The place of the elementary calculus in the senior high-school mathematics : and suggestions for a modern presentation of the subject

Preface

Even a little less than a decade ago, the trained teacher of high-school mathematics had the pleasure of teaching real mathematics as found in such subjects as trigonometry, solid and plane geometry, and advanced and elementary algebra. It is doubtless true that this kind of work was inclining too much toward formalism and that the needs of the learner were not being given sufficient attention. The strong reaction against formalism in general in education during the decade just past made the position of any genuine mathematics in our high schools very precarious. This threatened elimination of all real mathematics in our high schools was viewed with regret by those of us who had had the pleasure of starting many a pupil with a vigorous mind upon his mathematical way; and a number of prominent educators who are especially interested in the teaching of mathematics have been working toward a readjustment of the mathematics curriculum so that it will fit into and fulfill its mission in the changed conditions as found in our schools to-day.

The following study is the author's contribution toward this attempted readjustment. The kind of mathematics that should be taught in the first six school years has been definitely settled; the main points of the work in mathematics that should be included in the junior high-school period have been agreed upon; and the author hopes that his study will be of some value in helping us to formulate the content of the mathematics curriculum of the senior high school.

The author takes pleasure in acknowledging his indebtedness to Professor David Eugene Smith for his wise and constructive criticism and helpful guidance in the preparation of this study.

Noah Bryan Rosenberger

Acting Head of Department of Mathematics, Dickinson College, Carlisle, Pa.

July i, 1921

Table Of Contents

Page

Introduction i

I. A Study of the Status of Mathematics in the Schools Abroad

Introduction

A study of the nature of the courses in mathematics found from time to time in our high schools will show marked changes in the kind and extent of the topics included in these courses. These changes will be indicated in a general manner in the discussion which follows. In colonial times, in the schools corresponding approximately to our high schools, the arithmetic consisted of operations with integral numbers and of some work in fractions and in proportion. A few decades later, when our high schools first began to make their appearance, arithmetic had become more difficult and included such topics as stocks and exchange, circulating decimals, alligation, cube root, and permutations and combinations. Then it was realized that it would be better not to teach so much of the difficult parts of arithmetic but more of the simpler parts of some of the higher branches in mathematics. Hence, during the latter part of the nineteenth century, algebra and geometry became a part of the high-school mathematics. 1 These were followed a number of years later by the introduction of trigonometry.

In our present decade there is taking place an extended reorganization in our school curricula. One of the phases of this reorganization, namely, the junior high-school movement, has caused a radical change in the content of the mathematics curriculum. It has brought into the courses in mathematics in the seventh, eighth, and ninth school years the simpler parts of geometry, algebra, and trigonometry. This makes necessary an entire rearrangement of the courses in mathematics in the tenth, eleventh, and twelfth school years, or the senior high-school period. Such subjects as demonstrative geometry, algebra, and trigonometry are assured of a place in the mathematics of the senior high school. However, these subjects will not now be sufficient to meet the needs of pupils who wish to elect courses in mathematics in each of these three years. These needs will probably be met, as similar ones have been in the past, by the introduction of the simpler parts of some of the higher branches

1 Paul Monroe (ed.), A Cyclopedia of Education, Vol. 3, pp. 263 and 267, New York, 1912; Florian Cajori, The Teaching and History of Mathematics in the United States, p. 9, Washington, 1890.

in mathematics ; and it is the aim of this study to show that among these should be included the simpler parts of the calculus.

GENERAL STEPS IN THE ARGUMENT

I. The first topic will be a discussion based upon a study of the status of mathematics in foreign schools corresponding to our high schools. This will show that in many countries the calculus is taught in these schools before the end of the year which corresponds to our twelfth school year. This study will also give us an outside standard whereby to judge the status of mathematics in our schools.

II. The second topic will be a discussion of the trend of mathematics in our public school system. This will show that there has been created an opportunity for including the elementary calculus in the senior high-school mathematics. The question then arises, since there is an opportunity for doing this, is the subject of sufficient importance?

III. The third topic will be a discussion of the important position occupied by the calculus in the mathematics structure. This will show that the calculus is one of the most important aids in applied mathematics and contributes largely to the field of pure mathematics. The conclusion will be drawn that, because of its importance, senior high-school pupils who are interested in mathematics should be given an opportunity to become acquainted with the elementary calculus, provided it is not too difficult.

IV. The fourth topic will give a historical survey of the natural growth of the calculus in the development of mathematics. This will show that a first course in the elementary calculus is well within the mental grasp of pupils in mathematics in the senior high school. This historical survey will also suggest suitable methods of teaching the elementary calculus to younger pupils.

V. The fifth topic will give some of the features of various textbooks on the elementary calculus that have been written for beginners. It will show that a number of professors and able teachers of mathematics do not consider the elementary calculus too difficult for younger pupils. This comparison of textbooks will also be of value in the selection of methods of presentation of the elementary calculus.

VI. The sixth topic will indicate the trend of American education

Introduction 3

in general. It will show that among many people educational activity does not cease at the close of their public school life. Of course these later mental pursuits follow many different channels. A person working in the electrical engineering field will be likely to take up studies along the lines of electricity and engineering. There are many such fields in which a further study of the calculus will be an advantage. The course in the elementary calculus in the senior high school will make it possible for the pupil who has taken it to continue further his study of the calculus without the aid of a teacher.

VII. The seventh topic will give suggestions for a modern presentation of the elementary calculus. These suggestions will show how to simplify the subject, make the topics definite, and lead the pupils to see the theory as a related whole.

Among the sources from which some of these suggestions have been derived is the historical survey of the calculus in the development of mathematics. A comparison of textbooks on the elementary calculus which have been written for beginners has also furnished a means for finding out some of the most promising methods.

An additional source for these suggestions is the author's own experience. For several years he gave a course in advanced algebra to the scientific group in the public high school of a third-class city in which was included a considerable amount of work in differentiation. This work in differentiation included the greater part of the course in the differential calculus for use in the senior high school as it is outlined in the discussion under this section. His experience in teaching the calculus to classes in college has enabled him to determine the explanations necessary to bridge over the gap between the student's understanding and the usual method of presenting the calculus in college. The general training that he received in teaching pupils of a different race and language in the Orient has also been an aid to him in making troublesome topics clear to American pupils.

The author's graduate studies in pure mathematics have furnished him with a criterion for developing the course in the elementary calculus from the standpoint of mathematical rigor; that is, the deductions made throughout the course are sound, although their proofs may not be given. On the practical side, he has had the benefit derived from the study of the mathematics included in a college course in electrical engineering. This benefit has been increased by his association with managers of a large steel plant and of a large electrical firm.

The author is also indebted for suggestions to co-workers at Teachers College to whom he had submitted his manuscript. 2 He also profited by the Round Table Conference in Mathematics where he presented a part of this manuscript. 3

2 Among these are Messrs. M. A. Norgaard, assistant professor of mathematics, Grinnell College; J. R. Clark, editor of The Mathematics Teacher; W. W. Rankin, professor of mathematics, Agnes Scott College, Georgia; R. Beatley, of the department of mathematics of the Horace Mann School for Boys; and Misses H. Calkins, of the faculty of mathematics of Knox College; P. Wiggins, of the Los Angeles City Schools; T. Schlierholz, of the Ben Blewett Junior High School, St. Louis; R. S. Nichols, of the Vail-Deane School; M. A. Weber, of the Saginaw East Side High School, Michigan.

3 Round Table Conference, Department of Mathematics, Teachers College, Columbia University, March 19, 1921.

I

A STUDY OF THE STATUS OF MATHEMATICS IN THE

SCHOOLS ABROAD THAT CORRESPOND TO OUR

HIGH SCHOOLS

AS INDICATED BY THEIR TEXTBOOKS AND SCHOOLS

The status of mathematics in the schools abroad that correspond to our high schools will be indicated principally by a discussion on the extent of work covered by these schools in the subjects of analytic geometry and the calculus. A comparison of the leading European textbooks on the elementary calculus shows that this subject is taught to younger pupils than is customary in our American schools. (A more extended discussion on a comparison of these textbooks is given in Section V.) The German textbooks on the elementary calculus which have been written for use in the Prussian secondary schools adhere rather closely to the scientific method. In France, the elementary calculus forms a part of their course in algebra. The French stress generalization rather than application. However, the English authors of this type of book have made a radical attempt to treat the subject from the point of view of the young student. The author of one of these recent English textbooks, who is himself a teacher of mathematics for engineering students, says in the preface, "Fortunately, the calculus is now beginning to be taught to boys of fifteen or sixteen in day and evening technical schools, and it will grow up with them in the same way as their trigonometry and arithmetic have done." l

In the Manchester Grammar School (England) the curriculum for the Mathematical Sixth includes algebra, trigonometry, analytic geometry, and differential and integral calculus. This work corresponds to the work in mathematics given in the freshmen and sophomore years in American colleges. 2

In the Lycees (France), the course in secondary instruction covers a period of seven years and is coordinate with the preceding pri-

1 John Stoney, An Introduction to the Differential and Integral Calculus. (Junior Day Technical School and Municipal Technical School, Smethwick, London.)

2 Classical and Science Lists of The Manchester Grammar School, for the Midsummer, IQ20. Furnished by Dr. I. L. Kandel, of Teachers College, Columbia University.

mary instruction which covers a period of four years. 3 The period of secondary instruction is divided into two cycles ; the first cycle consists of four years and the second one of three years. In the first year (which corresponds to our eleventh school year) of this second cycle, the course in algebra for the Latin- Sciences and the Sciences- Langues vivantes sections includes the following work in the calculus : the notion of a derivative, geometric signification of the derivative, the sense of the variation indicated by the sign of the derivative, applications to very simple numerical examples and in particular to functions in preceding studies. In the next year in these two sections, the calculus is extended and includes the calculation of derivatives of the simple functions and a study of rectilinear motion by means of the theory of derivatives. In the last year, the work in the classes in mathematics includes the derivative of a sum, product, quotient, of the square root of a function, of sin x, of cos x, of tan x, and of cotg x, the topics of maxima and minima, and of the area of a curve regarded as a function of the abscissa. The calculus in the special class in mathematics is equivalent to that of our usual college course in the subject.

The reform movement in mathematics in the Oberstufe of Germany had the function concept as its central idea. 4 The following divisions into groups shows how the schools reacted toward this idea.

a. Those schools which declined to adopt the idea of the function concept.

b. Those schools which believed in a late discrete development of the function idea: without the infinitesimal calculus; with the differential calculus; with the differential and integral calculus.

c. Those schools which believed in a gradual, continuous development of the function concept: without the infinitesimal calculus; with the differential calculus in the Prima (the last year) ; with the differential and integral calculus already in the Obersekunda (the second half of the year preceding the last year).

The nature of the work in the calculus in the hbheren Schulen in Germany is given under the next topic.

3 Delalain Freres (ed.). Plan D' Etudes et Programmes D'Enseignement dans les Lycees et Colleges de Garcons, p. xviii, Paris, 1907-1908. F. E. Farrington, French Secondary Schools, New York, 191 o.

Status of Mathematics in Schools Abroad 7

AS INDICATED BY THE COURSES IN ANALYTIC GEOMETRY AND THE CALCULUS IN DIFFERENT COUNTRIES

This discussion is based upon the reports of the International Commission on the Teaching of Mathematics. This Commission owes its origin to the International Congress of Mathematicians at Rome in 1908. Thus far eighteen countries have made reports. These are so extensive that they cover over 12,000 pages. 5

The work in these two courses, analytic geometry and the calculus, in the foreign schools for the years that correspond to our tenth, eleventh, and twelfth school years will now be considered. The material for this comparison has been selected from the United States Bureau of Education Bulletin No. 45, which gives in systematic form that part of the report of the International Commission on the Teaching of Mathematics that relates to curricula in mathematics. 6

THE YEAR CORRESPONDING TO OUR TENTH SCHOOL YEAR

The ages of the pupils in this school year (10) are approximately fifteen to sixteen years. In this year in the Realschule of Russia, both plane analytics and the elements of the infinitesimal calculus are included in the course in mathematics. In analytic geometry, the principal theorems are derived by means of rectangular coordinates, and the equations for the circle, ellipse, parabola, and hyperbola are derived both in rectangular and polar coordinates. Their infinitesimal calculus is of the nature of an introductory course in the calculus. The simpler formulas for the derivative are taught. The work also includes the topics of maxima and minima, and indefinite and definite integrals.

In this year (10) the analytic geometry is also taught in the Realgymnasium and Oberrealschule of Germany.

THE YEAR CORRESPONDING TO OUR ELEVENTH SCHOOL YEAR

The ages of the pupils in this school year (11) are approximately sixteen to seventeen years. An introductory course in analytic

5 Raymond Clare Archibald, The Training of Teachers of Mathematics, United States Bureau of Education Bulletin, No. 27, 191 7.

8 J. C. Brown, Curricula in Mathematics, United States Bureau of Education Bulletin, No. 45, 1914.

geometry is given in this year (n) in the Gymnasium in Austria, and a more difficult course in the Realschule. In the latter school, easy differential and integral calculus are introduced in order to simplify or make more intensive the knowledge of physics.

In this year (n) in the Oberrealschule in Germany, courses in analytic geometry and in the differential calculus are included in the work in mathematics, but the calculus is not taught in this year in the Gymnasium and the Realgymnasium. A course in analytic geometry is also included in this year in the curricula of the three main schools of Austria, of the Lycees of France, and of the Realschule and Gymnasium of Hungary, and of the Realgymnasium of Sweden.

THE YEAR CORRESPONDING TO OUR TWELFTH SCHOOL YEAR

The ages of the pupils in this school year (12) are approximately seventeen to eighteen years. In this year, analytic geometry is taught in the Gymnasium schools of Holland, and of Hungary. Also, in this school year which corresponds to our twelfth school year, both analytic geometry and the calculus are taught in Austria, Belgium, Denmark, England, France, Germany, Roumania, Sweden, and Switzerland. The work in these two subjects ranges from that of a simple introduction to a course equivalent to our first course in college.

Thus, it is evident that the United States is far behind other countries in the work that our high schools do in mathematics. 7 That this difference has not appeared more prominently heretofore in such fields as engineering is partly due to the fact that our larger quantity of superior native ability has been able to compensate this difference. Owing to the rapid expansion of all fields making use of mathematics, however, and to the stronger competition resulting from the closer relations of the nations of the world both as to the shortening of time in transportation and as to political interests, the need of more men with trained mathematical minds cannot be met by relying upon native ability alone but by giving this native

7 Even the conditions in our own colleges show that the calculus should be placed earlier in the curriculum since frequently engineering students are asked to apply its principles in physics and mechanics before these principles have been developed in tlrregular calculus course.

Status of Mathematics in Schools Abroad 9

ability mathematical training of the most careful sort. To give this training in a satisfactory manner, it must be begun in the senior high-school period. 8

The main reason that pupils in the schools abroad are able to accomplish this work in mathematics which throws them about two years ahead of our American pupils by the end of the twelfth school year is because of the more profitable arrangement of the courses in the mathematics curriculum in their schools. A more profitable arrangement has already been made in the mathematics curriculum of the American standard junior high school, and an attempt is now being made to do a similar thing for the mathematics curriculum of our senior high school. In the junior high school, the propaedeutic courses in algebra, geometry, trigonometry, and graphs lay the foundations for the senior high-school courses in additional algebra, demonstrative geometry, additional trigonometry, and the introductory work in analytic geometry. Thus, the foundation is laid for an introductory course in the elementary calculus in the twelfth school year. A scheme such as this would furnish an opportunity to our pupils who have the inclination and the native ability to compete favorably with the highly trained pupils abroad, and would aid us as a nation in maintaining a foremost place in the fields of applied science and of engineering.

8 "If the student who omits the mathematical courses has need of them later, it is almost invariably more difficult and it is frequently impossible for him to obtain the training in which he is deficient. In the case of a considerable number of alternative subjects a proper amount of reading in spare hours at a more mature age will ordinarily furnish him the approximate equivalent to what he would have obtained in the way of information in a high-school course in the same subject. It is not, however, possible to make up deficiencies in mathematical training in so simple a fashion. It requires systematic work under a competent teacher to master properly the technique of the « subject and any break in the continuity of the work is a handicap for which increased maturity rarely compensates. Moreover, when the individual discovers his need for further mathematical training, it is usually difficult for him to find the time from his other activities for systematic work in elementary mathematics." — Elective Courses in Mathematics for Secondary Schools, a preliminary report by The National Committee on Mathematical Requirements.

II The Trend Of Mathematics In Our Public School

SYSTEM

A REASON FOR THE REORGANIZATION IN THE ELEMENTARY-HIGH- SCHOOL PERIOD

The mathematics curriculum of the first nine school years has already adapted itself to the changed conditions due to the reorganization in the elementary-high-school period. "One of the main reasons for taking two years from the elementary-school period and adding them to the high-school period, thus dividing the elementary-high-school period of twelve years into a period of six years of elementary school and six years of high school, and then subdividing the high-school period into three years of junior high school and three years of senior high school, was to make possible a better adjustment of studies and school methods in the three years now coming to be quite commonly included in the junior high school." 1

JUNIOR HIGH-SCHOOL MATHEMATICS

The nature of the mathematics in the three years of the junior high school can probably be most simply illustrated by an outline of the contents of a suitable set of textbooks on junior high-school mathematics. There are a number of such sets of textbooks on the market. The following is the outline of the contents of one of these :

First Year

First half. Arithmetic.

Arithmetic of the home, the store, the farm; arithmetic of industry; arithmetic of the bank; material for daily drill.

Second half. Geometry.

Geometry of form; geometry of size; geometry of position; supplementary work; tables for reference.

Second Year

First half. Algebra.

The formula; the equation; the graph; negative numbers; algebraic operations; further uses of algebra.

1 United States Bureau of Education, Secondary School Circular, No. 6, July, 1920. Statement by Commissioner P. P. Claxton.

Trend of Mathematics 1 1

Second half. Arithmetic.

Arithmetic of trade, of transportation, of industry, of building; arithmetic of the bank, of corporations; arithmetic of home life, of the farm, of community life, of civic life; arithmetic of investments, of mensuration; material for daily drill; tables for reference.

Third Year

First part. Algebra.

The chief uses of algebra; addition and subtraction; multiplication; division; fractions; simple equations; quadratic equations; general review.

Second part. Trigonometry.

Functions of angles, with applications; trigonometric tables, with applications.

Third part. Demonstrative geometry.

Meaning of geometry; triangles; parallel lines; parallelograms; angle sums; areas; constructions; loci; review; table for reference. 2

According to the arrangement of the work in mathematics in the standard junior high school, propaedeutic courses have been given in algebra, geometry, and trigonometry by the end of the ninth school year. The nature of the subject-matter in these three branches differs from the customary textbook type. Instead of an unnecessary amount of time and space being given to puzzle problems, problems of mere complexity, and problems of a too advanced character merely to make the work exhaustive, the time and space are devoted to a simple, careful, and helpful introduction into these subjects.

Now that the mathematics curriculum of the junior high school has been made definite, it can be used as a basis upon which to build the mathematics curriculum of the senior high school.

SENIOR HIGH-SCHOOL MATHEMATICS 3

Mathematics Elective. It cannot be expected nor would it be advisable that mathematics should be required of all pupils in the senior high school. One of the reasons for this is that there are some pupils who have not minds capable of the amount of rigorous thinking required in mathematics. These pupils will not elect mathematics, and those who do elect mathematics will do so largely because they like the subject. For this reason, there will not be the need of arti-

2 Wentworth-Smith, Junior High-School Mathematics, New York.

3 Some of these ideas were expressed by Professor D. E. Smith in his lectures at Teachers College, February, 192 1.

ficial motivation. Hence, we can follow the shorter road into mathematical insight by making use of suitable material that may not have any practical application whatever, and the pupils will enjoy it. There will not then be the necessity for adapting mathematics to mathematically weak minds since these pupils will not take mathematics. This selection will give us the stronger type of mathematical mind and that strong kind of incentive which comes from pupils who have mental hunger for this particular line of study. These pupils will take up the subject for the love of it. Accordingly, so far as mathematics is concerned, our high schools will be better off than corresponding schools abroad, since there the selection of pupils is usually on the basis of wealth and not on the basis of ability.

Various Courses. Various courses in which this elective mathematics may run are vocational, commercial, industrial, and agricultural courses. Commercial mathematics is such an extensive subject that it is taken care of by itself in the commercial high schools. There this work covers four years. There is also a great deal of mathematics required in industrial work, as is shown by the numerous technical schools which exist. There is little mathematics needed in agriculture. Also, there will be some pupils who will want mathematics for informational purposes and some for college entrance.

The possible courses themselves will now be taken up according to the school years in the senior high school.

The Tenth School Year. In the tenth school year, what should the student have if he likes mathematics, or if he wants to go to college, or if he wants to enter industrial fields where he will find it advantageous to continue further his studies in mathematics of his own account? There is no question but that he will need to take additional work in algebra. This too would be the time for him to become further acquainted with those geometric ideas which later will make it possible for him to grasp more readily the mathematical conditions involving such ideas. At the same time, the course in plane geometry should be so arranged that it will serve one of its main purposes, namely, to furnish the best developed applications of logic that we have. Along with this work in plane geometry, there should be given at opportune times the related notions of solid geometry in order that the learner's mind may not

Trend of Mathematics 13

be unnecessarily restricted to a geometric world of two dimensions. Such an arrangement will also give the preliminary exercise in the development of three dimensional space perception. In fact, there is no longer a necessity for teaching solid geometry as a separate subject in our high schools, and there is a tendency in a number of our high schools toward the elimination of solid geometry as a separate subject. 4 Its elimination can be effected without the loss of its three main aims, the development of space perception, the extensive work in mensuration, and its introductory and almost intuitive treatment of incommensurables and limits. It has already been suggested that the first of these can be partly taken care of in the course in plane geometry. The more extended practice in the development of space perception, the extensive work in mensuration, and the treatment of incommensurables and limits can be presented in a much more satisfactory manner in a course in the elementary calculus adapted for use in the senior high school.

Hence, the two main courses in mathematics in the tenth school year should be algebra and demonstrative geometry. With the propaedeutic courses in mathematics in the junior high school as a foundation, the minimum time to be devoted to either of these two subjects need not be greater than half a school year.

The Eleventh School Year. The first half of this year could be devoted to elementary courses in statistics and descriptive geometry. This course in elementary statistics would be a valuable aid to the pupil in his general reading. It would enable him to comprehend more readily the concentrated information contained in statistical charts that are found so frequently in current articles. The course in descriptive geometry would aid the pupil in extending his knowledge concerning the properties of solids and develop skill in drawing in perspective. This kind of drawing is the one most used in future work.

The second half of the year could be devoted to work in trigonometry in addition to the numerical trigonometry given in the junior high-school mathematics. In the second half of this year, there should also be given a considerable amount of graph work. The finding of maxima and minima of quantities by means of graphs should be especially emphasized. In rectangular coordinates, the

4 Questionnaire to Determine the Place of Solid Geometry in the High-School Curriculum, sent out by the author, spring, 1920.

equations and some of the simple properties of the straight line, the circle, the parabola, the ellipse, and the hyperbola should be taught.

The Twelfth School Year. It is seen that there is room in the last year of the senior high school for an additional course in mathematics. The mathematics of the tenth and eleventh school years were preparing the pupil principally to enter one of two fields, either the field of pure mathematics or some special field of industry in which he would find it desirable to continue further some of his studies in mathematics. Is there one such course that will give the pupil desiring to enter the industrial field such additional introduction into the field of mathematics that he will be able thereafter to continue his further study in mathematics without the aid of a teacher? Some of the possibilities of such a course will now be discussed.