RECENT TENDENCIES IN THE TEACHING OF ELEMENTARY APPLIED MATHEMATICS

TENDENCIES IN MATHEMATICS REACHING 237

RECENT TENDENCIES IN THE TEACHING OF ELEMENTARYAPPLIED MATHEMATICS.

BY J. R. YOUNG,University of Nevada.

HISTORICAL STATEMENT.The mathematics given in the elementary and secondary

schools of the United States in the latter part of the eighteenthand the early part of the nineteenth centuries was largely of avery practical character.1 Arithmetic was recognized as a prep-aration for business and commerce. Even a superficial surveyof a few of the early texts brings conclusive evidence to supportthis statement. The first arithmetics used in America wereEnglish works. One of these which was most widely used wasHodder^s Arithmetic: or, That Necessary Art Made Most Easy,of which I have examined the twenty-third edition published atLondon, 1705. The chapters 17 to 23 are devoted to problemsof a distinctly commercial nature. The New and Complete Sys-tem of Arithmetic, by Nicholas Pike, published in America, 1788,was a practical arithmetic, the later problems dealing exten-sively with business relationships. The Scholars Arithmetic, orFederal Accountant, by Daniel Adams, the tenth edition of whichwas published at Keene, N. H., in 1822, indicates both by itssubtitle and by the character of the problems that it was intendedprimarily as a business arithmetic.

Surveying was introduced into the academies in the latter partof the eighteenth century, in response to the practical need con-sequent upon the conflicting land claims resulting from theRevolutionary War and the opening up of great tracts of virginterritory. It has kept its place in a large number of secondaryschools and in the colleges as a disciplinary subject since it haslost its practical raison d’etre. That the subject is taught withlittle attention to practical considerations is shown in the factthat many long and involved methods of solution are employedin the classroom which are never used by the practical surveyor..Algebra and geometry were developed quite fully as college

subjects before they were introduced into the secondary schools.Their field of application to practical affairs is much more cir-cumscribed than is that of arithmetic. These subjects wereconsequently developed and organized from the disciplinarypoint of view. With the growth of our commerce and the build-ing up of a class which could afford the-time to study subjects

1 Cajori, A History of Elementary Mathematics, p. 217; D. E. Smith, The Teaching of Arith-metic, p; 33.

238 SCHOOL SCIENCE AND MATHEMATICS

which were not of immediate practical value, there arose ademand for algebra and geometry in the academies and highschools. With the introduction of these two subjects and theextension and organization of the mathematics in the secondaryschools, the abstract, disciplinary, pure science ideal of the col-leges became dominant in the mathematics in the secondaryschools. The advanced arithmetics designed for the use of thesecondary schools became the models for the writers of textsfor the elementary schools. As a result, the arithmetics of thisperiod (roughly from 1830-90) began to lose their practicaldirection. The drill problem, the involved problem having norelation to the conditions of real life, and the puzzle problem2became prominent. The mathematical gymnastic was the orderof the day. The slight preponderance of problems dealing withbusiness over those of any other field is the only reminder wehave of the fact that the early arithmetics were intended pri-marily as a training for practical business life.

During the period characterized above, applied mathematicswas taught by the masters of apprentices and in a very limitednumber of technical secondary and higher schools. As early as1850, the apprenticeship system began to break down, and fromthat time to the present there has been a growing demand fora more practical training in the public schools. In the last tenyears, this demand has become very insistent. Mathematics,along with the classics and many of the other subjects, has beenplaced upon trial from the point of view of its practical value forthe mass of our people. Mathematicians and educators havenot been unmindful of this jury of the people. They have testi-fied freely against themselves, and some have even pronouncedtheir own sentences. They have established many new schoolsof a practical character, and have incorporated vocational coursesin the curricula of the regular schools. The point of view ischanging rapidly, but the practice is changing slowly.

Superintendent Morrison, in a very able article,3 outlinessome of the fundamental principles which should govern thereorganization of the subject of mathematics:

1. There is general dissatisfaction with mathematical instruction.2. Subject matter must function throughout the process of learning,

and it is impossible that the present mathematics should function even inthe hands of skilled teachers. �

2 It is of course true that this type of problem is found in limited numbers in some of theearlier arithmetics.

3 Morrison, Henry C., "Reconstructed Mathematics in the High School,"- in Part I, Thir-teenth Yearbook, National Society for the Scientific Study of Education.

TENDENCIES IN MATHEMATICS TEACHING 239

3. The courses of study in mathematics must be reorganized so as tomeet social needs.

4. Detailed, concrete aims, related to social needs, must replace theformal and disciplinary aims.

5. The courses in mathematics should be differentiated along culturaland technical lines so that they parallel the broad zones of adult activity.

This tendency toward applied mathematics in the non-voca-tional schools has already passed beyond the realm of meretheory. A number of textbooks have been written which em-body the practical point of view. Of these, the Young and Jack-son series4 may be taken as typical. In the preface, the authorsstate that the effort throughout has been to select problemswhich are practical, applicable to daily life, and within theexperience of the children. Each of the three books containsone or more sections upon Âpplications of Processes/7 InBook I the section on applications contains problems concerningareas, weights, marketing, sales checks, bills, volumes, and twogroups of miscellaneous problems. Book II contains threesections on business applications and one on practical problemsof measurement. In Book III there are extensive sections deal-ing with the applications of percentage, interest, and banking;forests, lumber, railways, ranches, plantations, and orangegroves; and commercial problems including ordering goods,recording sales, bills, receipts, and stocks and bonds.A further step in the direction of applied arithmetic is taken

by Burkett and Schwartzel in their Farm Arithmetic.5 Thistext is intended to be used in the last two or three years of theelementary school. The point of view of the authors, as statedin the preface and directions to the teacher, is of interest:^There is much in modern arithmetic that is of little value tocertain classes of pupils. Particularly is this true of the subjectmatter of many textbooks now in use in rural schools�country,town, and village. These books were made by city-people forcity children, and are, for the most part, admirably adapted tocity schools.?; "Arithmetic may be taught in terms of agricul-ture. On the other hand, agriculture may be taught in termsof arithmetic..^ This purpose of teaching agriculture througharithmetic is evidently kept in mind throughout, for the prob-lems are both^chosen and stated in such a way as to emphasizethe best agricultural methods. In addition, the arithmeticcontains many "Important Truths^ interspersed among the

4 Young, J. W. A., and Jackson. Arithmetic, Books I-III, Appleton, 1904.5 James Baldwin’s Industrial Primary Arithmetic, Ginn & Co., 1891, is an earlier example of

the same general type. In this book a special effort is made throughout to show the pupils thepractical uses of numbers.


problems. The following is typical: ^Hogs return to theirowner the greatest relative profit if sold at an age of from sixto nine months. They then weigh between 150 and 200 pounds.Hogs weighing from 300 to 400 pounds are usually sold at a loss.Only when feed is cheap and prices high can heavy hogs be pro-duced at a profit.^ There are many good illustrations and manyseries problems in which the pupil may identify himself with thefarmer in the latter^s determination to drain his field, spray histrees, etc., share in his. first discouragement through the lack ofimmediate improvement, and in his ultimate success and satis-faction in the increased production of orchard and field.The emphasis upon inventional and constructive geometry,

and the effort to combine the most practical aspects of algebra,geometry, and arithmetic in single texts for use in the elementaryschools are also, in large measure, a result of the demand forapplied mathematics.

COURSES OF STUDY AND METHODS OF INSTRUCTION.In Corporation Schools.�One of the recent developments in

the field of industrial education is"the school for apprentices con-ducted by a corporation to train competent workmen for itsseveral departments. These schools represent a very highdegree of specialization; and the aim of developing an efficientworkman dominates the school. Everything that cannot bemade to harmonize with this aim is eliminated. We may notefirst a few general statements as to the instruction in theseschools.In a description of the methods employed in the Browne and

Sharpe School, at Providence, R. I., the writer says:_ ^Theinstruction is conducted mainly without textbooks and by directmeans, working from the particular problem and its solution upto the principle. Practical problems are taken up as they wouldoccur in the shop. Such reference books and tables are at handas any progressive mechanic would have, and the students aretaught to use them in solving the problems. These problemsare worked in blue-print form, and the sheets are preserved bythe boys in suitable covers. The blue prints follow a carefulgradation in the difficulty of the problems, and the directionsgiven are explicit.6A bulletin of the School for Apprentices of the Lakeside Press,

Chicago, describes the course of study given in that school asfollows: "Mathematics is taught from the shop point of view,

6 Report, Commissioner of Education, 1913, I, 260. -’


and accuracy becomes not merely the idea of getting the answer,but is absolutely necessary when applied to practical work.Arithmetic is reviewed from the factory side. An applied arith-metic has been prepared to be used in connection with the reviewwork. The elements of algebra and geometry are taught, andwhenever possible the problems are applied to the trade."The outline of the course of study in mathematics for the

Lakeside Press School is given as follows: Ârithmetic, Applied�Review of fundamental work as outlined in applied arith-metic; the technical work of the text completed; supplementedby actual .problems constantly arising." Âlgebra, - Elements(Wells)�Applied work as far as possible."Mr. Sheldon, Supervisor of Apprentices, says that he finds

that the number of algebra problems which can be directly ap-plied to the printing industry is exceedingly limited, and that ingeometry the principal points of contact are found between con-structive geometry and the department of design.The course of the School of Apprentices of the Lakeside Press

may be taken as typical of the courses in those schools whichplan to give a relatively broad training. A number of theseschools teach no mathematics, except arithmetic. A few haveno special courses in mathematics, but teach the subject in con-nection with the various shop problems as they arise in theprogress of the apprentice.7

All of the arithmetics used in these corporation schools areapparently a direct outgrowth of the work in the shops forwhich the apprentices are to be trained. C. J. Hicks thus de-scribes the evolution of the textbook used by the HarvesterCompany: "For several years the shop class instructors havebeen preparing and using special arithmetic lessons, and thepresent textbook is a combination of these lessons and a directoutgrowth of the work at the McCormick, Deering, and Mil-waukee works."8

I have examined five of these applied arithmetics.9 Thesebooks are all "pocket editions." None exceeds a hundred pages.All of the authors of these books agree that a comparativelysmall amount of arithmetic well taught is all that is necessary for

7 Cf. Report of Committee No. VI, Intern. Math. Commission, and Bul. Bu. of Ed., 1912, No. 4,p. 35.8 Bulletin of National Association of Corporation Schools, No. P, November, 1914, p. 34.9 E. E. Sheldon, Applied Arithmetic, Lakeside Press; Colvin and Cheney, Machine-Shop

Arithmetic. New York Central Lines; Santa Fe System, Apprentice School Arithmetic; Eatonand Brady, Ludlow Textile Arithmetic; Handbook of Arithmetic and Geometry, Fore River Ship-building Company, Quincy. Mass.


workers in the industries. These texts differ somewhat as toorganization and points of relative, emphasis. In most of thetexts the shop problems are organized under the regular arith-metical rubrics,�Addition, Fractions, Ratio and Proportion,etc. In the Machine Shop Arithmetic, however, the organiza-tion is largely under shop captions, as Rules for Selecting ChangeGears for Screw Cuttings, Speed of Pulleys, Gears, Cutters,Drills, etc. There are differences also in the methods of ap-proach. The Ludlow Textile Arithmetic quite uniformly pre-sents under each topic definitions or rules, drill problems, andthen problems applied to the industry. The approach is con-ventional. On the other hand, in the Santa Fe System Arith-metic there are practically no rules or definitions, and there islittle organization to the problems that points to any specialmethod of approach. In general, however, a limited number ofdrill problems are first presented when a new topic is intro-duced. . One of the things which comes forcibly to the attentionof even the casual student of the Santa Fe arithmetic is that itwas printed entirely without consideration of the eyes of theapprentices. The print is exceedingly small. The LakesidePress arithmetic is the only one of the group which devotes anumber of sections to general,’relatively abstract problems,^thinking exercises,77 as the author of the text calls them.These problems are given in the first half of the book, the lasthalf being devoted almost entirely to problems of the industry.The Machine-Shop Arithmetic is distinctive in that it gives avery small number of problems in connection with each topic,while the explanations are relatively elaborate. The principleis emphasized throughout as the essential thing. A consider-able number of algebraic formulas are elaborated in this text.The book used by the Fore River Shipbuilding Company appren-tices is the only text in which there is a section on practicalgeometry. An outline of the table of contents may be of interestas shown on the next page.

Tables are given in the back of the book, including decimalsof an inch, squares, cubes, square roots, cube roots, logarithms,reciprocals, circumferences, and circular areas of numbers from1 to 1,000. The problems are almost all in terms of shipbuild-ing, and there is a minimum of simple definitions. The geome-try is largely constructive, and it is evidently intended to pre-pare for mechanical drawing or to supplement it.A few distinctive features of the methods used in these schools


may be noted. In the Santa Fe System Apprentice School andthe Harvester Company^s school, a single leaf textbook is used.Each pupil is given an outline of the course and a folder in whichto keep each sheet of the arithmetic as it is given to him. Inthis way, concentration is secured and the students may advanceindividually. Even the simpler problems are presented in termsof the industry. In this way, the apprentice gets accustomedto the applied mathematics point of view, and at the^ same time

Arithmetic. PageDefimtions��--��-�-�.�--��..�. 3Fractions.-.-------------.---1-------��..��... 5-26Measures��--��--------�---�---��.�-��26-27Involution and Evolution� ��

�� 28-35Mensuration.-��

�� 35-40Formulas�� ..^��40-42Miscellaneous Problems -. - - �� .-..-.....46-51

Practical Geometry.Lines and Angles--�� 51-57Circles..---.�� .

�� 57-59Polygons--�� 59-61Ellipse--.--..----�---------------.---------------.-�- 61-62Solids-...---------.----..-.------.�� 62-65Developments..--..-.------.---�� .�..� �.��65-69Measures (Tables).-----.---.-..--....-....�� .��69-71

builds up his shop vocabulary. In the Ludlow Textile Arith-metic/SL glossary is given in the front of the book to aid thestudent in his interpretation of the problems and in gettingacquainted with the vocabulary of the mills. In the Santa Feschool, one pha§e of the work deals with the replacement ofvarious articles and pieces of machinery in the shops. Theapprentice is given a list of such articles and asked to write outeverything that it would be necessary to know in order to replacethe missing articles. In this way, exact measurement and state-ment, names of manufacturing establishments, and excellentdrill in various types of shop problems are secured. This workhas proved to be exceedingly helpful and interesting to theapprentices. In general, the writers of texts for these schoolshave succeeded remarkably well in securing clear and simplestatements of the fundamental principles of arithmetic, andsome of the makers of more advanced texts could study theirrules and definitions with profit.

In Secondary Technical Schools.�The report of CommitteeNo. VI of the American Commissioners of the InternationalCommission on the Teaching of Mathematics gives a relativelyfull statement10 of the tendencies in mathematical instruction in

^ See Bulletin 1912, No. 4, Bureau of Education.


the secondary technical schools.The agricultural schools usually devote from six months to a

year to arithmetic, most of .the problems relating to the life onthe farm. About the same amount of time is given to algebraand geometry as in the other secondary schools. Advancedalgebra is occasionally taught, and trigonometry is given inabout one-fourth of these schools. The committee concludesthat the curriculum in many of the schools is not at all affectedby the special object of the school. About one-third of theagricultural schools report that the arithmetic work is morepractical, while one-fourth say t’hat the methods used have notbeen affected by the special function of the schools. Arithmeticis the only mathematical subject correlated with agriculture.Algebra and geometry are taught from texts after the traditionalmethod, and few schools make any effort to emphasize or developpractical applications of these subjects. Solid geometry istaught in about twenty-five per cent of the schools. It is evi-dent that many of these schools are following almost exactlythe traditional course of study for the secondary schools, butthe committee adds the hopeful statement that the curriculumfor these schools is still in a formative state.

(Continued in April.)

ALGEBRAIC DERIVATION OF THE LAW OF COSINES.

BY ALBERT BABBITT,University of Nebraska,

Let A, B, C and a, b, c be, respectively, the angles and thesides of the triangle ABC.A=180° ~(B+C).sin A=sin (B+C) =sin B cos C+cos B sin C, or- sin B � , sin C � , �

,� .

1 ==��rcos C +-�-rcos B, or, by the law of sines,sin A sin A } J v

1=�cos C+�cos B.a a

Hence, a =5 cos C+c cos B. (1)Similarly, we obtain,

6= c cos A+a cos C (2)c==acos B+6 cos A (3)

Multiply (1), (2), and (3) by a, 6, and (�c), respectively, andadding, we get,c2=a2+&2-2a& cos’C.

Similarly, we obtain,b2=a2+c2-2ac cos B.

�

a2==52+c2-26ccosA.

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RECENT TENDENCIES IN THE TEACHING OF ELEMENTARY APPLIED MATHEMATICS