初等数学教学的艺术
“他是我所有员工中最棒的数学家之一，但却不是一个好老师。”
“是的，他教的很好。他能在一个小时之内将一个40人的班级打理的井井有条。问题是他教授的数学知识十分肤浅。”
这些来自校长的批评并不罕见。一位校长通常从两个方面来评价他的职员：健全的学科知识和学科教学技能。
假设一位教师在大学里度过三年或四年的时间，取得过令人垂涎的数学第一，然后在大学或学院的培训部门呆过一年时间。那他能自称是一个合格的数学家和一个有经验的教师吗？当然不。
从大学四年的课程中获得的数学知识，并在此之上形成了令人羡慕的基础，但有多少的技能是可以从短短的四年时间中学得的？在最后的时间里，一个大学生几乎没被教授过关于物理，化学，工程，机械和结构，航空，天文学，生物学，统计等更高深的数学或已掌握超过最基本的元素主体哲学的课程是很平常的。假设获得一门应用学科所需的基础知识需要一年，在新成熟的毕业生可以自称是广泛意义上的数学家之前仍有很长的路要走。至于数学哲学，在经过五年的艰苦阅读后他可能仍然是一个婴儿。一位数学家的全面训练是长久的事，因为数学触及生活的许多方面。假如一位35岁的数学家已经获取了他所在学科的大致全面的知识，他就算是做的不错的了。单独地考虑数学物理：这是多大的领域啊。当然，这是一个经由物理老师犁和播种，最后数学教师能够获得大丰收的领域。这样的收获总会发生吗？
教学的技巧：训练
接着说到教学。一位数学培训学校的学生如何能在他培训的最后时间成为一位能干的老师？任何艺术的技能只有通过更多的训练才能获得，而教学的艺术尤其困难。尽管数学可能是所有学科中最容易教的，但一个数学教师能在30岁前对他的教学技能感到满意是十分罕见的。如果在35岁之前其他人就认为他是一位能干的老师，无疑这是十分幸运的。所有数学学科都充满着教学困难。而每一个困难又是可以通过很多方法解决的。每一种方法都值得去尝试，并花费很长的时间去实践它。至于在数学教学中清楚地表现出一个典型的必要性本身就是需要花半辈子时间去完善的艺术。
一些没有接受过正式培训的数学教师经常问我从培训学校他们能获得哪些好的东西。他们更倾向于主张很少的培训学校有排在前列的数学家，因此他们也不能有效地解决课程问题；即使培训学校能够打动当地的数学家成为他们的职员也起不了多大的作用，因为一个大学教授能够教我数学却不能教我如何教导学生。Sir Percy Nunn教授第一个承认这样的观点不是没有道理，但这些狭隘的观点当然不适用于所有的培训学校。
还有进一步的批评指出花费在心理学上的时间是“完全的浪费”，因为“它对于真正的课堂问题没有任何实践价值”，这也可能是合理的。我不知道，还有培训学校还存在着半世纪以前的无用的好内省的心理学。但是年轻教师应当令自己受到全世界心理学家所做的有价值的实验的认可。这些实验通常建立在大量从课堂中获得的数据的基础上。确实得到明确的结果到目前为止相当凑合；一个完整的身体心理主义已经建成的东西，可以称是“科学”。但老师不可忽视已经做的和正在做的工作，如果仅仅是因为它的一大部分与当今学校的实践有着非常密切的关系。接受过培训的教师的观点与未接受过的教师有明显的区别，前者可以轻易地从他的特殊课程的偏见中摆脱出来，无论是数学家的还是经典的。他了解到，在公共休息室里他会成为一个社会成员代表着所有课程，这所有的课程适应了人们深层次的精神需要，它们汇聚在一起就称为文明。培训学校尽力培养称职的能发挥更大效用的教师。它引领学生理解教育的真正意义和其与生活多边联系的重要性。告诉他们教育不只是教导。并且学生每天都与身受广泛意义的教育与技术性的教导影响的人接触。
一个广阔的前景之间的区别是，有的老师已经通过培训学院，而有的没有经过任何训练的老师仍旧坚持着它的偏见，他（很自然地感到自己）的特殊主体，无论是数学，经典，或其它的。他学会了在公共休息室里他将成为一个社区的代表课程的所有受试者的一员，而所有这些人，不是他自己的课题研究，回答了人类精神的深层次的需求，他们都在运动称为文明的伟大流基本电流”。教师培训学校尽他们最大的努力来培养能干的工匠，当然他们做的要比这更多。它使学生理解教育的真正意义，关系到许多东西方面的商业生活的意义。这表明他们更广泛的教育是多少比单纯的教学。此外，学生们每天与人深刻地反映了对教育在更广泛的意义和教学在技术上的联系。
在一个好的培训学院，初期的数学老师不仅教如何传达给学生一个算术，代数，几何知识，和其他人，也如何使自己，在最大可能的措施，在数字领域，积极的知识的冒险家空间，如何跟进的大师的工作数学认为，如何抓住些什么的精神和展望。他鼓励问题接受数学的价值，和查询，在批判精神，什么传统的课程是很重要的部分是什么部分只有传统的价值。他意识到许多教材中包含大量的无用木材，他是教：如何区分这种方法是否是健全的方法或不是。
然后，在一个良好的培训学院的学生能够获得一种困难，他会在他的教学实践的专家意见。在什么阶段，例如，应该“直觉”工作在几何让严格的证明？如何才能最好的方法是在比和比例的教学做了？”平行线理论处理的怎样了？培训学院可以坦率地告诉他，在他目前的状态，他很可能尚未安装学徒处理平行线理论除了在实证的方法，从理论的内在困难只能在经过长时间的认真学习现代几何的研究；只有他在一个位置，使逻辑与直觉，所以可以设计一个适合第六男孩处理形式。简要介绍，培训学院将指出流沙，以及坚固的磐石上，数学教学中的流沙中比比皆是。老师是不是受过训练的往往不仅陷入流沙，但把他的孩子们和他自己也一块带进去了。
可以说，中学教师的绝大多数，尤其是在会议*学校教师，是从未经受过训练的。这是真的，结果往往是这样的教师在他们的孩子的代价买了前几年的工作经验。但不总是。如果从大学直下的年轻人去当老师，这是作为一个观察者和学习者，在一个大的学校，数学是很好的教育，如果数学系主任有足够的闲暇去引导他读作为他的专业的朋友，今年的训练可能不亚于在培训学院，一些当局敦促它可能是优越的，因为学生老师花一年真正的医生而不是理论家。我并不试图解决这个问题。
对于一个刚从大学毕业的新手教师来说，还有一个选择就是加入一个大学校的工作人员，以坦率的承认他是一个新手，并请所有的帮助和批评，他可以从他的同事获得数学。让他邀请他的学生来听他的教导，并批评他，让他乞求被偶尔出现在他们的教训的特权。让他征求他们的建议作为一个合适的课程阅读。但让他没有想到的是，他是一个老师派从天堂直接纠正25老化石的观点和方法上他可能在公共休息室里找到：这样就麻烦了。
有一件事，没有受过训练的新手不应该做的是加入的员工的一个小的学校，他必须承担的数学教学的整体责任。首先，是不是学校公平；男孩们派去教。第二，它是不公平的他，对于他是如何学习他的工作吗？为什么不打开一个办公室，并成立一个咨询工程师而不是吗？他是主管做另一件。
过去，所有想要成为工匠细木工，瓦工，力学，和其他花费七年时间来学习的学徒，他们真的知道他们的生意；他们要成为熟练的工匠。制度是死的，他们的工作是大多是由机械来完成的。但我们却不能用机械地方式来教学；并且在教学精湛的技艺是只能通过大量实践获得的。
在数学课进行评估的价值是多少？我指的不是普通的事情，作为一个数学课程，每一个老师用一年的经验应该达到的效率、一流的管理，纪律，使用黑板，专业性提问和处理的答案，等等。
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Skill in Teaching: Training
Then as to teaching. How can a mathematical Training College student claim to be an efficient teacher at the end of his training year? Skill in any art can be acquired only by much practice, and the art of teaching is a particularly difficult art. Although mathematics is one of the easiest,- perhaps the easiest, of all subjects to teach, it is a rare thing for a mathematical teacher to be able to feel at all satisfied with his professional skill before the age of .30. He is lucky if other people adjudge him efficient before the age of 35. Every mathematical subject is full of teaching problems. Everyone of these problems can be solved in a variety of ways.
Every one of these ways is worth testing. And all this takes along time. As for lucidity of presentation a prime necessity in all mathematical teaching that is in itself an art which half a life-time does not seem long enough to perfect.
I have often been asked by mathematical: teachers who have not been formally trained what good they would have done by going to a Training College. They are inclined to argue that very few of the Training Colleges have on their Staff sa front-rank mathematician, and that therefore such Train in Colleges are not in a position to deal with the subject effectively; that even if the Training Colleges are able to impress into their service members of the mathematical staff of the local University, little real help is obtainable, for although ha University Professor can teach me mathematics, he cannot teach me how to teach boys mathematics '. The general content ion is not without its points, but the strictures certainly do not apply to all Training Colleges, as trained mathematical teachers who have been through the hands of, say, Professor Sir Percy Nunn are the first to admit. The further criticism that the time spent on Psychology at the Training Colleges is 'absolutely wasted ', since' it has no practical value in the solution of actual classroom problems', is, perhaps, rather more justified. It is possibly true that the almost useless introspective psychology of half a century ago still hangs about some of the Training Colleges; I do not know. But young teachers should make themselves acquainted with the valuable experimental work which is now being done by psychologists all over the world. These experiment are often based on masses of actual data derived from the classroom. It is true that the definite results obtained so far are rather patchy; a complete body of psychological doctrine has yet to be built up into something that may claim to be 'science '. But no teacher can afford to ignore the work that has been done and is being done, if only because such a large part of it has a very close bearing on present-day school practice.
One broad distinction between the outlook of a teacher who has been through a Training College and that of one who has not is that the trained teacher has usually had knocked out of him prejudice which he (very naturally ) felt for his own special subject ,whether mathematics, classics , or what not. He has learnt that in the Common Room he will become a member of a community representing all the subjects of the curriculum , and that “all these subjects ,not his own subject alone, answer to deep-seated needs of the human spirit ,all of them essential currents in the great stream of movement called civilization ”. The training College dose its best, of course, to turn out competent craftsmen, but it does much more. It leads its students to understand the real meaning of education and something of its significance in relation to the many-sided business of life. It shows them how much wider education is than mere teaching. Moreover, the students are day by day in contact with men who have reflected deeply upon both education in the broader sense and teaching in the technical sense.
In a good Training College, the embryo mathematical teacher is taught not only how to convey to his pupils a knowledge of arithmetic, algebra, geometry, and the rest, but also how to make himself, in the greatest measure possible, an active intellectual adventurer in the realms of number and space, how to follow up the lab ours of the great masters of mathematical thought, and how to catch something of their spirit and outlook. He is encouraged to question accepted mathematical values, and to inquire, in a critical spirit, what parts of the traditional curriculum are really vital and what parts have only a conventional value. He is made aware that many of the textbooks contain a considerable amount of useless lumber, and he is taught: how to discriminate between methods that are sound and methods that are otherwise.
Then again, in a good Training College the student is able to obtain expert advice on every kind of difficulty he may meet with in his teaching practice. At what stage, for example, should ' intuition ' work in geometry give way to rigorous proof? How can the best approach be made in the teaching of ratio and proportion? 'How is the theory of parallels to be treated? The Training College may have to tell him frankly that in his present state of pupilage he is probably not yet fitted to deal with the theory of parallels except in an empirical way, since the inherent difficulties of the theory can only be grappled with after a prolonged and careful study of the researches of modern geometry; only then will he be in a position to disentangle logic from intuition, and so be able to devise a treatment suitable for Sixth Form boys. Briefly, the Training College will point out quick sands, as well as firm rock, for in mathematical teaching quick sands abound. A teacher who is not trained will often not only get into the quick sands but take his boys with him.
boys with him.
It may be said that the great majority of Secondary School teachers, especially teachers in the Conference * schools, are not trained. That
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