by G.L. Davies, B.Sc.
Volume 75, no. 11, 1964, pgs. 264-271
The following paper, 'Knowledge of the Universe', concludes those presented in the Parents' Review, from the recent C.M.C.A.
[Charlotte Mason College Association] Study Course at Hethersett.
In this talk, I am going to attempt to show how Knowledge of the Universe can contribute towards the development of the individual person. First I shall deal briefly with science and geography and at rather greater length with mathematics, the subject which most interests me.
Science and Geography
The child is himself part of the universe and of course he is curious about his surroundings. He wonders about pieces of apparatus he uses every day, e.g., the telephone, an electric torch or a clock. Naturally he would like to learn of the construction and functioning of these common pieces of equipment. He watches the sun rise and set, feels the warmth of the rays on his skin, hears the noise of thunder and wonders at the ebb and flow of the ocean. It is not surprising that he would like an explanation of these natural occurrences. He may be interested in the prevention and cure of disease and in the working of his body. Curiosity is mingled with aesthetic appreciation as he watches a butterfly probing into a flower cup, a young bird trying out its wings, sunshine showers and rainbows.
News of further achievements in modern science stimulate still more his
appetite for knowledge. The child's desire to understand is compelling
and so the teacher's task in teaching science should be rewarding right
from the beginning. Obviously scientific knowledge should stem from
personal observation. In this activity all the senses should be brought
into play. For instance the mental image a child forms of a flower
should be much more than Ôa head with five pink heart-shaped
petals.' He should smell the fragrance of the flower, handle its stem
and leaves to learn their texture and even taste the nectar if the
flower is not noxious. Observation is a very individual activity. Each
child will make his own mental picture of the subject and will enjoy
exchanging knowledge with his colleagues. All can make some
contribution to the common pool of discoveries.
Any apparatus used should be suited to the particular needs of the individual child. If he says, for instance: 'The butterfly we looked at yesterday had knobs at the ends of his feelers, but this moth's feelers seem to end in a different way, I should like a closer look,' then the obvious step is to make the closer look possible by providing a hand lens. If a model is used for illustration it should be designed simply so that the essentials of construction are immediately apparent. If the child is presented with detail before he is able to appreciate it, he becomes overwhelmed and discouraged.
The keeping of a Nature Note Book is an excellent means of encouraging scientific self-expression. However, I have found that some children need guidance as to which subjects to observe. In the winter months for instance a child may complain: 'But what is there to write about? Everything is covered with snow.' Of course there are also the enthusiasts who fill several Nature Note Books each term but generally it is as well to discuss with the class at the beginning of the month some likely subjects for observation. For example, in a snowy January the children may be told to look for birds searching for food, footprints in the snow, evergreens and certain constellations.
When learning science subjects, the child must understand the great importance of drawing and painting from personal observation, and of making clear diagrams. Written accounts must be graphically illustrated. Indeed there is much to be said for the reverse procedure, i.e., starting from the illustration and writing round it.
Many of our modern authors have considerable scientific knowledge. Most children are enthralled by stories based on Natural History and do not need coaxing to read such literature. This year my Form II children enjoyed reading Rowena Farne's Seal Morning. Some took out Gerald Durrell's The Drunken Forest from the form library. These authors have great sympathy for the animal world and by their writing they communicate their compassion and understanding to the reader. Such reading certainly helps the child to look beyond the fact.
We must not neglect the history of science. Older children should have some knowledge of how, when and where important scientific advances were made. They should realise what social conditions have apparently favoured scientific progress and how the advent of war seems to accelerate development in certain fields whilst denying humanity the ability to enjoy the fruits of successful labour. Biographies of certain scientists can be a source of inspiration to older pupils. One book of interest to children of about 14 years of age is Madame Curie, written by Eve Curie.
Now a few remarks about the teaching of geography. I speak mainly from the experience afforded me by trying to put into practice the P.N.E.U. programmes up to IIIB. First of all there is no doubt that what could be termed 'the personal approach' is the method which best appeals to a young child. Children like to hear about the lives of other boys and girls and so the books used in the early years should be about very young people.
The second point of significance is that at first a young pupil becomes more readily interested in a child who leads a life contrasting with his own; and so we teach first about Tooktoo, the Eskimo boy who has to cope with Arctic conditions and then about Bombo, who lives in the Congo, whose life is influenced by an equatorial environment.
Local geography is difficult to present to a small child. Being
accustomed to his surroundings, naturally he sees nothing remarkable in
them. His eyes are blurred by the detail which surrounds him and the
teacher must help him pick out the essentials. Possibly these problems
are parallel with those experienced in the teaching of recent history.
It is difficult to find suitable literature to use in the teaching of
local geography. Most 'handbooks' are too factual to interest young
However most children travel by road a great deal these days and father or mother should try to interest them in the road map, perhaps giving them the responsibility of directing the journey. If the children are told beforehand what they may see, they will notice not only roads, but also rivers and bridges, factories and other buildings in the towns, crops in the fields, woodland and grassland.
It is the preparation for the journey which is so important. This applies equally to longer journeys abroad. I know many parents are disappointed after having travelled extensively and frequently, that their children seem to benefit so little from an educational viewpoint. If adequate preparation for the journey is not carried out in the classroom beforehand, the young traveller will remember only the details which affected his personal life.
Perhaps he will remember with nostalgia the menu on Fancy Dress Night, diving for pennies in the ship's pool, the Egyptian 'magician' at Port Said or sleeping in the 'top bunk.' If on the other hand the child reads in the classroom beforehand about the places he will pass through, the journey can add living detail and colour to the knowledge he has acquired previously from books.
It is all too easy to teach geography from a utilitarian approach. How Man has partially harnessed the forces of Nature, using his achievement for his own comfort and advantage, is only part of the subject. The aesthetic side of the subject should not be ignored. The words 'lagoon,' 'atoll,' 'coral reef,' 'savannah' and 'volcano' should mean much more to the child than bald geographical terms. There should be time to paint a 'geographical scene,' perhaps of nomads in the desert or of fishing in the North Sea. Some children will want to write verses. Here are some lines written very quickly by a child:
The shaded jungle of the East,
Silent in its noonday peace.
The cunning leopard in its tree,
Legs and tail dangling free.
Hosts of spiteful monkey tribes,
Resting from their jests and jibes.
The sinuous snake beneath its stone,
Curved and sleeping all alone.
The written geographical account need not be entirely factual. Children
will enjoy making up stories about regions studied. Some of my Form II
pupils wrote very vivid stories about an imaginary prospector making a
'lucky strike' south of Hudson Bay.
Sometimes fiction can contribute useful background material to geography lessons. A short time ago a P.N.E.U. programme included Dry River Farm, intended to accompany the geography of South Africa. This year when studying with Form II the geography of Australia, I used James Vance Marshall's novel Walkabout in the same way. The story tells of two American children who miraculously survive a 'plane crash in the Northern Territories and of their subsequent wandering. The author most sensitively describes the topography, flora and fauna with vivid detail.
This is a very exciting time indeed for those involved in the teaching of mathematics. There have been almost revolutionary changes during the last few years in the way the subject is presented to the child. Now there is no point in change for its own sake. If a system is working well it is pointless to abandon it, or even to tamper with the details. However it is indisputable that a fresh approach was needed to the teaching of this subject.
Consider how mathematics was introduced to children a few years ago. To pupils of Junior School age only arithmetical ideas were presented. Algebraic and geometrical ideas were thought to be beyond the understanding of young children. Great emphasis was laid on mechanical work and learning by rote. It was only after several terms' labouring with pages of mechanical 'sums' based on the four rules and with 'tables' that problems were introduced. Even then, skill in problem work was expected only from the more able pupils.
What was the result of this preoccupation with mechanical 'sums'? It was true that a few pupils attained an almost machine-like accuracy and came to enjoy the subject in which they could achieve such perfection. However, the majority managed only partially to understand the elementary mathematical concepts. There were some, indeed, who gained no real knowledge of the subject; who remained completely mystified by the language of number and size. The unfortunate pupils in this latter category usually developed an early antipathy toward mathematics. Everyone who has been involved in the teaching of 'Secondary Stage' maths will have encountered the child, otherwise able, who has no understanding of Number.
In 'Secondary Stage Mathematics' the curriculum was rigidly divided into algebra, geometry and arithmetic, very little attempt being made to encourage the pupil to think of these branches as one subject.
Now computation itself will not help to develop the complete person. In
the adult world, most computation has been mechanised. Most commercial
firms invest in a computer, or in a share of one, and the shop
assistant will use a more modestly designed computer, or perhaps a
'ready-reckoner.' This does not mean that we can abandon the study of
To use a computer the operator must understand the significance of 'place' value, and know whether to add, subtract, multiply or divide. The basic ideas of mathematics must be understood if we are to be able to follow modern science, but undue emphasis on 'mechanical drill' will not achieve this end. The old system of teaching gave rise to the commonly-held belief that a real grasp of and an interest in mathematics was for only the very few. Surely this state of affairs rightly gave cause for concern.
Having, I hope, established that change is coming about through necessity, next we must consider those who are responsible for teaching the subject to a class or to an individual child. Charlotte Mason observed that mathematics should be taught by 'those who know' and elsewhere she remarks that much depends in this particular subject on 'the skill of the teacher.' This does not mean that only a specialist can teach elementary work well.
Indeed, sometimes someone with no specific qualifications at all is better able to help the young pupil, because she may possess an insight into the difficulties commonly encountered in trying to learn the subject. Mr. Sealey, who has done such excellent work on the teaching of junior school mathematics in Leicester, agrees that much depends on the skill of the teacher. In fact he suggests that in the next few years many teachers will abandon the junior school text-book as we know it today, i.e., a book containing sets of graded examples through which every member of the class works systematically.
He considers that 'assignment cards' made specially to suit the needs of each child should become the main-spring of mathematics at this stage. For the time being, I imagine most teachers will prefer to retain the text-book, at least as a guide. However there are other ideas which would be easy to introduce at once. I shall speak later about the ones which have appealed to me.
First let us consider how the mathematical knowledge may be achieved.
As in all other subjects there can be no learning without
understanding. It is fallacious to reason that the child who finds the
subject difficult must learn by rote; that only the bright child can be
expected to understand. There is nothing more tedious and exacting than
the task of trying to memorise sets of facts which have not been
properly comprehended. The teacher must patiently guide every pupil
towards appreciation of fundamental concepts. It is useless for a pupil
to perform mechanically pages of addition 'sums' until he understands
that the process means the collecting together of groups. This type of
activity is just as pointless as reading without understanding.
If the child is to succeed in becoming 'mathematically literate' the teacher must treat him as an individual. Different minds do not learn new concepts always in the same way. Each child takes his own unique path to knowledge. It is important that each child should be allowed to work at his own pace. Miss Mason wisely remarked that the tortoise should not be expected to keep pace with the hare. If the teacher sets, say, five examples for the class and insists that every child must complete them satisfactorily before they all proceed with the next activity, obviously the more able pupils will become bored and frustrated. A talented child may even come to dislike the very subject in which he could excel if taught in a more imaginative way.
The teacher should avoid telling the children facts they can find out for themselves. Every child has come into contact with mathematics during his everyday routine outside school. It is from these individual experiences that the teacher should proceed. For instance it would be quite wrong to approach the geometry of line and shape by displaying before the class a set of grim-looking wooden prisms, pyramids and cones. The child will have observed the spiral spring under his Jack-in-a-box, the spherical head of the dandelion clock, the conical church spire, the parabolic jets of water in the fountain and the hexagonal cells in the honey-comb. The teacher must build on these experiences.
In the very early years apparatus plays an important part in mathematics lessons. Of course equipment must be well-designed and its use properly planned. I do not propose to discuss apparatus in detail here, but before passing on, mis-use and over-use of equipment should be considered. The teacher should give out apparatus knowing exactly what concept the children are expected to learn from its use. When this object has been achieved, the apparatus should be discarded. It must be kept in mind that pieces of mathematical equipment are merely analogues; that they do not precisely represent the ideas we wish to present.
For instance the cardinal idea of 5 is not exactly represented by 5 dots on a domino, nor by 5 boys in a line. The group 5 is not itself connected with either particular objects or with the position of these objects. If a child continues to use a particular piece of equipment after understanding the idea which he is intended to grasp, the apparatus becomes of no more use to him than is a casual plaything. Various games may be used to teach mathematical facts, but once these facts have been remembered, the games should be discontinued having served their purpose.
In Forms I and II the P.N.E.U. classes use the 'Alpha Arithmetic Books' which are so designed that the child can progress through them largely through his own efforts. Of course he refers to the teacher when in difficulty. The child works at his own individual pace and the teacher's task is mainly tutorial. When work on a particular topic has been taught, such as 'area,' it is a good plan to ask each child to compose a set of original examples on the particular topic.
He should be asked to provide solutions and working on a separate sheet of paper. Children of equivalent abilities are asked to exchange examples and to find solutions. The teacher has no need to check unless there is disagreement over the answers. This is very valuable creative work. If the pupil is able to compose successfully a set of original examples with solutions, he has assimilated without doubt the mathematical idea behind the particular topic. A useful and interesting set of problems may be kept by the teacher for future use.
Occasionally the children should be asked to write a mathematically-based 'written account.' They could write about shapes of containers they have seen in the Supermarket, the measurements of the school building, the dimensions and uses of different wheels or speeds of different vehicles. They will enjoy doing their own research and describing their finding.
Each child can be asked to keep a particular form room graph. Suitable subjects are temperature, number of bottles of milk consumed daily, attendance and pocket money expenditure. The children should know of the many different types of graphs and in what circumstances they are employed. 'Pie' graphs and different types of 'bar' graphs may be cut out of newspapers and displayed on a 'Mathematics Board.' The Board may also be used for puzzles the children find interesting, and for questions the pupils cannot themselves answer, e.g., a child may write 'Can anyone explain to me what father is doing when he is opermuting 8 from 11" in the football pools?'
All children should be introduced to some type of mathematical
literature. The Form Library for Form II could well contain books such
as Thyra Smith's The Story of Numbers which is mainly historical and
Think of a Number, by Grace Moss, which deals with the behaviour of
Number. Children who are interested in Number will find these books
very absorbing. The child who has reached sixth form mathematics may
like to read Lancelot Hogben's Mathematics for the Million. Although
this later book is nearly 30 years old now, as an authoritative and
comprehensive History of Mathematics, I believe it is still unsurpassed.
Learning how mathematics developed will make the subject 'come alive.' The child will find some details of the History of Number very curious and intriguing. For instance most pupils would be surprised to learn that until 1812 some Treasury Accounts were still recorded on Tally Sticks. Even though this primitive method of bookkeeping was then abandoned, it was not until 1842 that it was decided that the Sticks had cluttered up the Palace of Westminster for long enough. These valuable government records were then used for fuel in the furnaces of the heating system.
Though not every pupil can hope to become a mathematician, if we try to teach the subject imaginatively there is no reason why most children should not successfully learn to understand the basic language of mathematics.
Proofread by Judy Elliot
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