INTUITIVE AND ANALYTIC THINKING
直觉思维和分析思维
MUCH has been said in the preceding chapters about the importance of a student's intuitive, in contrast to his formal, understanding of the subjects he encounters. The emphasis in much of school learning and student examining is upon explicit formulations, upon the ability of the student to reproduce verbal or numerical formula. It is not clear, in the absence of research, whether this emphasis is inimical to the later development of good intuitive understanding-indeed, it is even unclear what constitutes intuitive understanding. Yet we can distinguish between inarticulate genius and articulate idiocy-the first represented by the student who, by his operations and conclusions, reveals a deep grasp of a subject but not much ability to "say how it goes," in contrast to the student who is full of seemingly appropriate words but has no matching ability to use the ideas for which the words presumably stand. A careful examination of the nature of intuitive thinking might be of great aid to those charged with curriculum construction and teaching.
在前面的章节中，已经大量探讨了学生直觉思维的重要性，学生的直觉思维和他们所接触学科的正式形式相反。学校学习和学生考试，非常注重清晰的系统陈述以及学生用言语或数字形式进行复现的能力。着重这些方面，是否会妨碍以后直觉理解的健全发展呢？由于缺乏研究，并不清楚。——的确，连直觉理解究竟由什么构成还不了然。不过，我们仍然能够从表达思想不清楚的天才中区分出清楚表达思想的白痴——表现为前者的学生，通过运算和结论，显示出对学科的深刻领会，但不大能够“说出它是怎样的”。这跟另一种学生不同，后者看上去充满恰当的词汇，却没有相应能力去利用这些词汇所代表的观念。对于承担课程编制和教学的人而言，仔细地考察直觉思维的本质的可能大有助益。
Mathematicians, physicists, biologists, and others stress the value of intuitive thinking in their respective areas. In mathematics, for example, intuition is used with two rather different meanings. On the one hand, an individual is said to think intuitively when, having worked for a long time on a problem, he rather suddenly achieves the solution, one for which he has yet to provide a formal proof. On the other hand, an individual is said to be a good intuitive mathematician if, when others come to him with questions, he can make quickly very good guesses whether something is so, or which of several approaches to a problem will prove fruitful.
数学家、物理学家、生物学家和其余人士在他们各自的领域里，都强调直觉思维的价值。例如，数学中的直觉概念是从两种不同的意义上来使用的；一方面，说某人是直觉地思维，意即他花了许多时间做一道题目，突然间他做出来了，但是还须为答案提出形式证明。另一方面，说某人是具有良好直觉能力的数学家，意即当别人向他提问时，他能够迅速作出很好的猜测，判定某事物是不是这样，或说出几种解题方法中最终证明有效的一种。
The development of effectiveness in intuitive thinking is an objective of many of the most highly regarded teachers in mathematics and science. The point has been repeatedly made that in the high school plane geometry is typically taught with excessive emphasis upon techniques, formal proofs, and the like, that much more attention needs to be given to the development of students who have a good intuitive feel for geometry, students who are skillful in discovering proofs, not just in checking the validity of or remembering proofs with which they have been presented. There has been very little done, for example, on the use of diagrams as geometrical experiments as in Hilbert and Cohn's Geometry and the Imagination, in which visual proof substitutes for formal proof where possible. Similarly, in physics, Newtonian mechanics is typically taught deductively and analytically. In the judgment of many physicists, at least, there is too little attention to the development of intuitive understanding. Indeed, some have suggested that improving the use of intuitive thinking by teachers is as much a problem as improving its use by students.
直觉思维效果的发展是许多在数学和自然科学方面受到高度尊敬的教师们的一个目标。人们反复地指出，在中学平面几何学教学中的典型是，过分强调了对技巧、形式的证明；对几何学有着良好直觉感的学生，以及有本领发现证明方法而不只是去验算向他们提出的证明是否确实或记住这些证明的学生，需要多多注意他们的发展。例如，在希尔伯特和科恩《几何学与想象》一书中，采用图解进行几何学试验，尽可能用视觉证明替代形式证明，象这样的试验，实在做得很少。同样，在物理学中，牛顿力学也是典型地按演绎和分析方法进行教学。许多物理学家认为，至少而言，对直觉理解发展的的关注太少了。确实，有些人已经指出，改进教师对直觉思维的运用，如同改造学生对直觉思维的运用一样，是有待解决的问题。
Yet, as one member of the Conference put it, it is wrong to look at intuition as "all ala mode and no pie." The good intuiter may have been born with something special, but his effectiveness rests upon a solid knowledge of the subject, a familiarity that gives intuition something to work with. Certainly there are some experiments on learning that indicate the importance of a high degree of mastery of materials in order to operate effectively with them intuitively.
然而，正象会议中的一位成员所指出，把直觉看做“徒有形式而无内容”是错误的。直觉好的人可能生来有点特殊，但这些特殊性的效果如何，则有赖于对学科的牢固知识；熟悉学科知识能让直觉有所作为。真的，一些关于学习的实验表明，为了有效地用直觉方法运用材料，精通材料是重要的。
Those concerned with the improvement of curricula in physics and mathematics particularly have often cited as one of their important aims the use of procedures that will contribute to the improvement of intuitive thinking. In their attempts to design such procedures, there has been a question of the kind of systematic psychological knowledge that would be of help. Unfortunately, little systematic knowledge is available about the nature of intuitive thinking or the variables that influence it. What seems most appropriate at this point, therefore, is an attempt to outline the kinds of research which, if even only partially carried out, would begin to provide information useful to those concerned with the improvement of particular courses or, more generally, of the curriculum as a whole. What kinds of questions do we need the answers to?
同改进物理课程和数学课程特别有关的人们，经常把有助于改善直觉思维的程序的运用列为他们的重要目标之一。在他们试图设计这样的程序时，就发生了哪种系统的心理学知识有用的问题。可惜，关于直觉思维的性质和影响直觉思维的可变因素，合用的系统知识极少。所以，在这里最恰当的做法，似乎是试图勾划出一些研究工作的轮廓，这些研究工作，即使只进行一部分，也将开始提供与特殊课程改革、或者更一般地与整个课程改革有关的人们以有用的资料。我们要求回答的到底是些什么问题呢？
Questions about the nature of intuitive thinking seem to center upon two large issues: what intuitive thinking is, and what affects it.
关于直觉思维的性质问题好象集中在两个大的题目上:什么是直觉思维T 影响直觉思维的又是什么？
One can say many more concrete things about analytic thinking than about intuitive thinking. Analytic thinking characteristically proceeds a step at a time. Steps are explicit and usually can be adequately reported by the thinker to another individual. Such thinking proceeds with relatively full awareness of the information and operations involved. It may involve careful and deductive reasoning, often using mathematics or logic and an explicit plan of attack. Or it may involve a stepby- step process of induction and experiment, utilizing principles of research design and statistical analysis.
人们对分析，思维，可以说出比直觉思维多得多的具体情况。分析思维是以一次前进一步为其特征的。步骤是明显的，而且常常能由思维者向别人作适当报道。在这类思维进行的过程中，人们能比较充分地意识到所包含的知识和运算。它可能包含仔细的和演绎的推理，因为它往往使用数学或逻辑以及明确的进行计划。或者，它也可能包含逐步的归纳和试验过程，因为它利用了研究设计和统计分析的原理。
In contrast to analytic thinking, intuitive thinking characteristically does not advance in careful, welldefined steps. Indeed, it tends to involve maneuvers based seemingly on an implicit perception of the total problem. The thinker arrives at an answer, which may be right or wrong, with little if any awareness of the process by which he reached it. He rarely can provide an adequate account of how he obtained his answer, and he may be unaware of just what aspects of the problem situation he was responding to. Usually intuitive thinking rests on familiarity with the domain of knowledge involved and with its structure, which makes it possible for the thinker to leap about, skipping steps and employing short cuts in a manner that requires a later rechecking of conclusions by more analytic means, whether deductive or inductive.
直觉思维与分析思维迥然不同，它不是以按仔细的、规定好的步骤前进为其特征的。的确，它倾向于从事看来是以对整个问题的内隐的感知为基础的那些活动。思维者虽然得到答案（不管正确还是错误），但他对其间的过程究竟如何，却很少知道。他难以说明是怎样获得答案的，而且他也许不知道他所回答的问题情况是什么样子的。直觉思维总是以熟悉牵涉到的知识领域及其结构为根据，使思维者可能实行跃进、越级和采取捷径，多少需要以后用比较分析的方法——不论演绎法或归纳法，重行检验所作的结论。
The complementary nature of intuitive and analytic thinking should, we think, be recognized. Through intuitive thinking the individual may often arrive at solutions to problems which he would not achieve at all, or at best more slowly, through analytic thinking. Once achieved by intuitive methods, they should if possible be checked by analytic methods, while at the same time being respected as worthy hypotheses for such checking. Indeed, the intuitive thinker may even invent or discover problems that the analyst would not. But it may be the analyst who gives these problems the proper formalism. Unfortunately, the formalism of school learning has somehow devalued intuition. It is the very strong conviction of men who have been designing curricula, in mathematics and the sciences particularly, over the last several years that much more work is needed to discover how we may develop the intuitive gifts of our students from the earliest grades onwards. For, as we have seen, it may be of the first importance to establish an intuitive understanding of materials before we expose our students to more traditional and formal methods of deduction and proof.
我们认为，应该承认直觉思维和分析思维的相互补充的性质。一个人往往通过直觉思维对一些问题获得解决，而这些问题如果借助分析思维将无法解决，或者充其量也只能慢慢解决。这种解决，一旦用直觉方法获得，可能的话，就应当用分析方法进行检验；同时，把它们看作这种检验的有价值的假设。的确，直觉思维者甚至可以发明或发现分析家所不能发现的问题。可是，给这些问题以恰当的形式体系的，也许还是分析家。可惜，学校学习中的形式主义已经或多或少贬低了直觉的价值。过去几年来一直在从事设计课程，尤其是从事设计数学和自然科学课程的人，都坚决相信需要做更多得多的工作去发现，我们怎样才有可能从最早年级起便开始发展学生的直觉天赋。因为，正象我们已经看到的，在我们向学生揭示演绎和证明这种更传统的和更正式的方法以前，使其对材料能有直觉的理解，可能是头等重要的。
As to the nature of intuitive thinking, what is it? It is quite clear that it is not easy either to recognize a particular problem-solving episode as intuitive or, indeed, to identify intuitive ability as such. Precise definition in terms of observable behavior is not readily within our reach at the present time. Obviously, research on the topic cannot be delayed until such a time as a pure and unambiguous definition of intuitive thinking is possible, along with precise techniques for identifying intuition when it occurs. Such refinement is the goal of research, not its starting place. It suffices as a start to ask whether we are able to identify certain problem-solving episodes as more intuitive than others. Or, alternatively, we may ask if we can learn to agree in classifying a person's style or preferred mode of working as characteristically more analytic or inductive, on the one hand, or more intuitive, and, indeed, if we can find some way to classify tasks as ones that require each of those styles of attack. It is certainly clear that it is important not to confuse intuitive and other kinds of thinking with such evaluative notions as effectiveness and ineffectiveness: the analytic, the inductive, and the intuitive can be either. Nor should we distinguish them in terms of whether they produce novel or familiar outcomes, for again this is not the important distinction.
至于直觉思维的性质，到底是什么呢？很清楚，无论是把特定的解决难题的活动认作直觉的，或者是确实地鉴别直觉能力本身，都并不容易。根据可以观察到的行为来下一个精确的定义，在目前是我们力所不及的。显然，我们不能等到可以给直觉思维下一个纯粹而不含糊的定义，并对出现的直觉有鉴别的精密技术时，才来研究这个题目。这种精密性是研究的目标，而不是据以进行研究的起点。如果要探究我们是否有能力鉴定某些解决难题的活动比别的更为直觉，那么这种精密性就可作为起点。或者，另一种做法是，我们可探究能否同意把一个人的作风或偏爱的工作方式从性质上划分为更加分析或归纳的还是更为直觉的，而且更进一步，探究我们能否找出一个将任务进行分类的方法，看每个任务需要哪一种工作方式。确实很清楚，要紧的是不使用像有效或无效这类评价概念来混淆直觉思维和其他类型的思维，因为分析的思维、归纳的思维与直觉的思维都可能有效或无效。也不应该根据能否产生新的或熟悉的成果来区分这两种类型的思维，因为这不是重要的差别。
For a working definition of intuition, we do well to begin with Webster: "immediate apprehension or cognition." "Immediate" in this context is contrasted with "mediated"-apprehension or cognition that depends on the intervention of formal methods of analysis and proof. Intuition implies the act of grasping the meaning, significance, or structure of a problem or situation without explicit reliance on the analytic apparatus of one's craft. The rightness or wrongness of an intuition is finally decided not by intuition itself but by the usual methods of proof. It is the intuitive mode, however, that yields hypotheses quickly, that hits on combinations of ideas before their worth is known. In the end, intuition by itself yields a tentative ordering of a body of knowledge that, while it may generate a feeling that the ordering of facts is self-evident, aids principally by giving us a basis for moving ahead in our testing of reality.
关于直觉的界定说，我们正好先用韦伯斯特的解释:“(直觉就是)直接了解或认知。”按“直接”这个词是同“间接”——靠正式的分析法和证明法为中介所获得的了解或认知——相对而言的。直觉是一种行为，通过这种行为，人们可以不必明显地依靠其分析技巧而掌握问题或情境的意义、重要性和结构。直觉的正确或错误最后取决于通常的证明法而不是取决于直觉本身。然而，直觉的形式能很快产生假设，且在知道观念组合的价值之前，便发现观念的组合。最后，直觉本身可以产生一类知识的一个试验性的组织，同时可以造成一种感觉，使我们觉得那些事实如此组织起来是不言而喻的；它对我们的帮助主要在于给我们提供一种根据，使我们得以在检验现实的过程中前进。
Obviously, some intuitive leaps are "good" and some are "bad" in tenns of how they turn out. Some men are good intuiters, others should be warned off. What the underlying heuristic of the good intuiter is, is not known but is eminently worthy of study. And what is involved in transforming explicit techniques into implicit ones that can be used almost automatically is a subject that is also full of conjecture. Unquestionably, experience and familiarity with a subject help-but the help is only for some. Those of us who teach graduate students making their first assault on a frontier of knowledge are often struck by our immediate reactions to their ideas, sensing that they are good or impossible or trivial before ever we know why we think so. Often we turn out to be right; sometimes we are victims of too much familiarity with past efforts. In either case, the intuition may be weeks or months ahead of the demonstration of our wisdom or foolhardiness. At the University of Buffalo there is a collection of successive drafts of poems written by leading contemporary poets. One is struck in examining them by the immediate sense one gets of the rightness of a revision a poet has made-but it is often difficult or impossible to say why the revision is better than the original, difficult for the reader and the poet alike.
显然，直觉的跳跃有些是“好的”，也有些是“不好的”，这是按照它们的结果如何来定的。有些人是良好的直觉者，有些人则应预先提醒他们才行。根据什么来发现良好的直觉者呢？这点还不知道，但显然是值得去研究的。再者，把外现的技术转换成几乎能够自动地运用的内隐的技术，到底包含些什么，这也是个充满着猜测的课题。毫无疑问，经验和熟悉该学科是有些帮助的，但是，它只对有些人有帮助。我们中有的人教研究生初次突击新的知识领域时，在我们感到他们的观念是好的或是微不到的，或是浅薄的，可是又不知道我们为什么这样想之前，往往为我们对他们的观念的直接反应所震惊。结果常常证明我们是正确的；但有时候我们却成为过分习惯于过去的努力的受害者。这两种情形中的直觉，都可能走在我们显示聪明或蛮勇之前好几周或好几月。在法布罗大学，收藏了当代第一流诗人所写的诗稿和修改稿。令人震惊的是，在对诗稿进行研究时，立刻感觉诗人的修改是正确的——不过，经常难以说出或说不出为什么修正的比原来的好；而要做到这一点，无论对读者还是对诗人来说都是困难的。
It is certainly clear that procedures or instruments are needed to characterize and measure intuitive thinking, and that the development of such instruments should be pursued vigorously. We cannot foresee at this stage what the research tools will be in this field. Can one rely, for example, upon the subject's willingness to talk as he works, to reveal the nature of the alternatives he is considering, whether he is proceeding by intuitive leaps or by a step-by-step analysis or by empirical induction? Or will smaller-scale experimental approaches be suitable? Can group measurement procedures involving pencil and paper tests be used to provide a measure? All of these deserve a try.
的确很清楚，描绘测量直觉思维需要一定的程序或工具，而且应该大力发展这种工具。我们不能在现阶段预见在这个领域里将要用什么研究工具。例如，我们能否依赖受试者在工作中显出思维选择性时的谈话来判断他到底是靠直觉的跳跃，还是靠一步步的分析，抑或是靠经验的归纳来进行思维的呢？采用小规模的实验方适合适吗？ 能否用团体测量程序包括笔和纸的测验来提供某种测度呢？所有这些都值得试一试。
What variables seem to affect intuitive thinking? There must surely be predisposing factors that are correlated with individual differences in the use of intuition, factors, even, that will predispose a person to think intuitively in one area and not in another. With respect to such factors, we can only raise a series of conjectures. Is the development of intuitive thinking in students more likely if their teachers think intuitively? Perhaps simple imitation is involved, or perhaps more complex processes of identification. It seems unlikely that a student would develop or have confidence in his intuitive methods of thinking if he never saw them used effectively by his elders. The teacher who is willing to guess at answers to questions asked by the class and then subject his guesses to critical analysis may be more apt to build those habits into his students than would a teacher who analyzes everything for the class in advance. Does the providing of varied experience in a particular field increase effectiveness in intuitive thinking in that field? Individuals who have extensive familiarity with a subject appear more often to leap intuitively into a decision or to a solution of a problem-one which later proves to be appropriate. The specialist in internal medicine, for example, may, upon seeing a patient for the first time, ask a few questions, examine the patient briefly, and then make an accurate diagnosis. The risk, of course, is that his method may lead to some big errors as well-bigger than those that result from the more painstaking, stepby- step analysis used by the young intern diagnosing the same case. Perhaps under these circumstances intuition consists in using a limited set of cues, because the thinker l{nows what things are structurally related to what other things. This is not to say that "clinical" prediction is better or worse than actuarial prediction, only that it is different and that both are useful.
影响直觉思维的可变因素究竟有哪些？肯定会有带倾向性的因素，这些因素和直觉运用中的个别差异有关。这种倾向性因素甚至使人倾向于对某个领域而不对别的领域进行直觉思维。关于这些因素，我们只能举出一系列的推测。如果教师直觉地思维，学生的直觉思维会不会更发展？也许包含着简单的模仿，也许包含着更复杂的过程。如果学生从未见到他的长辈有效地运用直觉的思维方法，他似乎未必会发展或相信自己的这种方法。凡是乐意猜测班上提出的问题的各种答案，而后对他的猜测作严格分析的那种教师，恐怕比预先给全班分析一切的教师，更易于养成学生这种思维习惯。给某个领域提供各种经验，会增进那个领域内直觉思维的效果吗？多方面熟悉某门学科的人们，往往更能凭直觉一下子作出决断或解答疑难——后来证明这个决断或解答是恰当的。例如，内科专家第一次看某个病人，他可能问问病情，简短地检查一下，旋即作出准确的诊断。当然，也许会冒风险，风险就是他的办站可能导致大的错误——大于年轻实习医师诊断同样病人时使用较费力的、一步步的分析而得出的结论。在这种情况下，直觉大概是利用一套有限的线索，因为思维者知道什么事情在结构上同其他什么事情有关系。这不是说，“临诊”的预料比保险统计的预料来得好，或来得差。只不过是说，两者有所不同，并且，两者都是有用的。
In this connection we may ask whether, in teaching, emphasis upon the structure or connectedness of knowledge increases facility in intuitive thinking. Those concerned with the improvement of the teaching of mathematics often emphasize the importance of developing in the student an understanding of the structure or order of mathematics. The same is true for physics. Implicit in this emphasis, it appears, is the belief that such understanding of structure enables the student, among other things, to increase his effectiveness in dealing intuitively with problems.
关于这一点，我们可以问，在教学上强调知识的结构或连结性，能否促进直觉思维。那些从事改进数学教学的人，经常强调发展学生对数学的结构或顺序的理解的重要性。就物理学而论，也有这种情况。看来这种强调中含有这样的信念：结构的理解能使学生从中提高他直觉地处理问题的效果。
What is the effect on intuitive thinking of teaching various so-called heuristic procedures? A heuristic procedure, as we have noted, is in essence a nonrigorous method of achieving solutions of problems. While heuristic procedure often leads to solution, it offers no guarantee of doing so. An algorithm, on the other hand, is a procedure for solving a problem which, if followed accurately, guarantees that in a finite number of steps you will find a solution to the problem if the problem has a solution. Heuristic procedures are often available when no algorithmic procedures are known; this is one of their advantages. Moreover, even when an algorithm is available, heuristic procedures are often much faster. Will the teaching of certain heuristic procedures facilitate intuitive thinking? For example, should students be taught explicitly, "When you cannot see how to proceed with the problem, try to think of a simpler problem that is similar to it; then use the method for solving the simpler problem as a plan for solving the more complicated problem?" Or should the student be led to learn such a technique without actually verbalizing it to himself in that way? It is possible, of course, that the ancient proverb about the caterpillar who could not walk when he tried to say how he did it may apply here. The student who becomes obsessively aware of the heuristic rules he uses to make his intuitive leaps may reduce the process to an analytic one. On the other hand, it is difficult to believe that general heuristic rules -the use of analogy, the appeal to symmetry, the examination of limiting conditions, the visualization of the solution-when they have been used frequently will be anything but a support to intuitive thinking.
传授各种启发式程序，对直觉思维有什么作用呢？正如我们已经觉察到的，启发式程序实质上是达到解决难题的一种不严密的方法。启发式程序常常使难题解决，但它提不出解决难题的保证。此外，算法也是解答问题的一种程序，如果进行得准确，它保证会经由一定的步骤发现问题的解答办法——只要这个问题有解答之道。当算法的程序不明了的时候，往往可用启发式程序，这是启发式程序的优点之一。而且，即使有合用的算法时，启发式程序的进度也往往更快。传授启发式程序会促进直觉思维吗？例如，应否明确地教学生，“当你不能够看出如何着手解题时，不妨想想同该问题相似的但较为简单的问题；然后用解决较简单问题的方法作为解决接复杂问题的计划。”这样行吗？抑或应该引导学生学到这样一种技术，而不要他向自己实际上说出那一番话？当然，这里可能用得上古代的一个谚语：当毛虫试图说出它如何走的时候，它便寸步难移了。逐渐被迫地认识到用启发式规则进行直觉飞跃的学生，可能会把这个过程简约为分析的过程。另一方面，经常应用一般的启发式规则——利用类比，使用对称，考查有限条件，使解法形象化——肯定是对直觉思维的支持。
Should students be encouraged to guess, in the interest of learning eventually how to make intelligent conjectures? Possibly there are certain kinds of situations where guessing is desirable and where it may facilitate the development of intuitive thinl{ing to some reasonable degree. There may, indeed, be a kind of guessing that requires careful cultivation. Yet, in many classes in school, guessing is heavily penalized and is associated somehow with laziness. Certainly one would not lil{e to educate students to do nothing but guess, for guessing should always be followed up by as much verification and confirmation as necessary; but too stringent a penalty on guessing may restrain thinking of any sort and keep it plodding rather than permitting it to make occasional leaps. May it not be better for students to guess than to be struck dumb when they cannot immediately give the right answer? It is plain that a student should be given some training in recognizing the plausibility of guesses. Very often we are forced, in science and in life generally, to act on the basis of incomplete knowledge; we are forced to guess. According to statistical decision theory, actions based on inadequate data must take account of both probability and costs. What we should teach students to recognize, probably, is when the cost of not guessing is too high, as well as when guessing itself is too costly. We tend to do the latter much better than the former. Should we give our students practice not only in making educated guesses but also in recognizing the characteristics of plausible guesses provided by others-knowing that an answer at least is of the right order of magnitude, or that it is possible rather than impossible? It is our feeling that perhaps a student would be given considerable advantage in his thinking, generally, if he learned that there were alternatives that could be chosen that lay somewhere between truth and complete silence. But let us not confuse ourselves by failing to recognize that there are two kinds of self-confidenceone a trait of personality, and another that comes from knowledge of a subject. It is no particular credit to the educator to help build the first without building the second. The objective of education is not the production of self-confident fools.
为了学习怎样在最后作出聪明的推测，是否应该鼓励学生去猜想呢？也许有某种情境，其间猜想是相宜的，并且可以促使直觉思维向合理程度发展。真的，可能有一种猜想，需要予以仔细的培养。可是，在学校的许多班级里，猜想会受到严重处罚，而且不晓得什么缘故，还同偷懒联系起来。可以肯定，谁都不愿意只教学生猜想而不干别的，因为继猜想之后总会根据需要尽量作出证明和认可的；而对猜想处罚过严，会压抑学生任何种类的思维，使之只能辛辛苦苦工作，而不敢进行偶然的飞跃。当学生不能立刻说出正确答案时，他们进行猜想难道不比目瞪口呆好吗？直截了当地说，应该给学生一定的训练，使之认清猜想的似合理性。在自然科学和普通生活中，我们常常被迫根据不完全的知识去行动；我们不得不去猜想。按照统计学的判断理论，根据不充分数据而进行的活动，必需考虑到概率和代价两方面。我们应当教学生识别的也许是，什么时候不去猜想的话，会代价过高；同样的，什么时候猜想本身的代价又太昂贵。我们认为做后者比做前者好得多。我们应否让学生从事两方面的实践，即不但进行有训练的猜想，而且还要认请别人提出的那些好象合理的猜想的特征——知道某答案至少在范围上是恰当的，或者知道那不是不可能的而是可能的？我们感觉到，一般说来，如果使学生懂得，在真理和缄默之间可以有各种选择，这或许会给他的思维以很大的好处，但是我们切不可因自己不能鉴别两种不同的自信而把自己弄糊涂了——一种自信是个性的特征，另一种自信则是由于对一个学科具有知识才产生的。对教育者来说，只帮助学生树立前一种自信而不树立后一种自信，就算不了有什么特殊功绩。教育的目标决不是生产具有自信的傻瓜。
Yet it seems likely that effective intuitive thinking is fostered by the development of self-confidence and courage in the student. A person who thinks intuitively may often achieve correct solutions, but he may also be proved wrong when he checks or when others check on him. Such thinking, therefore, requires a willingness to make honest mistakes in the effort to solve problems. One who is insecure, who lacks confidence in himself, may be unwilling to run such risks.
可是，借助于发展学生的自信和勇气去培养有效的直觉思维，似乎还是可能的。凭直觉思维的人，常常可以获得正确的解答，但当他自行核对或接受别人检验时，也可能被证明是错了。所以，这类思维需要在努力试解难题过程中有甘愿犯真正的错误的精神。凡是动摇的和缺乏自倍的人，也许不愿意胃这样的风险。
Observations suggest that in business, as the novelty or importance of situations requiring decision increases, the tendency to think analytically also increases. Perhaps when the student sees the consequences of error as too grave and the consequences of success as too chancy, he will freeze into analytic procedures even though they may not be appropriate. On these grounds, one may wonder whether the present system of rewards and punishments as seen by pupils in school actually tends to inhibit the use of intuitive thinking. The assignment of grades in school typically emphasizes the acquisition of factual knowledge, primarily because that is what is most easily evaluated; moreover, it tends to emphasize the correct answer, since it is the correct answer on the straightforward examination that can be graded as "correct." It appears to us important that some research be undertaken to learn what would happen to the development of intuitive thinking if different bases for grading were employed.
许多观察指出，在工作中，当要求决断的情境的新颖性或重要性提高时，分析思维的倾向也就增强。或许，当学生看出错误结果似乎过于严重以及成功结果又似乎太无把握的时候，他会抱住分析程序不放，尽管这些程序可能是不合适的。根据这些理由，人们可能怀疑，学校现行的奖惩制度在学生看来是否实际上有抑制运用直觉思维的倾向。学校中等第的规定特别强调获取事实的知识，主要就是因为这类知识最容易进行评价；而且，它倾向于强调正确答案，因为在简便的考试中，凡属正确的答案，都可以评为“对的”。在我们看来，重要的应该是从事某种研究工作，以便查明若是采用不同根据划分等第时，直觉思维的发展将会发生什么情况。
Finally, what can be said about the conditions in which intuitive thinking is likely to be particularly effective? In which subjects will mastery be most aided by intuitive procedures followed by checking? Many kinds of problems will be best approached by some combination of intuitive and other procedures, so it is also important to know whether or not both can be developed within the same course by the same teaching methods. This suggests that we examine the mode of effective operation of intuition in different kinds of fields. One hears the most explicit talk about intuition in those fields where the formal apparatus of deduction and induction is most highly developed-in mathematics and physics. The use of the word "intuition" by mathematicians and physicists may reflect their sense of confidence in the power and rigor of their disciplines. Others, however, may use intuition as much or more. Surely the historian, to take but one example, leans heavily upon intuitive procedures in pursuing his subject, for he must select what is relevant. He does not attempt to learn or record everything about a period; he limits himself to finding or learning predictively fruitful facts which, when combined, permit him to make intelligent guesses about what else went on. A comparison of intuitive thinking in different fields of knowledge would, we feel, be highly useful.
最后，关于能使直觉思维特别有效的条件，我们能够说些什么呢？在哪些学科的掌握中，用直觉程序，并继之以检验，会最有助益？解决各种各样问题的途径，最好借助直觉程序和别的程序的结合来进行，所以，知道是不是两种程序在用同样教学法的同样课程内都能得到发展，也是重要的。这就提出要我们去考查在不同领域内直觉的有效运转方式。我们听到人们最明确地谈论象数学和物理学这些演绎和归纳的形式装置(formal apparatus) 高度发展的领域里的直觉。数学家和物理学家使用“直觉”一词，可能反映他们对自己专业训练的力量和严肃性的确信之感。然而，其他的人几乎也同样使用直觉，或者用得更多。单以历史学家为例。历史学家在探索他的学科时，大量地依靠直觉程序，因为他必须选择有关联的事物。他并不试图查明或记录某一时期的全部事情；他自己只限于去发现或预知有成果的各种论据，这些论据结合起来，就能使他明智地猜想还发生过什么别的事情。我们觉得，各种不同知识领域里直觉思维的比较研究，肯定非常有用。
We have already noted in passing the intuitive confidence required of the poet and the literary critic in practicing their crafts: the need to proceed in the absence of specific and agreed-upon criteria for the choice of an image of the formulation of a critique. It is difficult for a teacher, a textbook, a demonstration film, to make explicit provision for the cultivation of courage in taste. As likely as not, courageous taste rests upon confidence in one's intuitions about what is moving, what is beautiful, what is tawdry. In a culture such as ours, where there is so much pressure toward uniformity of taste in our mass media of communication, so much fear of idiosyncratic style, indeed a certain suspicion of the idea of style altogether, it becomes the more important to nurture confident intuition in the realm of literature and the arts. Yet one finds a virtual vacuum of research on this topic in educational literature.
我们早已顺便指出过，诗人和文学评论家的职业要求他们有直觉信心：需要在没有特定的和意见一致的标准下，着手选择一个形象或者表述一个评论。对一位教师、一本教科书、一部示范影片来说，要为培养判断的勇气提供明确保证，这是困难的。说不定，大胆的判断产生于人们对于感动人的、美的、俗气的等等事物的直觉。在象我们这样一种文明里，有这么多的压力使我们对大众化的传播工具的判断趋于一致，还有这么多的对个人特有的表现风格的害怕，而且确实还有对凤格观念一概怀疑的，因此，训练文艺界有信心的直觉，就变得更重要了。可是，人们发现在教育文献中关于这个课题的研究，实际上于真空状态。
The warm praise that scientists lavish on those of their colleagues who earn the label "intuitive" is major evidence that intuition is a valuable commodity in science and one we should endeavor to foster in our students. The case for intuition in the arts and social studies is just as strong. But the pedagogic problems in fostering such a gift are severe and should not be overlooked in our eagerness to take the problem into the laboratory. For one thing, the intuitive method, as we have noted, often produces the wrong answer. It requires a sensitive teacher to distinguish an intuitive mistake-an interestingly wrong leap-from a stupid or ignorant mistake, and it requires a teacher who can give approval and correction simultaneously to the intuitive student. To know a subject so thoroughly that he can go easily beyond the textbook is a great deal to ask of a high school teacher. Indeed, it must happen occasionally that a student is not only more intelligent than his teacher but better informed, and develops intuitive ways of approaching problems that he cannot explain and that the teacher is simply unable to follow or re-create for himself. It is impossible for the teacher properly to reward or correct such students, and it may very well be that it is precisely our more gifted students who suffer such unrewarded effort. So along with any program for developing methods of cultivating and measuring the occurrence of intuitive thinking, there must go some practical consideration of the classroom problems and the limitations on our capacity for encouraging such skills in our students. This, too, is research that should be given all possible support.
科学家慷慨地给予那些博得“直觉者”称号以同仁们的热情赞扬，这是直觉在自然科学中是一种有价值的商品和我们应该努力在学生中培养直觉的主要证据。就艺术和社会学科的情况而论，直觉也一样有着强大的优势。不过，从教授方法上看，要培育这类才赋是个困难的问题。我们热切地希望不要忽视这个问题，而要对这个问题进行实验研究。我们已经注意到直觉方怯往往会产生错误答案这个事实。它要求敏感的教师将直觉的错误——有趣的错误的飞跃——同愚笨的或无知的错误区别开来；它还要求教师对具有这种直觉的学生予以赞可，同时加以纠正。彻底了解该学科以致能够轻易地超出教科书，这是对中学教师的高度要求。有时候的确遇到这种情况，即学生不仅比他的老师更有才智，而且知识更加丰富，并且能够提出他自己不能解释的解决问题的直觉途径，这些途径，教师简直弄不明白，也不能仿做一遍。要教师给这样的学生以正确的奖励或纠错，这是不可能的，也许忍受这种无奖励的努力的正是我们才赋比较高的学生。所以，在计划使培养和测量直觉思维现象的方法发展起来的同时，一定要切实考虑课堂教学问题以及在鼓励学生发展这类技能上我们能力的局限性。这也是应该尽一切可能予以支持的研究课题。
These practical difficulties should not discourage psychologists and teachers from making an attack on the problem. Once we have obtained answers to various of the questions raised in this chapter, we shall be in a much better position to recommend procedures for overcoming some of the difficulties.
这一些实际困难不应该成为心理学家和教师们攻克这个问题的阻力。一旦我们取得本章举出的各种问题的答案，我们就将处于更好的地位，来介绍克服某些困难的程序。
