Digitized by the Internet Archive in 2012 with funding from LYRASIS members and Sloan Foundation http://www.archive.org/details/researchinconstrOOamer Bjfl M p <; RARN Superintendent of iichools EAST HARTFORD, COPlkCTIClT Department of Superintendence THIRD YEARBOOK RESEARCH IN CONSTRUCTING THE ELEMENTARY SCHOOL CURRICULUM PUBLISHED BY THE DEPARTMENT OF SUPERINTENDENCE OF THE NATrONAL EDUCATION ASSOCIATION OF THE UNITED STATES 1 20 1 Sixteenth Street Northwest, Washington, D. C. February, 1925 THE DEPARTMENT OF SUPERINTENDENCE OFFICERS, 1924-25 President, William McAndrew, Superintendent of Schools, Chicago, 111. First Vice-President, Payson Smith, State Commissioner of Education, Boston, Mass. Second Vice-President, John J. Maddox, Superintendent of Schools, St. Louis, Mo. Executive Secretary, Sherwood D. Shankland, 1201 Sixteenth Street Northwest, Washington, D. C. Executive Committee: Randall J. Condon, Superintendent of Schools, Cincinnati, Ohio. Frank W. Ballou, Superintendent of Schools, Washington, D. C. Frank D. Boynton, Superintendent of Schools, Ithaca, N. Y. M. G. Clark, Superintendent of Schools, Sioux City, Iowa. THE 1924-25 COMMISSION ON THE CURRICULUM Edwin C. Broome, Superintendent of Schools, Philadelphia, Pa., Chairman. John L. Alger, President, Rhode Island College of Education, Providence, R. I. Frank W. Ballou, Superintendent of Schools, Washington, D. C. Mrs. Susan M. Dorsey, Superintendent of Schools, Los Angeles, Calif. John M. Foote, Rural School Supervisor, State Department of Education, Baton Rouge, La. Charles H. Judd, Director, School of Education, University of Chicago, Chicago, 111. Harold O. Rugg, Lincoln School, Teachers College, Columbia University, New York City. Zenos E. Scott, Superintendent of Schools, Springfield, Mass. Frank E. Spaulding, Dean, School of Education, Yale University, New Haven, Conn. Paul C. Stetson, Superintendent of Schools, Dayton, Ohio. A. L. Threlkeld, Assistant Superintendent of Schools, Denver, Colo. H. B. Wilson, Superintendent of Schools, Berkeley, Calif. John W. Withers, Dean, New York University, New York City. FOREWORD "To bring together the elements for the construction of a suitable cur- riculum for the boys and girls of American public schools," was the purpose of the Department of Superintendence in its appointment of the Commis- sion on the Curriculum. The 1924 Yearbook of the Department laid the basis for this program through its statement of general educational aims and objectives, through its survey of current curriculum practice, and through its proposed machinery for cooperative effort in curriculum revision in a local community. When the Commission on the Curriculum met in July, 1924, the ques- tion was, What can a national commission do that will be of greatest help to superintendents of schools who are faced with the problem of curriculum revision? The commission decided that the most needed service was the collection and analysis of outstanding research studies in each of the sub- jects of the elementary curriculum. It was known that many such studies of high value had been made which should be considered in the revision of curricula ; but they were, for the most part, inaccessible because of their technical form, or because of their publication in isolated monographs or magazines, or because of their fragmentary distribution. A review of selected scientific studies in one volume, together with an analysis and summary of them in terms easily understandable, was the service for sci- entific curriculum construction which the Commission on the Curriculum decided to render. The work of carrying out the plan was delegated to the Division of Research of the National Education Association. The time allotted was less than six months. The studies to be reviewed were of a technical nature and in fairness to their authors could only be reviewed by specialists thoroughly familiar with each subject. It would have been impossible for any one person to do the work in the limited time. The leadership of the Nation in the curriculum field was sought out. Twelve subcommittees were appointed, one for each elementary school subject. A search for material was made in universities, colleges, and school systems throughout the country. As far as possible all published and unpublished elementary curriculum studies were collected. Because of the time limit and the fact that the reports of all subcom- mittees had to be included in one volume, the studies reviewed are selected and not comprehensive. In some fields much more research has been done than in others. This fact explains why certain subjects are given greater space in the Yearbook. Parts I and II were reduced to a minimum in order to give as much space as possible to the review of research studies. The Department of Superintendence is most appreciative and deeply indebted for the nation-wide cooperation which it has received. The Division of Research has bent every effort to make the Yearbook possible. Seventy-five curriculum specialists have given gratis of their time, training, and experience in analyzing the studies collected. Through their unselfish efforts this great piece of cooperative research was made possible. Their most important reward will be in the testing out and application of their findings, to the end that the boys and girls of our public schools may have a curriculum built on scientific principles and American ideals. CONTENTS PART ONE— NEED OF REVISION Chapter Page I. A cooperative Plan for the Revision of the American Elementary School Curriculum 9 PART TWO— VARIATIONS IN CURRICULA II. Possible Variations in .Curricula to Meet Community and Individual Needs 17 PART THREE— CURRICULA PROBLEMS AND THEIR SCIENTIFIC SOLUTION III. Arithmetic 35 IV. Spelling 110 V. Reading 152 VI. Handwriting 205 VII. The Social Studies 217 VIII. Language and Grammar 278 IX. Elementary Science 297 X. Health and Physical Education 303 XI. Home Economics 32Q XII. Industrial Arts 329 XIII. Art Education : 337 XIV. Music 354 PART FOUR— ADDENDA Report of Secretary 367 Members of the Department of Superintendence 372 Index to Authors of Research Studies . 422 [5] PART I A Cooperative Plan for the Revision of the ^American Elementary School Curriculum [7] PART I.— A COOPERATIVE PLAN FOR THE REVISION OF THE AMERICAN ELEMENTARY SCHOOL CURRICULUM CHAPTER I THE PUBLIC SCHOOL curriculum has a large share in determin- ing the Nation's present and future progress. Through its public schools, America is endeavoring to pass on to its youth all that has been found best in the past: the knowledge, habits, skills, attitudes, and ideals most useful to individual and national life at present ; and in addi- tion, it attempts to supply youth with the necessary tools, understanding, foresight, and incentive for the future advancement of its people. Knowl- edge and skill, disseminated among the masses of the people, according to a leading economist, constitute a nation's greatest wealth. The habits of a people determine the degree of faith that other nations accord it; and the idealism of its citizens is a nation's salvation. Since the public school is designed to disseminate knowledge, develop right habits, and implant proper ideals, it may rightly be termed the life insurance of civilization. The public school curriculum determines what boys and girls are taught in school. With the exception of the personality of the teacher, no other educational factor equals it in importance. The elementary school cur- riculum lays the foundation of every child's education ; and it constitutes all the school training of about three fifths of America's children. Why the Elementary School Curriculum Needs Revision The elementary school curriculum is a more vital factor in the develop- ment of national life today than ever before. In pioneer days, ordinary living at home and in the community, directed by insistent family needs, provided a fairly adequate preparation for life. Then the task of the American public school was mainly to supplement the life of the home, church, and community by adding the Three R's. As time went on, the public school was called upon to supply more and more formal education. Many new courses were added to the traditional subjects. Unfortunately, there was often lack of synthesis of the whole, since additions and modi- fications were made without plan or system ; and the result in many in- stances is "the amorphous product of generations of tinkering." Unless school practice is abreast or ahead of the time, it cannot be a directive power. There has been too much of a tendency for convention and tradi- tion to determine curricula, rather than practical, present-day life demands. An active principle which is now coming to be recognized in the selection of content is that of utility in a broad sense. Will the inclusion of a par- ticular topic or subject increase the effectiveness of the individual? Will it influence life for good? Will it build character? Today we are using a pragmatic philosophy. [9] 10 Department of Superintendence Pruning out the dead wood, selecting minimum essentials from total possible content, adding supplementary material — tinkering with the cur- riculum — will no longer answer the need. A complete revision is de- manded by both educators and laymen. Both groups demand a new cur- riculum, expressive of the changed conditions of modern civilization and reshaped in the light of our better understanding of child life and the learning processes. The Need for Curriculum Revision — What the Layman Sees The layman too often views the public school curriculum from these two angles: (1) growing costs, and (2) lack of thoroughness in essen- tials. The first viewpoint is sane, providing that it is positive and not negative. When larger numbers of children are educated and given the better and richer education which civilization demands, the cost is certain to increase — it cannot be otherwise — and the cost must be met. More careful selection of curriculum materials should be made, but it is doubt- ful that it will or should result in reduced expenditures. The second viewpoint from which the layman considers the need for curriculum revision is a supposed lack of thoroughness in essentials. There may be some justification for this attitude. The only reliable evidence, however, such as the Springfield tests and the Boston tests of 1845, has demonstrated that most public schools of today are securing better results in the so-called "fundamentals" with a heterogeneous mass of children than the schools of half a century ago secured with selected groups. The argument for "cutting out fads and frills" is sometimes only a plea for the old-time, traditional school program, which would be an anachronism in present-day life. The American school must expand until it offers whatever is necessary to develop intelligent American citizenship. The Need for Curriculum Revision — What the Educator Sees The educator also sees the need of curriculum revision from two angles : (1) the expansion of the field of knowledge, and the need for a dis- criminating selection of what is best suited to meet present social demands ; and (2) provision for character building, including the development of broad interests, liberal views, and the establishment of right social and ethical attitudes. The limited time that the average child spends in school in contrast to the vast amount of knowledge which one might acquire, the complexity of modern life with, its varied demands, and the necessity of developing individual talents are conditions which lead educators not only to restudy and analyze the aims and objectives of education, but also to demand a complete revision of the curriculum to express the new concep- tions of education. The educator is seeking as the major products of his enterprise strong character and right conduct, built not on precept, but fashioned through years of right thinking during lesson hours and prop- The Third Yearbook 11 erly controlled behavior in school activities. Teachers not only want their pupils to meet the standard tests in academic achievement but, what is more important, to meet successfully the tests of life, even to the point of temptation. For this reason, although teachers may rightly expect that character development will be an important by-product of all good instruc- tion, they have definitely in mind the moral qualities which they wish to develop in their pupils, such as: honesty, industry, self-control, courtesy, unselfishness, service, appreciation of beauty, openmindedness, cooperation, responsibility, sympathy, desire for improvement, adaptability, courage, initiative, thoroughness, self-judgment, thrift, faith in mankind, and rever- ence. Every true teacher hopes to develop these qualities through life situ- ations as they arise. When the educator fully comprehends that the objectives which the school must serve today are health, training for a vocation, citizenship, worthy home membership, the profitable use of leisure, and ethical char- acter, he keenly feels the need of a thorough revision of the present public school curriculum. These viewpoints of the layman and the educator relative to the need for curriculum revision can and must be harmonized into a working agree- ment. Present Activity in Elementary School Curriculum Revision The aim of the modern educator includes the aim of the layman, thor- oughness in essentials, and goes far beyond it. The teacher strives to have his pupils spell more correctly the words in common use, use the arith- metical processes necessary in modern life with more speed and accuracy, and read more and with greater rapidity and comprehension. In addition, the teacher would have his pupils appreciate the rights of others, and respect the law; in other words, he would have him establish a proper relationship between himself and society. The public-spirited layman cherishes these same objectives for the public school. He, too, would have the curriculum revised so that the above objectives may be attained. The result is that curriculum revision is in progress in all parts of the country. Several of the national societies and foundations, as well as university and state committees, are engaged in studying the problem of the elementary school curriculum. Among these are the National Society for the Study of Education, the American Asso- ciation of University Women, The Commonwealth Foundation, The American Association for the Advancement of Science, the California Cur- riculum Committee, and Teachers College of Columbia University. Many state, county, and city courses of study have been rebuilt recently, or are in the process of revision. Representative examples of large city school systems where courses of study have been revised recently are: Berkeley, California; Los Angeles, California; Denver, Colorado; Balti- more, Maryland; Detroit, Michigan; Cincinnati, Ohio; and Trenton, 12 Department of Superintendence New Jersey. In response to a request for recent elementary courses of study, 200 representative cities, counties, and states replied that they had published courses of study since 1920. This is indicative of the vast amount of activity in curriculum revision in all parts of the country. Should Curriculum Revision be Done Independently or by Cooperation? Some of the work in curriculum revision is the result of months of care- ful study and the pooling of leadership in the field of each subject. In some cities, however, the work has been done hastily and unscientifically; in every city there has been much duplication of effort. When each com- munity attempts to solve all its curriculum problems in isolation, the results are likely to be unsatisfactory and expensive. When scores of cities work independently, there is an unfortunate waste of time and effort in the du- plication of bibliographies, in the independent collection of data through repeated questionnaires and other forms of inquiry, and in the formulation of major principles. As a matter of fact, by the process of independent action, only those school systems which can afford to expend thousands of dollars on curriculum revision can make independently a comprehensive and satisfactory study of the problem. Curriculum making is becoming a specialized, technical task, which requires : ( 1 ) the study of social needs, including those peculiar to a local community, (2) analyses of the abilities and interests of growing children, (3) experimental study of the learning process, and (4) an understanding of the local school system, the temper and capacity of its teachers and administrators; in other words, what can be expected of them in helping in the development and putting into effect of a new course of study at a given time. One curriculum specialist has wisely said: "To make a radical selection and reorganization of the best in a civilization is not something that can be lightly undertaken or easily accomplished. Conferences and the pronouncements of those in official positions are quite inadequate. Two methods are needed in curriculum making: (1) the subjective method of expert teaching opinion, and (2) the objective method of expert analysis of social and psychological needs." Both processes should work together. Contribution of Research to Curriculum Revision During the past ten years, methods of determining what is useful and what is not have been devised in part, and they are still in process of development. Some important, results have already been obtained. Part III, Curricular Problems and Their Scientific Solution, is the work of sixty-nine curriculum specialists and subject specialists. It pools the most important research studies available and presents bibliographies that have already been tested out and found helpful by a number of local com- munities, and suggests problems which local committees have faced, but which require further research for their solution. The Third Yearbook 13 A National Committee Should Present Raw Material Rather Than a National Curriculum for General Acceptance A strictly uniform American curriculum for the elementary schools is hardly possible, because of the great variations in social, industrial, and economic conditions. Furthermore, it may not be desirable, because of our democratic ideal of recognizing local needs and conditions and of con- sidering individual differences of children. There are three elements in every course of study : ( 1 ) the general core that meets nation-wide re- quirements; (2) the part that has to do with the local community, which may even vary for different communities in the same city; and (3) adjust- ments for individual children and varying groups of pupils. In its Second Yearbook, the Department of Superintendence set up the machinery for devising, revising, and supervising a curriculum in an indi- vidual city. This machinery was based on the idea of arousing the interest and securing the cooperation of the local staff, and, at the same time, of assembling ideas from the outside. The Commission on the Curriculum has deemed it advisable to apply the same method to the work it has in hand. In Part II of this Yearbook, various ideas as to possible variations in the curriculum to meet community and individual needs are presented ; and accompanying each statement, general principles to guide local course of study committees in the adaptation of a curriculum to their particular community and to special groups of pupils are suggested. A Central Agency Could Render Invaluable Service as a Clearing- house for Curriculum Research The research studies here reviewed are illustrative of the kind of work which is essential to scientific curriculum revision. More of these are needed. The fact that practice should keep pace with research also calls for the testing of the findings presented in Part III. In many instances research at the present time is ahead of practice. The Commission on the Curriculum presents this Yearbook as the first comprehensive effort to develop a cooperative plan for curriculum revision. The commission believes that there should be a central agency, continuing from year to }^ear, which would act as a clearing-house for school systems which desire to cooperate in the work of curriculum revision. In other words, provision should be made for the exchanging of bibliographies, for rendering available research studies as soon as they are completed, and for the interchanging of the findings of local communities, as classroom teachers test out certain content and procedures. Just as scientifically trained workers in our laboratories furnish us with the bulk of scientific literature, so in the future more and more of our educational contributions should come from teachers, administrators, and research workers trained Scientifically to observe children and analyze the needs of society. 14 Department of Superintendence It is the nature of education to be ever progressing. Continuous, con- structive, scientific study must be carried on, if the public school curriculum is to keep pace with social demands and to become a contributory element to more intelligent and abundant living. The Commission on the Curriculum proposes that for the school year 1925-1926 as many school systems, as will, shall agree to cooperate in testing the findings of the research studies reported in this Yearbook, that new studies be undertaken, and that the results be placed at the disposal of the schools of the country through the Department of Superintendence as a central agency. PART II Possible Variations in Curricula to Meet Community and Individual Differences [15] CHAPTER II PART II. POSSIBLE VARIATIONS IN CURRICULA TO MEET COMMUNITY AND INDIVIDUAL NEEDS 1 MOST of the present elementary school curricula are based upon the theory that the course given in the first six grades covers the so-called tool subjects of learning; that these subjects are fundamental to the education of all children and should consequently be given approximately in the same degree to all normal children ; that dif- ferences in aptitudes and life plans should be recognized at about the age of twelve ; and that elective work should begin in the junior high school and should continue in increasing degree through high school, college, and university. Although this means that all children, except atypicals, shall take the same subjects during the first six grades of the elementary school, it does not mean that the content of each subject shall be the same for all chil- dren, or that the method of approach to a particular subject or topic or the time allotment will be the same. For, notwithstanding in each subject of the elementary school curriculum there is a general core which is uni- versal and belongs to the nation, there is additional material which must be adapted to community and individual needs. The General Core of the Elementary School Curriculum Belongs to the Nation For the drill subjects, Part III would seem to indicate that research will have soon determined what the general core of the elementary school curriculum should be — that is, the irreducible minimum for all normal children in various years of work the country over. Charles M. Reinoehl, in his Analytic Survey of State Courses of Study for Rural Elementary Schools, 2 lists topics which are common to nearly all present state courses of study. In his opinion, these topics, with some modifications, include much information of which every American has need. He shows that there are social and economic problems national in scope. There are problems peculiar to a state, but typical of problems in other 1 This section is a summary of statements of general principles for the differentiation of curricula obtained for this Yearbook from those who have had practical experience in the adaptation of education to the needs of individual children. Among those consulted were: Velda Bamesberger, director of research, Okmulgee, Okla. ; Orville G. Brim, professor of education, Ohio State University; Kenyon L,. Butterfield, president, Michigan Agricultural College; M. G. Clark, superintendent of schools, Sioux City, Iowa; H. M. Corning, superin- tendent of schools, Trinidad, Colo.; Virgil K. Dickson, director of research and guidance, Berkeley and Oakland public schools, Calif.; John M. Foote, rural school supervisor, State Department of Education, Baton Rouge, L,a. ; Armand J. Gerson, associate superintendent of schools, Philadelphia, Pa.; S. Monroe Graves, superintendent of schools, Wellesley, Mass.; W. J. Osburn, director of educational measurements, State Department of Public Instruction, .Madison, Wis.; and Carleton W. Washburne, superintendent of schools, Winnetka, 111. 2 Reinoehl, Charles M. Analytic survey of state courses of study for rural elementary schools. U. S. Bureau of Education, Bulletin, 1922, No. 42. 116 pp. [17] 18 Department of Superintendence states. There are problems truly representative and suggestive in char- acter, which with proper adaptation are fully as valuable as those of wider application. There is also a considerable body of subject-matter requir- ing drill, of which every normal American child has need. Reinoehl's study is intended to indicate the content of the core of the rural elementary school curriculum. The studies reviewed in Part III are applicable to communities of every sort. However, even if research should determine what the minimum essentials in curricular content should be, every course of study committee is still faced with the problem of additional adaptations for its particular school system. Two Guiding Principles for Curricular Adaptation What are the general principles for differentiation which should guide course of study committees? Two guiding principles invariably present themselves. The first principle has to do with education s material, the individual child, for whom educational activities are being planned. The second principle has to do with the child's environment. Nature and nurture are important factors to be taken into consideration when planning a cur- riculum. Differentiation of Curricula to Fit Individual Needs The fact that pupils differ in ability to master the work offered in the ordinary curriculum in any subject is well known to any one familiar with the teaching of children. Hence in fairness to the rights of each child, it is riot only impractical but impossible to teach all children in the same man- ner or to give them the same content. Ability grouping and the curriculum problem — One superintendent who has made a special study of the adaptation of curricula to individual chil- dren in his school system states that about six or eight children in every ten can work with satisfactory profit with the same educational projects. Two of every ten children can usually achieve more satisfactory results if given an opportunity with projects more advanced than that of the so-called "average six" of each ten. There are also frequently two of each ten who cannot, or at least do not experience' the same or equivalent success as do the others in the group. The proportions may vary, but the differences remain. Curriculum making, therefore, has specifically to do with the regular grade. work of children of average ability. It also has to do with ungraded classes of children who are below the average group of the class in actual results secured as indicated by the grades obtained by each in school. In addition there is the rapid promotion class, or the class capable of a wide scope of comprehensive offerings, but with the possibility of early promo- tion from grade to grade held in abeyance. In larger communities admin- The Third Yearbook 19 istrative action is imperative in dealing with the problem of the physically handicapped, as well as of children with emotional instability. These pupils may rank well in intellectual performance but, because of physical disabilities or emotional instability, they cannot profit from the standard curriculum unless many adjustments are made. Similarly, special courses must be arranged for the predelinquent. Nu?nber of ability groups — The size of the school usually determines the number of ability groups that it is administratively possible to organize. Even when there are as many as five or six ability groups, there will be considerable variation between individual members in any one group. The larger the number of groups, the smaller should be the variation in ability between individual students. What are the general principles that every course of study committee attempting differentiation of material should recognize? This question is one which deserves further study. In the absence of any better evidence, these principles are tentatively proposed for discussion. Major principles of curricular differentiation for ability groups: 1. Methods of teaching and standards of attainment differ for groups of different levels of ability, but there are real possibilities for achievement in any class in the public school. If possible, teachers should find at least one socially useful thing at which even the very limited x pupil can succeed, train him in this, and let him experience honest success. 2. Minimum essentials must be clearly stated for all, but the funda- mentals set forth for the lowest group must be included as a part of the material for the highest groups. It is important in a democracy that cer- tain information shall be the common heritage of all. In so far as they are able to acquire it, all children should have the best of the social heritage of the race. 3. The aims, objectives, and outcomes for all subjects should be defi- nitely set forth for the different ability groups. For the most part, there should be a common background for these aims. The methods and em- phasis are necessarily different, as are the immediate objectives and out- comes, but the same ultimate aim of education should guide all teaching. The same philosophy of education is back of whatever differentiation is made. 4. Definite standards of attainment must be indicated for all levels. Much study is needed to determine the amount and quality of work which we rightly can expect from pupils of different levels of ability. 5. The dull pupil should have extensive opportunity for drill. Contrary to general belief, the slow thinker is not necessarily the accurate thinker. Nor does the child who learns slowly necessarily have a more retentive memory than the bright child. x The term "limited" is used to designate those pupils who do not seem to have the ability to do the work of the standard elementary school curriculum. 20 Department of Superintendence 6. Limited pupils usually differ from superior pupils in their lack of ability to project themselves into new situations. The superior child with his quick imagination and ready powers of generalization does ,this very easily. On the other hand, the limited pupil usually cannot reason out general principles or methods of attack but has to learn largely by imita- tion and successive experiences. To illustrate, according to W. J. Osburn, a dull child who has learned 8 from 13 will probably be confused by the 48 from 53 which occurs in the exercise 534 divided by 6. To a gifted child this presents little diffi- culty. Dull pupils must have direct teaching on all of the important ele- ments of instruction, whereas gifted pupils will probably master the ma- terial after one tenth of it has been taught. There are more than 1200 combinations in arithmetic which everyone must know in order to carry on ordinary computation with integers ; yet we know gifted pupils can attain a reasonable mastery of this material by the presentation of 180 combinations; that is, the 45 principal combinations in each of the four processes. This is not true of the limited pupil. 7. The content of the course of study for the slow group should stand the test of practicability. A large majority of slow pupils leave school early, not because of economic necessity, but because the traditional cur- riculum does not meet their needs and interests. In order to meet their needs, the work must be of such a character that it will be especially ap- plicable to their every-day life needs. Local course of study committees must be guided by what they know almost certainly lies ahead of certain groups of students. 8. The content of the course of study for the gifted group must be en- riched, not necessarily by more difficult or more extensive material of the same type, but by a wider and more varied choice. Certain additional material may also be presented, as the beginning of foreign language study, and other work which will provide for initiative, organization, and devel- opment of the many-sided interests of the superior child, for which there is little time in the average class. 9. The course of study should make provisions for the rate of progress which may be expected in the various ability groups. The gifted child should probably be allowed to complete the elementary course in six years or less; for in addition to enrichment of courses, there is sometimes advan- tage in acceleration in the elementary grades when much of the work is in drill subjects. The normal group should finish the six grades of the ele- mentary school in six years ; and the under-average should be expected to make normal progress as far as possible, within their own group. In each grade the atypical group should be expected to acquire academic knowledge only in so far as their capacities permit, without excessive expenditure of time. 10. Suggestions as to methods and procedure best adapted to the dif- ferent levels should be included in the course of study. Critical investiga- The Third Yearbook 21 tions should be made of our present methods and their relative effectiveness in reaching pupils of varying ability, social background, and environment. 11. It is generally conceded that the keynote in the education of a child of limited mental ability is the establishment of specific habits. 12. The differentiated subject-matter for different groups should be elastic in content, so that it can be modified to meet individual needs as well as group needs. For example, children from foreign-speaking homes may have the same mental ability as other children in their group, but they have a different social background, which the teacher must consider in her approach to the subject-matter which she wants her pupils to acquire. Individual Instruction and the Curriculum Problem The viewpoint is pretty largely accepted that the needs of individual children can be far better met when children are classified on an ability basis determined by such factors as seeming innate mental capacity, teacher's estimate of ability, previous accomplishment, industry, health, and social maturity, than they can in a heterogeneous class. The larger the number of groups, the more nearly homogeneous can each group be made. In every instance, however, there will still be considerable variation between individuals in any one group. Hence, some argue that individual instruc- tion is the only plan whereby the curriculum can really be made to fit all children. The Winnetka Plan of Meeting Individual Differences Carleton W. Washburne, superintendent of schools of Winnetka, Illi- nois, discusses this point as follows: Children differ as individuals, not as artificially formed groups. The children of high intelligence quotient will usually do better work than children of low intelligence quotient ; children who are good in one subject are more likely than not to be good in other subjects; but the exceptions to these rules are numerous in every class or school and the variations within each group of ability are far wider than the variations between the central tendencies of any two ability groups. Where children are allowed to advance individually at their own natural rates, as in Winnetka, for example, it has been found that twenty-four per cent of the children who are advanced in reading are only average or actu- ally retarded in arithmetic and that sixteen per cent of the children who are advanced in reading are average or retarded in language. The same sort of thing is true of the children who are advanced in arithmetic or in language, although the percentages of the children average or retarded in the other subjects is somewhat lower in these cases. Similarly, twenty-five per cent of children of high intelligence quotient, 123 to 166, tend to progress in school at an average rate or a little below average, while forty-nine per cent of children of low intelligence quotient, 60 to 100, tend to progress at an average rate or a rate above average. 22 Department of Superintendence If children are divided into three ability groups as the result of an intel* ligence test, the high group containing the quarter with the highest intel- ligence quotients, the low group containing the quarter with the lowest intelligence quotients, and the middle group containing the half of the children between the two extremes, it is found in Winnetka, Illinois, that the difference in median progress of the three groups amounts to only about two months' work per year; whereas the range from the poorest child to the best in any one of the groups is from eight to ten times as great. The median child of the gifted or high group saves only about a month a year, while the median child of the low groups loses only about two weeks a year. Yet the slowest child in the gifted group loses two weeks a year, while the fastest one gains seven and one half months — that is, does almost two years' work in a year. The range between the slowest and fastest children in the middle group and low group is even greater than that in the gifted group. The slowest child in the low group, for example, does only a little over one third of a grade's work in a year, while the fastest child in the low group does the work of a grade and one half. Consequently, the organization of the curriculum to fit children should be an organization that permits each child to spend the amount of time he needs to master each phase of that curriculum. If the curriculum is to be fitted to the individual children rather than to ability groups, the following general principles may be laid down: 1. The knowledge and skills which every child must master shall be clearly specified. The knowledge and skills should be those which will be used by every child in his life. They should be determined by scientific investigations of social needs, not by tradition or committee judgment. They will be the common essentials — those facts and skills which are essential to all of us in common. 2. Opportunities for the mastery of these common essentials should be given to every child. This inevitably means differing amounts of time for differing children. But it also means that all the children get the habit of mastery and that all the children master those things which are essential for all. 3. Provision shall be made for the valuable and educative use of the varying amounts of surplus time of the more rapid workers. Too often plus assignments are given to such children in an effort to occupy their time. Such assignments are frequently mere padding. It is not necessary that the time a child saves in one subject should be spent in educational work on that same subject. It is often wiser to allow him to progress as rapidly as his ability will permit until he has a large enough unit of saved time to enable him to undertake some worthwhile project in line with his interests or needs. The Third Yearbook 23 Time and Content — Two Factors in Differentation to Meet Individual Needs The methods of meeting individual needs in children through more homogeneous grouping and through individual instruction accept as the bases of adjustment the modification of time that children of different ability require to complete the course of study together with modification in content. Many would agree that both the factors of time and content should be modified. Limited pupils, even when given more time, do not always grasp the same subject-matter as do gifted children. In curriculum construction, we are faced with the fact that all individuals cannot ulti- mately learn the same things. Differentation of Curricula to Fit Community Needs The discussion thus far has dealt with education's material — the indi- vidual child. The second general principle of curriculum differentiation has to do with the child's environment. In rural districts the objectives of elementary education are not different from those in the city, but the means of reaching them and materials available differ from those in large centers of population. Some communities are cosmopolitan in their makeup. They contain within their limits all sorts and types of school environ- ments and the superintendent soon finds that a curriculum which would fit one district in his city will not completely fit another district. The ultimate purpose of elementary education is the same everywhere, but the content and method through which it is achieved varies. Major principles for community curriculum differentiation — 1. The materials of instruction must be approached in terms of the child's interests and experience. The experiences may be pre-school or contemporary with his school life. Since the environmental conditions surrounding pupils must be made the basis of approach to larger experiences to be developed through the curriculum, local course of study committees, the elementary school principal, and the individual teacher must be responsible for the approach to subject-matter. This approach will be a means to an end, not an end in itself. To illustrate, agriculture and farm life experience may be used as the starting point of elementary rural education ; they need not, and oftentimes should not, be the ultimate goal. 2. Sheer economy in learning requires that the educational resources of local life be used. For example, museums and art galleries are sources of art lessons for the city child. Autumn coloring of woods, sunsets, and mountains may be the rural art teacher's approach. The elementary sci- ence teacher in the city could profitably have his pupils study gas meters; rural children would probably find the Babcock milk tester of more direct interest. The important educational resources of the community not pre- sented in the curriculum must be organized for instructional purposes so 24 Department of Superintendence as to contribute to the education of the child. Here again local course of study committees, the elementary school principal, and the individual teacher must be responsible for the educational use of materials of a local nature not found in the general curriculum. 3. The details, sequence, and emphasis of subject-matter must frequently vary with community needs. To illustrate, the manual training teacher may discover that pupils in the slum sections of a city will be interested in learning to make household repairs. The child from a wealthy home would probably have no opportunity to use this knowledge if he acquired it. He would be interested, however, in learning to make certain toys which money cannot buy. In the end, both children will have learned the same fundamental principles of manual training and the use of the same tools. The mere going through a process after it has been learned and ceases to have educational value should always be guarded against. 4. The course of study should not be based on traditions of the past or the customs of a community, although both of these should be taken into consideration in order that they may be evaluated. We need constantly to challenge our educational program if this program is to keep abreast of the progress made in social and economic organization. 5. A survey should be made of every community before planning its course of study to determine in so far as possible: (a) Physical boundaries and nature of the community; (b) economic interests ,and relations of these interests to social and spiritual values; and (c) social assets and liabilities of the community, including such items as communication, trans- portation, means of play and recreation, intellectual and spiritual leader- ship. 6. Curricula for both urban and rural communities should be developed with specific reference to the "lacks" or "needs" of urban or rural life in general and to each community in particular. 7. It is of vital importance that the community know, through parent- teachers' associations, school officials, etc., the reasons underlying the or- ganization of the course of study and its relation to the community. Out of a free and frank discussion of this phase of the problem will come the best suggestions as to: (a) The use of a community as source material for the elementary schools; and (b) the ways in which the elementary schools may better serve the community. The public school must seek constantly to interpret itself to the adult community. Unless it does this, its efforts are bound to be hampered. The adult public, not school officials, is the court of final authority. Hence the public must come into possession of the evidence. 8. Local school systems within states should be given a certain amount of freedom to test new educational theories. Progress in education, like progress in social and industrial development, comes largely through the ability of individual units to experiment, dif- ferentiate, and in various ways modify both the method and , the material of their curricula. The contributions of Rochester, Gary, Winnetka, and The Third Yearbook 25 countless other schools have come through their opportunity to differentiate more or less established curricula. 9. The outstanding character of a school doing a valuable piece of indi- vidual differentiation becomes a challenge first to schools in its immediate neighborhood and finally throughout the county, the state, and the nation. The purpose, living within a community, must find its initiative or motive within itself, not from without. A system superimposed may become an attitude, but never assumes sufficient importance to be character. External conformity may not mean inner acceptance. The Final Test of Differentiation To summarize, although there is a general core of subject-matter which should prevail throughout the nation in all courses of study for the first six grades, each local course of study committee and individual teacher must be responsible for the approach to the subject-matter, for the use and adaptation of the materials of the curriculum for community needs, for organization for instructional purposes of the educational resources of the community not presented in the curriculum, and for introducing material which will meet the needs peculiar to a given community. When these differentiations are wisely planned and carried out, com- munity interest and appreciation result from the service which the school renders the community. In turn the public gives better support, both financial and moral, to the work of the public school, so that the school is enabled to render still larger service ; in other words, the initiative of the individual community is aroused. To achieve this, we must have in addition to differentiation of curricula to meet the larger community needs, a flexible curriculum in the hands of an administrative and teaching force sensitive to the interests and needs of individual children ; for, after all, the problem of differentiation of courses of study is one of constant adaptation of education to the indi- vidual. N PART III Curricular Problems and Their Scientific Solution [27] SUBCOMMITTEES OF THE COMMISSION ON THE CURRICULUM Arithmetic Guy M. Wilson (Chairman). Professor of Education, School of Education, Boston University, Boston, Mass. John R. Clark, Editor and Business Manager, The Mathematics Teacher, National Council of Teachers of Mathematics, 425 W. 123rd St., New York City. F. B. Knight, State University of Iowa, Iowa City, Iowa. Garry Cleveland Myers, Cleveland School of Education, Cleveland, Ohio. W. J. Osburn, Supervisor of Educational Measurements, State Department of Edu- cation, Madison, Wis. John C. Stone, Montclair State Normal School, Upper Montclair, N. J. Clifford Woody, Director, Bureau of Educational Reference and Research, Uni- versity of Michigan, Ann Arbor, Mich. Art Education Leon L. Winslow (Chairman) Director of Art, Department of Education, Balti- more, Md. C. Valentine Kirby, Director of Art, Department of Public Instruction, Harris- burg, Pa. Walter H. Klar, Associate Professor of Normal Art, Carnegie Institute of Tech- nology, Pittsburgh, Pa. Ethelwyn Miller, Teacher Training Department, John Herron Art School, Indian- apolis, Ind. C. Edward Newell, Supervisor of Art and Hand Work, Department of Public Schools, Springfield, Mass. Wiliam H. Varnum, Associate Professor of Applied Arts, University of Wisconsin, Madison, Wis. Jane Betsy Welling, Director of Art Education, Toledo, Ohio. Elementary Science Eliot R. Downing (Chairman) Associate Professor of Natural Science, School of Education, University of Chicago, Chicago, 111. Charles W. Finley, Lincoln School, Teachers College, Columbia University, New York City. George W. Hunter, Knox College, Galesburg, 111. Edwin E. Slosson, Director of Science Service, Washington, D. C. Walter G. Whitman, Editor and Manager, General Science Quarterly, State Nor- mal School, Salem, Mass. Health and Physical Education Thomas D. Wood (Chairman) Professor of Physical Education, Teachers College, Columbia University, New York City. A. L. Beaghler, Director of Health Education, Denver Public Schools, Denver, Colo. Walter W. Davis, Seattle Public Schools, Seattle, Wash. William H. Geer, Director of Physical Education, Harvard University, Cam- bridge, Mass. Frederick W. Maroney, M.D., Department of Health Instruction, Atlantic City, N. J. R. C. McLain, Assistant Supervisor of Health Education, Detroit Public Schools, Detroit, Mich. i [28] Home Economics Henrietta W. Calvin (Chairman) Director, Division of Home Economics, Board of Public Education, Philadelphia, Pa. Ellen M. Bartlett, Supervisor of Home Economics, Department of Education, San Francisco, Calif. Emma Conley, Supervisor of Homemaking Education, University of the State of New York, State Department of Education, Albany, N. Y. Ula M. Dow, School of Household Economics, Simmons College, Boston, Mass. Wilhelmina Spohr, Teachers College, Columbia University, New York City. Mabel B. Trilling, School of Education, University of Chicago, Chicago, 111. Industrial Arts F. G. Bonser (Chairman) Professor of Education, Teachers College, Columbia University, New York City. R. J. Leonard, Director, School of Education, Teachers College, Columbia Uni- versity, New York City. Ella M. Nevell, Supervisor of Industrial Arts, Los Angeles, Calif. Clara P. Reynolds, Director, Fine and Industrial Arts, Seattle Public Schools, Seattle, Wash. W. E. Roberts, Supervisor of Manual Arts, Board of Education, Cleveland, Ohio. Language and Composition W. W. Charters (Chairman) University of Pittsburgh, Pittsburgh, Pa. James F. Hosic, Teachers College, Columbia University, New York City. J. W. Searson, Professor of English, University of Nebraska, Lincoln, Nebr. Music Jacob Kwalwasser (Chairman), Head of the Department of Public School Music, State University of Iowa, Iowa City, Iowa. C. E. Seashore, Dean, The Graduate College, State University of Iowa, Iowa City, Iowa. Peter W. Dykema, Director, School of Music Education, Teachers College, Colum- bia University, New York City. Will Earhart, Supervisor of Music, City Public Schools, Pittsburgh, Pa. Max Schoen, Department of Education and Psychology, Carnegie Institute of Technology, Pittsburgh, Pa. Penmanship Frank N. Freeman (Chairman) School of Education, University of Chicago, Chi- cago, 111. Mrs. Theodocia Carpenter, Strong Junior High School, Grand Rapids, Mich. Myrta L. Ely, Supervisor of Penmanship, Department of Education, St. Paul, Minn. Harry Houston, Supervisor of Penmanship, Board of Education, New Haven, Conn. Lena A. Shaw, Supervisor of Handwriting, Board of Education, Detroit, Mich. Joseph S. Taylor, District Superintendent of Schools, New York City. H. C. Walker, Supervisor of Writing, St. Louis, Mo. Reading William S. Gray (Chairman) Dean, School of Education, University of Chicago, Chicago, 111. Frank W. Ballou, Superintendent of Schools, Washington, D. C. Ernest Horn, College of Education, State University of Iowa, Iowa City, Iowa. Frances Jenkins, Assistant Professor of Education, University of Cincinnati, Cincin- nati, Ohio. [29] S. A. Leonard, University of Wisconsin, Madison, Wis. W. W. Theisen, Assistant Superintendent of Schools, Milwaukee, Wis. W. L. Uhl, Department of Education, University of Wisconsin, Madison, Wis. Laura Zirbes, Teachers College, Columbia University, New York City. Social Studies Harold O. Rugg (Chairman) Lincoln School, Teachers College, Columbia Uni- versity, New York City. M. G. Clark, Superintendent of Schools, Sioux City, Iowa. Earle U. Rugg, Head, Division of Education, Colorado State Teachers College, Greeley, Colo. Spelling Ernest Horn (Chairman) College of Education, State University of Iowa, Iowa City, Iowa. Leonard P. Ayres, Vice President, Cleveland Trust Company, Cleveland, Ohio. B. R. Buckingham, Director, Bureau of Educational Research, Ohio State Uni- versity, Columbus, Ohio. John M. Foote, Rural School Supervisor, State Department of Education, Baton Rouge, La. Helen R. Gumlick, Supervisor, Primary Grades and Kindergarten, Denver Public Schools, Denver, Colo. W. F. Tidyman, State College, Fresno, Calif. C. W. Washburne, Superintendent of Schools, Winnetka, 111. The summer school class at the University of Oregon, conducted by Carleton W. Washburne, helped in collecting and reviewing research studies. Those so assisting were: Florence E. Anderson; Wave Anderson; Edward Anderton ; Edna Biles; LaMoine R. Clark; Cora E. Coleman; Lyndsay L. Eastland; Almeda J. Fuller; E. H. Hedrick; Guy L. Lee; C. L. McFaddin; Delphie M. Taylor; J. K. West; L. Lee Williams; and F. C. Wooton. [30] INTRODUCTION TRADITION and opinion, unsupported by facts, are no longer suf- ficient bases for including any subject or topic in the public school curriculum. The material available for instruction is so vast that no one person can hope to master more than a small portion of it. How is selection to be made? The subjects in the elementary school curriculum are those intended for universal study. They must meet the tests of social usefulness and individual development, achievement, and happiness. Some of the research studies reviewed in this section represent attempts scientif- ically to determine what subject-matter is socially useful in present-day life. Others seek to learn what is of greatest interest to children; for, other things being equal, material should be taught at the moment when the use for it is apparent. Hence, the studies of the interests of children are important to curriculum makers. It is generally accepted that the traditional curriculum for each subject in the elementary school must be modified to satisfy specific objectives and definite outcomes formulated in the light of present-day standards. The immediate needs of the child, the future demands of his adult life, and his capacity for education are all determining factors. The drill subjects, where the content is definite, have been dealt with more or less completely. For these subjects the minimum essentials in functional content should be tentatively determined soon by research workers. In this section of the Yearbook, the outstanding research studies which have been made up to date are reviewed. In some subjects the research work which has been done is fragmentary. This largely explains why certain subjects are treated so much more completely than others, although space allotment in the Yearbook was to some extent a determining factor. It was necessary to limit the number of studies reviewed in those . sub- jects where extensive research has been carried on. The methods of attack in these research studies represent a conscientious striving for the scientific solution of curricular problems. The technique of the investigations in some instances is crude. In the absence of absolute measurement, concensus of expert opinion as well as of mass opinion, has been resorted to. Even this procedure, however, gives an approximation which is far superior to the personal opinion of a few individuals. The findings of some investigations are incomplete. They should serve as starting points for further work, not as bases for statements of uni- versality of results. These studies are the first forward steps and may be more useful as pointing the way rather than as presenting conclusive results. Perhaps their greatest service will be to show the vastness of the task of bringing the elementary school curriculum up to date and keeping it there and to emphasize the need for research to blaze the way for practice and to check results actually obtained. ; c. r 3ii 32 Department of Superintendence At the beginning of each chapter are listed problems which confront course-of-study committees in outlining that subject. In so far as possible these questions are answered by a review of scientific studies bearing upon them. The number of unanswered questions, however, shows that at the present time the outstanding need in the field of curriculum construc- tion is the making of hundreds of technical studies. This section of the Yearbook pools most of the important existing au- ricular studies. Here is made available much material of high value which hithertofore has been inaccessible because of its technical form and presenta- tion, or because of its fragmentary distribution. The analyses of the studies which follow should have these four advantages : " Many mooted curricular questions are answered ; Material is collected for a later program of evaluation; Information is presented that is at least useful in the form of suggestions ; Connecting links are furnished between principles and practice. The public school recognizes as its first aim the development of char- acter through fixing right habits of conduct. Such habits are fundamental to moral integrity. Preachments of the moral virtues too often miss their mark. But when a child in real life over a period of years finds himself in a situation in which it is made pleasant to be honest, to cooperate ef- fectively with his neighbors, to reverence things which should be reverenced, progress has been made toward developing his character. Research workers and school administrators recognize the importance of character development through the curriculum. Progress in this field has been more rapid during the past year than ever before. School systems in revising their courses of study are shaping their whole school programs, so that every subject taught and every school activity engaged in make their proper contribution toward the development of character. The reader will probably note the absence of any report of researches in the important subject of character-building. The members of the Commis- sion recognize the paramount importance of this subject, and they are aware of much of the splendid work that is ,being done, both directly and in- directly, by both public and private schools toward character-building. In many cities, notably, Buffalo, Philadelphia, Salt Lake City, Los Angeles, and Chicago, there are well organized courses in use. In other cities, a course in character-building comprises a definite element of the course in civics, or citizenship. In practically all schools of the country, principals and teachers are consciously and definitely seeking opportunities for the development of good moral habits through all studies, and especially through organized play, games, athletics, and through all group activities in which pupils participate. Progress in this field has been more conspic- uous since the war than at any time in the history of our schools. Un- fortunately, we find little that can be classified as scientific research in this field. To quote Professor Edwin D. Starbuck: "There are a' good many The Third Yearbook 33 studies that try to be scientific in the matter of character education, but they surely have little right to be dignified by the term, research." The Indiana Survey, by Professor Ahearn, is suggestive of what may be, and needs to be, done in this field. Several studies are under way. Professor May, of Teachers College, and Professor Starbuck are just now engaged in making a bibliography of such studies. The members of the Commission would feel remiss in their duty to the children of our schools, if they did not urge all school officials who are en- gaged in revising their courses of study to shape their programs so as to make definite provision for character-building. CHAPTER III ARITHMETIC Guy M. Wilson, Professor of Education, Boston University, Boston, Massachusetts, Chairman A RITHMETIC is not an end in itself. It is a tool. To meet the L\ demands of social utility, three phases of arithmetic need attention : •*■ •*■ (1) The basic experience which is necessary in order to make manipulative work meaningful; (2) the mastery for automatic reproduc- tion of the useful number facts; (3) training in application to life and business situations. The curriculum demands for the above program are so simple that there is no reason why it should not yield to complete analysis and scientific determination. Much scientific progress has been made. The problems needing solution may be listed with reasonable completeness ; on some there are much data. The following is submitted as a representative statement of the prob- lems on which further scientific evidence is wanted. Major Curriculum Problems in Arithmetic 1. Specifically, what is the purpose of arithmetic; what is its relation to the major objectives of education? 2. How much emphasis should be placed upon the building up of ex- perience before proceeding with the manipulation of symbols? In other words, this is the problem of building up experience, so that number work shall be meaningful, rich in application, and actually useful instead of being merely formal memory work. 3. When should formal drill work begin and at what rate should it pro- ceed? In this connection, there is needed a set of criteria and a means of checking, so that drill may not proceed on a meaningless basis. 4. Is it possible to replace haphazard drill by a scheme of drill that is completely systematized, all inclusive, with the elements ranked on the basis of difficulty? This complete scheme should lend itself as a means for checking or testing the entire range of the child's ability on fundamental drill needs. 5. What is the proper unit of instruction in arithmetic? 6. What are the procedures or requirements necessary to place arith- metic on a 100 per cent motivated basis? This is a problem of making arithmetic grow out of pupil experience and real life situations that are thoroughly comprehended by the children. 7. What is the best plan for eliminating the traditional and useless in arithmetic by a positive procedure that shall carry conviction to teacher, pupil, and patron? [35] 36 Department of Superintendence 8. By what plan or procedure may teachers eliminate merely stock examples in so-called reasoning problems, substituting therefore life situa- tions worth while in themselves? 9. What are the reasonable limits of general arithmetic or arithmetic needed by the consumer group ; where does vocational arithmetic begin ; and what are the specific needs for arithmetic by the various vocational groups ? 10. Is it possible to have an elastic set of norms or standards of per- formance that shall include speed, difficulty, and accuracy, and make proper allowance for individual differences? 11. In usable form, what is the specific psychology involved in learning and remembering in the various phases of arithmetic work — drill, reasoning, and application to life situations? 12. How much of the failure in lower grade arithmetic may be ac- counted for by low ability in reading, and what adjustments in time schedule are necessary to make sure that reading ability is adequate to carry the necessary work in arithmetic? 13. How replace the formal textbook procedure by the actual figuring used in business? In other words, what is wanted is a method by which the children may learn of the actual figuring used in business and be con- scious themselves that they are figuring in practically the same form in which it is being done by adults. 14. How should we profitably use the time usually alloted to arithmetic ? Sixteen and two thirds per cent of all school time for arithmetic, as shown by previous studies, is obviously more than is justified. Shall the extra time be used as a vehicle, or excuse, for approach to other aims in educa- tion: for example, effective home membership through the study of the budget, the civic-economic aim through a study of the cost of strikes, etc., or shall the time devoted to arithmetic be reduced as made possible and desirable by the greater efficiency of present-day teaching and the broader view of present educational objectives? The Committee's Plan of Report The Committee sought to obtain a review of all important studies, pub- lished or unpublished, and to organize the findings as answers to definite problems. The references in the Judd-Buswell Commonwealth Fund bibliography on arithmetic were divided among the Committee for review, and the entire country was sectioned for the gathering of unpublished studies. Members of the Committee gave such unlimited time and effort, that finally, the chief handicap of the Committee is lack of space assign- ment to include its full report. The report takes up successively the following problems: I. When should formal arithmetic begin? (Stone). II. What are the socially useful processes? (Woody). III. According to what criteria should drill be organized?, (Knight). The Third Yearbook 37 IV. What is the best procedure for building up ability to handle concrete or reasoning problems? {Clark). V. What has psychology and experimental education to contribute? {Myers and Osburn). VI. What help from standard tests? {Wilson). VII. What is the method of curricular determination in arithmetic? {Wilson). For several reasons these problems will not receive a well-balanced treat- ment. The amount of available data varies, and, due to the manner of apportioning the work, some problems received much more attention than others. The chairman took the liberty of reducing his own work to a mere summary in order to make space for the carefully prepared work of Woody and Knight. I. When Should Formal Arithmetic Begin? The term "formal arithmetic," as here defined, means formal drill work and a definite attempt to teach for automatic reproduction the number combinations in the fundamental processes, such as simple fractions. Since the purpose of drill is to fix by repetition something previously compre- hended, it is evident that drill should follow, not precede adequate ex- perience and understanding. Moreover, there should be on the part of the child before he begins formal repetition a real feeling of need and clear vision of use and application — in short, there should be complete motivation. The psychology of interest is so well established by scientific evidence that this statement should go unchallenged. Obviously, there is a tremendous opportunity for individual differences in the date of beginning formal drill. Dr. Washburne at Winnetka, Illi- nois, and Miss Ringer at Longbranch, Washington, have shown how to adapt a drill technique to differences in interest and capacity. There are a few other considerations involved. The relation of reading is evidently one consideration. The total profit resulting from work in a lower grade is another consideration. Scientific evidence is meager. Three studies which have a bearing follow : STUDY NO. I. Taylor, Joseph S. "Omitting arithmetic in the first year," Educational Administration and Supervision. February, 1916. pp. 87-93. This is an experiment conducted in Public School No. 16, the Bronx, New York City during 1913 to 1915. During the first of these two years, half the entering children in the school did no work in arithmetic except counting. The extra time was devoted to reading. At the close of the second year of the experiment, children who had had two years of formal number work and those who had had but one year — the children being otherwise equivalent — were compared by the use of standard tests. A summary of the results follows: 38 Department of Superintendence 2A Tests Oral Written Classes with arithmetic in first year 81.0 72.6 Classes omitting arithmetic in first year 87.1 89.4 2B Tests Classes with arithmetic in first year 87.2 69.5 Classes omitting arithmetic in first year 90.7 61.0 From this summary it is evident that the classes which omitted formal number work during the first year of school and gave the extra time to reading were, all told, at an advantage by- the close of the second year. The quality and quantity of reading greatly improved, while the number work showed improvement — that is, in the 2A tests the children omitting number work in the first year were superior in both oral and written work, while in the 2B tests they were superior in the oral and inferior in the written. The totals, therefore, appear to favor omitting formal number work from the first year and giving the time to reading. This is the interpretation placed upon the results by Taylor, the author. STUDY NO. 2. Hackler, John Monroe. The relation between suc- cessful progress in mathematics and the ability to read and understand, and the factors that contribute to success or failure in mathematics. An un- published master's thesis in the library of the University of Chicago, Chi- cago, 111. September, 1921. Problem. The main problem, as stated in the title, is to ascertain whether ability in mathematics depends on ability to read, at least to a certain extent. Apparently, comprehension-ability in silent reading is the thing in the author's mind, since silent reading tests were used. Method. Tests were given in silent reading and intelligence, and teachers' estimates were obtained in mathematics and other subjects and correlated with each other. The experiment was conducted with 40 students in the preparatory department first-year class in mathematics in the Northeastern State Normal School, Tahlequah, Oklahoma, and with the pupils of Grades III to VIII inclusive, of the Tahlequah city schools. The class in the preparatory department corresponds roughly to a first- year high-school class, but is far more heterogeneous, containing a wide range of ages, and for this reason the correlation studies of this group are supplemented by an individual analysis. Conclusions and Results. The results of this investigation have not confirmed the original hypothesis to the extent expected. It has failed to show that ability to succeed in mathematics depends to any great extent upon reading ability. Results with the preparatory school class showed correlations of — (— .36 between the Kansas Silent Reading Test and teach- ers' estimates of ability in first-year mathematics by the "listed pairs" method, and -(-.60 by the Ranks' method. In the city schools positive correlations of from .22 to .70 were obtained between reading tests and teachers' estimates in arithmetic. The Third Yearbook 39 The author states, "Personal investigations and analysis of individual cases lead to the belief that reading ability is a factor in successful work in mathematics." STUDY NO. 3. The effect of a years drive on motivated reading in arithmetic for mixed drill tests. Unpublished study, University of Iowa, reported by F. B. Knight. Data: 111 pupils in grades 3, 4, 5, 6, 7, and 8. Public Schools, Rad- cliff, Iowa. Tests used: Arithmetic computation phase of Stanford Achievement Test. Time of testing: First test, September 15, 1923; Second test, May 15, 1924. Results: I. Whole group of 111 pupils: A. Beginning status 1. Average actual arithmetic, com- prehension age 138.5 months 2. Average theoretical arithmetic, comprehension age 144.6 months 3. Average loss in arithmetic com- prehension age due to faulty reading 6.1 months B. Status after one year of motivated drive on reading 1. Average actual arithmetic com- prehension age 165.3 months 2. Average theoretical arithmetic comprehension age 166.0 months 3. Average loss in arithmetic com- prehension age due to faulty reading .7 month C. Gain in arithmetic comprehension age due to improvement in ability to resist influence of isolated drill 5.4 months II. Group of 67 pupils who made errors due to reading : A. Average loss in arithmetic comprehen- sion age due to faulty reading. First test 10.0 months B. Average loss in arithmetic comprehen- sion age due to faulty reading after year's drive 1.0 month • C. Gain in arithmetic comprehension age due to improvement in reading 9.1 months 40 Department of Superintendence , % III. Gains during year in arithmetic comprehen- sion age as result of year's drive on moti- vated reading of arithmetic: A. Average gain of 67 faulty readers 29.7 months B. Average gain of 44 good readers. 21.2 months C. Excess gain in arithmetic comprehen- sion age of faulty readers over good readers 8.5 months IV. Summaries: . A. Gain in arithmetic comprehension age of faulty readers due to improve- ment in reading 9.1 months B. Excess gain in arithmetic comprehen- sion age of faulty readers over good readers 8.5 months V. Conclusion : The excess gain made in arithmetic comprehension age made by the original faulty readers over the original good readers was in the main due to their improvement in the specific type of reading needed in offsetting the influence of isolated drill — namely, ability to follow directions call- ing for several processes in mixed order. Conclusion. The above studies and data are obviously inconclusive. They do, however, -point to the desirability of more careful attention to reading and the entire omission of formal number work from the first grade. These conclusions are supported by much unscientific evidence from various parts of the country, some of which may be briefly recounted. Professor Stone reports that the demonstration and practice school in the State Normal College of Ypsilanti, Michigan, 1902-1911 began formal number work in the third grade, except counting and simple addition and subtraction, facts that arose in regular school activities. The public schools of that city began number work in the first year. Pupils entering the training school from the city schools in the fourth year showed a little advantage in written work at the beginning of the year, but by the end of the second semester there was no difference between those who had had number work for three and one half years and those who had had num- ber work for only one and one half years. Dr. H. L. Smith in the Bloomington (Illinois) Survey shows that chil- dren who had had no formal number work in grades one and two were not handicapped by the end of the third year in competition with children who had had number work regularly during the first and second year. II. What are the Socially Useful Processes? McMurry, as long ago as 1904, with keen insight, called the attention of superintendents to the fact that many processes taught in arithmetic had The Third Yearbook 41 little or no social value. His opinion was supported again and again, but then as now mere opinion carried less weight than scientific evidence. Fortunately, a method has been discovered for gathering such scientific data. The actual analysis of usage has become the basis for inclusion of processes found useful, or the exclusion of processes not used. Mrs. Gallaway's study deals with the needs of freshmen college girls who take a course in clothing. Obviously, with the basic arithmetic taught and with the evident need at hand, college women will master additional arithmetic as needed. The same statement applies to the study of Williams dealing with the mathematics needed in freshman chemistry. If chemistry calls for more arithmetic, then is the time to teach it. The other studies divide themselves sharply into two classes, according to viewpoint. Adams and Mitchell are dealing chiefly with ability to read numbers and understand them in general and not with manipulative ability; on the other hand, the studies by Charters, Hanus, Wilson, Wise, and Woody deal with manipulative skills demanded of adults. It should be noted that these two viewpoints show radically different results. For a century or more after the founding of Yale University, it was possible for students to enter without knowing the multiplication table, yet there is little doubt that the general reading ability called for in the Adams study was met. Doubtless also the multiplication table was learned, but the point is that there is no established and verified relationship between the general understanding required for reading ability and the automatic response required for calculating. The findings from these studies point strongly to the following conclusions : ( 1 ) That experience must in all cases precede formal work in calculating; (2) that manipulative or calculating skill required is relatively small and not a matter of momen- tary occurrence to even the busiest of adults; and (3) many processes long retained in the textbooks in arithmetic have practically zero value. It is safe to say that most superintendents of schools today would agree to the complete elimination of the following processes: Compound numbers, addition, subtrac- Proportion. tion, multiplication, division. Ratio beyond the ability of fractions to Greatest common divisor and least com- satisfy. mon multiple beyond the power of Partnership with time. inspection. Longitude and time. Long confusing problems in common Exchange, domestic and foreign. fractions. Apothecaries' weight. Complex and compound fractions. Troy weight. Reductions of denominate numbers. Table of folding paper, surveyor's table, Cases two and three in percentage. table of foreign money. Annual interest. Much of mensuration — trapezoid, trape- Compound interest, except savings. zium, polygons, frustrums, spheres. Partial payments. Cube root. True discount. , The metric system. 42 Department of Superintendence The curriculum in arithmetic should make no provision for formal drill on the above processes. This does not preclude some "informational" attention to them, or some special attention whenever one of them is in- volved in a "life situation" studied in detail by the pupils. A summary by Woody of the scientific evidence follows in order of ap- pearance : 1. Computational uses of arithmetic — The findings on the computa- tional uses of arithmetic are so different from those on informational or general reading usage that it seems wise to separate the studies. The com- putational uses are much simpler, and according to Woody (Study No. 14), they are being further reduced by the uses of commercial calculating machines. By separating the studies into these two divisions they show great consistency as to conclusions. STUDY NO. 4.* Wilson, Guy M. Connersville course in math ematics. 1911; republished by Warwick and York, 1922. Baltimore, Maryland. Problem. To get the sentiment and goodwill of the business men of Connersville in the construction of the course of study in arithmetic in the public schools of that city. Method. The following questionnaire was submitted to the business men for consideration: (1) At present the school time in the grades is divided approximately as follows: Reading, 26 per cent; arithmetic, 16 per cent; language, 12 per cent ; history, 1 1 per cent ; geography, 8 per cent ; spelling, 7 per cent ; music, 6 per cent; drawing, 6 per cent; writing, 4 per cent; physiology, 4 per cent. Does this appeal to you as a proper division of school time in the grades? (2) Stated differently, English receives 49.2 per cent of the school time in the grades; the three R's receive 65.24 per cent; the fundamentals (the three R's plus geography and history) receive 86.09 per cent; the special subjects (physiology, music, and drawing) 13.91 per cent. Does your ex- perience suggest any change of emphasis? (3) Following are some topics we, as boys, studied in arithmetic in the grades. Check ( V ) the topics for which you have had considerable use during the past six months. Cross (X). the topics for which you have had little or no use during the six months : Troy weight Foreign exchange Longitude and time Compound proportion The surveyor's table True discount The greatest common divisor Cases 2 and 3 in percentage The least common multiple Compound interest Complex fractions Partial payments Cube root Partnership Compound fractions The studies are numbered consecutively throughout the chapter for convenience in reference. The Third Yearbook 43 May some of these topics be omitted from our arithmetic work without material loss? Which ones? (4) Following are some topics we did not study in arithmetic as boys: Saving and loaning money Keeping simple accounts Mortgages Investing money Modern banking methods Bonds as investments Building and loan associations Real estate as investments (cheap rentals, good residence property, business blocks, farm lands as investments) Marks of a good investment (It is estimated that the get-rich-quick concerns fleece the American people out of $60,000,000 a year) Taxes, levies, public expenditures Profits in different lines of business Check (V) an Y of the above that appeal to you as worthy of a place in present-day arithmetic work. (5) Connersville is developing important industrial and commercial interests. To what extent should those interests have an influence in shap- ing a course of study for our schools, especially in arithmetic? (6) Suggest some feasible plan of bringing the schools into closer rela- tions with the industrial, commercial, and business interests of our com- munity. Give some figures and problems from your own business. Findings. 1. On questions (1) and (2) few suggestions were made. Of these, most recommended reducing the time given to drawing and increasing the time given to writing and spelling. 2. On question (3) the business men reported which topics had been little used and suggested that they be eliminated from the course of study, There was meager support for cube root, compound proportion, cases in percentage, and the surveyor's table, and the majority favored eliminating these and putting the time on the fundamentals. 3. On question (4) all agreed that every topic should be included in the course of study. 4. On question (5) all agreed that the schools and the course of study should take into consideration the industrial and commercial interests of the community. 5. On question (6) nothing of consequence was submitted. STUDY NO. 5. Jessup, W. A. and Coffman, L. D. The supervision of arithmetic, Chapters I and II. New York, Macmillan, 1916. pp. 1-38. Problem. To determine the attitude of city and county superintendents toward the elimination of certain questionable subject-matter and the intro- duction of certain other new subject-matter. Method. The following letter was sent to all superintendents of schools in cities of 4000 population or over, and to every sixth county superin- tendent in the United States : 44 Department of Superintendence Underscore once the subjects which should receive slight attention ; underscore twice the subjects which should be eliminated. Apothecaries' weight, troy weight, furlong, rod in square measure, drachm, quarter in avoirdupois weight, surveyor's tables, foreign money, folding papers, reduction of more than two steps, long method of finding greatest common divisor, lowest common multiple, true discount, cube root, partnership, compound proportion, compound and complex fractions, cases in percentage, annual interest, longitude and time, unreal fractions, alligation, metric system, progression, and "aliquot parts. Others Underscore subjects which should receive more attention than is usually given : Addition, subtraction, multiplication, division, fractions, percentage, interest saving and loaning money, banking, borrowing, building and loan associations, investments, stocks and bonds, taxes, levies, public expendi- tures, insurance as protection and investment, profits in business, and public utilities. Findings. 1. Replies were received from 52 per cent of the superin- tendents of the city schools, and from 24 per cent of the superintendents of the county schools. 2. There was a majority of those reporting in favor of increasing the emphasis on the four fundamental operations, and upon phases of arithmetic involving savings and loaning of money, taxation, public expenditures, fractions, insurance, and percentage. 3. Between 10 and 50 per cent of those reporting favored increased emphasis on stocks and bonds, levies, banking, interest, building and loan associations, and profits. 4. Over half of the superintendents reporting favored eliminating: apothecaries' weight, rod in square measure, drachm, quarter in avoirdupois weight, compound proportion, unreal fractions, alligation, and progression. 5. From 8 to 50 per cent of all the replies favored elimination of all other topics included in the second half of the questionnaire. STUDY NO. 6. Chase, Sara E. "Waste in arithmetic." Teachers College Record, Volume 18, No. 4, September, 1917. New York, Bureau of Publications, Teachers College, Columbia University, 1917. pp. 350- 371. Problem. 1. To determine waste through the measurement of the per- manent effects resulting from the teaching of mensuration and through ascertaining the extent to which various terms and operations involved in mensuration were used in life outside of school. 2. To determine the waste due to the pupil's unfamiliarity with the language and conditions of given problems. Part I. Method. Test I. Sheets of paper, upon which were drawn a square, a rectangle, a rhomboid, a triangle, and a circle, were given with The Third Yearbook 45 a ruler to the following groups of students who were told to determine the exact area of each figure : Group I. One hundred and twenty-three pupils in Grades IX to XII. Group II. Thirty-eight pupils in Grade IX who had just spent three months in mensuration. Group III. Thirty pupils in Grade XI in a technical high school three months after they had completed mensuration in connection with plane geometry. Group IV. Fifteen adult students at Teachers College, Columbia Uni- versity. Findings. In Group I Forty-four per cent could not find any of the areas. Ninety-three per cent failed on the rhomboid. Fifty-five per cent failed on the square. In Group II > Twenty-four per cent could not find any of the areas. Ninety-three per cent failed on the rhomboid. Thirty-two per cent failed on the square. Even in Group IV there was considerable failure. Test II. As another test, the following lists of areas and volumes (1) area of square, rectangle, triangle, circle, and rhomboid; (2) entire surface of a cube, prism, pyramid, cylinder; (3) volume of a cube, prism, cylinder, were given to the following groups of people: (a) Pupils in school, (b) teachers of drawing, sewing, and manual training, (c) one hundred forty men and women engaged in various occupations. These different groups were asked to cross out all items in the given list which they had not used in their life out of school. Findings. Seventy-eight per cent of all the pupils crossed off every- thing. Teachers of manual training said that board measure was needed to compute cost, but no other areas were needed in this work. Teachers of other subjects answered that neither areas nor volumes were needed for their work. Of the workers, 68 per cent of the women and 30 per cent of the men crossed off everything. TABLE i.— PERCENTAGE OF MEN AND WOMEN USING DIFFERENT ITEMS IN ARITHMETIC Area of Men Women Entire surface of Men Women Volume of Men Women 66% 61 42 32 11 27 26 8 12 3 Cube Prism Pyramid Cylinder 38 20 9 • 24 8 5 5 Cube Prism Cylinder Square root of number. 39 18 36 41 9 Rectangle Triangle Circle Rhomboid 8 9 12 46 Department of Superintendence Part II. Method. Vocabulary tests, patterned -after the Thorndike Vocabulary Tests were made from words in textbooks in arithmetic and given to pupils in appropriate grades. Findings. Results show a woeful lack of knowledge of the vocabulary used in the written problems of arithmetic. Below is a list of words and the percentage of pupils in each grade failing to use each word in a sentence to illustrate its meaning: Percentage of pupils in Grade III Grade IV G rade V 80 40 90 40 20 60 95 24 The grocer 20 o The contractor 80 Children in the city do not know what is meant by "bin, head of cattle, loading grain, etc." The article makes a plea for the use of concrete situa- tions true to life and within the child's experience. STUDY NO. 7. Hanus, Paul H, and Gaylord, Harry D. "Courtis arithmetic tests applied to employees in business houses." Journal of Edu- cational Administration and Supervision, Volume 3, No. 9, November, 1917. pp. 505-520. Problem. To find out how the results achieved on the Courtis Research Test, Series B, by 446 employees of one of the largest trust companies and one of the largest department stores in Boston compared with those achieved by the pupils in Grade VIII in the Boston public schools. Method. The Courtis tests were given to the employees in groups of approximately 35 individuals and the results obtained compared with the achievements in Grade VIII in the Boston public schools. Findings. 1. In addition, the median number of problems solved by the employees was 22.87 per cent; the median accuracy, 88.6 per cent. By the pupils in Grade VIII, the median number of problems correctly solved was 12.9 per cent; the median accuracy, 77 per cent. Many of the em- ployees finished the test before the end of eight minutes. 2. In subtraction, the median number of exercises solved by the employees was 19.3 per cent with a median accuracy of 85.3 per cent correct; by the pupils in Grade VIII, 12.9 per cent exercises correct with an accuracy of 90 per cent. It is interesting to note that the pupils in Grade VIII were approximately five per cent more accurate than the employees. 3. In multiplication, the median number of exercises solved by the employees was 13.8 per cent with a median accuracy of 64.8 per cent; the median number of exercises correctly solved by the pupils was 11.8 per cent with a median accuracy of 82 per cent. Note that the 'pupils solved almost as many problems and were much more accurate. The Third Yearbook 47 4. In division, the median number of exercises solved by the employees was 11.1 per cent with an accuracy of 84.8 per cent; the median number of exercises by the pupils, 12.2 per cent correct with a median accuracy of 94 per cent. Note that the pupils worked more problems and were more accurate. 5. Analysis of the results showed that the accountants in the department store made much higher scores than the salesgirls, and that bookkeepers in the bank made higher scores than those in the listing department. It is possible that the former groups were more highly selected, but the fact that their speed in addition was much greater in proportion than that in other operations suggests that the greatest efficiency results from greater practice rather than selection of higher initial ability. Relatively greater practice in multiplication and division on the part of the school children of Boston accounts for their higher percentage of accuracy. STUDY NO. 8. Wilson, G. M. "A survey of the social and business use of arithmetic." A preliminary study. 16th Yearbook, Part I. Na- tional Society for the Study of Education. 1919. Wilson, G. M. "The social and business uses of arithmetic." Teachers College Contributions, No. 100. New York, Teachers College, Columbia Univ. pp. 1-62. Problem. To determine the nature of the arithmetic used by adults in their social and business relations. Method. Study based upon the analysis of 14,583 problems contributed by 4068 different persons representing 155 different occupations. These problems were collected through the cooperation of superintendents and teachers in the public schools of the east and the middle-west, who asked the children in Grades VI, VII, and VIII to get from their parents, each night for a period of two weeks, a statement of the uses which they had made of arithmetic during the day. The statements of the problems encountered, thus collected, provided the data upon which this study was based. The author contended that the data were extensive and varied enough to be representative of adult social and business life. Findings. 1. Eighty-five per cent of all problems involved the use of money in either the buying or selling of goods. Labor and wages, interest, rent, and insurance covered two thirds of the problems involving money, but not involving buying or selling. Problems not involving money involved hours of labor, measurement of capacity of bins, cisterns, cribs, tanks, estimates of building materials, etc. 2. Of all the problems involving buying and selling, 45.9 per cent related to food; 17.9 per cent to clothing. For all occupations the ten topics involved the greatest number of times in the different problems were: groceries, dry goods, labor, milk, making change, meat, eggs, clothing, butter, and fuel. 3. Multiplication, addition, subtraction, division, and fractions reported constituted 90.6 per cent of all the problems. Little use was made of pro- 48 Department of Superintendence portion, decimals, apothecaries' weight, square root, partial payments, and troy weight. When these topics were involved the problems were too complicated for the elementary grades. 4. In the problems involving addition only two and one half per cent of the problems had more than four places in the largest addend. Almost all of the multiplication problems had either one or two place numbers in the multiplier. In division 39.6 per cent of all the problems had one number in the divisor; 43.4 per cent had two-place numbers in the divisor. 5. The most commonly used fractions had denominators of halves, thirds, fourths, fifths, and eighths. 6. The following processes did not appear in the adult figuring reported : greatest common divisor, least common multiple, long confusing problems in common fractions, complex and compound fractions, reduction in denominate numbers, tables of folding paper, surveyor's measure, foreign money, compound numbers, longitude and time, compound interest, annual interest,, exchange, true discount, partnership with time, ratio, mensuration involving trapezoids, trapeziums, polygons, frustrums, spheres, cube roots, or the metric system. STUDY NO. 9. Mitchell, H. Edwin. "Some social demands on th< course of study in arithmetic." Seventeenth Yearbook of the National Society for the Study of Education, Part I. Bloomington, 111., Public School Pub. Co, 1918. pp. 7-17. Problem. To determine the relative importance of the various content elements of the course of study in arithmetic through ascertaining the frequency of their occurrence and the manner of their use. Method. Study based upon the arithmetic appearing in a standard cook book, in the payrolls of a number of artificial flower and feather factories, in marked-down sales advertisements, and in a general hardware catalogue. Findings. Study I — From the cook book : ( 1 ) The numbers occurring were very small regardless of whether they were integers or mixed numbers. (2) The most commonly occurring fractions had denomi- nators of 2, 3, 4, and 8. Study II — From the factory payrolls: . . (1) The word "dozen"- and fractional parts of twelve were used very much. The commodities were made and sold by the dozen. (2) To compute earnings the arithmetical processes in- volved: addition of all possible fractions with 12 as a dozens ; of decimals in terms of dollars and cents ; cation by a fractional, integral, or mixed number of denominator to integers and mixed numbers; multipli- addition of United States money. The Third Yearbook 49 (3) The fractions emphasized were twelfths, fourths, thirds, and sixths. Study III — From marked-down sales advertisements: (1) Discount notes with the exception of halves and thirds were expressed in percentages. (2) The most common practices were to use 10 per cent, 20 per cent, 25 per cent, one third, and one half off. Study IV — From the general hardware catalogue: (1) Many articles were sold by the dozen, thus necessitat- ing ability to calculate using fractions and mixed num- bers involving 12 as the denominator of the fraction. (2) Fractions with denominators of 16, 32, and 64 were used. Many of these fractions were merely descriptive, although the dealer had to read and understand them. (3) Quantities tabulated under avoirdupois weight were descriptive of the weights of articles, but addition of weights was necessary in determining freight rates. (4) The terms used frequently were: inches, feet, yards, ounces, pounds, dozen, gross, quarts. (5) The units often used in selling were: per each article, per dozen, per gross, per hundred, per pound, per foot, per hundred feet, per square foot, per hundred square feet. STUDY NO. io. Wise, Carl T. "Survey of arithmetical problems arising in various occupations." Elementary School Journal, Vol. 20. October, 1919. pp. 118-136. Problem. To test over a more varied area, the Wilson Survey method of determining the social needs for the arithmetical processes. Method. The problems of adults, 7345 in number, were gathered from the States of Texas, Iowa, Illinois, Wisconsin, California, and Missouri. The classification used showed the processes and their difficulty. Findings. 1. Differences in the classifications of problems from the city and rural districts and also differences in the classifications of problems from different parts of the country were negligible. 2. Eighty-five per cent of all problems classified involved only the four fundamental operations, or combination of fundamental operations. 3. The fractions commonly used were 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, and 1/8. These constituted 93.9 per cent of all fractions which occurred. 4. Problems involving common weights and measures occurred very frequently. 5. There were very few problems in compound interest, compound proportion, insurance, plastering, painting, masonry, and bank discount. 50 Department of Superintendence 6. No problems were received involving taxes, investments, stocks and bonds, equation of payments, foreign exchange, apothecaries' weight, alliga- tion, annual interest, compound and complex fractions, folding paper, troy weight, or the metric system. STUDY NO. ii. Noon, -Philo.G. "The child's use of numbers." Journal of Educational Psychology, Volume 10, November, 1919. pp. 462-467. Problem. 1. What arithmetical knowledge and power should be ac- quired by a child at the end of Grade VI. 2. What should be the content of the arithmetic course in Grades IV, V, and VI ? Method. The author went to 12 classes of Grades IV, V, and VI, and two immature classes of Grade VII in a choice residential section of Boston and asked the children to tell what numbers they make use of or notice outside of school. Items reported were accredited to the lowest grade mentioned. Summary: A. Games: (1) Calling or reading numbers : (a) Football signals; (b) spots on dice; (c) spots on dominoes; (d) number on sled to indicate size; (2) Counting marbles, children in games, tops, stripes, etc. B. Going to store: (1) Knowledge of quantity (pound), (2) number of articles, (3) finding cost, and (4) counting change. C. Other instances of reading numbers: (1) Telephone numbers, (2) automobile numbers, (3) policeman numbers, (4) numbers on fire engine, (5) numbers on police engine, (6) dates on almanac. D. Other instances of counting: (1) Money belonging to child, (2) postage stamps, (3) steps in dancing, (4) number of Christmas presents. Conclusion. Reading of numbers and counting include nearly every item recorded above. Below Grade VII there is no felt need for problems in arithmetic. These emphasize mechanical work up to the sixth grade. Then is the time for applied arithmetic. Before Grade VI, emphasize the four processes (addition, subtraction, multiplication, division), common fractions, and decimals. STUDY NO. 12. Woody, Clifford. "Types of arithmetic needed in certain types of salesmanship." Elementary School Journal, Volume 22, No. 7, March, 1922. pp. 505-521. Problem. The investigation was undertaken to gain a reliable index of arithmetic needed by the clerk in selling goods and by the consuming public in purchasing goods. Method. The study consists of an analysis of 4661 bills of sale, represent- ing a total value of $41,560.67 obtained from three large stores in Seattle, a wholesale and retail hardware, a wholesale and retail grocery, and a The Third Yearbook 51 large department store. In this study a problem was defined as any situa- tion demanding calculation. Oftentimes one situation demanding more than one set of calculations was listed as more than one problem. As the author collected data, he interviewed managers concerning needs for arithmetic and concerning the arithmetic used. Findings. 1. Nine hundred forty-five of the 4661 bills of sale required no computation other than in the payment of bills. (This was true with regard to the vast majority of the business conducted in one of the firms, as bills of sale were made only in case the purchases were to be delivered.) 2. There was no use of decimals save in connection with United States money. This was to be expected since calculations involved only buying and selling. 3. Of the problems in addition encountered none had more than five places in any addend, and few had more than six addends. The most com- mon type of problem in addition consisted of adding two three-place numbers. 4. Few problems in subtraction, other than making change, were en- countered. In the grocery only one of the 206 bills of sale involved any subtraction, other than making change. When subtraction was involved, usually the problems had but three or four numbers in the minuend. 5. In multiplication, the vast majority of the problems had but two or three-place numbers in the multiplicand and one or two-place numbers in the multiplier. 6. By far, the most common problems in division had two or three-place numbers in either the dividend or the divisor. Division as a process was used less than addition or multiplication. 7. The most commonly used fractions had denominators of 2, 4, 6, and 12, although 3, 5, 9, 10, 16, 24, and 144 were used to some extent. The most common unit of sale determined very largely the fractions used. Dozen or gross were often employed and thus the fractions with denomi- nators of 12 and 144 were used. Such fractions were rarely reduced to lowest terms. There was much multiplication of a two, three or four- place number by a simple fraction, or by a mixed number having one, two, or three places and a fraction. There was little use of addition, subtraction and division of fractions. 8. Denominate numbers were encountered as units of sale, but no reduc- tion into larger or smaller units took place. 9. Discount was widely used, especially in the hardware store. The most common rates of discount were 5, 10, 20, 25, 30, 35, 40, and 50 per cent. Inquiry revealed that discount tables were used and that little calcula- tion was employed. 10. Interviews with the managers of these stores to supplement the analysis of the bills of sale revealed the following significant points: (a) The goods were marked in terms of the decimal system to make calculations easy; (b) Goods were sold by convenient units or groups of units, and were not usually sold by using a simple unit or a multiple of the total cost of a 52 Department of Superintendence large amount of goods; (c) Measuring devices, weighing devices, calcu- lating machines, calculating devices, such as discount books, price tables, cash registers, central cashiers, etc., have reduced the amount of arithmetic used by clerks to a minimum. Oftentimes the manipulating of a mechani- cal device and reading numbers was all the arithmetic needed by the clerks in selling goods. If there was no central cashier, in addition to the ability to read numbers, the clerks had to be able to make change. 11. While the evidence indicated that the clerks under consideration made little use of arithmetic, the author argued that the consuming public needed arithmetic for its own satisfaction in checking the accuracy with which the calculating and measuring devices were manipulated. He argued for much emphasis on mental arithmetic — that is, calculating without pencil or paper, and for much emphasis on quickly estimating or approximating a correct answer to the problem involved. STUDY NO. 13. Woody, Clifford. Results obtained from giving the Courtis arithmetic test to members of the Rotary and Kiwanis Clubs. Unpublished study on file in the office of the Bureau of Educational Refer- ence and Research, University of Michigan. Problem. To determine the level of efficiency in addition and multiplica- tion existing in highly selected groups of our adult society and to make comparison with the level of efficiency attained by the children in Grade VIII. Method. The addition and multiplication tests of the Courtis Research Tests in Arithmetic, Series B, were given to 203 adults belonging to Rotary and Kiwanis Clubs in seven different cities of Michigan. The results achieved were compared with the Courtis Standards of Achievement for the children of Grade VIII. Findings. 1. In general, the members of the Rotary and Kiwanis Clubs attempted more exercises than the children in Grade VIII. The median number of problems attempted by the members of the clubs was 19.5 in addition and 13.7 in multiplication; the general standard of achievement for the children in Grade VIII is 11.6 in addition, and 10.2 in multiplica- tion. 2. There was much variation in the median number of exercises attempted by the various clubs. In addition, the median scores ranged from 13.0 to 24.1 exercises attempted; in multiplication, from 8 to 16.5 exercises attempted. 3. In general, the members of the Rotary and Kiwanis Clubs were more accurate in addition and less accurate in multiplication than the children in Grade VIII as indicated by the Courtis general standards. The median percentage of accuracy for the 203 members in. addition was 82 per cent cor- rect; in multiplication, 71 per cent correct. The corresponding figures for the general standards of the Courtis Tests in Grade VIII are 76 and 81 per cent, respectively. The variation of the median percentages of accuracy in The Third Yearbook 53 addition among the different clubs was from 71 per cent to 86.5 per cent correct; in multiplication, from 64.6 per cent to 79.6 per cent correct. It is interesting that the median percentage of accuracy in multiplication in each club was less than the general standard for Grade VIII. 4. In general, the median percentage of accuracy in the clubs was higher in addition and lower in multiplication than in Grade VIII. Note. This study suggests that the level of efficiency existing in these processes is largely conditioned by the social demands made upon the differ- ent individuals. Although the different investigations indicate that multipli- cation as a process is used in business and social life more than addition, it is ventured that the problems in addition in the Courtis Tests approximate the type of problems solved in business and social life more closely than the problems in multiplication do. Furthermore, much practice in addition is gained in exercises in which the predominant process is multiplication. Thus it seems safe to suggest that the club members achieved relatively higher scores in addition than in multiplication because relatively greater practice of the type measured by the test in addition is encountered in the routine of everyday business life. STUDY NO. 14. Woody, Clifford. The use of calculating and meas- uring devices in business and their influence on the use of arithmetic. Un- published study made in cooperation with Leander Beach, 1923. On file in the office of the Bureau ofi Educational Reference and Research, Uni- versity of Michigan. Problem. To find out to what extent calculating and measuring devices were used in business houses and to determine something of the nature of arithmetic utilized in carrying on business transactions. Method. The managers of 40 business houses in Ann Arbor were per- sonally interviewed concerning the use of mechanical devices, short-cut methods, and the nature of arithmetic utilized by salesmen and consumers. This list of business houses included : three banks, five drug stores, six grocery stores, four drygoods stores, three bakeries and restaurants, two hardware stores, two shoe stores, three men's clothing stores, two book stores, one electric shop, one wall-paper and paint store, one jewelry store, one office outfitting store, one five-and-ten cent store, one news company, one tea and coffee house, one meat market, one candy store, and one phonograph store. The same questions were asked each firm in accordance with a specially prepared questionnaire. Findings. 1. Each of the business houses made much use of mechanical calculating devices. Each firm had at least one cash register, and some had as many as 10. Almost all firms had at least one adding machine, and the banks had from six to eight machines. The banks used bookkeeping machines. Computing scales were used in all grocery stores, all meat markets, all drug stores, and in all candy shops. In one grocery store two cheese cutters were operated on the computing plan. 54 Department of Superintendence 2. Many tabular devices were utilized in order to reduce the amount of tabulation. The banks used bond and interest tables. The news company used a "price" table to tell the price of the different magazines or combina- tions of magazines. The drug stores used "percentage solution" tables whereby it was possible to find the different amounts of a chemical to make a solution of a particular strength. The hardware had "screen" tables for finding the areas of screens of different lengths and widths, and discount tables for determining the cost price from the list price in wholesale cata- logues. The book stores used a "printer's" table for determining the num- ber of sheets per pound according to the quality. In fact, almost every firm used tabular devices of some kind. 3. Several "short-cut" methods were discovered. In computing interest at. six per cent when the interest book is not used, point off two places in the principal and multiply by half the number of months. In figuring the selling price on an article when there is a discount, the method is to subtract .the discount from 100, point off two places in the remainder and multiply .by the original price. A system of alligations was used by the druggists. 4. The goods were always marked to avoid complicated calculations. Firms. are now. avoiding marking in fractions and employ the unit of decimal system to a great extent. They also avoid the sliding scales and sell at so much ..per unit or multiple of this unit, or group of units. An illustration of the unit or group of units plan is: selling six bars of soap for 25 cents but charging 35 cents for eight bars. ,5. Many, of the managers insisted that neither their employees nor the purchasers did much actual calculation. In certain stores like restaurants, and ice cream stores where articles were sold in combinations the clerks learned the sale, price for the combination of articles: for example, they learned the regular price for a sandwich, coffee, and a piece of pie was 45 cents and did not have to add the price of the different items to get the total. The. managers estimated that only about 29 per cent of the buyers ever add the cost of different articles purchased to find the total amount purchased, and that only about 66.6 per cent of their customers ever check the change given them. ' 6. These business men insisted that the school needs to emphasize the foUr fundamentals. They felt there is great need for "mental" arithmetic. Some emphasized the need for training involving interest, discount, and the metric system. The particular needs emphasized by the business men de- pended largely on their type of business. 7. The business men agreed that the greatest number of mistakes made in the buyer's calculations involve making change. The buyer does not understand the "make-change" method and gets lost. 8. The outstanding conclusion drawn from this investigation is that little calculation in arithmetic was actually done. The Third Yearbook 55 STUDY NO. 15. Charters, W. W. Department store arithmetic. Study reported in Charter's Curriculum Construction. New York, Mac- millan, 1923. pp. 231-236. Problem. To determine what operations in arithmetic are important for salespeople. Method. Study consisted of analyzing the arithmetic involved in the. use of several thousand sales checks. In addition, multiplication and divi- sion 7337 charge checks' were examined ; in subtraction 4304 cash checks were analyzed. The analyses consisted of detailed accounts of the number of addends, the number of figures in the addends, the number of figures in the minuend when subtraction was involved, the number of times a par- ticular amount of money was offered in payment for a bill of goods, the number of places in the multiplicand and multiplier when multiplication was involved, the number of places in the dividend and divisor when divi- sion was involved, the size of the fractions encountered, etc. Findings. The reader is referred to Charter's book for a detailed sum- mary, but the main points are set forth in the paragraph quoted: "In this one vocation of department-store selling the chances are nine out of ten that no problem in addition will be more complicated than the addition of four-place addends ; no subtraction is used in making change, and if it were, only forty-five out of one hundred subtraction facts would be used ;, the chances are ninety-seven out of one hundred that in multiplication the multiplier will be 12 or less and the multiplicand three places or less; in fractions, only eleven denominators are found, all being under 10 except 10', 12, and 16, and by a simple device of using the thumb decimals disappear." STUDY NO. 16. Moore, Ernest C. Minimum course of,, study. Chapter I. Arithmetic, pp. 1-41, 1923. At the beginning of this chapter, three studies for the purpose of ascertain- ing the minimum essentials of arithmetic were reported. \ Study I. Problem. To find out how much arithmetic is used in everyday life. Method. A questionnaire was sent to men of various vocations asking them certain questions concerning their use of arithmetic. ' Typical ques- tions folloAv: Do you personally have occasion to add columns of 2, 3, 4, 5, v 6, or more figures in width ? Do you personally have occasion to compute simple interest, etc. Findings. From the 799 replies to the questionnaire, it was evident that the minimum essentials should include : ( 1 ) Addition — five addends, five figures in width; (2) multiplication — the multiplicand, five figures in width, the multiplier, four figures in width; (3) division — the dividend, five figures in width, the devisor, four figures in width ; (4) fractions — halves, thirds, fourths, fifths, eighths, with tenths; (5) decimals of three places; (6) simple interest; and (7) percentage. 56 DEPARTMENT OF SUPERINTENDENCE Study II. Problem. To find the amount of arithmetic used in everyday life with regard to subject-matter. Method. A second questionnaire, listing various topics in arithmetic, such as addition of fractions, subtraction of fractions, simple cash accounts, drawing to scale, etc., was sent to the persons mentioned above with the request that they relate to the children in the home all the uses made of arithmetic each day for ten consecutive days and have the children record such uses by placing an X on the question blank after the enumerated topic. Findings. On the basis of the 314 replies to this questionnaire, it was concluded that the following should be included in the list of minimum essentials: (1) Simple cash accounts, or family expense accounts; (2) cash, checks, or bills; (3) addition of fractions; (4) multiplication of fractions; (5) subtraction of fractions; (6) banking; and (7) division of fractions. Study III. Problem. To find out the amount of arithmetic needed by the employees in fifty business firms of Los Angeles. Method. The following questionnaire was sent to fifty leading busi- ness firms in that city: (1) How much arithmetic should young people know when they enter your employment? (2) In what arithmetic work do you find them weak or unsatisfactory? (3) What suggestions do you make that may assist in correcting mistakes? (4) So far as it comes to your attention, what work in arithmetic is being taught that is of little or no value in your business? Findings. From the 24 replies received, the following answers were obtained: (1) The arithmetic needed on entering the firms should include addition, multiplication, division, decimals, subtraction, percentage, and fractions; (2) The weaknesses manifested were in accuracy (mentioned by 18 of the 24 replying), decimals, addition, multiplication, fractions; (3) The suggestions made for correcting the mistakes were : accuracy, short- cuts, mental arithmetic, and "teach the why"; and (4) In response to what should be eliminated, the replies were: no topics, higher mathematics, algebra, all except the four fundamentals, decimals, interest, and discount. Although the above questionnaires are suggestive, the nature of some of the answers given suggest that they should not be taken too seriously, for they do not smack of serious thought. It may be noted that Study II, based upon actual usage instead of opinion, shows a much simpler demand. Tabulation of usage is now conceded to be the most effective procedure in checking needs. STUDY NO. 17. Hansen, Einar A. The arithmetic of salespersons' tally cards. 1924. Unpublished study, Department of Education, Univ. of Iowa. Problem. To determine what additions salespersons use in totaling their daily sales: 1. How many of the cards have but one sale listed; how many two, three, etc. 2. How many sales are totaled in each column of sales on the cards? The Third Yearbook 57 3. What is the size of the final sum as measured in terms of the number of digits involved? 4. How many columns of sales are added by the salesperson? Method. 1324 tally cards, the cards on which sales are entered for pur- poses of record, from four large downtown stores, a bookstore, a music store, a department store, and one selling athletic goods, besides one fairly large drygoods store on the west side of the city, were used. The frequencies of occurrence of the four characteristics listed above were tabulated. The findings are recorded in tabular and graphic form. Findings. 1. Twelve per cent of the total number of cards fall in the group having one or two sales. The curve falls rather regularly from this point to two and one per cent toward the latter end, where it rises again to six per cent for those having fifty or more sales. 2. The number of sales totaled in the various columns was determined partially by the type of card used, depending upon whether the fifty possible items were arranged in four, three, or two columns. Thirteen-item columns ranked first with fifteen per cent, 12 and 2 — item columns second, with eleven per cent each. The preponderance of 2-item columns was due in part to the number of additions of sub-totals into final totals. 3. The number of digits involved in the final totals ran in the following order: four digits, 73 per cent; three digits, 17 per cent; five digits, 10 per cent. Single entry cards were excluded from this tabulation. 4. The number of columns added varied first, with the type of card used ; second, in actual count, the one column leading with 44 per cent ; two columns having 27 per cent, three columns 10 per cent, and four columns 9 per cent. With the exception of the tally cards from the bookstore, the sales con- sidered are the records of cash sales only and therefore do not represent all the sales made by these salespersons for a single day. 2. General informational or reading uses of arithmetic- — The arithmetic needed by any individual for actual computation is obviously limited, and actual computational practice is being further curtailed by the rapid intro- duction of calculating machines. But the same limitations do not apply to number concepts in general and the ability to interpret the number concepts met with in general reading. The suggestion naturally follows that in many phases of arithmetic, manipulation should be omitted and the time devoted to informational concepts and the general understanding of busi- ness situations. In time, we may have some phases of arithmetic dealt with in an arithmetic reader with little or no figuring required. STUDY NO. 18. Camerer, Alice. "What should be the minimum in- formation about banking?" 17th Yearbook of the National Society for the Study of Education, Part I. Bloomington, 111., Public School Pub. Co., 1919. pp. 18-26. Problem. To discover what should be taught concerning banking. 58 Department of Superintendence Method. An analysis was made of 35 replies to a questionnaire sent to bank employees in the states of Illinois, Indiana, Iowa, Kentucky, Missouri, Nebraska, North Dakota, Oregon, and Texas, relative to the facts in bank- ing that all people in a community should know. A similar analysis was made of the replies sent in by parents of the pupils in the elementary school of the University of Iowa. The instructions were to mark important items "XX", less important items "X", and to cross out those of no importance. The items reported in the replies were arranged according to their im- portance as determined by the bank employers and comparison made with their importance as determined by the parents. The results are reported under the findings. Findings. The bank employees determined the following rankings on twenty of the fifty-five items, the most important being placed first : (1) How to write a check, (2) how and why to fill out a stub, (3) when a check should be cashed, (4) how to stop payment on a check, (5) how to sign your name when indorsing a check, (6) what to do if a check is lost, (7) how to indorse a note, (8) how to write a negotiable note, (9) how to indorse a check in full, (10) how to use a bank book, (11) how to make a deposit slip, (12) how to find interest, (13) importance and purpose of savings banks, (14) importance and purpose of commercial deposit banks, (15) how to use a promissory note, (16) responsibility of maker if note is lost, (17) certified checks, (18) how to open an account, (19) when notes are void, and (20) legal rate of interest. Parents of the children in the elementary school placed 15 of the 20 items named above in their list of the 20 most important items, although the order selected was not quite the same. STUDY NO. 19. William, L. W. "The mathematics needed in freshman chemistry." School Science and Mathematics, Volume 21, No. 7. October, 1921. (Reported in Charter's Curriculum Construc- tion.) Problem. To determine what mathematics is necessary to carry suc- cessfully a course in freshman chemistry in which the textbook was Noyes, A Textbook in Chemistry. Method. Study consisted of analyzing both the expository body and the problems of. the text, and recording all words or expressions which were distinctly mathematical or implied mathematics and the nature of the different operations involved. All exercises were worked out by the simplest methods, and both ' the operations and the quantities were tab- ulated.' Findings. See the original article or the account in Charter's Curric- ulum Construction for detailed summary. 1. In the body of the text, 124 mathematical concepts were used 1156 times, of which per cent and volume comprised nearly one half. The Third Yearbook 59 2. Seventy-three different denominate numbers were used," the most common of which w r ere : degree, gross, and liter. 3. Fifty-five different fractions were used, but the denominators in all but 43 of the fractions were less than 10. Of these 43 denominators, eight denominators were less than 100, but 24 ended in hundreds or thousands. 4. Complicated decimals were very frequent. The one-place, two-place, and three-place were most frequent, but one decimal had eight places. There were usually two integers and one or two places, but at times the decimals were more complicated. Oftentimes decimals were mixed with fractions. 5. Chemical equations occurred very often. There were nine cases of substitution in formulas and seventy-seven cases of ratio and proportion. 6. The arithmetic at times was complicated. In addition there were problems consisting of adding an integer and three decimals, two integers and three decimals, and three decimals. In subtraction the most difficult problem was taking a six-place decimal from another. In multiplication at times it was necessary to multiply a four-place number by a six-place number. Mixed decimals had to be multiplied together. Complicated decimals at times had to be divided by one another. Thus the arithmetic needed in chemistry was complicated, but the algebra simple. STUDY NO. 20. Gallaway, Mrs. T. T. Mathematics needed in a freshman course in clothing. Unpublished study reported in . Charter's Curriculum Construction, pp. 241-43, 1923. Problem. To determine what mathematics is necessary to carry success- fully a freshman college course in clothing in which the text used is Baldt's Clothes for Women. Method. Study consisted of analyzing both the expository body and the problem of the text, and recording all words or expressions which were distinctly mathematical or which implied mathematics, and listing the nature of the different operations involved. All of the . exercises were worked out by the simplest methods, and both the operations and the quantities were tabulated. Findings. See the account in Charter's, Curriculum Construction for a detailed summary. 1. It is necessary to understand 111 mathematical concepts correctly to read this text intelligently. Of these, 43 were geometrical terms, chiefly included in mensuration. All other terms were arithmetical. Chief among these terms were: line, equal, center, measure, width, length, and point. 2. Denominate numbers such as appear in the ordinary arithmetic oc- curred and in addition, the two units of measure head and skein. The most common units of measure were : inch, dollar, yard, head, and year. 60 Department of Superintendence 3. The most common fractions used were: halves, quarters, sixths, eighths, tenths, twelfths, and sixteenths. There were a few cases of decimal fractions. 4. The integers occurring were mostly one-place integers if expressed in words instead of figures. If the integers were expressed in figures, the majority of them had less than four figures and none of them had as many as five figures. 5. Roman numerals were used considerably, although none of them had more than two places. 6. In the chapters on drafting and the use of patterns much use was made of algebraic equations: for example, "A. E. equals one-sixth neck measure plus three-eighths inch." The solution of similar problems in- volved: (a) Translating the algebraic equation expressed in words into a symbolic numerical equation, (b) multiplying a whole or mixed number by a fraction, (c) subtracting a whole or a mixed number from a whole or mixed number, (d) adding a fraction to a whole or mixed number, (e) finding a fraction of a fraction, (f) reducing a mixed number to an im- proper fraction, (g) reducing an improper fraction to a whole or mixed number, (h) reducing yards to inches, (i) drawing a line of a given length, (j) reducing inches to yards, (k) division of whole numbers, mixed num- bers, or fractions, (1) drawing a line parallel to a given line, (m) draw- ing a line perpendicular to a given line, (n) drawing a circle or the arc of a circle, having given the center and radius, and (o) reading geometrical figures by means of letters representing points. STUDY NO. 2i. Smith, Nila B. "An investigation of the uses of arithmetic in the out-of-school life of first-grade children." Elementary School Journal, Volume 24, No. 8. April, 1924. pp. 621-23. Problem. To determine the type of arithmetic used by 500 first-grade pupils in the Detroit public schools in their out-of-school life. Method. The data were obtained by having 100 regular room teachers interview five selected children each morning for 25 consecutive days con- cerning their out-of-school activities and record facts concerning the arith- metic used in connection with these activities. The interviews during the first week were for the purpose of familiarizing the teachers and pupils with the technique of the investigation and were not in the data under consideration. In the investigation each teacher itemized daily for each pupil the content of the situation in which arithmetic was used, the num- bers involved, the operations involved, whether the operation was per- formed by the child or not, the classification of the activity according to types., and .the number of times the particular type of activity was used. Much use was made of code numbers as aids in tabulating the data. The Third Yearbook 61 Findings. The results are shown in these two tabular summaries: TABLE 2.— RELATIVE FREQUENCY WITH WHICH SITUATIONS INVOLVING ARITHMETIC OCCURRED Activity Per cent Transaction in stores 30.0 Games involving counting 18.0 Reading Roman numerals on clock 14 . Reading Arabic numerals in finding pages in a book 13 . Dividing food with playmates and pets (fractions) 6.0 Depositing money in and drawing money from toy banks 5.0 Playing store 3.0 Measuring distance 2.2 Using calendars 2.0 Running errands 1.2 Setting the table 1.2 Buying and selling tickets 1.1 Acting as newsboy 1.0 Measuring in sewing 1.0 Counting in rhymes and jingles .5 Reading house numbers .2 Investments (made for them) .1 Measuring in manual training .1 Measuring height , . .1 Measuring objects .1 Reading numbers on book in hall .1 Reading numbers on ticket .1 100.0 TABLE 3.— RELATIVE FREQUENCY OF ARITHMETICAL OPERATIONS Process Per cent Addition 35.0 Counting • 23 . Subtraction 20.0 Reading Arabic numbers 6.0 Measuring 5.5 Comparison 4.3 Reading Roman numerals 3.1 Multiplication ' 2.0 Division 1.1 100.0 STUDY NO. 22. Adams, H. W. The mathematics encountered in general reading of newspapers and periodicals. Unpublished Master's thesis. Department of Education, University of Chicago. Reviewed by Franklin K. Bobbitt in Elementarv School Journal, Volume 25, No. 2. October, 1924. pp. 133-143. Problem. To discover the mathematics employed in the news, special articles, editorials, advertisements, legal notices, market reports, sporting pages, etc., in one issue, 20 newspapers and magazines, widely read by the general public. Method. All pages of one issue of the following newspapers and maga- zines were analyzed : the Chicago Herald and Examiner, the Chicago Evening American, the St. Louis Post Dispatch, the Springfield, Missouri, Republican, the Springfield Leader, the Lebanon Rustic, the Pathfinder, the Furrow, the Dearborn Independent, the Springfield Laborer, the Cosmopolitan, the Woman's Home Companion, the American Magazine, 62 Department of Superintendence the Household, the Pictorial Review, the Woman's World, the Literary Digest, the Modern Priscilla, McCall's, and the National Geographic Magazine. Findings. 1. Dates were numerous, oftentimes giving the year, the month, the day, the hour, and the minute. 2. There were found 3378 street addresses in all sorts of combinations, e.g., 1918 Broadway, Department 909; and 4711 North Clark Street. 3. A total of 1713 telephone numbers were discovered. They were expressed in a great variety of ways- 4. Numbers varied from such expressions as "50 words" to "255,000,000 packages of breakfast food." In all there were 21,619 such expressions. Most of them had two-place numbers, but many had nine or more places. 5. Roman numerals occurred 148 times. They were used most frequently to designate the volumes of papers or books, chapters of books, or sections of books. Most of the Roman numerals were below a hundred. 6. References to United States money varied from one cent to $100,000,- 000,000. However, most of the amounts were less than $100. Reference to foreign money contained the following terms: German marks, Italian lire, Japanese yen, Indian rupees, English pounds, English three-pence, English guineas, and English half-crown. 7. Only six fractions out of 3000 had a denominator larger than 16. The largest denominator was 10,000,000. 8. Decimals without the use of United States money occurred frequently. The smallest decimal without the use of United States money was .000,012, and the largest one was .97. Most decimals had less than four places. 9. A great variety of percentages were found, e.g., 1%, 1%, 2, 2%, 3, 33-1/3, 99-44/100, 177, 700, 1,000, and 1,300. 10. Simple ratios were found in a variety of ways, e.g., 30 to 1, 1 in 1,000,000,000, a 5-3-3 ratio, fifty-fifty, etc. 11. Almost every conceivable unit of measure was used. The different units usually found in the following tables of measure were used: dry measure, liquid measure, linear measure, square measure, avoirdupois meas- ure, measure of time, measure for electricity, and various miscellaneous measures as degree, calorie, volt, cord, dozen, teaspoon, tablespoon, etc. 12. Few graphs were used. 13. There were numerous mathematical terms or expressions used as: cedar chest 40x18x18, 40 per cent surtax, 90 miles-an-hour-gale, a 12 horse-power engine, etc. 14. No arithmetic problems of the textbook type occurred, although mathematical materials were presented so as to enable the reader to solve almost any problem which might occur to him concerning the situation described. 15. There was virtually no reference to algebra, geometry, and trigo- nometry in the magazines analyzed. The Third Yearbook 63 III. According to What Criteria Should Drill be Organized? The committee has not succeeded in getting a summary of studies bear- ing upon this point. Kirby, Practice in the Case of School Children, Teach- ers College, Columbia University, 1913, showed the large values resulting from systematic drill. Thorndike has. given data bearing upon the same point. The trend of evidence is that in the intermediate grades, when drill is confined to processes that are useful and well understood, regular time spent upon systematic drill gives returns unequalled by any other type of procedure. The findings are in favor of systematic drill rather than incidental drill. Another finding, on which the evidence is not so satisfactory as one could wish, is that attempts to do number work in the lowest grades, par- ticularly in grades one and two where the processes are so frequently not comprehended by the children, results in the development of errors which become a strong handicap in later work. This will be noted more fully under Section IV of this report. The chief contribution under this section is a definite and complete plan worked out by F. B. Knight for organizing the drill work of the fifth and sixth grades. The summary of this report is presented first for the con-, venience of the reader. Dr. Knight has shown, in a masterful way, and much more thoroughly than present space permits to be shown, the surpris- ing inefficiency and haphazard procedure which exists in drill work; the possibility of formulating drill to meet definite specifications; and the necessity of systematic drill if automatic memory results are to be had and permanently maintained. Summary of Report on Organization of Drill Introduction: Two uses of drill: 1. Drill as a teaching instrument; 2. Drill to maintain skills after instruction is passed. The following is a list of processes to be drilled under Use 2, with suggested times at which drill on each process should begin. DRILL ORGANIZATION FOR MAINTAINING ACQUIRED SKILLS Specification I. Drill should be on the entire process. (a) Various forms or types (b) Frequency of each number combination in drill (c) Location of number combinations in all positions (d) Computation difficulties present (e) Illustrations from texts on appearance of unit skills in fraction drills (f) Comment on the distribution of practice among the several unit skills in- volved in the division of fractions (g) Drills can be built to specifications Specification II. Drill should come frequently in small amounts. (a) Distribution of time for drill on a single process (b) Two opposing theories of time distribution (c) The cold storage theory and its weaknesses (d) Illustrations from texts on time distribution of drill on division of fractions (e) Comments on the illustrations of fraction drill 64 Department of Superintendence Specification III. Each drill unit should be a mixed drill (a) Isolated and mixed fundamental types of organization (b) Illustrations of the isolation theory (c) Illustrations of the mixed fundamental theory (d) Examples in each process to contain calculated practice (e) Advantages of the mixed fundamental type of drill Specification IV. Drills should have time limits. (a) Speed a secondary consideration (b) Wrong types of time standards (c) Correct use of time standards involves product measures Specification V. Drills should have accuracy standards. (a) The shortcomings of the 100 per cent accuracy standard for a list of ex- amples (b) Time and accuracy standards combined (c) Three factors of time and accuracy standards (d) A definite technique described (e) Advantages of the suggested technique (f) Use of the progress chart (g) Cost of the suggested technique is justified Specification VI. Examples in a unit of drill should be in the order of difficult}'. (a) Range of difficulty should be commensurate with range of abilities (b) Easy examples should include all processes Specification VII. Drill units should include verbal problems. Problem solving should not be isolated. Specification VIII.. Drills should facilitate diagnosis. (a) Remedial work should be specific (b) Waste of indiscriminate drill (c) Suggested standards of technique facilitates remedial work (d) Hard aspects as well as total examples should be present in drills (e) Teacher's cue for remedial work Drill in Fifth and Sixth Grades For the purpose of facilitating discussion, let us assume by agreement that drill or practice has two functions: First, to aid in learning new pro- cesses, this use of drill is the instructional use; second, to help maintain a skill after the class and the teacher have gone to the trouble of building up a new skill. This second use of drill is drill for permanency. The present discussion is concerned with the organization of drill material in the fifth and sixth grades, the purposes of which are to keep permanent the skills that have been built. Listing the Skills The list of skills to be kept permanently up to standard as a result of drill is more or less arbitrary. It should be limited to useful processes as developed in Section II above. On this basis, one is justified in listing at least the following skills: Addition, subtraction, multiplication, and division of whole numbers. Addition, subtraction, and multiplication of small fractions; occasionally division of simple fractions Addition, subtraction, multiplication, and division of decimals as required in United States money The Third Yearbook 65 Computing simple areas and simple percentage The time of entrance of these skills into the course of study is conditioned by local requirements and should be largely governed by individual interests and local needs. Since the topic under discussion is drill for the permanence of holding skills, it goes without saying, that if the topics listed above are to be drilled to maintain skill in them at the time listed, they must have been taught previously to those times. Specifications of Proper Drill Organization The next pages contain a description of the organization of drill which serves the purpose of maintaining skill in a process after it has been taught and after the instructional aspect of the class work is no longer concerned with that skill but with some new process. Examples are given supporting the specifications for organization of drill which are believed to be superior. Examples of contrasting types of drill organization are on occasion inserted not for purposes of criticism but to make the specifications, present in the report, clear through contrast with opposite types. Specification I. Drill Should Be On the Entire Process (a) Various forms or types— After instruction on a process is completed, the drill given to it should practice in some calculated fashion all the various types of that process or all that have been taught. Such drill should con- tinue throughout the elementary school. By way of illustrations: Example A Example B Example C 345 768 427640 567 54 392769 879 879 4318 407 807 915043 215 423 All the drill in the addition of whole numbers given after the process of whole numbers has been mastered from the standpoint of instruction should not be like Example A above. There should be some drill upon examples like B and C, since the ability to neglect gaps in the column is an ability that does not come by magic, can just as well be practiced, and is a type which the demands of life often present. In the present drill services, we note a lack of presenting the pupil with addition examples of irregular outline. The list of the items which go to make the total process of addition, sub- traction, multiplication, or division of whole numbers is a list on which there is as yet no common agreement. Following this paragraph, there are included the two analyses of processes, subtraction of whole numbers and division of fractions. Some such analyses as these should be used to check drill provisions. Drill provisions are weak to the extent that they slight important units of skill. They are also weak to the extent that they pro- 66 Department of Superintendence vide unnecessary over-practice or wasteful practice upon units of skill which possess no particular difficulty. Obviously, drill is good to the extent that it provides practice upon each type of the total skill in proportion to the difficulty of each type. Analysis of Subtraction of Whole Numbers in Terms of Learning Process I. As to Form of Stating Example: A. Number written with figures 1. Indicated subtraction, as 6 — 2 = 4 Additive subtraction, as 6 — ? = 4 6 2. Column subtraction, as 2 Unit of Skill Number 1 la II. 3. Words used 9 "minus" or "less" or "take from", as 9 minus 4 B. Numbers written with words 1. Indicated subtraction, as Four — two 2. Words used, as Four "minus", "less", "take from" C. Pictures used 1. Indicated subtraction, as 5 5 - S 5 2. Words used, as 5 5 5 minus Q As to Procedure: A. No borrowing or carrying 1. One digit, as number — 1 digit number. 6 — 4 = 2. Two digit number — 1 digit number When difference is 1 digit, as 12 2a 3 4 5 When difference is 2 digits, as 18 -6 10 3. Two digit numbers less two digits, as 48 —24 4. When the remainder is zero, as 5. More than two digit number, as 6. Zero in subtrahend, as 463 —102 6 —6 16 —6 483 —121 463 —120 11 12 13 14 The Third Yearbook 67 7. Zero in both subtrahend and minuend, as 40 —20 15 S. Gaps in column, as 483 —21 16 B. Borrowing or carrying: 1. Two digit number — one digit, as 46 —6 17 2. Two digit number — one digit Zero in minuend, as 40 —6 18 3. More than two digit number (a) Borrowing units column, as 423 —117 19 (b) Borrowing other than units column, as 463 —178 20 (c) Borrowing two consecutive columns, as 482 —197 21 (d) Borrowing two not consecutive columns, as 4236 —1718 22 4. Borrowing zero in minuend, as 420 —128 23 5. Borrowing two zeros in minuend, as 4200 —1267 24 6. Borrowing more than 2 zeros, as 43000 —12675 25 7. Borrowing zero in both not final, as 4306 —1204 26 8. Borrowing zero final, as 248000 —162000 27 9. Subtraction unequal number digits, as (a) Where zero is subtracted from last left number, as 4862 —732 28 68 Department of Superintendence (b) Where zero is not subtracted, as 1467 —835 29 (c) Last left numbers are equal, as 635 —604 30 C. Ability to check 31 D. Ability to copy for work as 43728 — 39162 = 43728 —39162 32 Analysis of Division of Fractions in Terms of the Learning Process Unit of Skill As to the Form of Stating the Example Number A. Fractions written in figures 1. Indicated divisions, as 3/8-4-1/2 1 2. Words "divided by," as 4/5 divided by 1/3 2 3. Complex fractions, as 3/5 1/3 3 4. Division indicated by parentheses, as 1/4)1/12, or 1/2)4/5 4 B. Fractions written with words 1. Indicated division, as two sevenths-4-one sixth 5 2. Words "divided by" used, as one eighth divided by three fifths ' 6 I. As to Procedure A. Nature of terms — Expression of all terms as fractions 1. Unit fraction-4-unit fraction, as 1/8-4-1/9 7 2. Unit fraction-4-other proper fraction, 1/3-4-2/5 8 3. Unit fraction-4-improper fraction, as 1/9-4-7/4 9 4. Unit fraction-4-mixed number, as 1/2-4-2 3/5 10 5. Unit fraction-:-whole number, as 1/4-4-7 11 6. Other proper fraction-4-unit fraction, as 3/4-4-1/7 12 7. Other proper fraction-f-other proper fractions, as 3/4-4-2/7 13 S. Other proper fraction-4-improper' fraction, as 3/8-4-4/3 14 9. Other proper fraction-4-mixed number, as 7/10-4-1 2/3 15 10. Other proper fraction-4-whole number, as 4/9-4-8 16 11. Improper fraction-4-unit fraction, as 7/5-4-1/3 17 12. Improper fraction-4-other proper fraction, as 5/4-4-7/8 18 13. Improper fraction-f-improper fraction, as 5/2-4-8/3 19 14. Improper fraction-4-mixed number, as 9/7-4-1 1/5 20 15. Improper fraction-4-whole number, as 7/5-4-3 21 16. Mixed number-4-unit fraction, as 1 5/6-4-1/2 22 17. Mixed number-4-other proper fraction, as 3 8/9-4-4/5 23 18. Mixed number-4-improper fraction, as 5 2/5-4-13/4' 24 19. Mixed number-4-mixed number, as 4 2/9-4-7 1/3 25 The Third Yearbook 69 20. Mixed numbers-whole number, as 3 7/8S-5 26 21. Whole numbers-unit fraction, as 8-=-l/7 27 22. Whole number-mother proper fraction, as 9S-3/5 28 23. Whole numbers-improper fraction, as 6S-7/3 29 24. Whole numbers- mixed number, as 4S-3 5/8 30 25. Whole numbers-larger whole number, as 7S-15 31 B. Change of S- to X 32 C. Inversion of divisor 33 D. Cancellation 1. No cancellation possible, as 3/5X4/7 34 2. Single cancellation a. One number, a factor of the other, as 3/5X7/9 35 b. Two numbers with a common factor, as 6/7X5/S 36 3. Double cancellation a. In each case one number a factor of the other, as 3/4X8/9 37 b. In each case two numbers with a common factor, as 8/9 X 15/18 38 4. Rec. One case of each type of cancellation, as 3/10S-8/9 39 5. Reduction cancellation a. One number a factor of the other, as 2/8 40 b. Two numbers with a common factor, as 8/10 41 6. Incomplete or continued cancellation a. One number a factor of the other, 8/9S-1 1/3S-1 1/15 42 b. Two numbers with a common factor, 1 37/40s-2 3/16S-1 19/25 43 E. Multiplication , 1. Neither factor unity in numerator or denominator, as 3/8X5/6 44 2. One factor unity in numerator, as 1/7X5/8 45 3. One factor unity in denominator, as 3/5X4 . 46 4. Both factors unity in numerator, as 1/4X1/6 47 5. Both factors unity in denominator, as 10X3/5 48 F. Analysis of quotient 1. Quotient a whole number a. Cancellation complete — irreducible as 7/1 49 b. Cancellation incomplete — reducible as 15/3 50 2. Quotient a proper fraction a. Cancellation complete — irreducible as 2/3 51 b. Cancellation incomplete — reducible (1) Numerator a factor of denominator, as 3/15 52 (2) Numerator and denominator having a common factor as 8/10 53 3. Quotient an improper fraction — reduce to mixed number a. Cancellation complete — fraction irreducible, as 13/4 54 b. Cancellation incomplete — fraction reducible, as 21/6 5 5 (b) Frequency of each number combination in drill— A consideration of the nature of drill material second only in importance to the calculated appearance of every type of example is the matter of the frequency with which each number combination occurs. < 70 Department of Superintendence FIG. i - o \ 2 3 4- 5 fo 7 & 9 T 1 2GI- ?H2 aw 1 36 146 I7| &o lo6 73 71 I 258 2 84 46 3H 149 142 91 39 57 50 75 755 3 o 35 49 48 36 27 152|124 569 9 Ol 33 30 28 32 20 26 2; C 19 »3 301 c