而不是直接告诉你解题方法或技巧。另外我觉得这本书中所使用的例题有许多并不适合我们中国的学生，我指的是高中生和大学本科生。一方面，我觉得其中有些例题太富于想象力或者天马行空了，而我们这些学生在现行的教育体制下，获得了充分的基础和解题技巧，想象力则因此而受到了压抑，涉世浅点染亦浅，涉世深机械亦深。就像我们中国的学生学到了很扎实的武功套路，而现在波利亚就像张三丰般的告诉我们无招胜有招，这很让人摸不着头脑而难以理解。在这里，我希望我所说的不会成为一些人用以来抨击我国教育制度的弊端，事实上，美国人非常崇拜我们的中学教育，我们教育的问题，在于太重视关于学习的教育，而忽略了关于怎样生活怎样做人的教育。另一方面，或许会有人说这本书那正好提升我们的想象力，补我们的短，那我要提醒的是，我们阅读这本书的目的乃是获得解题能力的提升，这才是这本书的优点，而太过于想象力（我说的是对于中国的学生的大多数）则是它的缺点，如果我们硬跟着它的缺点死磕而忽略吸取它的精华，那就真的太傻了。就例题方面来说，我推荐欧阳维城的《初等数学解题方法研究》，这本书我初中时看的，获益匪浅，而且这本书比波利亚的书更具体，两本书参考来读，效果会更好的

怎样解决一个问题

在问题解决的背后，思维过程是有规律可循的，这正是波利亚在做的事情。在我读到《怎样解题》之前，从未思考过解题背后的思维过程规律，只是觉得很奇怪——为什么别人一下子就能想到解题方法，而我却要想好久。当我根据这张解题表细想我解决一道数学题的过程，我发现好像真的是这样。在最终找到解题方法之前，有一段关于此题的联想过程——或想到很多关于这道题目的东西，也就是波利亚所说的“寻找已知和未知的联系”，只是我从来不知道要刻意去这么做。联想，寻找联系，是解决问题最关键的步骤。会做题和不会做题的差别，或许就在前者可以联想到相关的题目、定理、结论并加以应用，后者却不能把不同的东西联系起来。所以，怎样解题，怎样提高解题能力，关键可能在于回顾这个步骤。如果能够养成回顾的习惯，总是去思考是否有更简洁的答案、问题的本质是什么、如何利用这个答案，可以对问题有更深的认识，也就是抽象出问题的本质——而本质更容易被回忆起来，更容易被应用于另一个问题。如果有一种万能的办法，或者说是程序，按照程序一步步去运行，然后就能得到问题的答案，那每一个人都是天才。可惜没有。所以，《怎样解题》更加不能帮你做到这一步，但它为你指出了一路捷径，一条思维的捷径。

一本指导方法的书，你还需要动手做

这本书只是给了我们一些解题的指导方法，其实更重要的是去实践这些方法，于是大家就需要去做POLYA和另外一个作者合写的两卷本：Problems and Theorems in Analysis, Volume I，II。我相信如果大家可以认真做完的话，那你的解题水平真的会有质的飞跃。我很后悔之前没有好好地花时间把它做完。国内的徐利治和王文华老师，他们也编过一本很好的习题集：《数学分析的方法及例题选讲》

foreword by J.H.Conway

（不知道为什么中文版没有这篇前言。中间有一段Polya的生平这里省略了，有兴趣的可以自己谷歌一下。）"How to Solve It" is a wonderful book! This I realized when I first read right through it as a student many years ago, but it has taken me a long time to appreciate just how wonderful it is. Why is that? One part of the answer is that the book is unique. In all my years as a student and teacher, I have never seen another that lives up to George Polya's title by teaching you how to go about solvingproblems. A. H. Schoenfeld correctly described its importance in his 1987 article "Polya, Problem Solving, and Education" inMatheraatics Magazine. "For mathematics education and the world of problem solving it marked a line of demarcation between two eras, problem solving before and after Polya." It is one of the most successful mathematics books ever written, having sold over a million copies and been translated intoseventeen languages since it first appeared in 1945. Polya later wrote two more books about the art of doing mathematics, Matheraatics and Plausible Reasoning (1954) and Mathematical Discovery (two volumes, 1964 and 1965). The book's title makes it seem that it is directed only toward students, but in fact it is addressed just as much to their teachers. Indeed, as Polya remarks in his introduction, the first part of the book takes the teacher's viewpoint more often than the student's. Everybody gains that way. The student who reads the book on his own will find that overhearing Polya's comments to his non-existent teacher can bring that deskable person into being, as an imaginary but very helpful figure leaning over one's shoulder. This is what happened to me, and naturally I made heavy use of the remarks I'd found most important when I myself started teaching a few years later. But it was some time before I read the book again, and when I did, I suddenly realized that it was even more valuable than I'd thought! Many of Polya's remarks that hadn't helped me as a student now made me a better teacher of those whose problems had differed from mine. Polya had met many more students than I had, and had obviously thought very hard about how to best help all of them learn mathematics. Perhaps his most important point is that learning must be active. As he said in a lecture on teaching, "Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems." It is often said that to teach any subject well, one has to understand it "at least as well as one's students do." It is a paradoxical truth that to teach mathematics well, one must also know how to misunderstand it at least to the extent one's students do! If a teacher's statement can be parsed in two or more ways, it goes without saying that some students will understand it one way and others another, with results that can vary from the hilarious to the tragic. J. E. Littlewood gives two amusing examples of assumptions that can easily be made unconsciously and misleadingly. First, he remarks that the description of thecoordinate axes ("Ox and Oy as in 2 dimensions, Oz vertical") in Lamb's book Mechanics is incorrect for him, since he always worked in an armchair with his feet up! Then, after asking how his reader would present the picture of a closed curve lying all on one side of its tangent, he states that there are four main schools (to left or fight of vertical tangent, or above or below horizontal one) and that by lecturing without a figure, presuming that the curve was to the fight of its vertical tangent, he had unwittingly made nonsense for the other three schools. I know of no better remedy for such presumptions than Polya's counsel: before trying to solve a problem, the student should demonstrate his or her understanding of its statement, preferably to a real teacher, but in lieu of that, to an imagined one. Experienced mathematicians know that often the hardest part of researching a problem is understanding precisely what that problem says. They often follow Polya's wise advice: "If you can't solve a problem, then there is an easier problem you can't solve: find it."  ………………………………………………How to Solve It was written in German during Polya's time in Zurich, which ended in 1940, when the European situation forced him to leave for the United States. Despite the book's eventual success, four publishers rejected the English version before Princeton University Press brought it out in 1945. In their hands,How to Solve It rapidly became--- and continues to be ---one of the most successful mathematical books of all time.