Great Throughts Treasury

"Bad teaching is teaching which presents an endless procession of meaningless signs, words and rules, and fails to arouse the imagination."

"Children are born wanting to think and discover for themselves; rote teaching systematically destroys their powers of thought; the older the students are, the less curiosity and intelligent interest you can expect."

"An activity engaged in purely for its consequences, without any pleasure for the activity itself, is likely to be poorly executed."

"Anyone who has ever been gripped by enthusiasm for a subject or a hobby knows the infinite difference between the way the mind works in such a situation and the way it works when we are engaged in something that does not appeal to us."

"Energy then is morally neutral, it is good or bad depending on the direction it takes. Our task is to provide legitimate outlets for it. The provision of such outlets is important in three ways - for society, for the individual and for the learning of subjects. For any society that wishes to remain civilized this is the highest priority; only by providing satisfactory activities for all citizens can delinquency, vandalism, crime and violence become exceptional rather than normal. This is a priority of which our dominant institutions seem totally unaware. For the individual, finding a satisfactory outlet for energy means a sense of fulfillment and escape from frustration. For the learning of subjects it is the driving force without which little will be learned."

"Education consists in co-operating with what is already inside a child's mind. A good teacher will nearly always be able to make his points simply by asking a class questions, by making the class realize clearly what they already know "at the back of their minds."

"Education is essentially the direction of mental energy. Children have abundant energy looking for an outlet. If adult society provides a satisfactory outlet, hobbies develop into professions and adults find life in their work. If adult society fails in providing an outlet, a double disaster occurs. The child has no energy or enthusiasm for work; and the child's energies are left to find an outlet at random. Society has then abdicated its duty to educate."

"Complete success would mean that every individual felt, "I enjoyed the mathematics that I had time to learn. If I ever need or want to learn some more, I shall not be afraid to do so.""

"Children live in the present. They feel that at any moment something tremendously exciting may happen. Successful teaching makes them feel that something tremendously exciting has happened. Preparation for the business worries of adult life does not meet this specification. Mathematics teaching is practical and purposeful only if it enables children to do better something they desperately want to do here and now."

"Children want to know things, they want to do things. Teachers do not have to put life into them: the life is there, waiting for an outlet. All that is needed is to preserve and to direct its flow."

"I enjoyed the mathematics that I had time to learn. If I ever need or want to learn some more, I shall not be afraid to do so."

"I have for a long time believed that the thought processes of very young children closely resembled the thought processes of genius."

"In discovering something for ourselves, we have a sense of freedom and conquest. In memorizing something that another person tells us and that we do not understand, we are slaves."

"I would like to emphasize that the activity approach does not mean every lesson must take place in the playground. All that is necessary is that there should be enough activity to make the mathematical work meaningful and purposive."

"Most teachers who are honest with themselves are forced to admit that, in the main, the class would make the same progress if the teacher were not there at all, the clever remaining clever and the dull remaining dull."

"My concern has always been to promote joy in mathematics for all-ability pupils, and the driving force has been a feeling that must be shared by anyone who has derived great pleasure from any subject - intense regret that the subject in question is so often presented in a way that produces dislike, worry and a sense of failure."

"One cannot expect Miss Jones to revolutionize her outlook overnight. One has to provide her with a steady stream of material, which she can introduce into her existing syllabus as enrichment. In the course of years, she will enlarge her repertoire. All the time, the new material should be as close as possible to what she already knows. This is how revolutions are made; not by taking one big step, but by taking many little steps quickly, one after other."

"In mathematics, if a pattern occurs, we can go on to ask, ‘Why does it occur? What does it signify?’ And we can find answers to these questions. In fact, for every pattern that appears, a mathematician feels he ought to know why it appears."

"It is clear that a great increase in the quality of life would occur if we could improve the teacher-pupil ratio. In the present economic climate of the world, such an aspiration appears hopeless. However, most official economic thinking relates to an age long dead. It is concerned with greater efficiency of production. But it was evident in 1930, and is still more evident in the age of the micro-chip, that the problem is not to produce but to distribute. The main social problem is to keep people occupied; the main economic problem is to spread incomes so that people who need things can afford to buy them."

"It is quite natural that if a child of limited intelligence can only do one subject, that subject should be arithmetic. The judgments involved in thinking, "Is this the right block? No, that one's too long," and later associating the various blocks with 0, 1, 2, ...., and 9 are much simpler than those required to learn the 26 letters of the alphabet and the eccentricities of English and American spelling."

"Mathematicians, it is often said, tend to be musical. It is less well known that problems arising from music have played an important role in the discovery of fundamental mathematical ideas. Questions about the vibrations of a piano string led to a fierce controversy that forced mathematicians to clarify their ideas about area, continuity, and the convergence of series."

"It is widely recognized that the weakest students can be very unhappy if their special needs are not met. It is often not recognized that the ablest too can suffer acutely, if they are captive in a lockstep class and have to work, at what seems to them a snail's pace, through material they could have disposed of quickly when they were several years younger. The root of the trouble lies in the concept that education is something done to a pupil by a teacher. This is entirely untrue, at any rate for mathematics. Young mathematicians are hungry for knowledge and nothing delights them more than to be given the opportunity to read ahead on their own. The strongest students will then reach topics far beyond anything that a school curriculum could possibly contain or a school teacher be expected to expound. Even those, who are slightly above the level the curriculum envisages, will benefit from the relief of boredom and the extra knowledge acquired."

"In the rhythm of rote learning all the emphasis is on the answer. In the rhythm of research, the emphasis is on the two items: understand what the problem is and solve the problem."

"It is well known that there exists a mathematical theory of musical sounds. Very few people know that music has made a contribution to mathematics. In fact a suggestion made by a musician led to a total revolution in the way mathematicians approach their subject. No doubt, mathematicians would have made this step forward sooner or later if the musician had not given this hint, but as a matter of historical record, that was how it happened."

"The best way to learn geometry is to follow the road which the human race originally followed: Do things, make things, notice things, arrange things, and only then reason about things."

"The aim of scientific education, as I see it, is to produce workers in all departments of knowledge in their due proportions; to encourage communication between workers in different fields, and also between them and the general community; to raise the intellectual standards and intellectual interests of the whole country."

"Only by providing satisfactory activities for citizens can delinquency, vandalism, crime and violence become exceptional rather than normal"

"The commonest complaint from the abler pupils is that if they finish a set of exercises ahead of the rest of the class they are simply given more of the same kind, which they find very boring."

"The appeal of arithmetic to infants is usually self-evident and recognising unusual mathematical maturity is not difficult. The unjustified fears of some educationists about allowing children to forge ahead, needs discussion and recognition of the need for young mathematicians to work in depth and at speed."

"The essential quality for a mathematician is the habit of thinking things out for oneself. That habit is usually acquired in childhood. It is hard to acquire it later."

"The first duty of a teacher is not to talk but to listen; to try to understand the direction the energies of each pupil are taking and not to expect activity in places that Energy has yet to reach."

"The effectiveness of education depends on adaptability to individual differences. Rigid syllabuses and lock-step teaching ignore the real forces in the mind that make for development and learning and the resentment of the gifted when drill continues after a skill has been learned is very often evident. The problem is how to get practice of necessary skills without causing such resentment, a problem often neglected by some modern mathematics' reformers."

"The practical value of mathematics lies in the fact that a single mathematical truth has a multitude of applications. If children can handle numbers with confidence and enthusiasm, they will be able to apply arithmetic to any situation that later life may bring."

"The main task of any teacher is to make a subject interesting."

"To master anything from football to relativity requires effort. But it does not require unpleasant efforts, drudgery. The main task of any teacher is to make a subject interesting."

"The topics and treatment of the mathematics syllabus should be determined by the following principles: a. The course must be enjoyable and generate steadily increasing enthusiasm in the pupils, b. It should develop independence and activity of mind, curiosity, observation, and confidence, c. It should make pupils familiar with the basic ideas and processes of mathematics."

"We are all of us imprisoned in our habits. The essential of education is to foster correct habits. It is easier for a mentally defective child to develop the rhythm of research than it is for a normally intelligent adult who has been subjected to fifteen years of parrot learning."

"We can go a stage beyond talking about things and drawing pictures of things by arranging for the actual handling of things. There is evidence that this greatly increases the proportion of the population capable of learning mathematics and this evidence is on a mass scale."