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completed the course of a municipal elementary school and be 16 years of age.
The length of the normal seminary course is three years, but efforts are successfully being made to extend it to four years. The aim of the mathematical course is closely adjusted to the demands of the future career of the students, i. e., the curriculum of the lower elementary schools. The course is as follows:
(1) Theoretic arithmetic with solution of problems. Five to eight hours a week are devoted to the subject according to the previous preparation of the students.
(2) Principles of algebra. Here there is much divergence, both in standards, which vary from simple equations to quadratic equations, progressions, and logarithms, and in time which ranges from one to five hours a week.
(3) Six to eight hours a week are given to geometry, the program in which is identical with that of the secondary schools (given below) with the exception that more attention is given to surveying.
The two upper classes have instruction in the methods of teaching arithmetic and geometry, one hour a week being given to each. The graduating class has practice teaching in a two-class school.
The aim of the work of the normal institutes is defined in the same way as that of the normal seminaries. The standards are accordingly higher, both for this reason and because the preparation of the students has been more extensive. The following is an outline of the mathematical course:
(1) Theory of arithmetic (one-half to three hours a week). (2) Algebra up to binomial theorem, recurring decimals, etc. (seven to eight hours).
(3) Geometry (four to five hours) follows the program of the secondary schools, viz: Straight lines; angles; parallels; triangles, quadrilaterals and polygons in general; circumference. Fundamental problems of construction and numerical examples. Measurement of straight lines and angles. Proportionality of segments. Similarity of triangles and polygons. Numerical relation of the sides of a triangle. Regular polygons. Limits. Length of circumference and calculation of it. Simple problems of constructions and numerical applications to each figure studied. Relative position of lines and planes in space. Dihedral and polyhedral angles.
Regular polyhedrons-areas and volume of prisms and pyramids-round bodiescylinder, cone, and sphere, and calculation of their area and volume.
(4) Plane trigonometry with a little of spherical trigonometry (one or two hours). This subject, however, is not prescribed and may be introduced optionally.
The upper classes have instruction in methods of teaching mathematics in the municipal elementary schools, to which the graduating class is also assigned for practice teaching.
The regulations recommend that the method of teaching in the normal seminaries should be didactic without neglecting, however, the importance of developing habits of independent work. In the normal institutes the method of the recitation is employed with drill reviews about every term. The textbooks play a secondary part and are used merely for purposes of reference. The same books are used in the secondary schools.
The staffs of the normal seminaries are recruited from among university graduates and in mathematics from those who have graduated in the faculty of science. But since the supply of these is short, graduates of the normal institutes are not infrequently appointed. In the normal institutes the instructors are exclusively university graduates. It is objected that such training is not adequate qualification for normal school positions, since the university graduates have no acquaintance whatever with the elementary schools and have had no pedagogic training, while the courses of the normal schools are not definitely prescribed. It is claimed, however, that these disadvantages are offset by the introduction of a broad and scholarly spirit into the training of the future teachers.
Since the supply of normal school graduates is small, candidates for teaching positions are also accepted who pass examinations in the theory and practice of the work of the elementary schools without any special training in one type or other of the normal schools.
The system of normal schools of Sweden belongs to that class which limits itself somewhat closely to the actual needs of the future teacher. Two types of schools are maintained—the regular four-year normal school for men and women, maintained by the State, and institutions for training infant school teachers (Kleinschulseminar), attended almost entirely by women, and giving a course varying in duration from one to two years.
Candidates for admission to the normal schools must be between 16 and 26 and must pass an entrance examination on the subjects of the elementary school curriculum. Since the number of candidates is usually greater than the number of vacancies, this examination is practically competitive. Preparatory institutions have been established to cram students for these examinations. Some of the students are graduates of “Real schools.” Students from the universities who desire to teach in elementary schools are admitted to short courses, usually of one year's duration. The instruction in the normal schools is given by university-trained specialists who have the same qualifications as candidates for teaching appointments in secondary schools.
The mathematical course of the normal school is at present still unsatisfactory, for the reason that it does not furnish a much more extensive course than that found in elementary schools. The regulations bearing on the subjects date back to 1886, with some amendments in 1894, and do not contemplate an extensive course. Their prescriptions, however, are regarded as minimal, and several normal schools have adopted broader standards. Thus they have introduced algebra up to arithmetical and geometric progressions, geometry of as high a standard as that taught in the secondary schools, and a little trigonometry, branches which are not required by the regulations. Four hours a week are devoted to mathematics in the first three years and two hours in the fourth. Professional work is begun in the second year with observation and practice teaching under the direction of the specialist instructors.
The following general outline of the course in mathematics represents the letter rather than the spirit of the present day; the latter is being strongly influenced by modern reform tendencies:
Exercises to attain speed and accuracy in handling the four operations with integral numbers and fractions, with applications and problems on the most important practical needs; the knowledge of theory underlying these operations; the theory of numerical equations of the first degree with one unknown; mental arithmetic, especially with the numbers from 1 to 100; extraction of square and cube root; and introduction to bookkeeping.
Principles of geometric bodies, the general properties of the most important plane figures, with the method of their measurement and calculation.
The institutions for training infant-school teachers (Kleinschulseminar) are governed by the regulations of 1897. The scope of the mathematical course is even more restricted than of that given above. It includes: Speed in the four operations with integral numbers and fractions, and their application to simple practical problems with exercises in mental arithmetic.
While geometry is not referred to in the regulations, it has been introduced in several institutions.
The whole question of the normal-school curriculum has been under discussion for several years, and it is highly probable that changes will be made that will bring the curriculum into conformity with the demands of recent reform movements.
The educational system of Switzerland presents an interesting parallel with that of the United States. So far as the training of teachers is concerned, the system is entirely local, though slight aid is given by the Federal Government. Since, as in the United States there is a great variety of students without the minimum requirements usually established under systems that are centrally organized, it is practically impossible to present a comprehensive account of the work of the schools.
The normal schools are public and private; some are independent institutions, others are departments of secondary or "middle" schools; a few are coeducational; others admit only one sex. few instances special short courses of two or three semesters are
organized for graduates of the “Real schools” and gymnasiums, but as a general rule the normal schools are intended for students entering between the ages of 14 and 16 from a higher elementary school (Sekundarschule). With a few exceptions the duration of the courses is four years. The divergence of standards of attainment is very great; some institutions offer the same standard of work as the gymnasiums, while others hardly surpass those of the higher elementary schools. In the best schools the mathematical subjects include arithmetic and algebra, geometry, surveying, theory of projections, commercial arithmetic and bookkeeping, and mathematical drawing. The total number of hours devoted to these subjects ranges from 10 to 28 a week, with an average of about 19 for the four classes.
At least three types of courses can be distinguished: (1) Those limited to a review of the elementary school courses and instruction in method; (2) those which aim to give a deeper and better command of the mathematical work of the elementary, higher elementary, and “middle” (secondary) schools with an emphasis on practical and mental arithmetic--the idea of number, numeration, common and decimal fractions, and general insight into the content side of practical arithmetic; (3) those in which mathematics is taught as in the gymnasiums as a cultural subject with training in mathematical thinking
In general, the courses in geometry confine themselves to the study of surface and volume. But attempts are being made to extend this and to include the theory of projections, conic sections, mathematical geography, and physics.
The methods of instruction naturally vary with the scope of the courses, so that while drill is emphasized in the lower stage the higher stage introduces a scientific treatment of the subject.
Under these circumstances it will perhaps be most useful for the purpose in hand to present an outline of one of the fullest courses, premising, however, that this represents the goal to which most of the normal institutions (for men at any rate) aspire rather than conditions widely prevalent. The course given is that of the normal school of the Canton of Zurich, at Kusnacht.
Aim of mathematics course: To give training in clear comprehension of space and number in the use of critical and logical processes of numerical and measurable quantities; to develop ability in terse, clear, and logical expression; to train in the recognition of the quantitative relations and problems of practical life, nature, and technology, and to solve these with insight and accuracy by calculation and drawing.
FIRST YEAR (five hours). (a) Arithmetic: Review of rule of three and simple bookkeeping; profit and cost accounts. Mental arithmetic. Short methods. Ratio and proportion.
(6) General arithmetic: Operations of the first and second stages with general numbers; rational numbers; powers with integral exponents.
(c) Algebra: Equations of the first degree with one unknown. Problems.
(d) Plane geometry: Review of ideas of space and the establishment of fundamental ideas. Lines, circle; measurement of distance and angles; parallels and central and axial symmetry; congruence. Translation and rotation. Construction of triangles. Principles of general and special quadrilaterals, secants, tangents, and circumscribed triangles. Comparison, computation, and measurement of the areas of triangles and polygons.
SECOND YEAR (six hours). (a) Bookkeeping: Accounts current with payment of interest and use of interest tables; elements of double entry limited to simple commercial processes.
(6) Arithmetic: Extraction of square and cube root. Irrational numbers. General in volution. Prime numbers.
(c) Algebra: Equations of first degree with several unknowns.
(d) Plane geometry: Similarity of plane figures. Scale drawing. Regular polygons and measurement of circle. Construction of simple algebraic expressions. Line and circle as loci. Methods of solving problems in planimetric construction. Practical exercises in use of simple instruments. Application to surveying of small plats of land.
(e) Trigonometry: Definition of the functions of acute angles; study of right angles and equilateral triangles. Problems with use of numerical values of functions.
THIRD YEAR (five or six hours). (a) Arithmetic: Theory of common logarithms. Arithmetical and geometric progressions. Compound interest and annuities.
(6) Algebra: Solution and theory of equation of the second degree with one unknown.
(c) Trigonometry: Principles of obtuse angles, triangle and geometric application. Continuation of definitions. Problems, especially of triangulation. Physics and solid geometry. General definitions and chief principles of goniometry. Construction of trigonometric expressions and examples of the trigonometric analysis of geometric constructions.
(d) Solid geometry: Relative positions of objects in space, and especially parallels and perpendiculars. The idea of projections, measurement of distances and space, angles, symmetry. Uniquely determined construction of trihedral and polyhedral angles. The sum of the sides and angles of convex polyhedrons. Euler's theorem on polyhedrons and regular polyhedrons.
(e) Theory of projections: Inclined parallel projections as illustrative material. Representation of points, straight lines, plane polygons, and simple objects in ground plan and elevation. Folding. The ellipse as projection of a circle and its focal determination.
FOURTH YEAR (six hours). (a) Arithmetic: The principal notions of permutations. Elements of probability applied to insurance.
(6) Coordinate geometry: Rectangular and polar coordinates in plane and space. Graphic representation of the simplest functions of one variable. Graphic solution of numerical equations.
(c) Solid geometry: The sphere and its related surfaces. Plane sections of these surfaces. Spherical triangles. Areas and volume of simple solids and their parts. Application to determination of weight.
(d) Theory of projections: Representation of polyhedra and elementary curved solids in ground plan and elevation. Plane diagrams of the surfaces of solids. Making of models. The chief map projections. Simple problems of intersection with appli