DEPARTMENT OF THE INTERIOR BUREAU OF EDUCATION BULLETIN, 1920, No. 1 THE PROBLEM OF MATHEMATICS IN SECONDARY EDUCATION A REPORT OF THE COMMISSION ON THE REORGANIZATION OF SECOND- ARY EDUCATION, APPOINTED BY THE NATIONAL EDUCATION ASSOCIATION WASHINGTON GOVERNMENT PRINTING OFFICE 1920 DEPARTMENT OF THE INTERIOR BUREAU OF EDUCATION BULLETIN, 1920, No. I THE PROBLEM OF MATHEMATICS IN SECONDARY EDUCATION A REPORT OF THE COMMISSION ON THE REORGANIZATION OF SECOND- ARY EDUCATION, APPOINTED BY THE NATIONAL EDUCATION ASSOCIATION WASHINGTON GOVERNMENT PRINTING OFFICE 1920 ADDITIONAL COPIES OP THIS PUBLICATION MAY BE PROCURED FROM THE SUPERINTENDENT OF DOCUMENTS , GOVERNMENT PRINTING OFFICE WASHINGTON, D. C. AT 5 CENTS PER COPY REPORTS OF THE COMMISSION ON THE REORGANIZATION OF SEC- ONDARY EDUCATION. The following reports of the commission have been issued as bulletins of the United States Bureau of Education and may be procured from the Superin- tendent of Documents, Government Printing Office, Washington, D. C., at the prices stated. Remittance should be made in coin or money order. Other reports of the commission are in preparation. 1915, No. 23. The Teaching of Community Civics. 10 cents. 1916, No. 28. The Social Studies in Secondary Education. 10 cents. 1917, No. 2. Reorganization of English in Secondary Schools. 20 cents. 1917, No. 49. Music in Secondary Schools. 5 cents. 1917, No. 50. Physical Education in Secondary Schools. 5 cents. 1917, No. 51. Moral Values in Secondary Education. 5 cents. 1918, No. 19. Vocational Guidance in Secondary Schools. 5 cents. 1918, No. 35. Cardinal Principles of Secondary Education. 5 cents. 1919, No. 55. Business Education in Secondary Schools. 10 cents. 1920, No. 1. The Problem of Mathematics in Secondary Education. 5 cents. COMMITTEE ON THE PROBLEM OF MATHEMATICS IN SECONDARY EDUCATION. William H. Kilpatrick, chairman, professor of education, Teachers College, Columbia University, New York, N. Y. Fred R. Hunter, superintendent of schools, Oakland, Calif. Franklin W. Johnson, principal University High School, University of Chicago, Chicago, 111. J. H. Miunick, assistant professor of educational methods, University of Pennsylvania, Philadelphia, Pa. Raleigh Shorling, Lincoln School, 646 Park Avenue, New York, N. Y. J. C. Stone, head of department of mathematics, State Normal School, Mont- clair, N. J. Milo H. Stuart, principal Technical High School, Indianapolis, Ind. J. H. Withers, superintendent of schools, St. Louis, Mo. THE REVIEWING COMMITTEE OF THE COMMISSION ON THE REORGANIZATION OF SECONDARY EDUCATION. (The Reviewing Committee consists of 26 members, of whom 16 are chairmen of com- mittees and 10 are members at large.) Chairman of the Commission and of the Rcvieicring Committee: Clarence D. Kingsley, State high-school supervisor, Boston, Mass. Members at large: Hon. P. P. Claxton, United States Commissioner of Education, Washing- ton, D. C. Thomas H. Briggs, associate professor of education, Teachers College, Co- lumbia University, New York, N. Y. 3 MATHEMATICS IN SECONDARY EDUCATION. Members at Ivrn? Alexander Inglis, assistant professor of education, in charge of secondary education, Harvard University, Cambridge, Mass. Henry Neumann, Ethical Culture School, New York, N. Y. William Orr, senior educational secretary, international Y. M. C. A. com- mittee, 347 Madison Avenue, New York, N. Y. Willialu B. Owen, principal Chicago Normal College, Chicago, 111. Edward O. Sisson, president University of Montana, Missoula, Mont. Joseph S. Stewart, professor of secondary education, University of Georgia, Athens, Ga. Milo H. Stuart, principal Technical High School, Indianapolis, Ind. H. L. Terry, State high-school supervisor, Madison, Wis. Chairmen of Committees: Administration of Secondary Education Charles H. Johnston, professor of secondary education, University of Illinois, Urbana, 111. 1 Agriculture A. V. Storm, professor of agricultural education, University of Minnesota, St. Paul, Minn. Art Education Royal B. Farnuin, president, Mechanics Institute, Rochester, N. Y. Articulation of High School and College Clarence D. Kingsley, State high- school supervisor, Boston, Mass. Business Education Cheesman A. Herrick, president, Girard College, Phil- adelphia, Pa. Classical Languages Walter Eugene Foster, Stuyvesant High School, New York, N. Y. English James Fleming Hosic, Chicago Normal College, Chicago, 111. Household Arts Mrs. Henrietta Calvin, United States Bureau of Educa- tion, Washington, D. C. Industrial Arts Wilson H. Henderson, extension division, University of Wisconsin, Milwaukee, Wis. (now Major, Sanitary Corps, War Depart- ment, U. S. A.). Mathematics William Heard Kilpatrick, associate professor of education, Teachers College, Columbia University, New York, N. Y. Modern Languages Edward Manley, Englewood High School, Chicago, 111. Music Will Earhart, director of music, Pittsburgh, Pa. Physical Education James H. McCurdy, director of normal courses of ' physical education, International Y. M. C. A. College, Springfield, Mass. Sciences Otis W. Caldwell, director, Lincoln School, and professor of edu- cation, Teachers College, Columbia University, New York, N. Y. Social Studies Thomas Jesse Jones, United States Bureau of Education, Washington, D. C. Vocational Guidance Frank M. Leavitt, associate superintendent of schools, Pittsburgh, Pa. Deceased Sept 4, 1917. CONTENTS. Page. Membership of the committee on the problem of mathematics 3 Membership of the reviewing committee of the commission 3 Letter of transmittal 7 Preface 8 I. Introduction 9 II. The demand for an inquiry . ^ 9 III. Analysis of the situation 11 1. The problem of presentation 11 2. The several needs for mathematics 14 3. Comparative values 15 4. Formal discipline 16 5. The needs of the several groups 17 6. Selecting mathematical ability 20 IV. Suggestions as to courses . 21 1. The work of the junior high school 22 2. Trade mathematics 23 3. Preliminary engineering 23 4. For the specializes 23 5 LETTER OF TRANSMITTAL. DEPARTMENT OF THE INTERIOR, BUREAU OF EDUCATION, Washington, October 11, 1919. SIR: One of the committees of the Commission on the Reorgani- zation of Secondary Education, appointed by the National Educa- tion 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 recom- mendations in regard to the reconstruction of the courses of study in this subject in the high schools. The committee has, therefore, con- fined 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 intelli- gently 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 ex- pected 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. P. P. CLAXTON, Commissioner. The SECRETARY OF THE INTERIOR. 7 PREFACE. The Commission on the Reorganization of Secondary Education finds itself confronted with problems of great difficulty in recom- mending a reorganization of the mathematical studies of the sec- ondary school. Antecedent to new courses, there should be an agree- ment among psychologists and educators such as has not yet been reached. It seems, therefore, that the best service that the commis- sion 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. KINGS:LEY, Chairman of the commission. 3 THE PROBLEM OF MATHEMATICS IN SECONDARY EDUCATION. I. INTRODUCTION. Few subjects taught in the secondary school elicit more contra- dictory statements of view than does mathematics. What should be taught, how much of it, to whom, how, and why, are matters of dis- agreement. There is every variety of position. A conservative group would keep substantially unchanged the customary content and divi- sion into courses, and find the hope of improvement in a more ade- quate 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 mathe- matics 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 im- possible. 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 conclusions. Within the time allotment available to the committee there seemed then only the choice between no report and an admit- tedly preliminary report. The committee has chosen the latter alter- native, 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 solu- tion of the problem, and (3) certain tentative suggestions for ex- perimentation to develop new and better courses. 1 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 reconstitut- ing secondary mathematics is demanded from a variety of considera- tions. 1 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. 155900 20 2 9 10 . , . t MATHEMATICS IN SECONDARY EDUCATION. 1. It] is be in^ insisted as never before that each subject and each fern JS^t tJje subject jivstify itself; or, negatively, that no subject or item be retained in any curriculum unless its value, viewed in rela- \ tion to other topics and to time involved, can be made reasonably \ probable. No longer should the force of tradition shield any sub- ject from this 1 , 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. f 2. Moreover, a growing science of education has come to place appreciably different values upon certain psychological factors in- \volved, 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 ma- terial 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 mathe- matics, in arithmetic to an appreciable extent, more in algebra, most of all in geometry. AVith the scientific scrutiny of the conditions under which " trans- fer" of training takes place, the inquiry grows continually more insistent as to whether our mathematical courses should continue un- changed, 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 ANALYSIS OF THE SITUATION". 11 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 pres- 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 consider ation, 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 " logi- cal" 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 any- thing 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- 12 MATHEMATICS IN SECONDARY EDUCATION. trol of subject matter. This reversal of method, coupled with a dis- trust of the theory of discipline, has thus not only reduced grammar to a small fraction of its former self, but has, besides, greatly re- arranged and rewritten the study. Keeping before us the demand for economy in learning and effec- tive control of subject matter, what can we say about method? How does learning in fact take place? (1) Eepetition is a factor in learning known to all. (2) An inclusive " set " which shall predis- pose the attention, focus available inner resources, and secure repe- tition is a necessary condition less commonly considered. (3) The effect of accompanying satisfaction to foster habit formation is a third factor to be noted. 1 These three factors are necessary, then, to adequate consideration of the problem of method. It accords with these considerations and with undisputed observation that, other things being equal, any item is more readily learned if its bearing and need are definitely recognized. The felt need predisposes at- tention, calls into play accessory mental resources, and in proportion to its strength secures the necessary repetition. As the need is met, satisfaction ensues. All factors thus cooperate to fix in place the new item of knowledge. The element of felt need thus secures not only the learning of the new item, but it has at the same time called into play the allied intellectual resources so that new and old are welded together in effective organization with reference to the need which originally motivated the process. Lest some should fear that by need is here meant a mere " bread and butter demand," the committee hastens to say that it is psycho- logic and not economic need which acts as the factor in learning. Economic need may indeed be felt; and, if so, may then serve to in- fluence learning; but there is nothing in the foregoing argument to deny that a purely "theoretic" interest might not be as potent as any other to bring about the learning and organization of subject matter. To speak of the bearing and netd of any new material is to imply the presence and functioning of already existent purposes and in- terests. From this consideration thus related to the foregoing the committee believes that, speaking generally, introductory mathe- matics ordinarily conceived as separate courses in algebra, geome- try, and trigonometry should be given in connection with the solving of problems and the executing of projects in fields where the pupils already have both knowledge and interest. This would make the study of mathematics more nearly approximate a laboratory course, in which individual differences could be considered and the effective devices of supervised study be utilized. The minimum of 1 Tlv i behaviorist psychologist by definition rejects the subjective connotation of " satis- faction." If we had access to the actual psychology involved, possibly the difference of statement would in effect disappear. ANALYSIS OF THE SITUATION. 13 tlie course might well in this way be cared for in the recitation period, reserving the outside work rather for allied projects and problems in which individual interests and capacities were promi- nent factors. The significant element in this conception is the utilization of ideas and interests already present with the pupils as a milieu within which the mathematical conception or process to be taught finds a natural setting, and from which a need to use the conception or process can as a consequence be easily developed. Where this state of affairs exists, the bearing and felt need utilize the laws of learning as was discussed above, and the mathematical knowledge or skill is fixed in a manner distinctly economical as regards both present effort and future applicability. As was stated at the outset, this suggested procedure reaches beyond the questions of economy of learning and application con- trolling though these here are to the question of content. The pro- cedure here contemplated makes definite demand for an appropriate introductory content. To work along this line there must be made a selection of conceptions and processes which can serve the pupils as instruments to the attainment of the ends set before them in the proj- ects or problems upon which they are at work. This instrumental character becomes then the essential factor in any introductory course. It is these instrumental needs and not " logical " intercon- necteclness which must give unity to such a course. A content thus instru mentally selected will, on the one hand, be free of the old formal puzzles, the complex instances, the verbal problems which in the past have wasted so much time and destroyed so much potential interest; and will, on the other, run across the divisions heretofore separating algebra, geometry, and trigonometry. A distinct advantage in the procedure here suggested is the better promise it holds out of meeting in one introductory course the needs of both those who will go on to advanced study in mathematical lines and those who will not. Where the basis of selection and procedure is instrumental, all can begin together. The future specializes in mathematics will as the course proceeds take increasing interest in the mathematical relationships involved and will stress this aspect in their individual problems and projects. Those whose tastes and aptitudes lead them elsewhere will in the meanwhile have had the opportunity to learn in practical situations some of the mathematical concepts and processes which they will later use in their own chosen fields. Their individual projects in the course can serve well as con- necting links between the mathematics taught and their later field of vocational application. After the introductory course has been completed, and differenti- ation has begun, the same principles still hold, though in the different 14 MATHEMATICS IN SECONDAKY EDUCATION". fields. Those who have chosen to continue the study of mathematics as such will find their problems or projects within the field of mathematics itself, quite likely examining anew in the light of wider acquaintance assumptions freely made in the earlier period. Eu- clid's system of axioms and postulates might here receive its first careful consideration. Those who had elected to prepare for engi- neering and the like might continue to find their mathematics in con- nection with problems or projects -devoted now particularly to a preliminary engineering content. Conceptions usually reserved for college analytics and calculus if not 'indeed already used in the introductory course can well have a place here. Their rich in- strumental character will justify their presence, even if they lack somewhat in relationship to a fully developed logical system. 2. The several needs for mathematics Among the multiplicity of specific occasions for using mathematics and among the various types of subject matter, there are certain possible groupings which promise aid in the determination of the mathematical courses. Without implying the possibility always of sharp differentiation, we may distinguish in the realm of mathematical knowledge (i) those items the immediate use of which involve a minimum of think- ing, as, for example, adding a column of figures, and (ii) those items which are primarily used as notions or concepts in the furtherance of thinking. It is clear that the distinction here is of the way in which the knowledge is used and not of the knowledge itself ; for any item of knowledge might at one time serve one function and at another time the other. It would still remain true, however, that certain groups of people might have characteristically different needs along the two lines. Under the first head we should include the mechanic's use of a formula, the surveyor's use of his tables, the statistician's finding of the quartile. The man in the street would call this the "practical'' usp of mathematics. Under the other head we should in- clude the intelligent reader's use of mathematical language by which he would understand an account of Kepler's three famous laws. Some may wisli to call this the "cultural" use of mathematics. The term "interpretative" might, however, more exactly express the dif- ferentiating idea. We may next ask whether there are different iable groups among' high-school pupils whose probable destinations or activities deter- mine within reasonable limits the extent and type of their future mathematical needs. In a democracy like ours, -questions of prob- able destination are of course very difficult. There must be 110 caste-like perpetuation of economic and cultural differences; and definite effort must be made to keep wide open the door of further study for those who may later change their minds. But differ- entiating choices are in fact made; and in view of the wealth of ANALYSIS OF THE SITUATION. 15 offerings on the one hand and of individual differences on the other, such choices must be made. Properly safeguarded by an intelligent effort to adopt social demands to individual taste and aptitude, these choices should work to the advantage both of the individual and of the group. The committee considers that four groups of users of mathematics may be distinguished : (a) The " general readers," who will find their use of mathematics beyond arithmetic confined largely to the interpretative function described above. (6) Those whose work in certain trades will make limited, but still specific, demand for the " practical " use of mathematics. (c) Those whose practical work as engineers or as students of certain sciences requires considerable knowledge of mathematics. (d) Those who specialize in the study of mathematics with a view either to research or to teaching or to the mere satisfaction of extended study in the subject. It is at once evident that these groups are not sharply marked off from each other; and that the needs of the first group are shared by the others. It is, moreover, true that the " general readers " represent a wide range of interest. The committee has taken all these things into account, and still believes that the division here made will prove of substantial utility in arranging the offerings of high-school mathematics. 3. Comparative values. Out of the conflict of topics for a place in the program there emerges one general* principle, already suggested in these pages, which is being increasingly accepted for guidance by students of education. In briefest negative terms : No item shall le retained for any specific group of pupils unless, in relation to other items and to time involved, its (probable} value can le shou-n. So stated the principle seems a truism, but properly applied it proves a grim pruning hook to the dead limbs of tradition. A final method of ascertaining such, comparative values remains to bo worked out; but the feasibility of a reasonable application of the principle will hardly be denied. In accordance with this, many topics once common have been dropped from the curriculum and more are marked to go. Thus our better practice has ceased to in- clude the Euclidian method of finding the H. C. F.. because the knowledge of this method is nowhere serviceable in life; and in secondary algebra itself little if anything else depends on it. In- deed, the H. C. F. itself might well go, as it is used almost exclusively in simplifying fractions made for the purpose. In a full discussion, many terms of the statement would need con- sideration. What constitutes value is probably the point where most questioning would arise. The committee takes this term in its broadest sense, specifically denying restriction to a " bread and but- 16 MATHEMATICS IN SECONDARY EDUCATION. ter " basis or other mere material utility, though affirming that re- munerative employment is normally a worthy part of the worthy life. What the statement then in fact demands is (i) that the value of the topic be not a mere assumption a positive case must be made out; and (ii) that the value of the topic so shown be sufficiently great in relation to other topics and to the element of cost (as regards time, labor, money outlay, etc.) to warrant its inclusion in the cur- riculum. This principle of exclusion seems especially applicable to those items which now remain merely as a heritage from the past and to those which have been introduced mainly to round out the subject or where the unity of the subject matter has been found in the con- tent itself and not in the relation of the content to the needs of the pupil. In offering such a principle for guidance, the committee considers that it is merely stating explicitly what has been implicitly assumed in all such controversies. The committee none the less believes that conscious insistence on the point is necessary in order to disclose whatever indefensible elements may be in our present program of studies. 4. "Formed discipline.''' A full discussion of this topic, of course, is impossible within the limits of this paper. Such a discussion is, moreover, for our purpose unnecessary, because we shall wish to use only the most general conclusion, in which there is substantial concurrence. We can thus avoid the niceties of elaboration, about which agreement has not yet been reached. The older doctrine as- sumed uncritically a very high degrefe of what we now call " general transfer " of training. Modern investigation, to speak generally, re- stricts very consi4erably the amount of transfer which may reason- ably be expected, and inquires strictly into the conditions of transfer. Under the older doctrine it was a sufficient justification for the re- quiring of any subject that pupils .should gain through it inerqased ability in the use of any important " faculty," because the'increase in ability was naively assumed to mean an increase in the equally naively assumed faculty itself and would accordingly be effective wherever the faculty was used. As pupils shoAV such an increase of ability in one or more " faculties " by the simple fact of learning any new subject, the convenience of this older doctrine for curriculum defense is evident. When this old psychological doctrine was first called in question by scientific measurement, the idea gained popular currency that all transfer was denied. No such claim has serious sup- port. The psychologists, however, have so far found it difficult to agree upon any final situation as to the amount of transfer which in any particular situation may be a priori expected. All agree, none the less, in greatly reducing the old claim both as to the amount and ANALYSIS OF THE SITUATION. 17 as to the generality of conditions under which transfer may be ex- pected. In accordance with these considerations the committee has not used the factor of " formal discipline " in determining the con- tent of the mathematical courses to be recommended in this report. 5. The needs of the several groups. With these several princi- ples and factors before us, we are now ready to consider more fully the needs of the several groups of users as distinguished above. [VVe are particularly concerned to ask whether or not their respective group needs are compatible with one introductory course to be taken in common; and if yes, when the differentiation from such a common course should begin. (1) The "general readers" This group will need to use in " practical " fashion but little of mathematics other than ordinary arithmetic. As general readers, however, they will still require a certain acquaintance with mathematical language and concepts. Just what terms, symbols, and concepts would meet the requirements of this group will have to be determined by extensive inductive studies. Assuming, however, ordinary arithmetic and mensuration, some items can be at once named as fairly certain to be included: How to interpret and evaluate a simple literal formula; the mean- ing and use of an algebraic equation of one unknown; the notion and use of negative numbers in such simple cases as temperature, latitude, and stock fluctuations; the simpler conception of space relations (inductively obtained) ; the notion of function (the depend- ence of one quantity upon another) ; the graph as a means of inter- preting statistical information, with such terms as average and median. (2) The group preparing for certain trades. Under this head the committee would group those whose use of " practical " mathe- matics is, while generally quite definite, still relatively small such, for example, as machinists, plumbers, sheet-metal workers, .and the like. The general run of the need here contemplated can be gath- ered from the requirements laid down for machinists in one of the more recent vocational surveys simple equations, use of formulas, measurement of angles, measurements of areas and volumes, square root, making and reading of graphs, solution of right triangles, geometry of the circle. Much practice would of course be necessary to make even this small amount of mathematics function adequately. It is at once evident that if no more algebra is needed than formu- las, simple equations and the graph, and no more geometry than is here suggested, then the ordinary high-school courses in these sub- jects are but ill-adapted to the needs of such pupils. It would seo.m to follow that this group of pupils has no need to follow courses in mathematics other than (i) arithmetic, (ii) the "interpretative" 18 MATHEMATICS IN SECONDARY EDUCATION. V (introductory) mathematics discussed above, and (iii) the special applications of these to the specific subject matter of their several specializations. This group might then well study in common with the preceding until the completion of the work there laid out. The presentation of this common course along the lines previously laid down (p. 11) would well harmonize the somewhat "diverse interests of the two groups. What little additional content and whatever practice in specialized application this second group might need could then be given either in a parallel or in a succeeding course (or courses) especially devised for that purpose. (3) The group preparing for engineering. This group will con- sist mostly of boys intending to study in engineering schools. In contrast with the two preceding groups, appreciably more mathe- matics is here needed. In contrast with the following group, there are here specific aims external to mathematics itself which define and limit the mathematical knowledge and skill needed. Although recognizing that the individual teacher will require a certain leeway as regards content in getting his class effectively to w r ork at any topic, we may still profitably ask as to the minimum content fixed for this group by its peculiar needs. The minimum mathematical content suitable for the use of this group can probably best be secured by working simultaneously along two lines: First, to ascertain inductively what mathematics the engineer needs (including experiment to find out what part of this can best be taught in the secondary school) ; second, to criticize the existing courses to see what they lack and what they include that is useless for this group. It is much to be hoped that necessary in- ductive studies and experiments along the first line of procedure may be vigorously pushed. The second in important respects waits for the first, but it is possible from certain inherent considerations at once to exclude some matter now customarily taught. Taking the customary high-school mathematics as a basis for com- parison, we find at least three principles of criteria for exclusion from the present offerings: (a) Exclude all those items which are not themselves to be directly used in practical situations or which are not reasonably necessary to the intelligent mastery or use of such " practical" items; (>) exclude all involved and complicated in- stances of otherwise useful topics or applications which do not serve to clarify the main point under consideration; (c) exclude all such proofs and discussions as do not in fact help the pupil to an intelli- gent use of the topic. It is probably correct to say that these exclu- sions relate to material introduced from considerations of theory rather than of intelligent practical mastery; from considerations of ANALYSIS OF THE SITUATION. 19 the pleasure that thcorizcrs (teachers mostly) get from the study of mathematics rather than from a conscious purpose to give that famil- iarity and grasp which the future practical man will need. Under the head of (a) topics excluded as not needed in this group the committee would mention such as the H. C. F. and the L. C. M. ; operations with literal coefficients (except for a few formulas) ; radi- cal equations; the theory of exponents, except the simplest opera- tions with fractional and negative exponents (these to be retained to give meaning to logarithms and the slide rule); operations with imaginaries; cube root; proportion as a separate topic (the simple equation suffices) ; the progressions. Among (b) excluded complex applications might be mentioned the following: All lengthy exercises in multiplication and division; fac- toring beyond the simplest instances of the four forms (i) ax-fay, (ii) a 2 b 2 , (iii) a 2 -j-2ab+b 2 , (iv) x 2 +(a+b) x+ab; all but the sim- plest fractional forms (the more complicated are in fact given to illustrate factoring); all radicals beyond ^/ab ^nd VaT^rT; simulta- neous equations of more than two unknowns; simultaneous quad- ratics (except possibly a quadratic and a linear) ; the clock, hare and hounds, and courier problems and the like ; the extended formal dem- onstrative geometry of our ordinary schools; most trigonometry be- yond the use of sine, cosine, and tangent in triangle work. (