Experin I II ourses in Secondary School Mathematics BY RALEIGH SCHORLING The Lincoln School of Teachers College, New York City TABLE OF CONTENTS PAGE INTRODUCTION PART I. IMPORTANT STATEMENTS AND CONCLUSIONS..... PART II. GENERAL DISCUSSION PART III. THE WORK OF THE INDIVIDUAL SCHOOLS.. 2. The University of Chicago High School, Chicago.. 3. The High School of Commerce, New York City. 4. The English High School, Boston.... 1 2 5 18 18 26 33 40 5. The Ethical Culture School, New York City... 44 6. The Horace Mann School for Girls, New York City. 7. The Lincoln School of Teachers College, New York City...... 8. The High School of the New Mexico Normal University, Las 69 9. The Parker High School, Chicago.. 10. The Stuyvesant High School, New York City.. 11. The University High School of the University of Missouri, 12. The University High School, Minneapolis.... 14. The Washington Junior High School, Rochester. Purpose and content of this report. The purpose of this study is to make generally available the experience of teachers conducting experimental courses in mathematics. Teachers will desire to know what is done by other teachers who have had greater freedom in the choice of materials and methods than is feasible in the typical public school. The value of this report is enhanced by the fact that the teachers in these schools, considered as a group, have wide experience, sound scholarship in mathematics, and unusual professional training. The material includes a detailed description of one or more courses in each of fifteen schools. Not all schools designated as experimental schools feel that they have made a real contribution to the teaching of mathematics. The search for new materials by the National Committee has been widespread. It is reasonably certain that no work considered unique by the persons doing it has been omitted. Form of the report. The report is presented in three parts. Part I aims to meet the needs of the reader who is interested only in a general survey of the important conclusions reached, but who does not desire to examine the evidence given in later parts of the report. Part II is for the reader who may wish to consider illustrations without going into the full details which constitute the basis for the statements in Part I. Part III is for the reader who has a genuine interest in considering more carefully some of the problems involved in improving the mathematics instruction of secondary schools. This constitutes the source material of the report. A list of names together with addresses is given on page 277 to whom credit is due for furnishing the facts. Presumably, those interested in a particular phase of the problem may get further information by writing directly to the individual concerned. Part I-Important Statements and Conclusions 1. A considerable number of schools in the United States claim to have undertaken some experimental work. Of these very few (probably not over ten) feel that they have made a significant contribution to the reorganization of secondary school mathematics. 2. There is very little, if any, work in mathematics that is experimental in a scientific sense. In general, the word "experimental" is used in a vague sense to indicate the use of materials or methods differing from those in current practice. 3. There are public school systems in which the administrators believe it highly desirable to incorporate experimentation as a regular and systematized feature of their programs of education. These officials believe that directed experimentation contributes to the efficiency of a school system. 4. The teachers of mathematics in the schools represented in this report constitute a group unusually well trained in mathematics and in professional courses. 5. Most of the efforts to improve mathematics at present are directed toward grades seven, eight and nine. Particular attention. has been given to the mathematics of the ninth school year. There are relatively few innovations under way in the later years of the senior high school. 6. No general agreement as to the details of a reorganization program is found in the practice of these teachers. There is an especially wide difference of opinion concerning the grade placement of materials of instruction. 7. In spite of a wide divergence in both theory and practice there is a marked agreement concerning important issues. These may well serve as a basis for a number of practical and constructive steps in completely reshaping the teaching of mathematics in American secondary schools. These generally accepted principles will be stated in the paragraphs that follow. 8. There is dissatisfaction with the results achieved in the arithmetic of grades seven and eight. It is felt that the time and money spent should result in (1) a higher degree of accuracy and speed in applying fundamental principles to fractions (common and decimal), (2) greater skill in dealing with percentage relations, and (3) more significant information about common business forms. 9. The pupils of the seventh and eighth grades lack the maturity and the social experience for much of the social-economic arithmetic that now is taught in these grades. Mortgages, stocks and bonds, bank discount, cost of plastering and the like, are in general not vital problems to twelve and thirteen year old girls and boys. The time spent on the application of arithmetic to business may well be decreased in the early years and the work continued later so as to capitalize increasing maturity, social experience, and mathematical power. IO. The mathematics of the early high school years must be considered worth while by the student on the basis of what it does for him here and now. His experience in other school subjects and life out of school should begin to mean more to him because he is taking a course in secondary mathematics. II. The algebra of the ninth school year as found in current practice is characterized by an indefensible amount of mere manipulation of symbols. The thinking processes involved are far removed from the way children get mathematical meanings. Spending from four to six weeks on factoring and special products, three weeks on "apartment house" fractions and all similar abstract and questionable units of work must cease. It is significant that everyone of the schools herein represented is concerned with the problem of modifying the mathematics of the ninth grade. 12. Ninth grade algebra has delayed to later courses the teaching of much material that is relatively simple, worth while and interesting to children. Graphic, statistical and trigonometric methods of solving problems are illustrations of these materials. So seldom are they employed in introductory courses that most high school graduates never learn to use these important tools. 13. The teaching of demonstrative geometry is recognized as one of the most serious problems. There has been much dissatisfaction with results generally secured. The teachers of mathematics in these schools are still convinced that the technique of thinking used in analysis (the method of attack in problem solving in demonstrative geometry) is of vital importance. But they also know that plane geometry was originally developed by and for adult philosophers. Current practice which devotes the seventh and eighth grades to arithmetic and the ninth grade solely to algebra, plunges the pupil into demonstrative geometry with inadequate preparation. The solutions. proposed vary widely but they indicate a clear recognition of the need for changing current practice. 14. The course for grades seven, eight, and nine should include arithmetic, intuitive geometry, algebra, and numerical trigonometry. Some schools advocate the addition also of a unit of demonstrative geometry. The important fact is that the courses for these years are to be enriched by new materials through the use of intuitive geometry and trigonometry. 15. It is possible and desirable to ignore preparation for college entrance requirements. These schools have an excellent record in the matter of preparing pupils for college, but they do not permit college requirements to shape their courses much before the twelfth school year. Part II-General Discussion This part of the report aims to amplify and to furnish illustrations for the corresponding statements in Part I. 1. Only a Few Experimental Schools Have Made a Systematic Attack on the Mathematics Problem. At the beginning of this study, a list of schools which were giving courses more or less experimental in nature was prepared. It was desired at the outset to include every institution in which the teacher (or teachers) of mathematics felt that their experiment had progressed sufficiently for other teachers to profit by a detailed account. On this basis the list of experimental schools to be included was reduced to a brief one. An experimental school must of necessity confine its constructive efforts to relatively few problems. Moreover, such schools are sometimes more concerned with becoming exponents of some philosophy of education or some educational ideal than in attacking a definite problem in a specific field. At any rate, there were very few experimental schools in which the teachers thought that they were teaching mathematics by other than conventional methods. Furthermore, it became clear that the report could not be complete without describing courses that are being taught in schools not generally considered experimental, as for example, Stuyvesant High School (New York City), East High School (Rochester, N. Y.), and the Cass Technical High School (Detroit, Mich.). The National Committee through its numerous cooperating organizations has solicited accounts of experiments in all sections of the country. The report includes a description of experimental courses in fifteen different schools. 2. Experimental Work in Mathematics Not Scientific. The word "experimental" is ordinarily used in a vague sense to indicate the use of materials or methods differing from those in current practice. But there is seldom an attempt to define the problem with precision or to measure the results; and rarely does anyone know when the experiment ends. |