the age for beginning to learn Arithmetic in distinct and specific lessons. As the instruction in number has been gradually changing for the better, so has the instruction in form. While diminishing in quantity it has improved in quality. At one time children were required to name from a card various plane figures, such as triangle, square, oblong, parallelogram, pentagon, hexagon, &c. They folded paper and learnt something about angles, lines and surfaces. By means of the Kindergarten gifts they were even initiated into three-dimensional geometry. In drawing lessons the terms vertical, horizontal and oblique were learnt and illustrated. Nearly all this pedantry has now been abandoned, for it is being recognised that better ideas of form are gained incidentally through handwork and drawing. There seems to be no department of elementary work where more uncertainty reigns than in number teaching in the infants' school. Many teachers of experience share my scepticism about the value of any formal kind of mathematical instruction given to children under seven years of age. Nearly all are agreed as to the necessity for experiment. Mention may be made of a few methods that are being adopted in some schools and are not, in my opinion, wholly successful. A triadic system is used extensively in School C, and to some extent in a few other schools. All numbers up to ten are to be arranged and visualised in threes, &c. e.g. 4 In other schools it is insisted on that the children should always arrange their numbers according to some fixed design. They are graphically represented in the form, say, of pips on a playing card or dots on a domino. Thus five must be always pictured the imagery of children into one mould is in my opinion ill advised, and in any case destined to failure. The investigations of Galton tend to show that each individual has his own characteristic mode of imagining numbers. Another prevalent practice of a more advanced stage consists in decomposing numbers so as to form a series of tense.g., 7+ 8 = ( 7 + 3) + 5 = 15 &c. 289 (28 + 2) + 7 = 37 = In actual practice the method is safe but slow. As a practical method it is more rapid than counting by ones and less rapid than reliance upon memory for the unit figure. For the logical justification of the statement 7 + 8 = 15, we must go back to the natural number series and count. The best prac tical expedient is to memorise the result 7+ 8 = 15. The step (73) + 5 is a temporary device neither logically nor practically ultimate. This completes my account of the Infants' Schools. MATHEMATICS IN THE SENIOR DEPARTMENTS. Before dealing with the senior departments, it may be well that I should explain the range of experience upon which my report is based. First, comes my knowledge of the schools in my own district; secondly, visits made to typical schools in other parts of the Metropolis; thirdly, conferences with my colleagues; and fourthly, an examination of the syllabuses and methods in use in 70 departments taken entirely at random from the various inspectorial districts. Of these 70 schools, 30 were boys', 30 girls', and 10 mixed departments. The list comprised both Council and Non-provided Schools; in fact, every variety of school except the Higher Grade and Central. Some of the results of my investigations are herewith tabulated :-- Geometry is taken as a separate subject in 30 cases, and an average of 35 minutes per week is devoted to it. Algebra, on the other hand, is only taken in 16 cases, with an average time of 60 minutes per week. These investigations confirm me in the view that my own district is fairly representative of the rest of London, and that any generalisations that I may make can be regarded as holding good of at least a considerable majority of the schools. It will be seen that Arithmetic still remains the one branch of Mathematics common to all the schools, and that Scheme B. is the root from which it springs. It is not, however, possible by a mere examination of the syllabus to form an estimate of the kind of work done in any given school. A far safer clue is to be found in the kind of test set by the Head Teacher. Every class teacher in the London service is supplied with a Report and Progress Book, which contains, among other things, the syllabus in Mathematics. The Head Teacher has to examine the class on the syllabus at least twice a year, and to record the results in the Report and Progress Book. It is this examination that mainly determines the teaching. It indicates the specific interpretation given by the Head Master to his syllabus. It serves as a goal towards which the teachers work, for they feel that their professional reputation is largely dependent on successful records. It is not, therefore, unusual to find two schools with almost identical syllabuses, but widely different methods of instruction. In the one case, the terminal test may consist of a small number of sums of fixed type; in the other case it may consist of a variety of problems each making some fresh demand upon the ingenuity of the pupil. The first kind of examination encourages example-grinding; the second kind encourages intelligent teaching. It will thus be seen that the influence of the Head Teachers in determining the method as well as the matter of instruction is paramount; and it is pleasant to record that they are rising to their responsibility. Indeed, far more intelligent enterprise is being shown in the devising of tests than in the drawing up of syllabuses. In Appendices IV. and V. will be found sets of examination. questions, which will illustrate better than any other data that I can supply the progress that has taken place in the mathematical instruction of some of the schools during the last six years. The first instance is a girls' school; the second, a boys': Two sets of tests are supplied in each case, one of each set being given in 1904 and the other in 1910. The other conditions of the examinations-the standard and the time of the year-were almost identical. A comparison of these tests will reveal the following tendencies: 1. Arithmetic is passing into Mathematics. The claims of Geometry, Mensuration and Algebra obtain at least some recognition. 2. Fixed conventional types of sums are being largely abandoned. 3. The examples are brought into close relation with the children's own everyday life. 4. An abundance of practical work is being introduced. It may be pointed out that School E. (Appendix V.) is a boys' school where Mathematics is exceptionally well taught. The percentage of marks in 1910 was 66 in the practical paper and 70 in the theoretical, which is remarkably good for an ordinary school attended by an average class of London boys. It must not be supposed that even five per cent. of the schools came anywhere near this mark. To show what can be done in a school of an entirely different type, I give in Appendix VI. a test set in School F in 1910. This is a girls' school, where the children. are abnormally poor, and where the special strength is literary rather than mathematical. The average percentage of marks was 60. I do not quote these as model tests, but as indications of tendencies. There is much in them that is open to criticism. The wording of some of the questions could be improved, and the boys' test in particular strikes me as being inordinately difficult. These are extreme cases; but they represent changes that are going on to some extent in all the schools. First let us consider the tendency to extend the limits of Arithmetic and to break down the barriers between the various branches of Mathematics. Some five or six years ago Algebra appeared as a separate subject on the time tables of a large number of schools, but it consisted of little more than an unintelligent manipulation of symbols. It was in no way connected with the Arithmetic; the need for a more general symbolisation than the arithmetical digits was never made clear; and the usefulness of the algebraic processes was left in obscurity. But Algebra as a separate subject has now largely disappeared, except in the Central Schools, and attempts are being made to graft a little of it on the Arithmetic. The attempts are laudable, but not very successful. I should not, for example, care to defend the algebraic questions in the representative tests I have appended. It is, perhaps, a debatable question whether Algebra should be taught at all in an ordinary elementary school. I am personally inclined to think that it should, and that it should have its starting point at problems leading to simple equations, or at mensuration formulæ such as: area of triangle b.h. In any case, the teacher should provide situations where a symbol like x would be a more potent instrument of thought than a symbol like 5, where, in fact, Algebra is more useful than Arithmetic. The teacher should see that the need for the new kind of symbol is felt by the pupil himself. I have rarely, as a matter of fact, seen this done, and the Algebra that is brought into the arithmetic lesson is generally of an adventitious kind which might just as well be left outside, = Geometry. Geometry, however, seems to me to be upon another footing. It has distinct claims of its own, apart from the use made of it in mensuration. Much practical geometry is absorbed in the practical arithmetic of most schools; but when it is taken as a separate subject (see page 10) it generally assumes the form of working problems with compasses and ruler. It is in fact sometimes classed with the art subjects and sometimes with the mathematical. The procedure in each case follows on the lines which were common in the days when the old South Kensington Art syllabus was taught in the schools. The teacher sets a problem and works it on the board, and the children copy it into their Drawing Books. This is the almost universal method, and it is unquestionably a bad method. If heuristic training is justifiable in any subject it is justifiable in this subject. Treated as a progressive series of puzzles to be solved by the child himself, the subject may be made of immense educative value. Unfortunately a fixed syllabus is generally given to the class teacher, and he is required to get through a certain number of problems in a given time. This is a mistake. It is not the master who should set the pace, but the class. Theoretical geometry is taken only in the Central Schools. The Teaching of Practical Arithmetic. Far and away the most significant change that has taken place during the past few years has been the introduction of what is known in the schools as practical arithmetic. The differentia of practical arithmetic seems to be the presence of the material objects dealt with. If they are not actually present, they must be represented by natural and not mere conventional symbols. If, for instance, a boy were asked to find the diagonal distance across a rectangular garden, 30 yards long and 25 yards wide, and proceeded to draw the garden to scale and to find the approximate length of the diagonal by measurement, that would be considered an exercise in practical arithmetic; for although neither the actual garden, nor the actual unit of measurement was used, yet each was adequately represented by things which made the same kind of perceptual appeal. Practical arithmetic aims, in fact, at presenting in full the evidence on which the conclusions are based, and testing the conclusions by fresh application to concrete experience. The advantage of it is that the pupil knows what he is doing; he is going through a bit of real experience. It is well known that a child may be able to multiply 6489 by 76 with ease and accuracy, and yet be unable to show its connection with ordinary life, or, indeed, to attach any meaning whatever to the process. The aim of the practical arithmetic is to remedy |