&c. Probably, however, most teachers will prefer to wait until a fair foundation of algebra and trigonometry has been laid. There seems, however, little doubt that the easiest method of approach is through the graphical work, and that the rate of increase of y compared with that of a is more readily understood by consideration of various curves previously obtained. The method of obtaining this rate by calculation should then be studied. The meaning of a horizontal tangent in application to maxima and minima opens up a large field of interesting work of practical utility and fortunately there is no lack of suitable simple engineering and physical problems to afford practice and arouse interest. At an early stage when differential coefficient is found the corresponding integral should be written down, so that the student constructs for himself a table of the more common differentials and integrals. Later the idea of the integral as a sum is introduced which leads immediately to the determination of areas, volumes, moments of inertia, centres of gravity, &c., and can be applied to innumerable examples of interest in physics and engineering. Even if the teacher finds himself unable to proceed any further in the study of the calculus, yet even this small amount will be found of very great value to the student and will probably stimulate him to further study, for there appears no branch of Mathematics which so whets the appetite as the elementary portions of the calculus. Those who, while approving of the ideals suggested, yet feel that the time will not allow their attainment, should endeavour accurately to estimate the amount of time which in the ordinary course is devoted to the portions suggested for omission and to complicated algebraical and trigonometrical relationships, &c., and they will probably be surprised to find the large proportion of time a student spends in the useless portions of Mathematics. Then, and probably not till then, will they be convinced of how much useful Mathematics a boy of 16-17 could have learnt.

LIONEL M. JONES.

MATHEMATICS IN SCOTCH SCHOOLS.

INTRODUCTION.

The principal object that I have set before myself in preparing this report has been to give an account of the present position of Mathematics in the schools of Scotland. I have tried to present the curricula of typical schools in such a way that a reader of the report may obtain a fairly clear idea of the general aim and scope of the mathematical teaching, and I have refrained from criticism of curricula or advocacy of merely personal views. The Scotch system, though by no means stereotyped, has nevertheless assumed a definite character, and an account of the system is likely to be of greater interest and value than the statement of individual opinions in regard to the methods or content of school curricula. Many valuable papers that discuss didactic questions of a more or less controversial character have been prepared for the International Commission on the Teaching of Mathematics, and there is, therefore, less reason for entering into debateable matters in this report.

To understand the position of mathematical instruction in the schools of Scotland it is essential to consider it in relation to the general organisation of the schools, and the following brief statement may, therefore, be useful.

The funds which are provided by Parliament in aid of school education in Scotland are administered by a Department called "The Scotch Education Department." In the Stateaided schools education is carried on under the Code of the Department for Day Schools, or, especially as regards the more advanced work, subject to Regulations contained in various circulars issued by the Department. Even when the schools are not in receipt of financial aid from the State (and the number of such schools is relatively small) the organisation and curricula conform, in the main, to the requirements of the Department, and a statement based, like the following report, on the system administered under departmental supervision will embody all the essential features of school education in Scotland.

The classification of schools adopted by the Department--a classification based solely on distinction of curriculum-is as follows:

Primary School.-A school, or a department of a school, giving an education based entirely upon English, to pupils who are, as a rule, below the age of 14. A Primary School may contain pupils, or small sections of scholars, who are being instructed on the lines of an Intermediate School.

Intermediate School.-A school providing at least a three years' course of instruction in Languages, Mathematics, Science,

and such other subjects as may from time to time be deemed suitable for pupils who, on entering, have reached the stage of attainments in elementary subjects indicated in Article 29 I. of the Code ("the qualifying stage").

Secondary School.-A school providing at least a five years' course of instruction as aforesaid beyond the qualifying stage (Article 29 I. of the Code).

In the Primary School the normal organisation for children up to the age of 12 consists of (a) the Infant Division, for children under 7 years of age; (b) the Junior Division, for children between the ages of 7 and 10; and (c) the Senior Division, for children between the ages of 10 and 12. This is the organisation laid down in Article 19 of the Code for schools receiving grants under the Code; but the education of children up to the age of 12 follows essentially the same lines in all Scotch schools, and the stage of attainments in elementary subjects reached at that age by pupils who have completed the course satisfactorily is called the "qualifying stage.'

With the completion of the qualifying stage divergence of curriculum begins, and pupils may take (i)a Supplementary Course, or (ii) an Intermediate Course, or (iii) a Secondary Course. The Supplementary Courses are designed for pupils who leave school at the age of 14 and are the most advanced work of the Primary School; the Intermediate and Secondary Courses are taken in the Intermediate and Secondary Schools.

It may be noted that the nomenclature in common use for the schools in Scotland is not that adopted in the Department's classification, but has gained currency on other than educational grounds. A school that receives grants under the Code for Day Schools is generally called a Public School; it is elementary if its curriculum does not go beyond the Supplementary Course, but higher grade if it provides an Intermediate or a Secondary Course. Such names as Grammar School, High School, Higher Class School usually indicate schools of the secondary type which receive grants under a Code of Regulations specially applicable to them, and, in almost every case, send forward their pupils to the Leaving Certificate Examinations conducted by the Department. As a rule, intermediate and secondary schools have primary departments and the work of the first three years after the qualifying stage in the secondary school is generally equivalent to the work of the same years in an intermediate school.*

PLAN OF THE REPORT.

The method adopted in preparing this report has been, except in the case of the primary school up to the qualifying stage, to select typical schools and to state as fully as possible

*In Nelson's Annotated Code (Edinburgh, Nelson and Sons), will be found the Official Codes, Regulations, Circulars, &c., bearing on Scotch schools.

the various courses of instruction. Teachers in these schools have kindly furnished the necessary details.

Copies of Leaving Certificate and of University, Preliminary and Bursary Examination Papers are given in the Appendices at the end of the report. In conjunction with the description of the courses these will, it is hoped, give a fair view of the principal features of mathematical work in schools.

The account of school work in science, as distinguished from that in mathematics, is not so full as I could have wished. I have, indeed, tried to give due prominence to the fact that the teaching of science is a very important part of every curriculum in Scotch schools, but I have not been able to go into the details. of science teaching. A report that would deal adequately with this side of school work demands a much greater familiarity with the methods and scope of elementary science than I possess and would, if it were to be thoroughly satisfactory, require special discussion. It is my hope, however, that readers of this account may obtain a fair idea of the attitude adopted by school authorities towards this branch of education.

The schools are taken in the following order :

I.—The Primary School up to the Qualifying Stage.

II. The Primary School: Supplementary Courses, with
Note on Continuation Classes.

III. The Secondary School, represented by three types:
School A, a Public Higher Grade School with a
five years' curriculum; School B, a Secondary School
with a seven years' curriculum; and School C, a
Science School.

The most important examinations in connection with the schools are those for the Leaving Certificate granted by the Scotch Education Department; a statement of the regulations relative to these examinations precedes the account of the work undertaken in the secondary schools.

I. THE PRIMARY SCHOOL UP TO THE QUALIFYING STAGE.

Pupils who have satisfactorily completed the course up to the qualifying stage are expected, besides attaining a fair proficiency in English subjects—

(1) To know the four rules of arithmetic as applied to whole numbers, easy vulgar fractions, and decimals to three places, and to be expert in applying this knowledge to the calculation, both mentally and on paper, of simple sums in money and in the common weights and measures.

(2) To be reasonably proficient in the other subjects included in the approved scheme of work of the class.

The work specified under (1) and (2) is that which comes within the range of this report; under (2) are included drawing and nature study.

In recent years the Department has adopted the policy of asking teachers and managers to formulate courses of instruction suitable to the particular circumstances of their schools, and does not prescribe in detail the syllabus of any school subject. As a guide, however, to the manner in which a subject may be treated it has issued a series of Memoranda on the teaching of various primary school subjects.* The Memoranda, it is expressly stated, "are not put forward as final or authoritative documents," but they embody, it is thought, the ideals and aspirations that animate the best schools; and few, if any, specific recommendations are made which are not supported by the actual experience of teachers of repute.

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(1)-Arithmetic.

A detailed syllabus of the arithmetical work of a selected primary school would probably be less satisfactory than a statement of its general aim; the following summary of the “Memorandum on the Teaching of Arithmetic" is, therefore, given as representing a very widespread ideal.

Arithmetic in the primary school is regarded as consisting, not of so many isolated rules, but simply of a few fundamental operations which may be applied in a great variety of circumstances. The work is divided into two periods. The first period is concerned mainly with whole numbers and occupies approximately four years of the pupil's school life. During this period the aim of the teaching is (1) the development of the pupil's interest and intelligence in solving simple concrete problems with a clear understanding of the processes employed, and (2) the gradual perfecting of the machinery for the addition and multiplication of abstract whole numbers (within certain limits) and, as a consequence, for the inverse operations of subtraction and division.

In the first period great importance is attached to oral (or mental) arithmetic, written work being considered as belonging specially to the second period. Concrete problems and the gradual introduction of the common units of everyday life are essential. The pupil must handle the coins and know their value and he must be practised in the use of measures till he can give a fair estimate of a length or a weight. He should also learn to state concrete problems by means of the four symbols +, -, x, and, conversely, to give concrete interpretations of such shorthand expressions.

The attainment of mechanical precision with abstract numbers is indispensable and the Memorandum gives various

* The Memoranda may be obtained from Wyman and Sons, London; they are all contained in Nelson's Code.