JOHN RICHESON, superintendent of city schools, East St. Louis, Ill.-In talks with aged people of sixty or more it is found that they incline to relate reminiscences in which the tragical or unpleasant predominates. Teachers of English have a very great opportunity of carrying pupils away from the sensational and morbid, as it is found in the daily papers, and to get them to think on the pleasant incidents of their lives. Thus the greatest results may come from the efforts of such teachers, in consequence of which they should be better paid.

MISS FRANCES WILDE, Kirkwood, Mo.—Essays may be corrected before they are written. Enthusiasm alone will not secure good results in a classic composition. As a class in physical culture gains grace thru constant practice, so in composition pupils can learn to avoid errors in English by what may be called composition gymnastics. Thus my pupils are asked to combine six or eight sentences into one complex, balanced, or periodic sentence. The effect of a little of this formal work each day is quickly seen in the essays. As much as possible must be done to forestall weariness on the part of the teacher. To gain the best results moral, mental, and physical poise are necessary to the teacher.

MISS HARRIET L. KEELER, assistant in Central High School, Cleveland, O.—At this time, when written work is being made the test in schools and colleges, it is good to hear someone emphasize reading aloud. A class that has read a text, as for example Macbeth, aloud, in a thoroly intelligent and intelligible manner, is quite as well prepared to meet the requirements of the Harvard entrance examinations as a class that has studied and interpreted it. Read, I pray you.

W. W. EARNEST, superintendent of city schools, Macomb, Ill.—As a teacher of 130 pupils for the past two years I have gained much experience in the work, and may well offer a suggestion, namely, that subjects for composition should possess natural interest and should be of such a character as to prevent the children from seeking aid from encyclopædias and other literary sources. Essays should be written once in two weeks, and no line should be drawn between the narrative and descriptive kind. If the subjects appeal to the students, the teacher will find no little ground for hilarity in reading them.

MISS NINA UPDIKE, St. Charles, Minn.-Ought the study of the history of literature to be taken up during the third and fourth years of the high school?

MISS F. R. PERRIN, St. Louis, Mo.-It seems that the study of literature can best be studied in combination with history.

MISS FLORENCE L. ROGERS, Burlington, Ia.-As we learn to know people best by conversing with them, so we learn to know authors best by reading them.

MISS ESTHER MILLS, Central High School, St. Louis, Mo.-No work of literature can be understood or enjoyed without an acquaintance with the author-his life, environment, and the spirit of his time. Teachers ought not to attempt to present a classic before their pupils without touching, incidentally at least, upon the life and personality of the author.

MISS L. B EVERETT, Selma, Cal.-According to the course of study in California, the history of literature is studied during the last year of high school. Experience has shown that it appeals only to a select few. Would it not be better if it were made an elective? Much of the work of the senior year in high school, like the history of literature, seems beyond the range of the average senior.

J. M. GAMBRILL, department of English and history, Baltimore Polytechnic Institute, Alberton, Md.-The history of English literature ought not to form part of a highschool curriculum as a formal study. A manual is used in the Baltimore Polytechnic Institute rather as a book of reference. It is used in connection with Ivanhoe for the specific purpose of getting pupils interested in Scott thru a knowledge of his characteristics and preferences, whereby the boy will soon desire to know what Scott wrote. The study

of history of literature formally presented to the immature pupil would be as undesirable as studying the philosophy of history without previously having pursued the study of history.

MISS DIANA SIME, De Forest, Wis.—How many are in the habit of teaching literature chronologically?

MISS LILLIAN GRAVES, St. Louis, Mo.-In Mary Institute it is the custom to teach Chaucer during the first year and Shakespeare during the second, and in this way to follow the development of English literature.

JAMES M. MILLER, Chester, W. Va.-Some pupils are interested in the history of literature and some in literature alone. Is it not possible to teach them so that the class will be interested in both?

MATHEMATICS CONFERENCE

WHAT STUDY OF MATHEMATICS IS NEEDED BY THE MAN OF AVERAGE EDUCATION FOR PRACTICAL LIFE?

JOHN S. FRENCH, PROFESSOR OF MATHEMATICS, JACOB TOME INSTITUTE, PORT DEPOSIT, MD. In what follows I propose to consider in general the part played by mathematics in the training of the average man; this to be followed by a consideration of the different topics as to their places in the curriculum, and as to methods most effective in bringing about the results defined by their various functions in the achievement of the general object.

Ever since Xenocrates called it the handle by which to get hold of philosophy, mathematics has played an important (altho somewhat exaggerated) part in schools as a mental discipline. With the invention of analytic geometry by Descartes, in 1637, there came a revolution in methods when the so-called geometric method was superseded by algebra. This, together with the infinitesimal calculus laid the foundation of modern mathematics, which, in the hands of a remarkable array of talent, has shown itself indispensable in scientific research, not only in effecting a co-ordination of the facts gained by observation into a systematic theory, but also in anticipating certain results to be obtained from experiment. Thus mathematics has come to play another important rôle in education in preparing certain students for future special work.

Lastly, and what is universal in our age of push and among a people purely industrial and commercial, it must at once be recognized that the technique of mathematics is indispensable to the employee whose habits of accuracy, facility of process, and capacity of application are essential to the success of the industry, and fundamental to the administrator whose powers of classification and organization explain the existence of so many colossal enterprises.

It appears, then, that mathematics has a threefold function to perform, and in the immediate attainment of the objects noted above, which must appear of tantamount importance to a liberal training, I venture to say there is no field so fertile, so rich in resources, offering so many inducements to the prospector-i. e., the genuine teacheras is contained in mathematical study.

It develops the powers of observation and retention more readily than any other science. From a utility standpoint, its "practical examination" in the battle of Manila, where, as Dr. Hodge so aptly expresses it, “the battle was won by having woven into the warp and woof of the brains of those gunners the mathematical truth that it is only waste of time and ammunition to pull the trigger of a cannon until it is aimed mathematically straight," its requirement in all construction work-these are only a few of the many

attestations of the intimacy of mathematical education to every human interest, individual and national.

It is a wonderful thing to realize that man has acquired a wider knowledge of the world in the last 250 years than in all the preceding centuries. We may, I think, ascribe the great strides of the last two and a half centuries to three sources; viz., the elimination of the church as a dominating influence in education, the invention of mechanical aids and appliances, and the application of mathematical principles to the systematizing of natural phenomena. And I dare say that, referring to the future, the one overshadowing criterion of the successful advance in scientific circles will be the efficiency in mathematical research which leads to experimental discovery, which in turn admits of a commercial phase by putting into practice with profit the practical application of this discovery.

It seems almost evident, then, that the education of our youth, threefold in purposenamely, for the advancement of scientific knowledge in the study of nature's laws, for the development of our mechanical and industrial resources, and for the upbuilding of principle in the maintenance of a people rich in material wealth and all that goes to make high character-is conditioned by a mathematical training.

To the uncritical observer, in view of a certain discontent among pupils toward mathematical study which tends to make it ineffective, and because of certain well-defined efforts to subordinate it in the secondary course to the study of natural science, it might seem that the importance of the study of mathematics had been overemphasized. This ineffectiveness is due, as I see it, to three causes: misjudgment as to the immediate object of mathematical teaching, its improper arrangement in the curriculum, and the lack of a teaching force adequate to its demands.

Altho the chief aim of secondary education is cultural, we must not ignore the fact that the necessities of the strenuous life demand a cultural training leaning as much as possible on the material side. Mathematics is a classic in the history of cultural education, yet I believe its fitness today in the curriculum is best demonstrated by making practical utility its chief end. Again, the subject-matter of mathematics is peculiarly adapted to the logical order of development, but I am firmly convinced that up to the period of adolescence, in which is the dawn of the powers of logical reason, only those parts of number and form should come into the teaching which are most receptive to the child in the course of mental growth.

The inefficiency of the teachers of mathematics is one of the serious problems for solution in our educational work. It seems to come from a tendency to rely too much on the text-book, which in turn is due primarily to a narrow knowledge of the subjects, on the one hand, and a lack of appreciation of the student's powers of comprehension, on the other. Again, no subject has suffered more than mathematics from the narrow and rigid prescriptions from institutions of higher learning whose dominance has to a great degree stripped the secondary institution of its independence. Just as long as the measure of a successful completion of requirements in a subject is the ability to pass collegeentrance examinations, just so long will our methods of instruction remain narrowed down to and into the rut of machine processes, with a corps of instructors whose abilities need not extend beyond the field of the drillmaster.

For mathematics to be successfully taught in the secondary school, it must be freed from the yoke of the narrow college-entrance requirements. It must be in the hands of instructors whose knowledge of the various branches is such as to bring into use the methods of instruction adapted to the ends sought. They must have such an appreciation of the interrelations of the different branches as to bring before the students that most powerful conception that the symbols of each with its methods of operation are in many instances only different modes of reaching substantially the same conclusion; and, what is more important, they should develop, by example, in the students the power of discernment as to the proper method to use in concrete application. With practical utility as a basis,

instruction should be conducted in such a manner as to make evident the practical usefulness of the different parts, at the same time bearing in mind that these parts together go to make up a systematic course.

In regard to the subordination of mathematics to physics in the secondary curriculum, I must say I am not in sympathy with such an extreme step. Physics, it seems to me, if made subject to a development based on logical sequence and mathematical formulæ, is stripped of its virility for the boys in the middle and early teens. Like Clerk Maxwell, when a boy, they are interested in the "go" of things, and the laboratory should be their text-book.

I am, however, heartily in sympathy with a correlation of the two; in fact, the converging to one common ground of the laboratory experiment and the applied mathematical development will conserve in the pupil an interest which is bound not to wane, and at the same time will cause him to scent the possibility of the intimate connection between light, heat, and electricity. What better stimulus than this is there for the development of independent thought-the true goal of education? To make this correlation most effective, we should include in the group manual training and mechanical drawing.

We come now to the consideration of the various branches of mathematics with regard to their places in the curriculum.

The subjects of which the curriculum is made up may be classified into general and special. Under the classification general I shall include all subjects regarded as fundamental in a liberal education; those subjects classified as special have to do, primarily, with preparation for some future vocations. For convenience in illustration, let us assume that the electives, which constitute the special class, are offered by the group system, and, in accord with general opinion that this system is found properly in the last two years of the, at present existing, high-school course.

I am of the opinion that from the time when the pupil's capacity is maximal for fixing in mind the fundamental operations for small numbers up to the introduction of electives, some form of mathematics should find place in each year's program. I make this statement on the ground that consecutive study adds greatly to efficiency in the study of mathematical technique, while its presence does not cause a displacement of other topics more suited to the needs of the pupil's mental condition.

In the grades the pupil should obtain a working knowledge of the fundamental operations of arithmetic, tested by his skill in using the tables and in applying them to continued multiplication and division; with an appreciation of algebraic notation as used for abridging and facilitating arithmetical processes; and with a true concept of geometric figures and a knowledge of certain mutual relations between them. All instruction in arithmetic should be centered about the fundamental operations, and the topics of factoring, fractions, percentage, interest, taxes, and simple denominate numbers should tend to give practical applications of these operations. The introduction of algebraic processes should be gradual and brought about in such a way as to make the pupil appreciative of the generalities of algebra. This introductory work should include the fundamental operations, factoring, H. C. F. and L. C. M. of monomials, fractions (in my opinion, this should be the beginning of the study of fractions, as a part of the number concept and numerical fractions should be taught as an application), the solution of all arithmetical problems by the equation so long as consistent with facility, and very elementary work with type-forms. The study of geometry should bring into use the metrical properties as usually taken up under mensuration.

The pupil enters the high school, then, well skilled in the operations of addition, multiplication, substraction, and division; with a knowledge of some of the simpler applications. This reduction to a minimum of intricate applications means simply a postponement of technical arithmetic to a later period when as a special subject it shall be taught to pupils for immediate use. He has become acquainted with the symbolic language of elementary algebra, and has thru its use come to appreciate algebra as generalized arith

metic, and, wherever feasible, has used it for solving arithmetical problems. Thus is obviated the difficulties found in beginning algebra in the usual way, and which are due to strangeness of expressions and methods. His knowledge of geometric figures gained in the study of form has freed him from embarrassment in later work, caused otherwise by gaining false notions, and he has learned to apply number to form.

The elementary algebra of the first high-school year should have for its immediate aim the mastery of linear and quadratic equations, paying particular regard to the applications where the possibilities of solutions and interpretations of results are most important in their practical aspects. This immediate aim in no wise detracts from the force of mathematical study as a mental discipline, nor does it impugn the ultimate aim of higher mathematical development in establishing a systematic theory for the extension of the number concept, because of the necessity of interpretations in solutions not expressible by the ordinary numbers of arithmetic, and the mastery of type-forms in the classification of algebraic expressions to facilitate and abridge operations, these may be shown in later work, and without any alteration of definitions to fit into a universal theory. Care should be taken not to bring into the work long and complicated applications of principles. Teaching should always be commensurate with the power of assimilation; otherwise it might cause mental indigestion.

Plane geometry, which is usually taught in the second year, owes its presence in the curriculum to the demand of college authorities. This is the only apology which I can offer for the presence of the subject employing the methods prevalent in current textbooks. As a mental discipline simply, it has no place in the modern curriculum. As to the general field of mathematics, its methods caused a stagnation of development for twenty centuries, the tremendous advance of late years being due primarily to the supersedure of algebraic for geometric methods. The chief purpose of geometry in the curriculum is a study of the properties of space, and it should be taught as a practical subject.

In contrast to the existing demonstrational and deductive methods, they should be observational and inductive. The one consists of a collection of facts obtained from observation and verified by experiment; the other is a series of properties whose right to fit into a definite system founded on definite assumptions is shown by demonstration. The first cultivates individual initiative in the discovery (to the pupil) of new truths-an acquirement attainable by all pupils; the second cultivates the power of exact reasoning-an acquirement attainable, from this source, by practically no pupil. The first sharpens by progressive development; the second dulls by a lack of evident usefulness.

The material of observational geometry should be such as will easily find its place in mechanical drawing and physics. It should include accurate familiarity with geometric concepts and properties of geometric figures, constructions with the ruler and dividers, experimental verification, and perhaps later demonstrations based on earlier experiences.

In connection with the observational geometry the study of logarithms should be taken up, with the end that it furnishes a tool by which all numerical computations involving involution, evolution, and continued multiplication and division, may be most readily solved. The elements of trigonometry should come into the third-year course as a general topic; the instructor should have constantly in mind the practical side of the subject, and the course should be supplemented by field work.

With the introduction of electives in the third-year comes a bifurcation in mathematics, as a special subject, designed to meet the needs of preparatory and nonpreparatory students. For the former the study of algebra should be continued, with the idea of developing technique. With this in view, particular attention should be paid to methods, and the full force of the formula for classifying should be taught. This course furnishes one of the best examples of the necessity of an efficient teaching force.

For those students who are preparing for a business career the study of merchantile arithmetic should be pursued at this time. It is here that there should come a thoro training in applications of arithmetical processes to the more advanced forms of merchan