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This scheme represents the general scope of the work in elementary mathematics. Variations will, however, be found, and some schools will take their pupils beyond these standards. Thus, algebra may be taken up to binomial theorem, and geometry may include the ground covered by Euclid VI and XI (1-21). Mensuration is not infrequently added in many schools, and use is generally made of the opportunity for practice in logarithms. Trigonometry, again, usually forms part of the mathematical course in secondary schools for boys. Mechanics, statics, dynamics, and hydrostatics appear more rarely as part of the usual mathematical curriculum, although they may be included under the subjects of science. The general tendencies of the reform movement in mathematics is to do away with non-essentials and thus to find time for trigonometry or some other branch, while on the side of method more attention is given to training in mathematical thinking in place of mere imitation of processes.
The preliminary examination for the elementary school teachers' certificate, which is conducted by the board of education, consists of two parts. Part I, which is a qualifying test, includes arithmetic among the subjects of examination. In Part II, elementary mathematics forms one of eight optional subjects. The scope of the requirements in arithmetic and elementary mathematics is stated as follows:
PART I. ARITHMETIC.
Excluding Troy weight, apothecaries' weight, practice, ratio, proportion, except by the unitary or fractional method, stocks and shares, true discount, foreign exchange, scales oi notation, recurring decimals and complicated fractions, and square and cube roots.
Candidates may be asked to find the square or cube roots of numbers that can readily be expressed as the product of the squares or cubes of small factors.
The metric system will only be applied to measuring length, area, and volume.
As a rule, the questions set will not involve long operations or complicated numbers, and the answers to money su ms will not be required beyond the nearest penny.
The papers will be sufficiently long to allow candidates some latitude in the selection of questions, but no limit will be placed on the number of questions which may be attempted.
PART II. ELEMENTARY MATHEMATICS.
The papers set will be sufficiently long to allow candidates some latitude in the selection of questions, and will permit, therefore, of latitude in the teaching schemes. Candidates will not be limited in the number of questions which they may attempt, nor will they be expected to answer the whole paper.
Every candidate should be provided with a ruler, graduated in inches and tenths of an inch and in centimeters and millimeters, a small set square, a protractor, compasses furnished with a hard pencil point, and a hard pencil. Squared paper will be provided when needed.
As a rule, the questions set will not involve long operations or complicated numbers, and the answers to money sums will not be required beyond the nearest penny.
Excluding Troy weight, apothecaries' weight, true discount, foreign exchange, scales of notation, recurring decimals, and cube roots.
Candidates may be asked to find the cube roots of numbers that can readily be expressed as the product of the cubes of small factors.
Candidates must understand the principles of the metric system, and should be able to decimalize money readily.
Questions on stocks and shares will be of a simple character and will not involve a knowledge of brokerage.
The use of algebraic symbols will be permitted.
As far as and including simultaneous equations (one of which is linear) in two variables-least common multiple and highest common factor—by means of factors. Problems leading to the types of equations specified.
GEOMETRY-(1) PRACTICAL. Bisection of angles and of straight lines. Construction of perpendiculars to straight lines. Construction of an angle equal to a given angle. Construction of parallels to a given straight line. Simple cases of the construction from sufficient data of triangles and quadrilaterals. Division of straight lines into a given number of equal parts or into parts in any given proportions. Construction of a triangle equal in area to a given polygon. Construction of tangents to a circle and of common tangents to two circles. Simple cases of the construction of circles from sufficient data. Construction of a fourth proportional to three given straight lines and a mean proportional to two given straight lines. Construction of regular figures of 3, 4, 6, or 8 sides in or about a given circle. Construction of a square equal in area to a given polygon.
-(2) THEORETIC. Angles at a point.If a straight line stands on another straight line, the sum of the two angles so formed is equal to two right angles; and the converse.
If two straight lines intersect, the vertically opposite angles are equal.
(i) a pair of alternate angles are equal, or (ii) a pair of corresponding angles are equal, or (iii) a pair of interior angles on the same side of the cutting line are together
equal to two right angles, then the two straight lines are parallel; and the converse.
Straight lines which are parallel to the same straight line are parallel to one another.
Triangles and rectilinear figures. The sum of the angles of a triangle is equal to two right angles.
If the sides of a convex polygon are produced in order, the sum of the angles so formed is equal to four right angles.
If two triangles have two sides of the one equal to two sides of the other, each to each, and also the angles contained by those sides equal, the triangles are congruent.
If two triangles have two angles of the one equal to two angles of the other, each to each, and also one side of the one equal to the corresponding side of the other, the triangles are congruent.
If two sides of a triangle are equal, the angles opposite to these sides are equal; and the converse.
If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles are congruent.
If two right-angled triangles have their hypotenuses equal, and one side of the one equal to one side of the other, the triangles are congruent.
If two sides of a triangle are unequal, the greater side has the greater angle opposite to it; and the converse.
Of all the straight lines that can be drawn to a given straight line from a given point outside it, the perpendicular is the shortest.
The opposite sides and angles of a parallelogram are equal; each diagonal bisects the parallelogram; and the diagonals bisect one another.
If there are three or more parallel straight lines, and the intercepts made by them on any straight line that cuts them are equal, then the corresponding intercepts on any other straight line that cuts them are also equal.
Areas.-Parallelograms on the same or equal bases and of the same altitude are equal
Triangles on the same or equal bases and of the same altitude are equal in area. Equal triangles on the same or equal bases are of the same altitude.
Illustrations and explanations of the geometric theorems corresponding to the following algebraic identities:
k (a+b+c+ )=ka+kb+kc+...
a? -b2=(a+b) (a-b). The square on a side of a triangle is greater than, equal to, or less than the sum of the squares on the other two sides, according as the angle contained by those sides is obtuse, right, or acute. The differences in the cases of equality is twice the rectangle contained by one of the two sides and the projection on it of the other.
Loci.—The locus of a point which is equidistant from two fixed points is the perpendicular bisector of the straight line joining the two fixed points.
The locus of a point which is equidistant from two intersecting straight lines consists of the pair of straight lines which bisect the angles between the two given lines.
The circle. A straight line drawn from the center of a circle to bisect a chord which is not a diameter is at right angles to the chord; conversely, the perpendicular to a chord from the center bisects the chord.
There is one circle, and one only, which passes through three given points not in a straight line.
In equal circles (or, in the same circle) (i) if two arcs subtend equal angles at the centers, they are equal; (ii) conversely, if two arcs are equal, they subtend equal angles at the center.
In equal circles (or, in the same circle) (i) if two chords are equal, they cut off equal arcs; (ii) conversely, if two arcs are equal, the chords of the arcs are equal. Equal chords of a circle are equidistant from the center; and the converse.
a The tangent at any point of a circle and the radius through the point are perpendicular to one another.
If two circles touch, the point of contact lies on the straight line through the centers.
The angle which the arc of a circle subtends at the center is double that which it subtends at any point on the remaining part of the circumference.
Angles in the same segment of a circle are equal; and if the line joining two point subtends equal angles at two other points on the same side of it, the four points lie on a circle.
The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
The opposite angles of any quadrilateral inscribed in a circle are supplementary; and the converse.
If a straight line touch a circle, and from the point of contact a chord be drawn, the angles which this chord makes with the tangent are equal to the angles in the alternate segments.
If two chords of a circle intersect either inside or outside the circle, the rectangle contained by the parts of the one is equal to the rectangle contained by the parts of the other.
Proportion-Similar triangles. If a straight line is drawn parallel to one side of a triangle, the other two sides are divided proportionally; and the converse.
If two triangles are equiangular their corresponding sides are proportional; and the converse.
If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle, and likewise the external bisector externally.
The ratio of the areas of similar triangles is equal to the ratio of the squares on corresponding sides.
Elementary solids.-Candidates will be expected to be acquainted with the forms and simple properties of the cube, rectangular block, cylinder, and cone.
The board's preliminary examination or its equivalent is the main avenue of entrance to training college, but it must be borne in mind that a certain percentage of the elementary school teachers in England and Wales do not attend a training college, but may become certificated by passing the necessary examinations. The usual length of the course in the training colleges is two years. The best candidates, however, are usually selected for admission to the departments of education of universities, where they are required to take one or other of the courses leading to a degree, with education as one branch of the required work. A still higher type consists of those students who take a four-year course, the first three being devoted to some course leading to a degree and the last to purely professional work.
THE TRAINING COLLEGES.
The various ways by which a candidate can become a teacher in an elementary school have already been indicated above. It will be necessary here to deal only with the course provided in the twoyear training college and in the university departments of education. The two-year training colleges are institutions established by private and mainly religious bodies, and recently in an increasing number by local education authorities. They provide in the two years both academic and professional courses, and, up to the present, have placed more emphasis on the academic than on the purely professional side.
The standards of admission have, so far as mathematics is concerned, already been described. Elementary mathematics is continued as one of the required subjects of study, and here, again, the requirements of the final examination conducted by the board of education and leading to the elementary school teachers' certificate, indicate the scope of the subject. Two papers are given in the final examination. The first paper includes arithmetic and algebra, Part I, which all candidates are required to take; a few questions are also set in geometry, but women are not required to answer them unless they desire to obtain distinction. The second paper also contains questions in arithmetic and algebra, Part II, and in geometry; all men students must take this paper, but women only on the conditions just described. The following scheme of study is outlined in the board's regulations for students taking the final examination in 1913 and 1914:
EXAMINATION IN MATHEMATICS FOR ELEMENTARY-SCHOOL TEACHERS' CERTIFICATE.
ARITHMETIC AND ALGEBRA.
Part I.-Arithmetic, excluding Troy weight, apothecaries' weight, true discount, cube root, foreign exchange, and scales of notation. Questions on stocks will not involve a knowledge of “brokerage.” Candidates must be acquainted with the principles of the metric system. Algebraic symbols and processes will be generally permitted.
Algebra as far as, and including, simple equations of one unknown, with easy problems leading up to such equations.
Part II.-Quadratic equations of one unknown, simple simultaneous equations of two unknowns and easy simultaneous equations involving the squares of the unknowns and problems leading up to these equations. Arithmetical progressions; geometrical progressions to a finite number of terms; square root, highest common factors, and lowest common multiples, ratio, and proportion. Permutations and combinations. The binomial theorem for positive integral exponents. The use of logarithmic tables.
N. B.-Questions involving graphic methods may be set in both parts, and when such questions are proposed squared paper will be provided.
Every candidate must be provided with a ruler graduated in inches and tenths of an inch, and in centimeters and millimeters, a small set square, a protractor, compasses furnished with a hard pencil point, and a hard pencil.
Figures should be drawn accurately with a hard pencil.
Questions may be set in which the use of the set square or of the protractor is forbidden.
Any proof of a proposition will be accepted which appears to the examiners to form part of a logical order of treatment of the subject. In the proof of theorems and deductions from them, the use of hypothetical constructions is permitted.
Practical geometry.—The following constructions and easy extensions of them: Bisection of angles and of straight lines. Construction of perpendiculars to straight lines. Simple cases of construction from sufficient data of triangles and of quadrilaterals. Construction of parallels to a given straight line. Construction of angles equal to a given angle. Division of straight lines into a given number of equal parts. Construction of a triangle equal in area to a given polygon. Construction of tangents to a circle. Construction of common tangents to two circles. Construction of circumscribed, inscribed, and escribed circles of a triangle. Construction of a segment of a