Sawyer says ...

(from the various books of W. W. sawyer)

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 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.

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 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.

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.

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

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 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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

The depressing thing about arithmetic badly taught is that it destroys child's intellect, and to some extent, his integrity. Before they taught arithmetic, children will not give their assent to utter nonsense; afterwards, they will.

A high-school teacher I knew in America asked his students, "Do you like or dislike mathematics, and which teacher made you feel that way?" Nobody mentioned a secondary school teacher. Whether they were for mathematics or against it, the feeling had been created in primary school. The qualifications for teaching mathematics in primary school are:
(1) To enjoy mathematics,
(2) To expect the pupils to enjoy mathematics,
(3) To see mathematics as something you can discuss and arrive at conclusions,
(4) Knowledge adequate to the tasks at hand.