LETTER OF TRANSMITTAL.

DEPARTMENT OF THE INTERIOR,

BUREAU OF EDUCATION, Washington, October 11, 1919.

SIR: One of the committees of the Commission on the Reorganization of Secondary Education, appointed by the National Education Association, and several of whose reports this bureau has already published in the form of bulletins, undertook the study of mathematics in the high schools. As stated by this committee in the introduction to this report and by the chairman of the commission in the preface, the committee found itself unable to make final recommendations in regard to the reconstruction of the courses of study in this subject in the high schools. The committee has, therefore, confined its work to a preliminary report, presenting an analysis of the subject, and raising certain fundamental questions which must be answered before the reconstruction desired can be undertaken intelligently and with any certainty of satisfactory success.

I am transmitting this preliminary report for publication as a bulletin of the Bureau of Education, in order that in this form it may be accessible to students of education, teachers of mathematics, and directors of mathematics teaching in high schools. It is expected that it will give rise to such discussion and experimenting as will enable other committees to carry forward the work of the point of definite reconstruction of courses of study in this subject. for the several classes of high-school pupils.

Respectfully submitted.

The SECRETARY OF THE INTERIOR.

P. P. CLAXTON,

Commissioner.

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PREFACE.

The Commission on the Reorganization of Secondary Education finds itself confronted with problems of great difficulty in recommending a reorganization of the mathematical studies of the secondary school. Antecedent to new courses, there should be an agreement among psychologists and educators such as has not yet been reached. It seems, therefore, that the best service that the commission can at this time render is to present an analysis of the situation. This report, therefore, is submitted primarily for the purpose of stimulating discussion. It is hoped that the practical suggestions will also serve to direct experimentation in planning new courses for secondary school students of the various types here recognized. CLARENCE D. KINGSLEY,

Chairman of the commission.

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THE PROBLEM OF MATHEMATICS IN SECONDARY EDUCATION.

I. INTRODUCTION.

Few subjects taught in the secondary school elicit more contradictory statements of view than does mathematics. What should be taught, how much of it, to whom, how, and why, are matters of disagreement. There is every variety of position. A conservative group would keep substantially unchanged the customary content and division into courses, and find the hope of improvement in a more adequate preparation of teachers. To this limited reform an increasing number object, with little agreement, however, among themselves. Amid the conflict of opinions the committee on the problem of mathematics in secondary education believes that a reconsideration of the whole question is desirable.

To present the finished details of a working plan would have been most gratifying to the committee, but this has been judged impossible. The situation seems to force the limitation. To carry weight, such a detailed plan would have to be based upon a wider range of experiment than in fact exists. Only recently has there been serious effort to consider the problem of the proper content and arrangement of the courses in secondary mathematics. The pertinent experiments available for study do not as yet present a variety of type and testing sufficient to establish the necessary conelusions. Within the time allotment available to the committee there seemed then only the choice between no report and an admittedly preliminary report. The committee has chosen the latter alternative, and proposes to lay before the American educational public (1) some of the considerations that demand a fresh study of the problems involved, (2) some of the factors that bear upon the solution of the problem, and (3) certain tentative suggestions for experimentation to develop new and better courses.' It is but fair to say that few of the specific suggestions made are in fact new, many being already somewhere actually in practice.

II. THE DEMAND FOR AN INQUIRY.

An inquiry into the advisability of reorganizing and reconstituting secondary mathematics is demanded from a variety of considerations.

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It is gratifying to note that the Mathematical Association of America is pushing a program of study and experimentation along lines quite similar to those here discussed, 9

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1. It is being insisted as never before that each subject and each item in the subject justify itself; or, negatively, that no subject or item be retained in any curriculum unless its value, viewed in relation to other topics and to time involved, can be made reasonably probable. No longer should the force of tradition shield any subject from this scrutiny. A better insight into the conditions of social welfare, and the many changes among these conditions, alike make inherently probable a different emphasis upon materials in the curriculum, if not a different selection of actual subject matter. This calls for a review and revaluation, in particular, of all our older studies, mathematics not least.

2. Moreover, a growing science of education has come to place appreciably different values upon certain psychological factors involved, chief among which is that relating to "mental discipline." No one inclusive formulation of the older position can be asserted, yet on the whole there was acceptance of the "faculty" psychology with an uncritical belief in the possibility of a good-for-all training of the several "faculties." To the extremist of this school the "faculty of reasoning," for example, could be trained on any material where reasoning was involved (the more evident the reasoning, the better the training), and any facility of reasoning gained in that particular activity, could, it was thought, be accordingly directed at will with little loss of effectiveness to any other situation where good reasoning was desired. In probably no study did this older doctrine of "mental discipline" find larger scope than in mathematics, in arithmetic to an appreciable extent, more in algebra, most of all in geometry.

With the scientific scrutiny of the conditions under which "transfer" of training takes place, the inquiry grows continually more insistent as to whether our mathematical courses should continue unchanged, now that so much of their older justification has been modified. Possibly both purpose and content need to be changed.

3. Yet another demand for reconstruction is found in the now generally accepted belief that not all high-school pupils should take the same studies. The fact of marked individual differences has been scientifically established. The principles of adaptation to such individual differences, that is, to individual needs and capacities, is now widely accepted in the high schools of America. The exception calls for scrutiny. Traditionally, algebra and geometry have been required for graduation. Is this necessary or advisable? In this growing practice of differentiation and adaptation we have then a third reason for at least reconsidering the customary mathematics

courses.

4. A demand for reconsideration well worthy of our attention is found in the insistent question whether a content chosen to furnish

preparation for further but remote study does necessarily or even probably include the wisest selection of knowledge useful for those who do not reach that advanced stage of study. Whether all should learn first the more assuredly useful topics, or whether alternative courses should be offered, are proper subjects of inquiry. In either event we find in this consideration a fourth reason for studying anew the offerings of our high-school mathematics.

5. A fifth reason for reconsideration is found in the problem of method. Educators are studying now with new zeal the proper pre-entation of subject matter in all school work. Should not this study extend to secondary mathematics? Have we arranged the subject matter of that field in the best form for appropriation? Might it even be possible that mathematics should be reorganized in a way to run across customary lines of division? Or might this be true of some parts of mathematics for some groups of pupils and not be true of all? The proper answers to such questions are not at once evident, but certainly there is enough point in the inquiry to add a fifth reason for our proposed investigation.

III. ANALYSIS OF THE SITUATION.

1. The problem of presentation.-Far-reaching differences of method carry with them widely different organizations of subject matter, especially in introductory courses. From this consideration, at least, there are certain advantages in discussing as the first factor in the situation the problem of presentation.

The traditional school method has been that based upon the "logical" arrangement of subject matter. Thus our fathers studied English grammar before they took up composition, the "science" being "logically" anterior to the "art." The science, in this case grammar, began with a definition of itself and the analysis of the subject into its four principal divisions. Then came the definitions of the "parts of speech." It was a long-and generally dreary-road before the pupil could see any bearing of what he learned upon anything else. At length, after toilsome memorizing, there appeared within the subject itself a new variety of mental gymnastics which called forth from some a certain show of activity. In the end the survivors caught some glimpse of what it had all been about. But when they took up the "art" of composition, the "science" proved of small assistance. Somehow the "art" had to be learned as if it alone faced the actual demand.

From an implicit reliance upon this "logical" arrangement there has come a revolt, not yet universal, but still unmistakably at hand. The demand has now become insistent that in arranging subject matter for learning, consideration be given, not to "logic" as formerly conceived, but to economy in learning and effective con