3 THE TEACHING OF MATHEMATICS IN LONDON ARY Ten years have passed since the Board of Education ceased to prescribe a fixed syllabus of instruction in Arithmetic for elementary schools. During these ten years each school has had freedom to devise its own scheme, conduct its own examinations, and adopt its own methods of instruction. One would therefore naturally expect to find as the years advanced a gradual divergence from the Board's syllabus as a common point of departure, so that at the present day the syllabuses of the various schools would differ widely from one another and from the original common syllabus. As a matter of fact the divergence, with a few noteworthy exceptions, has been very slight. In the bulk of the schools the Board's Scheme B (see Appendix I.), which was the better of the two prescribed courses, still forms the framework of the mathematical curriculum. It has been enlarged by the inclusion of mensuration, and slightly rearranged so as to admit of the earlier introduction of fractions, but rarely has it been drastically altered. This conservatism is partly due to the fact that the Board continued, up till 1905, to publish Scheme B as a standard of proficiency for the labour certificate, and partly to the fact that most of the available text-books for elementary schools are even to this day based upon that scheme. It required therefore a head teacher of some force of character to overcome the inertia of the older system and strike out a line of his own. Some encouragement was given to initiative and experiment by the publication of the Suggestions to Teachers in 1905, for since that date departure from the normal course has been a little more frequent. TEACHING OF ARITHMETIC IN INFANTS' SCHOOLS. Instruction in the rudiments of number is given in the infants' school, on the organisation of which a few explanatory words are necessary, Ön account of exigencies of accommodation children who by age and attainments would normally pass into the lowest class of the Senior Department are in the majority of the London Schools retained in the Infants' Department as Standard I. These children are as a rule seven years of age at the middle of the Educational Year. The children of six, five, and under five years of age are known respectively as Grade III, Grade II, and Grade I or babies. The Standard I. children have for many years past been expected to deal with the four simple rules (see Appendix I.). At the end of the Educational Year a joint examination based upon this syllabus is held by the head teachers of the Senior and Infants' Departments with a view to the allocation of the examinees to suitable classes in the Senior School in the following year. This examiation has been a very important factor in determining the nature of the number instruction in the infants' school. It has served as the objective towards which the work in the grades has been directed. The examination has too often taken the form of a mechanical test, as in Appendix II., but there is an increasing tendency to set simple problems, as in Appendix III. Although this examination serves a useful purpose, its influence on the whole has been more baneful than beneficial; for it has been the custom in past years to work up to this examination right through the grades by the most direct route. "Sums" dominated the number lesson. Children began even in the babies' class to set down little sums, and gradually proceeded towards harder sums. In Standard I. practice in' setting out was essential. The four simple rules were always taken in the same order, and each rule assigned a particular place on the paper, so that at the final test the elements of novelty should be reduced to a minimum. But these shallow devices are happily obsolescent, if not entirely obsolete. In the great majority of the infants' schools serious attempts are being made now to impart a sound and intelligent knowledge of the fundamentals of number. And the clue to the instruction is being sought in the child's interests and aptitudes rather than in any logical arrangement of subject matter. No longer are lessons on number given to the babies, and paper sums have disappeared from Grade II. and all but disappeared from Grade III. Rarely is the chanting of the multiplication table heard in the school. The aim of the work is gradually changing. The teachers now try to foster an intelligent grasp of the relations between the first few natural numbers and a facile application of them to simple and familar things. And this they attempt to do by the analysis of number by the Grübe method. The teacher first fixes the notion of unity, and then deals with 2, 3, 4, &c., exhausting the possibilities of one number before proceeding to the next. Everything that can be done with 4 is done with it before 5 is taken up. Each child is supplied with objects sometimes sticks, sometimes beans, sometimes tablets, &c.-which he manipulates and arranges at the command of the teacher. The results discovered then become the basis of questions in mental arithmetic asked by the teacher and nearly always put in concrete form; for example:-Five little birds sitting on a tree; two fly away, how many are left? The teacher aims at getting the children by frequent repetition of the concrete process to memorise the result so as to avoid counting by units. In some schools numbers up to 5 are dealt with in Grade II., and numbers up to 10 in Grade III. In other schools the two grades go up to 10 and 20 respectively. These seem, to represent the maxima and minima. The attempts made in the bulk of the schools range between these limits. This system is the very keel and backbone of the instruction in number in the London infants' school-the decomposition. of small numbers, the manipulation of objects, and oral questions in the grades; supplemented by written sums in Standard I. Sometimes the teaching is limited to this, with no counting and no learning of tables beyond the number dealt with concretely. It may be claimed that this is a great improvement upon the older ciphering method, for it succeeds in giving a child a fairly intelligent notion of the relationships of the simple numbers which form the basis of our denary system, and it enables him to deal with common-sense problems. Still it is open to serious criticism:- 1. Unity cannot be dealt with alone. The notion only 3. Counting should not be reined in so as to keep pace with 4. The ratio or measuring aspect of number is inadequately 5. The various operations do not in general arise from any 7. Finally, I have very grave doubts whether number should These objections are not unfelt in our schools, and the bare Grübe system is accordingly being modified and supplemented. In one instance at least it has been entirely abandoned. It is a well-known fact that children will learn to count of their own accord. They do it at home, in the playground, and at their games. It is impossible to stop them. It may be that the counting is of that type which Mr. Raymont designates "spurious," a counting which is a mere rythmical sequence of sounds like "ena dena dina dust," from which no real inference is drawn as to the size of a group; still, it forms the basis of real counting. This spontaneous nature of children's counting is used by some head teachers as an argument for teaching the children in the grades to count up to a hundred; and by other head teachers, curiously enough, as an argument for not teaching it. I tested a school where no counting is specifically taught and found that about 60 per cent. of the children could count up to 20 before they were five years old, and about 70 per cent. could count up to 30 before they were six. On inquiry of the children themselves I found that they had, as a rule, learnt at home from their mothers, brothers, or sisters. I may point out that children incidentally learn to count when finding the page in the reading book. The question whether initial instruction in number should be based on counting or measuring is one which is agitating the minds of some of the teachers in infants' schools. A child naturally learns to count before he learns to measure. But he should learn to measure as well. In consequence of this belief Tillick's bricks are used in a few schools. They consist of rectangular prisms of equal cross sections, but standing at various heights to represent the first ten numbers. are compared by guessing, and the guesses are verified by measuring one brick against another. The purpose of this training, however, appears to be defeated in one or two instances by marking off the prisms into distinct units. A certain amount of simple measuring in inches by means of rulers is occasionally to be found in the grades, but generally speaking this work is confined to the standards. They There is to be observed a commendable tendency to relieve the tedium of the number lesson by the use of toys, such as soldiers, wooden animals and trees of the Noah's ark type, as substitutes for counters. Still more significant is the effort that is made in many of our schools to provide a motive for the various numerical exercises. A competition in skipping or in tossing a shuttlecock takes place before the class, the children counting and recording the score. A ring and hook game is engaged in, the class adding the numbers and comparing the results. Shopping with artificial coins is a device that is rapidly spreading. In at least two schools, rooms not needed for ordinary classwork are roughly fitted up as shops, and rice, sugar, tea and other commodities are weighed or measured in bulk and sold behind a counter by a child who acts as shopman to his companions who act as customers. But the most revolutionary of these experiments in providing a motive for calculation is to be found in School A. Here there is no formal instruction in number below Standard I. Number is learnt incidentally through the construction of objects. Articles or designs of various kinds are drawn, modelled, cut * The schools described as School A, School B, &c. are Public Elementary schools in the London area. out, or constructed; and the children are encouraged to converse about the things in such a way as to use numerical and quantitative terms. Even in Standard I. there is no written work except that notation is learnt and the answers to problems put down. No " sums are taught. The intelligence of these children when they pass into the Senior School is beyond question. The mathematics in the boys' department is exceptionally good all through. This leads up to the final bit of criticism, which strikes at the very root of number teaching in infants' schools. It raises the serious question: Is it worth while? Is there any ultimate gain to be derived from specific lessons in number given to children under seven years of age? However ambitious the class teacher may be, it is very rarely found that Grade II. children can deal with any degree of readiness with numbers beyond 5. These children, as a rule, receive one lesson of 20 minutes per day for five days in the week and forty-four weeks in the year, and yet they only master the numbers 2, 3, 4 and 5. Think of the progress that a child of the same age during the same period can make in motor co-ordination as shown in games, dancing, drawing, modelling, constructing and other activities; think of the number of words that he can thus add to his usable vocabulary; think of the number of songs that he can learn and the number of nursery rhymes. that he can memorise; think of the possible increase in his store of nature knowledge; and then compare any of these with his meagre achievements in Arithmetic. Does it not give rise to the suspicion that the child's mind is at that period not quite ripe for that particular kind of training? This suspicion is somewhat strengthened by the results of an experiment that has been going on for some years in School B. Here the number teaching begins in Grade III., and it has been found that at the end of two years (that is when Standard I. course is completed) the children are quite as advanced as they were when the number lessons began in Grade II. and extended over three years. The amount of time per day devoted to the subject. remained the same under both systems. Last July 176 of the children in the school were of Standard I. age, and were tested on the Standard I. syllabus. The examinees included all the laggards, the children retarded through ill-health, the migratory children from other schools-in fine, all who were qualified by age for transfer to the Senior School. One of the actual tests set is given in Appendix III. Alternative tests of equal difficulty were given to prevent copying. Out of the 176 children 84 got all four sums right, and altogether 77 per cent. of the sums were worked correctly. In the best of the three classes 90 per cent. of the sums were right. This result is well above the average, and when considered in conjunction with the result of the experiment in School A., it points to the necessity for further experiment in the direction of postponing |