The conduct of the class is quite different from that in Germany or in France. In Germany, for instance, the class method predominates. In the United States, the individual is the center of interest. This is seen in the arrangement of blackboards. One black-board suffices for the German or French schoolroom where the class as a unit develops a particular theorem. With us it is necessary to have a continuous board around the room, so that each pupil can draw his own particular figure from which to explain. The above typifies what has been, and is yet to a large extent, our common schoolroom practice. Nevertheless we have teachers who follow the German plan and still others who try to combine these two extremes. The dependence upon a text-book and the requirement of a considerable amount of home work prevent the class period from being a time for instruction. Hence our teachers in mathematics tend too much to “hear” recitations. We have much to learn

from Germany and France in this respect.1

Since 1871, when the University of Michigan established the system of accrediting high schools, the teaching of mathematics in the latter schools has been more largely influenced by the universities. We find teachers, therefore, more busily engaged in preparing students for the university than for their daily living. The large universities of the east have not generally adopted the accrediting system, several admitting only by examination. Hence there have been varying standards of admission. Uniform entrance requirements being desirable, the College Entrance Examination Board, made up of representatives of the various universities concerned, was organized in 1900. This will undoubtedly secure greater uniformity of work in the various preparatory schools and will also improve the teaching of geometry from the point of view of the university, but it does not necessarily follow that this will be best for the pupils.

Within the last few years a movement of another sort has gained some prominence in America. Professor Moore of the University of Chicago was the first prominent mathematician in this country to champion the cause of the "Perry Movement."

Chapter VII below contains much that pertains to the teaching of geometry in our schools.

'See above, pp. 126-128.

This position was taken by him in his presidential address1 before the American Mathematical Society, in December 1902. The result has been that in the Middle West many of the highs chools in touch with the University of Chicago have begun a reform in the teaching of mathematics. They seek to teach the various mathematical branches in close correlation, this to be brought about by making mathematics the tool for scientific investigation. We thus see the tendency. Geometry, for example, is not to be taught solely for its logical value. What can be done with it is the chief desideratum. As a result of the interest aroused by this new movement, five associations for teachers of elementary mathematics were organized in 1903 alone.2

The Central Association of Science and Mathematics Teachers presented three objects in its organization: 1. To promote better teaching of science and mathematics, especially in the secondary schools. 2. To obtain a better correlation of these subjects to each other and to the other subjects of the curriculum. 3. To bring the college and the secondary school into closer relations. The second of these objects indicates the keynote of the new movement already mentioned.

1 Moore, On the Foundations of Mathematics, Bulletin A. M. S., 1903, p. 424; also School Review, Chicago, Vol. 11, 1903.

2 The Central Association of Science and Mathematics Teachers, with headquarters at Chicago; the Association of the Teachers of Mathematics in the Middle States and Maryland; the Association of Mathematical Teachers in New England; the Association of Teachers of Mathematics in Washington (State); and the Association of the Teachers of Mathematics in Kansas. Since 1903 other States have organized similar associations. 3 This association stands for the "Chicago Movement."

CHAPTER VII

PRESENT PROBLEMS AND THEIR HISTORIC CONNECTONS

AN HISTORIC SURVEY

Certain features have characterized the development of geometry and its teaching that have a particular bearing on present problems and practices. In its geometric development, the race has passed through at least four well-defined stages. There was first the intuitive stage, in which geometric forms and principles were adapted to the needs of daily living, without, however, any principle of classification or generalization. A second stage was marked by an ability to recognize space forms per se, and to classify geometric notions sufficiently to be able to deduce and apply adequate, although not always exact, working rules. A third stage was reached when geometric relations were systematized into a logical sequence. Lastly, practice was again developed, this time controlled by logic.

In this order of development, it is to be observed that:

1. The practical1 preceded the logical.-The Egyptians, Babylonians, and Chinese had a knowledge of practical geometry before Greece invented its logic. Before the study of Euclid gained a foothold in the schools of Europe, the practical geometry of the Romans had been developed. With the Hindus and the Arabs, the practical was of prime importance even after the latter became familiar with the Greek learning. In later times, the various countries of continental Europe paid more or less attention to the practical, and Russia in particular was late in assigning a prominant place to logical geometry.

1 The term practical is used as heretofore with reference to the applications of geometry within the field of mathematics or in the related fields of science

2. The transition into the logical was not abrupt.-Thales based his deductive geometry on the practical work of the Egyptians. He and his school were interested in astronomy, and to them are assigned some practical problems in geometry. The few theorems assigned to Thales show what a small beginning was made in logical geometry at Miletus. Pythagoras was the first to sever geometry from the needs of practical life and make it an abstract science. The first Greeks who developed a logical geometry did not make, then, an abrupt transition into it, but were guided in part by practical considerations.

When geometry began to be taught in the medieval universities, it was confined to the learning of definitions with perhaps some practical applications. The logical work was begun with the teaching of Euclid. The early secondary schools which have been previously mentioned gradually introduced the logical, the first geometry being taught in connection with astronomy or geography. It seems to have been generally recognized that the logic of Euclid was difficult.

We thus see that the first development of logical geometry was not completely divorced from practical applications, and, in a greater or less degree, its first teaching in the early universities1 and secondary schools was not according to the "Elements."

3. The tendency to hold to the special has been marked.-We have seen that even the Greeks in their logical work found difficulties in proving a general theorem. Euclid, in some of his definitions, fails to recognize the concept of the general. The Italian practical geometries of later times illustrate this tendency to a marked degree. These books show but little the influence of the "Elements," and so illustrate what may be called natural tendencies.

4. Geometry has been taught more and more to younger students. -With the Greeks, the study of geometry was for mature minds. Plato, we recall, believed it should be studied between the years of twenty and thirty. The medieval universities taught Euclid in the higher classes. When our own universities were founded they taught geometry in the last year. In later times it was taught in the freshman year, and finally the high schools took up the work. Geometry is now taught in our high schools generally in the second year, when the pupil is about the age of fifteen.

'This was for a short period only in the universities.

In France and Germany the boy is doing logical work at least three years before this. With such a changed position of geometry in the course of study, one should expect to find a change of aim and method.

In this historic survey, we are also concerned with:

1. The sequence in the subject-matter of geometry.-Euclid did not follow the historic development in his sequence of subjectmatter, for the practical was entirely eliminated, and the Greeks began the development of the subject-matter of geometry in quite a different order from that found in the "Elements." The Pythagoreans studied the regular solids before plane geometry had received any great development, and they also studied proportion, which comes late in the "Elements." Definitions and axioms, which introduce the first book of Euclid, were given prominence by Plato and Aristotle about 200 years after Thales founded the science. The sequence in three of Euclid's definitions is not pedagogical. He defines point, line, and surface in the order named, placing first the most remote from experience. This order has prevailed to a large extent up to recent years. One marked exception is found in the geometry of Gerbert, where the above order is reversed, the definition of a solid being placed first. Modern practice adopts the order of Gerbert.

2. The sequence in the development of the mathematical subjects. Historically, arithmetic, geometry, trigonometry, and algebra have been developed in the order named, but not in a strict "tandem" order, for there has been considerable overlapping. Elementary geometry alone was systematized within 300 years, but even it, within a century past, has annexed a vast field, including, among other topics, the geometry of the triangle and circle, the theory of transversals, and radical axes.

3. The correlation of geometry with science.-Thales was an astronomer and employed some of his theorems in practice. One cannot say positively that there was any correlation between mathematics and science in this first development of the logic of geometry, but there can be little doubt that these early investigators were stimulated to further study of geometry through their interest in astronomy. The Athenian Greeks were interested in both science and geometry, but the former was divorced from the latter. The same relation existed between