England.

down rules for curtailing the work according to the degree of accuracy specified. A boy finds these rules troublesome-he would generally prefer to do a tedious straightforward piece of work, rather than take the trouble to think at the outset; the process never becomes habitual and remains therefore valueless. Such numbers never occur in the laboratory, and the problem there is not to save trouble, but to keep within the limits of honesty to get all that can fairly be got out of the measurements and to pretend to nothing more. Science masters ignore the point altogether and have been known to tell boys for instance that "densities must always be calculated to 3 places of decimals,❞ irrespective of the fact that their measurements may only justify one. The difficulty can easily be met if both teachers will take the trouble; the method needed is much simpler than the one generally given in Arithmetic books and taught by the Arithmetic teacher, but for want of joint consideration it has generally escaped attention.

It is plain that if substantial progress is to be made, the study of both subjects- Mathematics and Physics-and, if possible, actual experience in the school teaching of both, must become much more general than it is at present. On this point German practice is of interest: the Prussian "Prüfungsordnung" of 1898 still in force lays it down that "Pure Mathematics and Physics must always be taken together."

W. C. FLETCHER.

MATHEMATICS IN THE PREPARATORY SCHOOL.

Our most successful administrators have said that the young public school man, notwithstanding all that has been said and written about him to his disadvantage, is one of the best aids the country has in administering our Colonies. The administration of the Empire is to a great extent in the hands of those who have passed through the public schools. Thirty years ago a large percentage of boys entering public schools were educated at home in their early years, and recognised preparatory schools could be counted on the fingers of two hands. At the present moment there are 500 preparatory schools preparing boys for more than a hundred public schools, to thirty of which might be applied the name "the Great Public Schools." From this it will be seen that the vast majority of boys, who enter the public schools, receive their early training at the preparatory schools. As a result of this, there is a distinct break in the training, and an examination into the effects of this break, in any particular branch of study, is of great importance.

This paper deals with the effects of the break upon mathematical teaching, and may be divided into three heads :

(I) Effects due to public school scholarships offered to boys under 14 years of age.

(II) Effects due to change of schools.

(III) Recommendations.

As a help to the consideration of these effects, a questionnaire has been sent to the great public schools, and a separate one to the headmasters of 40 or 50 well-known preparatory schools, and I have been able to obtain much valuable information.

(I) At the outset, it may be said that it is a matter of regret that public schools find it necessary-and without doubt always will find it necessary--to offer entrance scholarships, for in an ideal system no scholarships would be offered to boys of thirteen years of age. In the keen competition which arises when 500 schools are doing work that might be done with half that number, there must be a certain number of masters who look to the publication of scholarship lists to give them a much desired advertisement. It is, however, a healthy sign that, without exception, teachers in public and preparatory schools are opposed to all forms of cramming.

(a) It might be supposed that many preparatory schoolmasters, in order to obtain scholarships, would cut off mathematical hours in order to win classical or modern language

scholarships, and vice versâ; this, however, is not the case as a general rule. In a very few cases, boys are given a little extra work; and sometimes, again, in the last month before examination, some "polishing" is done at the expense of a few hours which, under ordinary conditions, would have been given to other subjects. Most masters rightly let scholarship boys work separately from the rest of the school, in order that they may not be kept back by less proficient pupils. We may say

that in nearly all schools boys take their scholarships "in their stride." It should, however, be noted that my information comes from 40 or 50 well-known preparatory schools, and that, in the remaining 450, there may be masters who, in order to win scholarships at the public schools, resort to cramming.

(b) In preparing boys for Mathematical Scholarships there is always a danger of cramming. In former days, when bookwork formed a large proportion of the test in mathematical papers, there was a much greater danger of this. Unfortunately in this subject a boy, even at the age of 13, may show great facility (I might even call it a fatal facility) in mechanical work and in the reproduction of bookwork, and from that point, when the higher ranges of the subject present themselves, he may fail to maintain his early promise. During the last few years the papers set for mathematical scholarships have improved very greatly, and very few are set in which a boy could score well who had only a superficial knowledge and lacked real mathematical ability. Though it may be invidious to mention the papers of any particular school, where so many are admirable, I feel I must say that the papers set for the Winchester College election are, in the opinion of many, very nearly ideal. These papers do not demand a great range of reading-not so much as in the cases of the papers of many other schools-but they test, not mere ability to learn mechanical processes, but originality and power. In Appendix I. of this paper will be found the papers set at Winchester in the year 1909.

(c) It is necessary to examine the evidence derived from the public schools as to the mathematical knowledge expected of classical and modern language scholars. It is very clear that the mathematical pass papers do not count; they are looked at with a very tolerant eye. The questions asked of the Headmaster on this subject were:-(1) "Does the pass paper in Mathematics mean anything (i.e., count anything) in Classical and Modern Language Scholarship papers? (2) What percentage of marks as a rule do classical and modern language scholars get in mathematical papers which are not pass papers?" In most schools the pass paper is a very simple one and in several it counts "practically nothing," though in one important school an unusually bad paper in Mathematics caused a boy "to drop from 7th to 12th place in the scholarship list," and in another "the pass paper in Mathematics might differentiate two boys equal in Classics." In (2) the papers referred to are not pass

papers and classical boys as a rule seem to score anything from 0 to 20 per cent.

(d) It is important to consider, further, the ultimate success or failure of mathematical scholars. In former days of the purely classical public school, when modern sides were unknown, the boy whose bent was mathematical was under great disadvantage. To a question as to the success of these scholars in the general life of a public school and afterwards in life, the answer is almost invariably that they are successful; they take a good position in the school life, and many go on to the Universities as mathematical scholars. This is largely due to the fact that they are required to attain to a fair standard in other subjects before being allowed to specialise in Mathematics. Further, public schools do not look out for young mathematical prodigies, but are quite ready to take boys of undoubted ability, who have had the ordinary training obtained in a preparatory school.

(II) "Is there a break in continuity in mathematical teaching owing to the passage from the preparatory to the public school?" Ten years ago the answer to this question would have been "Yes." To-day, though the answer could not be "No," the break is much less pronounced, owing to the more definite agreement as to sequence. It will be worth while to look into the causes of this agreement.

(a) The chief factor has been, without doubt, the institution of the Common Entrance Examination for admission into the public schools. Instituted in 1904, and directed by a board of six, three drawn from public school headmasters and three from preparatory school headmasters, this examination is at the present time adopted by 52 public schools, which in fact include all the great public schools, with one notable exception. Before its institution each public school had its own particular examination, and preparatory school masters endeavoured to arrange that each of his pupils was prepared for the special papers he was likely to have at the school for which he was entered. It is not necessary to point out how much the work at preparatory schools has been unified as a result of this common examination. I find that in answer to a question to both public and private schools as to the utility of the examination, the answer has invariably been that as pass papers the mathematical papers are excellent, though there is a very general feeling that the time devoted to Mathematics is too short, and that alternative papers for boys of different standards would be more effective than alternative questions, and would ensure a more accurate estimate of a boy's capacity and capability.

(b) The co-operation of the Head Masters' Conference and the Association of Preparatory Schools, moreover, has brought about the issue of a schedule of mathematical work suitable

for boys from the ages of 9-16*; it will be noticed that the limits of age envelop the break from one school to the other. This schedule has been submitted to all the public schools and to all the members of the Association of Preparatory Schools.. Unfortunately a large number have either not troubled to look at it, or have not studied it with care. On the other hand, in nearly every case where it has been considered, the scheme is approved of, and in most cases adopted. From one great public school, however, comes the answer: "Not approved of and not adopted"; and from another important school: "We only agree with a small portion of it-to the bulk of it we are in strong opposition." These are the only strong dissentients, and in the latter case the master in charge of the mathematical department has strong views of his own on mathematical teaching; but, though he carries them to a very successful issue in his own school, his methods would not be generally possible in other schools.

(c) To the public schools the following question was sent :"Do you find boys coming from Preparatory Schools in any way uniform in their methods?" This question was a natural one, as, in a given list of 20 new boys at a public school, it would be observed that they were drawn from a dozen or even 15 different schools. The answers to this question are hardly satisfactory. One school, indeed, replies frankly: "No, and not expected." To the question was added: "This may not be the case in geometry. Are they uniform in arithmetic and algebra ?" It is a remarkable fact that, notwithstanding the revolution in the teaching of geometry in the last ten years, there seems to be more uniformity in geometrical teaching than in arithmetic and algebra. There are still teachers who regret the old Euclid with its rigid chain of reasoning, but it has been realised that the earlier propositions in that book were beyond the powers of the young boy, and should be left till later in the course. By assuming a certain number of facts, by aid of which easy riders can be proved, a young boy can be led on to reason according to Euclid's method, when he will find the propositions. come to him naturally and easily. The use of hypothetical constructions has done away with many of the old clumsy problems" of Euclid, and the so-called "geometrical drawing" strengthens his grasp of the subject. There is always a danger, especially in the case of teachers who are not mathematicians, of overdoing this branch of the work. I believe, however, that the majority of teachers would say that the change in the method of teaching geometry has been for good, and it will be found that, if the headmasters' schedule be adopted, the break in passing from school to school will entirely

This schedule will be found in Appendix II.; it is now sent out with information relative to the common entrance examination, and papers are set in accordance with the scheme.