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circle containing an angle of given magnitude. Division of straight lines into parts in any given proportion. Construction of a fourth proportional to three given straight lines and of a mean proportional to two given straight lines. Division of straight lines in extreme and mean ratio. Division of a straight line internally or externally into segments, so that the rectangle under the parts is equal to a given square. Construction of regular polygons in and about circles. Construction of a circle from sufficient data of the following character: (1) Radius given, (2) point on the circle given, (3) contact with a given straight line or circle, (4) contact with a given straight line at a given point. Construction of a rectilinear figure to a specific scale or of specified area, and similar to a given figure. Construction of a square equal in area to a given polygon. (In cases where the validity of a construction is not obvious, candidates may be required to indicate the reasoning by which it is justified.)
Illustration and explanation by means of rectangular figures of the following identities:
k (a + b +c+ .)=ka + kb + kc +
(a + b)2= a2 + 2ab + b?,
2ab + b2,
a? – b2 = (a + b)(a - b). Theoretic geometry.-Candidates should be acquainted with the fundamental propositions concerning angles, parallel straight lines, and the congruence of triangles, such as are contained in the substance of Euclid, Book I, Propositions 4–6, 8, 13–16, 18, 19, 26–30, 32. Easy deductions from these theorems will be set, and arithmetical illustrations will be included.
The substance of the theorems contained in Euclid, Book I, Propositions 33-41, 43, 47, 48; and Book III, Propositions 3, 14–16, 18–22, 31, 32, 35-37; Book VI, Propositions 1-8, 19, 20, 33, together with Propositions A and D. Questions upon these theorems, easy deductions from them, and arithmetical illustrations will be included.
In dealing with proportion it may be assumed that all magnitudes of the same kind can be treated as commensurable.
Candidates will be expected to be acquainted with the forms of the cube, the rectangular block, the tetrahedron, the sphere, the cylinder, the wedge, the pyramid, and the cone.
In addition to the ordinary prescribed subjects, students may also offer an examination in 2 optional subjects selected from 20 subjects, including mathematics. The scope of the examinations in optional elementary mathematics is shown in the following syllabus :
Elementary mathematics up to, and including: Geometry of lines, circles, and of the simpler solid bodies, but excluding conic sections.
Coordinate geometry of lines and circles.
Algebra: Progressions. The binomial theorem for positive integers. Logarithms and their use. Probabilities.
Plane trigonometry: The solution of triangles.
Mechanics: Friction. Virtual work. Center of gravity. Simple machines. Motion of pendulums and projectiles. Motion in a circle. Impulsive forces acting on elastic and inelastic particles.
The board of education, however, does not require training colleges to undertake the whole of this syllabus, nor is it intended that the examination shall be a test of the knowledge of the whole of this ground so much as a test of mathematical power.
The professional work is included in a course on principles of teaching and a course of six weeks' practice teaching. The board of education offers a number of suggestive syllabi on principles of teaching which include "Numbers and elementary mathematics; methods and apparatus; practical instruction and its relation to handwork; use of literal symbols and graphs by older children; mensuration and geometrical drawing.” Special provision is made for candidates who intend to teach in infants' schools. At the London Day Training College a course of lectures, extending over two terms, is given on general principles of mathematical teaching, with special reference to the more elementary parts of the subject. Demonstration lessons are also given by the members of the staff in the practice schools of the college. The system varies, and it is impossible to make any general statement of the amount of time given to the methods of teaching mathematics. Similarly, in the case of practice teaching, in which each student must engage for six weeks, there is no requirement that mathematics shall be taught; although it is very probable that some time will be given to this subject. But at best the time devoted to practice teaching is limited. It is becoming more usual now for the lecturer in mathematics of the college to include the methods of teaching in his course, to give demonstration lessons, and supervise the practice of the students in their field.
The majority of students who pass through a course of training for the elementary school positions attend the two-year training colleges. Provision is, however, made for three-year courses in training departments closely connected with the universities. The number who can avail themselves of these courses is limited, since, so far as is possible, it is proportioned to the total number of students in the respective universities. The three-year courses include both academic and professional subjects, and students are expected to have met the ordinary requirements for entrance to the universities. Since the students themselves select the group of academic subjects which they will study for their degree, there is no compulsion that mathematics shall be included. Hence the general standard of attainment in this field is the elementary mathematics required for the entrance examinations. A few students, of course, may include mathematics in their course or may even take an honors course, that is, specialize intensively in the subject. It is intended that in future the training of teachers at the universities shall be given in a fouryear course—three to be devoted to academic subjects and one to the professional. Among the professional subjects are included principles of education which cover instruction in the methods of teaching all the subjects of a public elementary school. The practical work consists of eight weeks of practice teaching in an elementary school. The three-year course, therefore, does not provide for higher attainments in mathematics than are found in the examination for the board's certificate; it does, however, afford better opportunities than the two-year course to those students who are interested and desire to carry forward their study of mathematics.
The training of teachers in France is entirely under the control of the State. There are 85 normal schools for men and 84 for women teachers. Students are admitted to the normal schools by a competitive examination. Candidates for these examinations must possess the brevet élémentaire, a certificate which is itself a qualification for teaching in the écoles maternelles, classes enfantines, and écoles primaires élémentaires. Candidates for the brevet élémentaire must be 16 years of age. The examination for this certificate consists of questions divided into three series. The mathematical questions are somewhat simple and consist (1) of a question in arithmetic and the metric system and the analytic solution of a problem covering thefour operations with integral numbers and fractions and the measurement of surfaces and volumes, and (2) of oral questions in arithmetic and the metric system. The exercise in drawing may, for boys, include geometric drawing of a simple object with ground plan, cross section, and elevation. The examination is based on the following standards, which may serve to indicate also the attainments of the elementary schools:
Review covering principles and theory of arithmetic, and short processes in mental and written work.
Primary numbers; the most important cases of divisibility.
As a general rule, the examinations show that the reasoning ability of the candidates is defective and indicate no powers of analytic thinking or acquaintance with simplified methods. Considering that the brevet élémentaire is accepted as a qualification for appointment as temporary teacher (stagiaire) in the schools mentioned above, it is felt that these standards are by no means too high.
The competitive examination for admission to the normal schools requires approximately the same standards. Since 1910 definite syllabi are issued for each school every four years in certain subjects, including mathematics, and it is hoped that as a result the work of the pupils will be less vague and more accurate than hitherto. The candidates come from rural schools, higher primary schools, and the
supplementary courses (cours complémentaires). The preparation and attainments of the candidates are accordingly not uniform. It is proposed, therefore, that the following standards be accepted as sufficiently satisfactory: (a) Detailed study of integers and decimals and their application in mental arith
metic. (6) Definitions, theories, and rules of the following operations in written work:
Addition, subtraction, and multiplication of integers. Definitions and rules
of the divisions of integers and the extraction of the square root. (c) The same for fractions and decimals. (d) Properties of sums, differences, products, and exact quotients. Simplified
processes. (e) Theory and rules of mental arithmetic. (f) Characteristics of divisibility by 2, 3, 4, 5, 9, 25. Tests of multiplication and
division by the excess of 9's. (9) Metric system. (h) Problems on the four fundamental operations. Percentage. Rules of interest
and commercial accounts. (i) Solution of simple numerical equations with one and two unknowns. In geometry the following outline is suggested:
Experimental study by folding, drawing and measurement, of the chief properties of the following figures: Straight line, plane figures, circumference, angle, triangle, parallelogram, rectangle, rhomboid, square, rectangular parallelepiped, cube, cylinder, cone, and sphere. The use of the rules, square, compasses, protractor, measuring gauge. Rules for measurement of surfaces and simple volumes.
Candidates for the competitive examination must be between the ages
of 16 and 18. The normal school course consists of three years. Since 1905 the first two years have been devoted to general instruction and the last year to professional work. The schedule in mathematics is as follows:
Normal school course in mathematics.
There is an examination for promotion at the end of each year. At the end of the second year the students must pass the brevet supérieur; at the end of the third year comes the final examination (l'examen de fin d'études normales).
The aims of mathematical instruction (including geometric drawing) in the normal schools are declared to be (1) intellectual training through habits of clear and precise thinking, logical analysis, discrimination of the true and false, and accuracy in reasoning; (2) to furnish a certain amount of useful and definite knowledge and to clarify and complete such knowledge as the students already have; (3) instruction in the methods of teaching arithmetic and geometry.
The course of study for the three years is as follows:
Practical and mental arithmetic, and algebra (one hour): Algebraic calculation; positive and negative numbers; operations limited to such applications as can be employed in the normal schools. Simple equation. Problems.
Geometry (2 hours): Straight lines, circumference; angles, triangles; rules of equality; parallels; parallelograms; chords and arcs of circles; tangents, measurement of angles; construction.
Straight lines and planes; parallel straight lines and planes; straight lines perpendicular to planes; dihedral angles; perpendicular planes; parallel planes. Introduction to trihedral angles. Definitions of simple polyhedrons, prisms, parallelepiped, pyramid.
Proportional lines. Similar triangles and polygon. Metric properties of triangles and circles. Exercises. Measurement of areas. Regular polygons. Length of circumference. Area of circle. Measurement of volume, parallelepiped, prisms, pyramid.
Methods employed for solution of problems or demonstrations of theorems. Methods of deduction. Analysis and synthesis. Examples.
Algebra (1 hour): Review. Quadratic equations with one unknown with simple problems. Compound interest and annuities.
Geometry (1 hour). Review. Cylinder, cone, sphere. Elementary properties of solids. Mensuration.
Arithmetic (2 hours): Theoretic arithmetic; proportions; rule of three; partnership; bonds; insurance; discount; average maturity; proportional division; problems of alligation and alloys.
Surveying (10 hours). Cusmography (10 lessons).
Method of mathematical sciences (mathematical deduction, definitions, axioms and postulates, propositions, examples). Methods of demonstration and examples. Outline of the development and progress of mathematical sciences (3 lessons).
Serious criticisms are brought against the present arrangement of putting algebra in the first year, to be followed by arithmetic in the second year. It is objected that (1) the students, through insufficient knowledge of arithmetic, do not see its connection with algebra, which in any case can not be studied without a knowledge of arithmetical terms; (2) mental work is based largely on such knowledge, while the practical arithmetic lacks the solid foundation which should be laid in the first year; (3) the study of proportional lines and equalities demands a knowledge of ratio and proportion which are not taken up until later. The program of the third year is further criticized as too ambitious; too much is attempted in too little time. It is suggested, for example, that, instead of teaching the history of