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cation to shading. Fundamental ideas of geometric perspective, especially of the vanishing point and foreshortening.

(e) Methodic and historical survey in the field of pure and applied arithmetic and the study of forms in the elementary school.

The certification of the students as teachers depends on a final examination which consists of two parts, general-academic and professional. Both parts of the examination may be taken at the same time, or the professional examination in theory and practice of teaching may be postponed for one or two years during which the candidates are expected to teach in an elementary school. No provision is made, however, for the supervision of the candidates during this period.


In dealing with educational topics concerning the United States it is difficult to do more than present a general picture of tendencies, since the standards in different parts of the country vary so greatly and since they are at different stages in their development. But this variety is perhaps no greater than in Switzerland and, in the matter of training of teachers, than in England. In some States the qualifications demanded of teachers in elementary schools are still no higher than the attainments of one or two years beyond the elementary schools, while at the other end of the scale teachers may have the qualifications of college graduates. The most general standard may, however, be said to be graduation from high school, with two years of normal school training. The mathematical attainments will thus have been acquired in the eight grades of an elementary school, four years of high school, and two years of normal school.

The work of the elementary grades covers the four fundamental operations, fractions, decimals, and percentage, with business applications. Mensuration and algebra are included in some systems in the eighth grade. There is more uniformity in the first six grades than in the last two. The variations in the seventh and eighth grades are due not only to differences in amount of subject matter taught, but also to the employment of the department system. Departmental teaching in these grades is quite common in the larger school systems and is being extended as rapidly as circumstances permit. Under this system teachers are employed who are not only more experienced, but have had special training for the work and are frequently college graduates. Where algebra and geometry are not introduced, more intensive work is given in the practical and business applications of arithmetic. There is a tendency in the better schools to emphasize thoroughness and practical utility. In the schools where algebra is taught, there is again some variation in scope and method; some courses are merely based on high-school texts; in

And even

others the subject is introduced as part of the work in arithmetic. The amount of geometry taught in the elementary schools has in recent years tended to decrease. The mensuration of geometric figures has always been included in arithmetic textbooks. The reform movement in geometry, however, while not widespread, aims to cultivate habits of geometric study and to train such powers of observation and generalization as pertain to geometric data.

In the high schools the curriculum in mathematics is determined in general by the admission requirements of the colleges, especially in those States which have a complete and State-wide organization of education, crowned by the university. This is true, also, in other States and even in the smaller communities, where few, if any, of the pupils plan to go to college, and where the local school committee disclaims the intention of following university guidance. where this is not true of other subjects of study, it holds for the mathematical curriculum, because, apart from the colleges, the subject is not defined by any other authority that the schools are willing to accept.

The ordinary mathematical course in high schools includes algebra, plane geometry, solid geometry, and trigonometry, and, less frefrequently, arithmetic. The algebra courses are usually divided into three groups—(1) up to quadratics; (2) quadratics up to the binomial theorem; and (3) an advanced course, including various disconnected subjects and intended to furnish preparation for further mathematical study. In plane geometry the ordinary sequence is that of Legendre, including geometric constructions and original exercises ("riders”). Under solid geometry are included the usual topics up to the surface and volume of the sphere. The course in trigonometry includes definitions, the formulas and their proofs, logarithms, and triangles with practical applications to surveying and navigation. Where arithmetic is included, not more than half a year

is devoted to it, usually at the beginning of the course, to make up any deficiencies in the elementary school preparation, or in order to insure facility and accuracy in routine operations. It must be noted that under the elective system no student in high school is required to take all the courses offered in mathematics, with the result that the mathematical equipment of the future teachers varies considerably both in scope and depth.


The institutions for training teachers for the elementary schools are in a preponderating majority normal schools maintained by the State. A few of the larger cities maintain their own training institutions, ard there exist in addition a number of private normal schools. More recently there have been established training classes in connection with high schools, for the purpose of training teachers for rural schools. The high-school training classes usually offer one year of professional study either as part of or at the end of the regular highschool course. There is great variation in the entrance requirements, in the quality and amount of the work offered, and in the length of the course.

With the exception of the high-school training classes, the training institutions as a rule offer a two-year professional course based on graduation from high schools. But, as usual, it is impossible to generalize, for some normal schools offer five-year courses beyond the elementary school; others one-year courses beyond the high school. The State normal schools with a two-year course beyond the high school may be taken as the type. They are generally supported by appropriations by the State legislatures. A small number of these schools are supported by a specified State tax, supplemented by appropriations by the State legislatures, and in a few cases they receive some income from the sale or rental of public-land grants. A few State normal schools have small permanent endowments. Tuition is free, except for slight incidental fees, to persons declaring an intention to teach in the State in which the school is located. With the

exception of a few schools in the South, the State normal schools are coeducational.

The curriculum of these schools includes academic and professional subjects. The quality of the academic work varies greatly, partly because of the variety of preparation of the students, partly because of the lack of training on the part of the teachers and the heavy programs that they too frequently carry. But the graduates of the better normal schools usually receive full credit for two years' work on entrance to college and universities. The professional studies include psychology, history of education, methods, observation and criticism of instruction, and practice teaching. All normal schools have facilities for practice teaching, either in schools under their immediate control or in schools of the regular public system. The practice or training school is usually under the charge of a member of the normal school faculty, whose task it is to coordinate the work of the normal schools and practice schools. The practice teaching of the students is done under his supervision and that of critic teachers.

There are as yet no definite qualifications required from normalschool teachers. Accordingly, while the majority of the teachers who have charge of mathematics are college trained, there is still a large number with only high-school or normal-school training. Most of the teachers of mathematics would prescribe work through the calculus as the minimum academic preparation for teachers of mathematics in normal schools; a few would include more advanced courses as the minimum. It is generally admitted, too, that such teachers should have had some professional preparation as well as experience in teaching in elementary and high schools. With the gradual extension of the scope of the work of the normal schools, as, for example, the

inclusion of analytics and calculus in the curriculum offered, there is a tendency also to require at least one year of graduate work from the teachers of these subjects.

It is usually assumed that students have had a high-school course in arithmetic of one term or semester, one year or more of algebra, and one year of geometry, usually plane. In some normal schools arithmetic is required of students “found wanting" in its subject matter. In many schools solid geometry, advanced arithmetic, advanced algebra, and trigonometry are offered as electives.

It is thus apparent that the teacher of mathematics in the first six grades has a knowledge of subject matter in formal mathematics far beyond any of the actual needs arising within these grades. That this knowledge of subjects beyond elementary arithmetic is effective in the teaching of the arithmetic, it would be very difficult to show. The formal and isolated character of the work in algebra and geometry, as they are usually taught, leaves them barren of any content values having a bearing upon anything which appears in the usual work of the first six grades. Even in the best normal schools there is little evidence that the work in algebra and geometry is any less academic than in the classical high schools. In the normal school "humanistic" values should certainly most fully reveal themselves. There is but little specific preparation for the teaching of seventh and eighth grade mathematics. There is a failure to realize the truth of the statement that “this (seventh) grade marks the approach of the time when the pupil should pass from mere control and manipulation to understanding and investigation,” and to provide the kind of instruction needed to lead the pupil out into the broader field of mathematical knowledge.

The following table shows for 64 schools the different courses given, the number of schools offering each course, and the average number of hours given to each course:

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Elementary algebra.
Plane geometry
Advanced algebra..
Solid geometry
Methods of teaching.
Plane trigonometry
Spherical trigonometry.
Theory of equations..
Differential equations.
Analytical mechanics.
History of mathematics.


4 25 23 3 1 11 1 8

63 77 80 96 87 52 48 57 48 70

The professional courses and work include a course in arithmetic methods or in arithmetic with some attention to methods in all normal schools, observation and practice teaching in some form in most of them, and a course in the history of mathematics in very few. The subcommittee of Committee No. I, that dealt with the preparation elementary branches in the preparation of the American report to the international commission, came to the conclusion that great differences in ideals and practice exist with reference to the following points: There is variation from method as a mere incident to subject matter to the use of full time for method in "methods" courses; from no use whatever of current literature on the teaching of mathematics to "very extensive" use of such literature; from no consideration of games and recreational devices to very careful consideration and testing of these; from no consideration of the course of study in arithmetic for the grades to the full development of such a course and the organization for presentation of certain of its units; from no mention at all of the history of the development and teaching of mathematics to the establishment of well-organized courses in this subject; from no observation of lessons in arithmetic in the grades in connection with methods courses to one observation lesson each week during the course; from no practice teaching at all required in mathematics to practice work in arithmetic for all in at least two grades; from no supervision of practice work and the teaching of mathematics in the training school by the teacher of mathematics to close, jointly responsible supervision with the grade supervisors; from positive discouraging of departmental teaching of mathematics in the grades to positive advocating of it in middle and upper grades; and from no differentiation in training for the grades to courses in detailed special methods for primary grades.

There are also evident the following points of general uniformity: Entrance requirements to methods courses in arithmetic are quite uniformly high-school courses of about one-half year's work in arithmetic, one year in algebra, and one-half year or one year in plane geometry. Some kind of methods or teacher's course in mathematics is found in all. Some form of observation work, either in connection with the course in arithmetic methods or with general method or practice teaching, is advocated by all. In general, departmental work below the seventh is discouraged. No school is fully satisfied with its present practice.

Suggestions for reform are of four types: Greater knowledge of subject matter; a reorganization of arithmetic material, giving it more vital relationship to the child's life and to social life; a more intelligent knowledge of the pedagogy of arithmetic; and a closer, more vital relationship between the department of mathematics and the teaching of arithmetic in the training school. Three ways

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