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ARITHMETIC AND ALGEBRA.

First year.

Prussia.

Saxony. Fundamental operations with integral numbers, Operations with vulgar and decimal fractions; decimals, and vulgar fractions; averages; percent- practical applications. age; profit and loss; taxes and customs; alloys; approximations.

Second year. Interest; profit and loss; discount; social arith- The four fundamental operations in algebra. Exmetic; alligation; insurance, especially workmen's tension of the idea of number to negative and comcompensation; notes and bills.

pound numbers.

Third year.

Introduction to algebra; fundamental operations Fractions, factors; proportions; equations of the with integral numbers; equations of the first degree first dergee with one unknown quantity; practical with one unknown.

problems solved by means of equations and pro

portions; powers.

Fourth year. Powers; roots; logarithms; equations of the first Roots; extension of the idea of numbers to irradegree with several unknowns.

tional and imaginary numbers; equations of the first degree with two or more unknown quantities: practical problems in arithmetic; introduction to logarithms.

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Second year. Parallelograms; trapezoid; regular polygons; Continuation of triangles; quadrilaterals and circle.

polygons; elements of the theory of the circle; constructions by geometric analysis.

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Fisth year. Solid geometry; constructions of algebraic ex- Plane trigonometry; solid geometry. pressions; trigonometric functions; mensuration of plane figures.

Sirth year. Arithmetic and geometry included in the one hour Conclusion of solid geometry. per week devoted to the methods of instruction in these subjects.

These programs form the core of the mathematical studies throughout the German normal schools. More extended courses are found in some of the normal schools of the smaller States of central Germany and of Wurttemberg. Thus the use of graphs has been introduced as early as the second year at Sondershausen and Cöthen and in the third year in Wurttemberg. At Hildburghausen and Gotha algebra is carried up to quadratic equations; in Wurttemberg the elective course in the sixth year includes the binomial theorem and some study of functions. In geometry Cö hen introduces the theory of projections in the third year, and solid geometry is begun in Wurttemberg in the fourth year. Trigonometry finds a place in tho programs of Coburg, Greiz, Cöthen, and Sondershausen in the fourth year, while spherical trigonometry is found in the fourth-year course at Cöthen, in the fifth year in Wurttemberg, and in the sixth year in Baden, with applications to geography and astronomy. Surveying with practical work in the open has been introduced in many schools, but it is objected that too many students are required merely to look on while the few gain the practical experience.

The suggestions and recommendations of the Deutscher Ausschuss für den mathematischen und naturwissenschaftlichen Unterricht will indicate at once the criticisms that are leveled against the work of the normal schools and the direction in which the reformers are moving. In its Vorschläge für der mathematischen, naturwissenschaftlichen und erdkundlichen Unterricht an Lehrerseminarien the committee sets up the following theses: (1) The mathematical course should be six, even seven, years in length. (2) Preparation for the teaching profession covers both academic and professional training; each is to be treated separately; the professional training should come at the close of the academic. (3) The six-year course should be given in one institution under one director and one staff with the same spirit throughout. (4) The normal school should in every way approximate more closely to the secondary schools, but should not attempt to prepare for the university. (5) Those subjects which are not of great value to all students should be limited in treatment, as well as those which can not be made up with the help of books, and more emphasis should be placed on training in the methods of independent study, the use of books, apparatus, etc., which are as a rule not accessible to the teacher in a small elementary school. (6) The students should be taught methods of study, trained in habits of thinking, and given a grasp of the broader phases of education rather than be crammed in subject matter and methods of the elementary school. (7) The normal school staff should be in a position to give instruction in all stages of the school, with full mastery of the subject. (8) Opportunity should be afforded in the last year of the course for specialization in some group of subjects or some special subject.

The committee recommends five hours a week in the first five years and two hours a week in the last, for mathematics. The problem at present is to secure not only logical training, ability to handle numbers, and preparation for teaching arithmetic and mensuration, but also adaptation to the problems of the modern school. Greater emphasis is necessary on the thought content of mathematics and its connection with the actual facts and needs of practical life. Ability must be developed to think mathematically about the phenomena of the world and the relation of mathematics to life, and to develop the idea of change and function in the most general sense. In general the standard set up in the Meran proposals for the secondary schools should be carried over to the normal schools with the addition that the future calling of the teacher must be borne in mind. The teacher, however, must be in a position to take a broad view of his subject from a higher standpoint. The Prussian program, for example, should be supplemented by the addition of spherical trigonometry, with application to mathematical geography and astronomy, and a little of conic sections. The study of complex numbers and infinitesimal calculus should be offered as an elective only. Algebra could well be begun in the first year. Arithmetic should deal with the problems of family, community, and State management, and economic life in general, with the emphasis as much on content as on arithmetical ability. The five hours of the course should be divided so that two hours a week are given to algebra, two hours to geometry, and one hour to arithmetic, but the different branches should be taught by one teacher.

COURSE IN MATHEMATICS RECOMMENDED BY THE COMMITTEE.

FIRST YEAR.

Arithmetic: Problems of household management and the vocations with applications of the rule of three and percentage.

Algebra: Algebraic formulas for the fundamental operations with integral and fractional numbers as an introduction to general arithmetic. Concrete representation of numbers by lines. Evaluation of algebraic expressions. Numerical solution of simple equations with one unknown. The use of brackets and their application to mental arithmetic and abbreviated methods.

Geometry. Straight lines and angles. Triangles and quadrilaterals, particularly parallelograms, trapezoid, and deltoid (motion, interdependence of the parts, symmetry, congruence, area). Measurement and plotting of lines and angles in the field. Geometric drawing in connection with field observations and the construction of simple triangles and quadrilaterals.

SECOND YEAR.

Arithmetic: Community and State management.

Algebra: The idea of relative magnitudes with practical examples, concretely illustrated by directed numerical line segments. Rules for calculation with relative magnitudes. Simple polynomials. Theory of proportion in connection with fractions. Pure and applied equations of the first degree with one or several unknowns. Simple inequalities.

Geometry.—Chief principles of the circle. Equivalence of figures bounded by straight lines. Computation of the areas of such figures and their transformation into equivalent figures. Approximate calculation of figures bounded by curved lines. Such constructions as are closely connected with the course. Applications of the theory of triangles and quadrilaterals to simple problems in surveying and measurement of altitudes in the field. Geometric drawing of the circle and straight line in ornamental design. Drawing of plane figures. Examples from surveying and technical sciences.

THIRD YEAR.

Arithmetic: The money market and international exchange. The decimal system. The simplest rules of divisibility. Decimal fractions. Approximations in working with fixed numbers.

Algebra: Graphic representation to illustrate ordinary relations. Graphic representations of linear functions and their application to the solution of equations. Powers with positive integral exponents. Graphic representations of the functions y=axn (parabolic curves), n=2, 3, 4. Powers with negative integral exponents. Graphic representation of the functions y=ax-n (hyperbolic curves), n=1 and 2. Roots.

Geometry.-Theory of similarity, and the circle. Measurement of circle. Practical exercises in mensuration. Geometric drawing to scale, especially of sketches made in the field by the students; drawing of curves and use of squared paper.

FOURTH YEAR.

Arithmetic.-Selected parts of commercial arithmetic.

Algebra.-Equations of the second degree with one unknown. Relation between coefficients and roots. Graphic representation of an expression of the second degree depending on a varying quantity. Graphic solution of equations of the second degree with one unknown, also by the intersection of a fixed parabola and a movable line, or of a fixed line and a movable circle. Equations of the second degree with two unknowns in simple examples, to be solved numerically and graphically.

Geometry.--Simplest propositions about lines and planes in space; plane representations and measurement of solids. Conic sections as plane sections of the right circular cone. Geometric drawing; introduction to descriptive geometry. Representation of simple solids in perspective, as well as ground plan and elevation. Plane sections of simple solids. Plane diagrams of the surface of such solids. Construction of ellipses. Spirals.

FIFTH YEAR.

Arithmetic.—Simplest ideas of insurance. Review of elementary school arithmetic, with observations on method.

Algebra.-Extension of the idea of powers. The function y=ax and its graphic representation. Idea and use of logarithms; simple method of calculating logarithms; representation of logarithmic functions. Theory and use of the slide rule. Arithmetical progressions of the first order. Geometric progressions and their application to compound interest and annuities. Comprehensive review of the extension of numbers to irrational numbers.

Geometry.Trigonometry in connection with constructive plane geometry. Graphic representation and calculation of trigonometric functions. Application of the measurement of triangles to practical problems in connection with independent measurement by the students in the field. Sine and cosine in spherical trigonometry and their application to fundamental problems of mathematical geography and astronomy; determination of time and position. Comprehensive review of the scientific structure of geometry. Geometric drawing as in third year, with the simplest penetrations and their practical applications; shading.

SIXTH YEAR.

Comprehensive review of the functions studied, with examples from geometry and physics, especially mechanics.

Geometry.--Constructive treatment of conic sections, with indications of their analytic representation.

With reference to the method of instruction in the normal schools the committee emphasizes the need of reform. The pupils should be discouraged from mechanical work and memorization. In place of the dogmatic and didactic teaching, which is a characteristic of the Prussian schools in particular, more encouragement should be given to independent and experimental work, especially in geometry. Great importance should be attached to oral expression and mental arithmetic. The students should be trained to construct their own problems, and should be restricted to problems of a practical character rather than the mere elaboration of fictitious exercises. More attention should be given to graphs and graphic representation, and to some extent to the history of important theories and principles. In the final year the instructor should be given free scope and should aim to give the students some acquaintance with the problems of higher mathematics and an insight into mathematical method to serve as a preparation for further study.

In the training of the students in the methods of the teaching, two principal types are again found. A number of States, following the practice of Prussia, assign instruction in methods of each subject to the specialist. Thus, while one teacher gives the general method and principles of teaching, the teacher of mathematics has charge of the instruction in the special methods of his subject. In the second type one teacher has charge of all the purely pedagogic work of the school and gives courses in both general and special methods. It is felt, however, that in both types when methods courses are handled by two instructors, there may be not only an absence of uniformity, but even that contradictory principles may be enunciated. This danger is all the greater in those States in which the specialist in subject matter has had no experience in the elementary school, while on the other hand it is feared that too often the special teacher of method may, for example, be inadequately trained in mathematics. To avoid these dangers a system of cooperation has been adopted in Saxony between the specialist and the teacher of general method. While the specialist throughout the course aims to show the development and interconnection of the different parts of the subject with special reference to the elementary

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