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or the education department attached to a university, and those who have been uncertificated teachers, have had some experience in schools, and have passed the certificate examination of the board of examination for teachers in elementary schools.

In 1911–12 the number of uncertificated teachers was 39,125; that of certificated teachers, 97,104, of whom only 55,497 had been professionally trained.

GENERAL CAREER OF CANDIDATES FOR TEACHING PROFESSION.

The most usual career of the certificated teacher, according to the standard at which the authorities are aiming, would be as follows: Entering the elementary school at the age of 6 or 7, he would remain and probably complete the course at the age of 12, when he would pass on to the secondary school. Here he would be a bursar or scholar maintained by public appropriations until 16 or 17, when he would become a student teacher for a year, combining practical training in an elementary school with academic training continued in the secondary school. At the close of this period he must pass the preliminary examination for the elementary school-teachers' certificate, or some equivalent examination, which admits him to a training college for a two-year course or the education department of a university for a three-year course. The training college gives both academic and professional training; the university education department gives the same and leads to a degree. Recently a new system has been introduced by which a student may devote three years in the university or college to academic subjects and spend a fourth year in the education department on professional work only.

ELEMENTARY SCHOOL MATHEMATICS.

The scope and character of mathematical teaching are changing rapidly in English schools, both elementary and secondary. In the elementary schools the subject has been broadened and now includes, in addition to arithmetic, some instruction in algebra to pupils in upper classes and those preparing for secondary schools, and geometry, covering practically the content of the first book of Euclid. The nature of the work in arithmetic is indicated in the following quotion from the Elementary School Code, 1912:

Arithmetic, including practical work in measuring and weighing, oral exercises, written exercises (which should be of a varied character and should not infrequently involve the application of more than one arithmetical operation), and, in the higher classes, practice in explaining the processes used. The principles and advantages of a decimal system of weights and measures should be explained to the older scholars, and the use of literal symbols in working simple problems may with advantage be taught in the higher classes. Practical instruction should be given in mensuration and should include drawing to scale, the older boys should learn the use of compasses and protractor, and such practical instruction should be correlated as far as possible with handwork.

Much of the old formal and mechanical drill in numbers is being replaced by more insistence on clear and systematic thinking about number relations in practical situations; reasoning rather than mere mechanical figuring, logical setting out rather than working by rule, are the newer aims of instruction in this field. The following scheme indicates generally the amount of arithmetic, the chief portion of the mathematics of the school, that will have been acquired by a prospective school-teacher in the elementary school period. The practical work and the work correlated with handwork, drawing, and domestic subjects are not brought out in the scheme, but must be taken into account:

FIRST YEAR.

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Introductory. First period. (Without notation or any ciphering abbreviations; e. g., the 4 in 43 must always

be spoken of as forty or four tens, never as four.) (i) Numeration, as far as 100. (ii) Relations to each other of the 10 primary numbers comprise:

(a) The different ways in which each of them may be made up of any

two which come before it.

(6) Those of the multiplication tables. (iii) Use of (a) to tell the sum or difference of two numbers, one of which is a primary. (iv) Addition and subtraction of two composite numbers. (v) Continued practice in making up the tables, using what has already been

remembered. Second period. (i) Continuation of above; increased speed to be looked for in addition and

subtraction, especially in (iii) above.. Counting and other practice with

round numbers up to a few tens beyond 100. (ii) Notation, and suitable exercises from what precedes, to be done in writing. (iii) Application of the foregoing to simple practical calculations (money, length,

and height, omitting fractional values, such as farthings). Nomenclature, none.

SECOND YEAR.

Chiefly experimental practice, oral and written, all operations without ciphering abbreviations. First period.

(a) Recapitulation.
(6) Practical calculations as before, occasionally with two steps.
(c) Introductory and experimental:
(i) Extension of numeration to 1,000.

(A few hundreds will be sufficient for all the practice necessary in

this period.) (ü) Continued practice with tables, looking for the results to be remembered,

especially the first few multiples of 12. (iii) Conversion of numbers expressed in tens to hundreds, and vice versa. (iv) Simple multiplication and division; i. e., performing calculations simi

lar to those of the tables with numbers outside the range of the tables. (The two forms of division to be kept separate until the children have had experience enough to recognize their identity when the working is set down in figures.)

Second period.

(i) Continuation of above.
(ii) Visual demonstration of the principle axb=bXa, and its use especially

for reckoning ten, twenty, or a hundred times a number.
(ii) Easy exercises introductory to long division, such as number of

thirties or twenty-eights in 200. (iv) Practical exercises in the four rules with simple numbers. (d) Ciphering: Additions, subtractions, and multiplications.

Nomenclature: Add, subtract, multiply (and unit, if required).

THIRD YEAR.

First period. (a) Recapitulation; all previous exercises to be frequently repeated at full length;

i. e., without technical nomenclature or ciphering abbreviations.
(6) Practical examples as before with larger, but in the main round, numbers.
(c) Preparatory and experimental:

(i) Leading to the long rules.
(ii) Easy examples of repeated multiplications and divisions, with numbers

and money.

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(iii) Written form of the four rules, numbers and money, multipliers and

divisors as a rule of not less than two figures. (iv) The standard units; pounds sterling and avoirdupois, day, year, yard. Second period.

(v) Submultiples of these, attention being called to their insufficiency for

the general purpose of measurement. (vi) Experimental practice in measurement (length and value) with fractional

remainders as }, }, }, }, }, $, to, }, , fractional notation being withheld

at first. (vii) Easy additions, etc., of such fractions, e. g., fd.+1d., 1-4 in.+ 1-4 in.

half of id., in. +i in., 3 in. X 5. (viii) Reductions to be worked visually without ciphering (e. 8., 178.=17 times

12 pence or 12 times 17d.). (ix) Simple examples of factorial reductions, e. g., 27 times 8d., 75 times 16,

9 times 16 in., 90 lbs.X7. (d) Ciphering, practice simple and compound rules and reductions.

Nomenclature, previous terms, divide, and words like third, in the sense of

the third part.

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FOURTH YEAR.

(a) Recapitulation as before; simple explanations of suitable matters to be asked

for in writing (6) Practical exercises rather more searching from a manual and from blackboard. (c) Preparatory and experimental:

(i) Further practice in reduction, showing the form of the rules.
(ii) Further practice in repeated multiplications and divisions and fac-

torial reductions.
(Children being shown the advantage to be derived from these by

examples.)
(iii) Return to general problem of measurement; expression of the size of

fractional remainder when a common measure is obtainable.
(iv) Conversion of fractional quantities into others of lower denomination (oral

demonstration only).
(v) Easy additions, etc., of fractional quantities.
(vi) Tables of weights and measures, capacity, right angle, degreos.
(vii) Further development of the long rules.

(d) Ciphering. All previous rules, reduction, with some practice in factorial

multiplication and reductions.
Nomenclature as before; terms relating to fractions (as numerator, denominator,

common measure, same name) to be withheld, and use of abstract fractional
numbers to be avoided.

FIFTH YEAR.

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First Period.

(a) Recapitulation as before. (6) Practical exercises in what precedes from a graduated collection and the black

board. (c) (i) Further discussion of fractions with examples of their use in the same way

as whole numbers for expressing relative magnitude, e. g., 30=21/2 times

12, 20=12|3 times 12, 21/2 times what 100? etc. (ii) Notation of fractions. (iii) Conversion of a fraction to an equivalent one with any given denominator

(by a rational process rather than by rule, e. g., 3/5=3/5 of twenty twentieths), occasional examples with fractional numeration (e. g., to

may assist rather than perplex the class. (iv) Practice in making up the four rules for fractional numbers; (least common

denominator, where required by inspection); the two purposes of division to be treated separately. Special attention is recommended to the case

which provides the rule for finding x from the datum times nx=A. Second Period. (v) Calculation of values, amounts, etc., which may be expressed by simple

fractional numbers in terms of quantities for which rates are given; e.g.,

40 cwt. at 58. for 12 cwt. (commonly called “proportion” sums). (vi) Easy calculations in the same way involving shop discounts at so much in

the shilling, rates and taxes at so many shillings in the pound, interest and discount at so much per £100, profits and losses expressed as simple

fractions of the outlay. (vii) The principle of practice:

(a) Short division preferable to compound multiplications, division by 12,

etc.

(b) Cost of n things at 1/3 of x sh.=1/3 of cost at x sh. (d) Ciphering practice; all previous rules, including simple fractions, bills of parcels,

and practice. First Period.

(a) Recapitulation; increased use of questions to be answered in writing. (6) Practice as before. (c) (i) (a) Rules for greatest common measure and least common multiple, whole

and fractional numbers. (b) General form of the rules for multiplication and division with fractional

numbers. (ii) Square measure and cubic measure. (iii) Practice in setting out the work of weight and measure problems with alge

braic symbols. (iv) The same where the quantities involved are fractional numbers. (v) Purpose and meaning of the term “per cent": 3 per cent of quantity defined

to mean three hundredths of it, and to find what per cent a quantity A is

of another B, we must find how many hundredths of B there are in A. (vi) Calculations involving percentage, profit and loss, attendances, strengths of

mixtures, interest, etc., to be worked by fractional rules.

SIXTH YEAR.

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Second Period. (vii) Further considerations of problem of measurement; failure of method of

vulgar fractions to meet the general case; decimal system of expressing

fractional quantities and its advantages. (viii) As a new system of numeration (cf. fifth year (c) (i) above).

(A still further extended system will be propounded when algebra is

begun, shortly.) (ix) Construction and practice of the fundamental rules with decimal fractions. (d) Ciphering; all previous rules, etc., processes, and elementary algebra, notation,

further extension of numeration to negative values, etc.

SEVENTH YEAR.

(a) and (b) as before, practice now largely preponderating. (c) (i) Banking and investments; explanation of terms and procedure; examples

involving stocks and discounting of bills. (ii) Limitations of decimal system as described; recurring periods. (iii) Square root; explanation of rule and exercises.

SECONDARY SCHOOL MATHEMATICS.

All candidates for the teaching profession must attend a secondary school recognized by the board of education as efficient for at least three years. The standards of attainment in mathematics in such secondary schools are fairly well defined by the requirements for the entrance examinations conducted by the various universities, which are accepted as equivalent to the preliminary examination for the elementary school teachers' certificate. The University of London requires for its matriculation examinations the following attainments in elementary mathematics: Arithmetic: The principles and processes of arithmetic applied to whole numbers and vulgar

and decimal fractions.
The metric system.
Approximations to a specified degree of accuracy.
Contracted methods of multiplication and division of decimals.
Ratio and proportion, percentage. Averages.

Practical applications of arithmetic.
Algebra:

Symbolical expression of general results in arithmetic,
Algebraic laws and their application,
Factors of simple binomial or quadratic expressions.
Equations of the first or second degree, and problems leading thereto.
Square root. Graphs of simple rational integral algebraic functions.
Arithmetic and harmonic progression.

Geometric progression.
Geometry:
The subjects of Euclid I-IV with simple deductions, including easy loci and the

areas of triangles and parallelograms of which the bases and altitudes are given commensurable lengths. (All proofs of geometric theorems must be geometric. Euclid's proofs will not be insisted upon.)

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