TRENDS IN PURPOSE AND CONTENT OF THE

HIGH SCHOOL MATHEMATICS COURSE

IN TEXAS

THESIS

Presented to the Graduate Council of the North

Texas State Teachers College in Partial

Fulfillment of the Requirements

For the Degree of

MASTER OF ARTS

By

Vena Mae Brantley, B. A.

90489Jefferson, Texas

June, 194L.

3 R

g, a qj)

90489

TABLE OF CONTENTS

PageLIST OF TABLES . . . . . . . . . . * . . . . * * . v

Chap terI. INTRODUCTION . . . . 1

Statement of the ProblemThe Purpose of the StudyScope of the StudySources of DataSome Related Studies

II. CONDITIONS EXISTING PRIOR TO 1923 . . . . 6

College Entrance Examination BoardThe Perry MovementAmerican Mathematical SocietyReport of International CommissionResulting CriticismThe Mathematical Association of AmericaConditions Existing in 1900Procedures UsedOpinions, Practices, and Criticisms of

EducatorsTheory Underlying: OpinionsSummary of Theories

III. RESULTS OF SURVEYS OF1923. . . . . . . . 35

Report and Recommendations of theNational Committee

Mathematics for Years Seven, Eight, andNine

Mathematics for Years Ten, Eleven, andTwelve

College Entrance RequirementsReport of Texas Educational Survey of

1923Summary of Theories Underlying Recommenda-

tions

ii

ChapterIV. CURRENT PRACTICE AND THEOI . . . . . . . .

Recommended Course of Study in Bulletin 243Recommended Course of Study in Bulletin 254Recommended Course of Study in Bulletin 293Recommended Course of Study in Bulletin 325Recommended Course of Study in Bulletin 379general Statement

Evident Procedures Resulting from Nationaland State Recommendations of 1923

Recommendations of Texas Authorities

V* SUAi .IAC " . . . .* * . . . * . * . . . . .

Page0 55

79

Changes as Covered in This StudyTexas' Requirements in Relation to

National RecommendationsTherein We Are Failing: Recommendations

BIBLIOGRAPHY * . . . . . . . . . . . . . . * . . . * . 90

iv

LIST OF TABLES

Table Page

1. Enrollment in Mathematics Courses, 1922-23 . . 49

2. Pupil Enrollment in High School Subjects . . . 49

V

CHAPTER 1

INTROIhJCTIOI

Statement of the Problem

The mathematics course in the American high school has

changed materially throughout the development of the school.

The past few years have brought radical changes, worthy of

detailed study. The purpose of this study is to examine

these changes, noting the trends and purpose of required

and elective courses offered in the high school mathematics

curriculum, in light of the theory and philosophy under-

lying these requirements. Special emphasis is to be placed

on the trends in Texas schools.

The Purpose of the Study

It is the purpose of this study to review in brief

the changes that have taken place since the turn of the

century in content of mathematical studies in the high

school and to examine theoretically the significance of

such changes.

Since the days of the earliest secondary schools,

mathematics has been recognized as an important study.

Its importance is regarded from a different standpoint

now, however, Mathematics is growing in importance. It

1

2

is the function of the school to equip children to be ef-

fective members of society and to be appreciative of our

culture.1

Hogben, not a teacher of mathematics, but a social

biologist, urges that larger numbers of persons should be-

come proficient in a wider range of mathematics. The

Rational Council of Teachers of Mathematics is not in

favor of forcing mathematics upon any who do not want

it. However, there are many students who do not know

what work they wish to follow on leaving school; so they

erroneously assume they will need no extensive work in

secondary mathematics. On entering college they find the

doors to desired fields closed because of lack of adequate

mathematical preparation.2

A familiar complaint is heard that the study of mathe-

matics has been unsuccessful, but the same may be said of

other subjects of study. 3

Pupils need to be led to see that acquaintance with

mathematics helps one live more intelligently in a scien-

tific world. An examination of texts used in numerous

technical fields, professions, and trad-es reveals their

dependence upon mathematics. 4

1 The Place of Mathematics in Secondary Educ n,Fifteenth Yearbook of the National Council of Teachers of.Mathematics, p. 35.

2lbid. 3lIbi d. M

3

If accuracy is an ideal for thinking, then it is to be

attained in mathematics.5

Mathematics searches for relationships, and, as Keyser

has said, everything has relation to everything else.

Mathematics certainly influences our philosophy. 7

Since mathematics has influenced our civilization to

the extent that it has, it seems reasonable that it should

occupy an important place in education. There are those

who urge that only a small amount of mathematics be re-

quired, and that only those with special inclinations go

further.8

The study of mathematics should help one to get to

the meaning of things

There is a definite trend now toward leading pupils

into new topics through their own experiences. 1 0

Better courses of study and better teachers will de-

crease the number of persons disliking the subject. 1 1

The effort to' make mathematics prominent in education

indicates a high levelfor secoxidary education, both in

ideals and achievement. If we but agree with Hogben that

5 lm

GWeyser, C. J., Male Philosoph anQ Other Bssa,pp. 94-95.

7The Place of Mathematics, 2. cit.8lbid. 9 Thi6. 10Ibid 111bi4.

4

mathematics is the mirror of civilization, we can find re-

vealed in its position in the school something of our con-

ception of education and more about our philosophy and

ideals.12

The present may be described as the Golden Age of

Mathematics.1 3

Scope of the Study

This study touches only briefly in discussion the

content of the mathematics course in the early American

schools. Most of the study is centered in general in the

program of the twentieth century schools and specifically

in the progress and trends of the schools of Texas.

Sources of Data

The trends in the development of the secondary school

mathematics course are considered in view of reports and

recommendations of the International Commission on the

Teaching of Mathematics, The Tational Committee on Mathe-

matics Requirements, National Council of Teachers of

Mathematics, Mathematical Association of America, National

Committee of the Mathematical Association of America,

American Mathematical Society, U. S. Bureau of Education,

Texas Department of Education, Texas Educational Survey

13abid.,:2I # ., . 51.

5

Commission, and various outstanding educators and textbook

authors.

Some Related Studies

Related studies have been made under auspices of the

various mathematical organizations of our country and state.

The National Council of Teachers of Mathematics has published

in the form of yearbooks reports and recommendations con-

cerning the study of mathematics in this and other countries.

The Council has pointed out significant changes and made

surveys that are of utmost importance to students and edu-

cators.

Other studies could be made dealing with each of the

individual "compartments" of mathematics, the changes,

purposes, and trends. Comparative studies should be made

dealing with the importance of general and integrated mathe-

matics.

CHAPTER II

CONDITIONS EXISTING PRIOR TO 1923

Before the advent of the twentieth century the purpose

of high school mathematics was to satisfy requirements for

entering college. Examinations were set by each college

for its own candidates irrespective of requirements of

other colleges, the needs of the secondary schools, or the

interest of the general public. The subject matter covered

by different examining bodies was fairly uniform, based

"upon a tradition that was generally known throughout the

country.

College Entrance Examination Board

The College Entrance Examination Board was organized in

1900.

It sought to unify the examinations and toprepare them with Trich greater care than wasusually the case with local efforts. It alsogave an opportunity for schools to be consultedby and become a part of a central organization,thus being represented in the preparation ofthe papers. While the. range of examinationssoon became that which was set by the committeeof the Atierican Mathematical Society, and was

1A General Surv f Prg s of a fl.s in Our

Hish Schools in the L t nty-Five Years, First Yearbook ofthe National Council of Teachers of Mathematics, 1926, p. 2.

6

7

rather indefinite as to limitations, the papersthemselves became more standardized and representedin general a better selection of material. Thetraditional still played a leading role, but atlast there "was some hope of modernizing the syl-labus and there was a feeling of assurance thatthis improvement would in due time be realized. 2

The Perry Movement3

Perry started a movement in 1901 to discard the pure

mathematics syllabus and introduce a new improved method

of mathematical teaching, with the idea that usefulness

must determine what subjects be taught and in what ways.

He did not advocate that the study of pure mathematics be

discarded, but that it should be limited to those whose

particular mental processes are suited for it. He wished

to alter the plan of teaching boys elementary mathematics

as if they were all going to be pure mathematicians. He

advocated letting the pupil discover even very simple

things for himself because that sort of discovery is of

most real value -to him, whether he is to become a mathe-

matician or not; it "becomes a permanent part of his mental

machinery. Educate through the experience already pos-

sessed by a boy; look at things from his.point of view;

2Ibid., pp. 2-3.

3Jobn Perry, "The Teaching of Mathematics," EducationalReview, 22III (1902), 158-181.

8

that is, lead him to educate himself."4 The study of the

philosophy of a subject is not necessary to learning of

the subject. Teachers need merely make suggestions and

answer questions and leave the pupil to find things out

for himself. Educate for citizenship, and the few who are

to become specialists in mathematics will not be hurt, but

if the aim is to educate all to be specialists, the un-

specialized citizens will be neglected.

American Mathematical Society

A historical starting point of the modern improvement

movement is attributed to E. H. Moore's presidential ad-

dress before the American Mathematical Society in 1902.

This was a part of the world-wide movement. 5

Report of International Commission

Some of the recent changes in mathematics have come

about as a result of recommendations made by the Inter-

national Commission, formed by the International Congress

of Mathematicians, held in Rome in.1908. It recommended

"that the function concept be made the unifying element

of all mathematics and that it lead to an introduction of

4I i., 165.165

The Reorganization of Maematis in Secondaryduca-don, Report of rational Committee, The Mathematical Associa-tion of America, Inc., (1923), p. ix.

9

calculus and statistics in the twelfth year." 6 This was

adopted immediately by the leading countries of Europe,

but the United States was not yet ready, The American

commissioners reported on their return that this country

was then at least two years behind other leading countries

in mathematical attainments.7

Resulting Criticism

Studies resulting from the movement started by the

commission have caused skepticism as to value of the study

of mathematics. Educators attracted wide attention by

their criticisms, some of which are:

Mathematics . . , is too mechanical andformal. The reasoning is deductive while prac-tical living demands induction. It is arti-ficial. The time at which it is studied, thetime devoted to it, the manner in which it istaught, the amount to be covered, are determinedby tradition, not by any consideration of theneed of the child. The subject is difficult.The average child is not mathematically in-clined. Algebra is kept up because it is easyto teach. Algebra is mysticism to the parent.Should the girl be made to study algebra? Thepresent dominance of mathematics is due to itsservice to colleges and universities who useit as seive for sorting out young men andwomen.

6John A. Swenson, "The Fewer Type of Mathematics

Compared with the Old," School Science and Mathematics,XXXVIII, 110.

71id.

8 A. R. Crathorne, "Required Mathematics," School andoiety, VI (July, 1917), 7.

10

These criticisms came from men who were not prejudiced

against mathematics.9

Mr. Crathorne then discusses each of the above criti-

cisms. He concludes by saying that if the high school

student is not required to study mathematics, he will have

had an injustice done him because he may find the door of

entrance to a profession for which he is otherwise fitted

closed to him because of lack of preparation.

The Mathematical Associationof America

It was soon realized that something must be done.

The Mathematical Association of America was formed in 1916,

and a committee, the now famous Rational Committee, was

appointed to plan a reorganization of mathematics for our

junior and senior high schools. This report was published

in 1923.14

Conditions Existing in 1900

Arithmetie.--It was believed that arithmetic had to

be hard in order to be valuable, and a necessary prerequisite

seemed to have been that the pupil hate it. This attitude

was thought to prove that arithmetic was good for the mind

91bid.

1oJohn A. Swenson, "The Newer Type of Mathematics Com-

ared with the Old," School Science and.Mathematics, XXXVIII1938), 11i.

11

and the soul. All phases of arithmetic that time allowed

were supposed to be introduced regardless of the mental

horizon of the child or his probable needs after leaving

school. I

Alpebra.--Algebra was regarded by many persons as a

"purely mathematical discipline, unrelated to life except

as life might enjoy the meaningless puzzle."12 The pur-

pose of all secondary mathematics seems to have been to

make mathematicians.

The textbooks contained several pages of definitions

and theory before an example was given. The "examples"

used in the texts did not relate to any condition that

might arise in daily life. Texts were arranged on the

theory that we must scientifically define all terms before

they can be used; however, they were defined, not as

needed, but in order for readiness at some future time.13

If the text gave an idea concerning the use of algebra,

it was "that it was a science in which letters were used

in solving the most impractical sort of number puzzles."1 4

There was no effort to develop real applications of

algebra. 1 5

1 1 David Eugene Smith, "A General Survey of the Progress

of Mathematics in Our High Schools in the Last Twenty-Five

Years," First Yearbook of the National Council of Teachers

of Mathematics, (1926 ,p.19.

12Ibi d. , p. 20. 13Ibd., p. 22. 4 b.

15_i_, p. 23.

12

Geomtry.--Intuitive geometry did not exist in 1900.

Demonstrative geometry consisted of one year of plane

geometry and a half year of solid geometry. The demon-

strations were mostly memorized, were without purpose,

and were looked upon generally as an intellectual grind. 1 6

Procedures Used

Workboks.--Teachers today are prone to regard the

workbook procedure as a newer method of work in any field

of study. A workbook for use in mathematics has been

found, published in 1813 in Keene, New Hampshire.17 The

title page asserts that the product is new in form and

method. Its purpose was for the use of the school master

and for greater progress of the scholar. The scholar was

expected to work the examples on the slate or waste paper

and transcribe them into the book, in the space provided.

Definitions were called for, and questions asked, with

space for the scholar to answer. The English system of

notation was used and numeration was carried to great

length. Instructions were given for reading numbers of

thirty-five digits. Complicated work in multiplication

and division, using six and eight digits, was given. Rules

1 6 Ibid., pp. 26-27.

1 7 Cecil B. Read, "A Century Old Arithmetic Workbook,"School Science and Mathematics, XL (1940), No. 6,pp. 516-517.

13

were given which were advised to be necessary for any person

to transact business; namely, reduction, fractions, federal

money, interest, compound multiplication, compound division,

single rule of three, double rule of three, and practice.

A comma was used instead of the decimal as we use it. Op-

erations which were apparently useful at the time were

extraction of square root and work with duodecimals.

Interesting insight is given into the social and eco-

nomic conditions of the time. A man's wages are quoted as

being two dollars a week. Problems such as the following

are given: reduce the currency of New York and North

Carolina to that of South Carolina and Georgia; reduce

either of these to that of New England; or to federal

money; or to shillings and pence. One problem asks the

age in seconds of a man twenty-one years old.

This book contained a combination of methods employed

by our present-day workbook and textbook. A workbook was

not available for second-hand use, probably due to the

foresight of publisher and author.18

Compartment mathematics.--The oldest type of mathe-

matics taught in the schools may be roughly classified as

18Ibd

14

"compartment" mathematics, because of its division into

the numerous branches of "water-tight" subjects.19

General mathematics.--About twenty-five years ago

general mathematics was introduced for the benefit of the

weak pupils who were unable to master the compartment type

of mathematics. For this reason most colleges and uni-

versities refuse to accept it for admission. 2 0

Action of Harvard UniversityIn 1916, it was found

necessary by the mathematics department of Harvard Uni-

versity to lower the passing mark in the entrance examina-

tion in algebra to forty per cent. It was evident that

something was wrong with the algebra paper when "more than

75 per cent of the candidates failed when judged by a

passing mark of 60, whereas more than 75 per cent passed

in all other subjects.,22

Opinions, Practices, and Criticismsof Educators

John Perry2-The general mathematics movement dates

19John A. Swenson, "The Newer Type of Mathematics Com-

ared with the Old," School Science and athemati s, XXXVIII1938), 107-112.

20 bid., 107.

21John G. Hart quoted in "College-Entrance Mathematics,"

School and Society, IV (October, 1916), 634-636.

22Ibid., 635.

23John Perry, "Discussion on the Teaching of Mathe-matics," British Association Meeting in Gasow, 1901,pp. 15-90.

15

back to John Perry in 1901. He favored education for

citizenship. He believed the study of mathematics by

everybody to be of utmost importance to our country, not

just for knowledge, but for producing scientific habits

of thought. He recognized that a fault of methods of

teaching was that the pupil was being taught as if he

were going to be a teacher himself. He said:

I believe that men who teach demonstrativegeometry and orthodox mathematics generally arenot only destroying what power to think alreadyexists, but are producing a dislike, a hatred,for all kinds of computation, and therefore forall scientific studj of nature, and are doingincalculable harm.

In discussion Hudson replied:

If mathematics were better taught, thestudy of all other subjects would improve.That is one of the main reasons for endeavor-ing to effec 5a reform in the teaching ofmathematics.

Mathematics must be taught as a process of reasoning. It

is sometimes taught merely as an exercise of memory. The

pupil should have understanding of the thing he is to learn.

Smith sums Perry's lecture in the following points:2 6

1. Mathematics is poorly taught in England.(America has followed England's practice.)

2. It is absurd to say that the study is useless.3. Mathematics produces higher emotions and gives

mental pleasure.4. Teachers have not considered aid rendered by

mathematics in studying physical sciences.

26Ibid., pp. 89-90.24 d. , p. 16. 25 P.32

16

5. Mathematics has been taught almost solely forexamination purposes.

6. Pupils are not taught to think in mathematics.7. Problems should be modern and practical for

the pupil.8. England and America should adopt the metric

system.9. England should abandon Euclid.

JohnDe ! -John Dewey has done more to change the

teaching of mathematics and all other school subjects than

any other living man. It was Dewey who started us teaching

children rather than subjects. He said, "Education is not

preparation for life, but is life."

Eliakim H. Moore The main problem in mathematics is

the unification of pure and applied mathematics. The

material and methods of mathematics should be enriched and

vitalized. One type of arrangement has it now that in the

first year of high school algebra is taught; second year,

plane geometry is taught; third year, physics is taught;

fourth year, more difficult algebra, solid geometry, and

review of all mathematics. These subjects are in water-

tight compartments, and only very late, or not at all,. is

the student allowed to see the connection between the

2 7 David Eugene Smith, "A General Survey of the Progressof Mathematics in Our High Schools in the Last Twenty-FiveYears," First Yearbook of the national Council of the Teachersof Mathematics,7(1926, p. 59.

2 8Eliakim H. Moore, "On the Foundations of Mathematics,"Bulletin of the American Mathematics Society, IX (1903),402-424.

17

different subjects. It is hoped that they may be organized

into a coherent four years' course.

The writer endorses the laboratory method of instruc-

tion because it permits freedom of individuality. It is

the best method for use with students preparing for college.

Changes must come about as evolution, not revolution.

Teachers need to be better trained, and should be given

greater freedom and responsibility. There should exist a

better state of cooperation between the college teachers

and secondary school teachers.

W. fD. Reeve,2-Reeve discusses changes and trends in

the teaching of mathematics since 1910.

College entrance examinations grew out of adesire to standardize the mathematical productof the schools. The result of leaving each schoolto determine what a pupil should know in orderto enter any higher institution of learning wasoften chaotic. However, the results have some-times been detrimental to the best interests ofmathematics. Teachers should be encouraged tohave a philosophy of their own and to teach thesubject as it ought to be taught rather than totry to prepare their pupils for one final ex-amination. This they will not do so long asthey are forced to follow a course of studywhich they have had no hand in making gg withwhich they are not in sympathy . .

29W. D. Reeve, "Significant Changes and Trends in the

Teaching of 'Mathematics," Fourth Yearbook of the rationalCouncil of Teachers of Mathematicsi7T29), pp. 132-150.

30 Ibid., p. 132.

18

Tradition has been a hard factor to overcomein modernizing the curriculum in mathematics, thedifficulty being largely a matter of clinging tothe hazy and invalid objectives used in achinggthe mathematics of many generations ago.

Undoubtedly the influence of tradition'or thefeeling of satisfaction with the status quo hasretarded improvement in gUr courses in mathematicsmore than anything else.

Too much standardization is done to subject matter

rather than consideration given stages of mental maturity

of the students. Radical changes will take place in teach-

ing in secondary schools if the teachers and supervisors

will become more interested in the subject "getting on" in

the student rather than the student "getting on" in the

subject.

Most colleges still cling to the curriculum of earlier

periods, and because of the dominance of the colleges over

the high schools, changes are difficult to effect. They

seem unable to conceive of education as having anything to

do with modern life.

The theory of mental discipline has long held sway as

an aim of instruction. One kind of algebra was thought to

be as good as another.

As Mr. Dooley would probably say, 'The onlything we need to do is to make mathematics hardso that childred will hate it. The3 gore they hateit the better it will be for them.

31 Iid 32Ibid., pp. 135-136. 33I_ ido. p. 137.

19

It was believed that mathematics should be studied for

the habits of logical thinking, precise and accurate work,

and that it would aid pupils in other subjects and establish

certain life habits. It has been found, however, that unless

the work is properly developed, none except the gifted chil-

dren are benefitted.

In some schools a list of "minimum essentials" has been

prepared. This is a worthwhile plan if the minimum is not

made the standard, for in this case the gifted pupils are

the most neglected.

Perhaps the biased attitude of many persons toward

mathematics arises in a dislike for the subject--a hatred,

let us say. This is not natural. The trouble probably lies

with the arithmetic of elementary school. First impressions

make a great difference.

Teachers have not been properly trained. Many have had

no knowledge of mathematical subject matter past high school

algebra and geometry.

It is an old story that the athletic coach isoften given a class in mailematics to justify hisemployment in the school.

The old material . . . was taught accordingto the theory that more of a topic was needed toprepare fg more of the same topic and all for itsown sake.

123 5 Ii|., p. 150.34I_ i.., p. 142.

20

Where the aim of education was formerly to develop scholars,

the aim is now to develop well-educated citizens.

It was suggested in 1910 that calculus be taught in the

high school. This idea has been the cause of some agitation.

There is some experimenting being done in various schools by

the most progressive teachers in teaching the fundamental

elements of differential and integral calculus.

Walter B. Ford--The rational Committee of Fifteen in

1909 recommended that a number of particular proofs in

geometry be made informal. We need to break away from the

formal geometry which still exists as a result of the desire

of the Greek people to rise above reality and indulge in

the realm of pure abstraction. That was the spirit of their

philosophy. We are too much bound by tradition. The formal

should be servant of the natural.

Cheesman A. Herrick -Certain educational ideas have

been called to serious question. Many leaders urge that

nothing be included in the curriculum just because it is

traditional or has disciplinary value, but those subjects

should be included which "have value from consideration of

present needs." 3 8 According to this, much of the algebra

3 6 Walter B. Ford, "The Future of Geometry," 5chScience and :athematIc, XIV (1914) , 485-490 .

3 7 Cheesman A. Herrick, "What High School Studies Are

of Most Worth," Scho _and Soc y, IV (1916), 305-309.

38Ibid., 309.

21

and geometry should disappear from our schools. Much of

the old, however, will remain and new application will be

given in the relation of vital present needs. Changes

should be made in view of this guiding principle:

Those high school subjects are of most worthto the individual pupil, which will best fit himfor meeting the many- ided demands of the lifewhich he is to live.

I. 0. WinslowlQ.-The original purpose of the secondary

school was to prepare a selected few for college and the

learned professions. So a curriculum adopted long ago

with such a special purpose fails to meet present needs.

"The high schools are no longer select. They have become

the schools of the people and must be conducted accord-

ingly."4 1

The difficulty is that the extended coursein mathematics that was once required withoutquestion, when there was room for it in themeager curriculum and when pupils of the selecttype could easily cope with it, has been re-tained in the requirements notwithstanding thechanged conditions that have rendered it un-profitable and inconsistent. After murmurs ofdoubt began to arise over the question of values,a defense was found in the Doctrine of FormalDiscipline, according to which mental disciplinein the line of intellectual effort that did notgive promise of practical application was held

39Ibid., 309.

40lsaac 0. Winslow, "How Much Mathematics Should BeRequired for Graduation from High School?" Education,xovXi (1916), 581-584.

41 Ibid., 581.

22

to have effects that would be transferale toother lines of more practical activity.

This theory has been exploded. Perhaps no subject should

stay in the curriculum that cannot offer higher claim than

the transferable effects of discipline.

Winslow believed that at least one year of mathematics

should be required of all high school students. He says

that young people ought to have that much respect for their

ancestors. Because algebra and geometry are commonly men-

tioned in conversation and literature that for social reasons

alone a passing acquaintance is desirable. One-half term

of each of algebra and geometry should be required for

graduation.

The writer of the article believes that no suitable

textbook has been published. He states that authors know

too much of their subject to present it in simplified form,

and that they do not know enough about children and the

common level of life. This statement concerning authors'

knowledge of subject matter invites disagreement from several

sources.

Harriet R. Piercet4-"It is claimed that in this age so

rich in objective life, we have mistaken literacy for

42 Ibid.

43Harriet R. Pierce, "The Value of Mathematics as aSecondary School Subject," School Science and Mathematics,

''I (1916), 780-788.

23

education, that we have forgotten that true education is

unfoldment, and have made it simply an outer shell . . . ."44

"Because we do not know what else to do with them, our schools

are turned to workshops. "45

Mathematics must justify its place in the curriculum.

If its study gives worthy attitudes and ideals, if the sub-

ject matter is valuable, and if the habits formed are worth-

while, then it may be given a place in the curriculum.

Mathematical habits are classified according to form,

content, and method. Under form are the mechanical habits

of neatness, orderly arrangement, accuracy, persistence,

attention, and memory. These are important in mathematical

as well as other work.

"Painstaking systematic arrangement must be associated

with clear thinking."46 Mathematics is the best study for

forming the habits of accuracy. The habits of persistence,

holding attention on the goal to be reached, leads to the

problem-solving attitude. The problem-solving attitude is

necessary for success in other subjects and in the affairs

of practical life. The study of mathematics helps to develop

the habit of attention. The American has been criticized

by the foreigner as being lacking in accuracy, as presenting

a go-as-you-please type of work, as showing an unwillingness

to concentrate the attention. "The study of mathematics is

4 5Ibi., 781. 4 6 i, 78344Ii. , 780.

24

also valuable in helping to form habits of logical memory." 4 7

"Among the habits related to the content of the subject are

the habits of economy of thought and clearness, brevity,

and precision in expression. Mathematics is the shorthand

language of abstract thought." 4 8

While general power is not necessarily gainedby the study of mathematics, the student may be soimpressed by the perfection of mathematical reason-ing, that an ideal is formed which gives a sendardby which his thinking is consciously tested.

Various types of thinking are necessary tosolve all the problems of life. There is onemethod for mathematical reasoning, another forthe physical sciences, which diffgs from thatof the biological sciences, . .

If there were no other reasons for subjecting all chil-

dren to the study of algebra and geometry, the aesthetic and

ethical values would justify their place in the curriculum.

S,, G. _i~);-

As taught in American high schools, much ofthe prevalent mathematics is simply a waste oftime. It is uninteresting to many--perhaps most--students, and certainly enlarges neither theirpowers nor their outlook. A well-devised coursein 'general mathgatics' should in part alleviatethis difficulty.

47I i .,785. 48Ibi., 786.

49I |,,787, quoted from J. W. A. Young. 5%TL. ,787.

51l. G. Rich, "Compulsory Mathematics," School andSoiey, V (1917), 290-291.

52Ibid., 291.

25

We tacitly assume that every high schoolstudent will go to college and there electanalytics and calculus. We prepare for thiswhile neglecting mathematical courses whichmight give us defnite and educationally de-sirable results.

Instead of learning to demonstrate theorems(mere memory-work in the case of most pupils),we should aim to make geometry, algebra, andadvanced ari hmetic keys for understanding ourenvironment. 4

'General mathematgs' courses are a logicalneed for high schools.

David Sneddenzd-Snedden advises that algebra not be

made obligatory, especially for girls who will leave school

early. However, he does not advise that the subject be

abandoned from the high schools.

H. E. Slaught,-The power of mathematics was revealed

in a most emphatic manner to the unsuspecting public during

the World War. Mathematicians were indispensable in.de-

termining the effectiveness of gun-fire and in developing

submarine detection appliances. "These two achievements

were of vital importance in determining the outcome of the

war."5 8 Men without mathematical training who with it could

5 3 l . 5 4 Ibid, 5 5Ibid.

56David Snedden, "Mathematics in Secondary Schools,"School and Society, VI (1917), 651-652.

5 7 H. E. Slaught, "Mathematics and the Public," FirstYearbook of the rational Council of Teachers of Mathematics,(1926), pp. 186-193.

58 Ibid., 189.

26

become officers in the army or navy begged for the oppor-

tunity to enter mathematics classes. "The war served to

elevate mathematics to a prominence not previously recog-

nized by the casual public."59 Classes in mathematics in

colleges have become more popular in the years immediately

following the war.

C. H. P. aPyo-Mathematics is sadly neglected and

does not flourish in the public schools.

The policy of teaching subjects so that students may

pass examinations "has had a disastrous effect upon the

loss of intellectual life in many schools."61 Teaching

for the sake of passing examinations lowers the standard

of education. "Examinations must follow teaching, not

lead it." 62

Mathematics is studied for mental training because:

1. It develops the imagination.

2. It trains the logical faculty, the "deduction

form" law.

3. It trains one in sense of style, accuracy, and

power of observation.

4. It trains for precision in the use of language.

59Ii.,9189.

60C. H. P. Mayo "The Position of Mathematics," Educa-tional Review, LVII (1919), pp. 194-204.

61Ibid.., p. 198. 62I id.

'Mayo believes that experience with material objects,

the realm of fact, has little contact with thought. It

hinders, even repels, intelligence. "The concrete ousts

the abstract; our tendency is to drill rather than inspire,

to give easy smatterings rather than to develop law."63

He believes it proper to let the concrete precede the

abstract, but does not want it to replace the abstract

altogether. Things of the sense are crowding out things

of the mind. He longs for the return of the Euclid regime.

The average boy needs little beyond preparatory school

mathematics, a little algebra and geometry. The mathe-

matics the boy does in compliance with a formula is of no

value mentally; he must think. Lectures should be given

on fundamental principles.

Let mathematics primarily be for educatingthe human mind and not merely for instruction,or let it make way for some other subject whichcan play a larger part in deepening the intel-lectual life and enfg cing the moral value ofintellectual effort.

Theory Underlying: Opinions

652. _W. Jeyers.--The courses in mathematics are too

greatly isolated from each other and from other fields. No

course gives consideration to the previous or succeeding

63 Id.., p. 201. 64Ibid., p. 204.

65G. W. Meyers, "Educational Movements and GeneralMathematics," School Science and Mathematics, XVI (1916),97-105.

28

courses. Courses should be graduated and progressive from

year to year, and regard should be given the maturing

abilities of the pupil.

The movement to determine psychologically the value of

high school subjects demands a new order of mathematics.

This demand points to general mathematics as having a sound

psychological basis. General mathematics is the only

feasible kind for use in the junior high school.

"t*.,. The thing fusion mathematics is attempting to

accomplish for mathematics is just what the leaders of all

the junior high school movement are urging be done with all

the studies of this part of the curriculum."66 The general

mathematics plan will allow the pupil to progress "as fast

as he can or as slowly as he must."67 ". * . General mathe-

matics facilitates the use of supervised study."6 8

It is very difficult to get a chanceto see any real correlating being done in actualmathematical teaching. We are told that manyteachers claim to be doing some correlating, butthat when their6 ork is inspected no correlationis in evidence.

B. W. Hobson?9 -Pure mathematics should be allowed to

661b., 101. 67 bi.

68 bid., 102. 69Ibid., 104-105.

70E. W. Hobson's presidential address before the mathe-matics section of the British Association at Sheffield, 1910."Tendencies of Modern Mathematics," tn Review, XL,524-530.

29

develop its own course rather than be a tool for the physical

sciences. Mathematics is constantly enlarging its field,

The certainty of mathematics depends upon assumptions which

it is the business of the philosopher, not the mathematician,

to investigate. Discoveries in mathematics have depended

upon the intuition of the investigator. The teaching of

mathematics should emphasize the principles underlying the

processes.

R. fD. CarmichaelTh-

It is customary to think of mathematics asseparated from the usual concerns of ordinarylife. As conceived by many people it is essen-tially a sort of monstrosity of the human mindwhich has come to the place it now occupies onlybecause there has been an unbroken chain of menfrom the remote past up to the present who arecharacterized by an unusual development of thatpart of their mental organisms in which such amonstrosity may appear. How widespread is thisfeeling concerning the very nature of mathe-matics is usually not realized. Many of thosewho have the feeling harbor it in their con-sciousness only in a dumb sort of way. Theyhave never so much as fashioned it into con-viction, but they unconsciously allow it todominate the attitude toward the science ofmathematics.

The student does not fail to learn mathematics because

it is hard, but because he does not see that "it has a value

for life and for character equal to the demand which it makes

upon his energy." 7 3

71?R. D. Carmichael, "Mathematics and Life--The Vitalizingof Secondary Mathematics," School Science and Mathematics,XV (1915), 105-120.

72 Ibid., 105. 7 Ibid., 106.

30

Everything we do should have an intimaterelation to life and it should be done becauseof that relation; for everything which we areor have is summed up in life, whether it be apresent or an eternal value; whether it be thehappiness of a service or the pleasure of amoment or the destiny of an immortal spirit,its roots agg laid deep in the life of dailyexperience.

Mathematics must be brought close to life.

Life does not consist alone in the practical,in getting around among the things of this ma-terial world. There is an esthetic element inman and it is profoundly related to the wholestructure of his character. There is also amoral nature in him and out of it spring thegreat motives of conduct and from it arise thegreat movements of progress. And there areeternal values which are not7 8 f today and to-morrow but are for all time.

The primary problem of the practical man is in learning

to be efficient. Mathematics and life

are connected in the most obvious and tangibleway in the realm of the practical. Here we haveto do with the ever-flowing stream of reality.We have to find the relations among things. Wemust know the laws which connect them. We areever concerned with their various qualitativeaspects. Fortunately for us, several of thegreat controlling elements here are, for prac-tical purposes, expressible in simple mathe-matical form so that by means of such expressionswe can see much further into them than we areable to sense by experience. when one looksabout him and sees how many relations are essen-tially mathematical in their nature, and when heobserves further how fundamental these are tothe vast ramifications and connections amongphenomena, it begins to come home to him withgreat force that the universe is mathematical,that God indeed geometrizes, as one of the

74 Ibid., 107. 75Ibid., 108.

31

ancients has said. Among all the ways in whichmen have been able to express the relations ofexternal phenomena there is none other which com-pares in vastness and consistency with the mathe-matical. Witness such great provinces of appliedmathematics as mechanics, celestial mechanics,theory of magnetism, theory of electricity, andtheory of the potential, to mention a few of theleaders. It is just to say here that the mechani-cal philosopher has here summed up into hisdoctrines the most-essential and far-reachingelements of these central subjects and that eachof them, under the magic of his touch, has beenexhibited as a single body of doctrine united toand depenggnt upon a few fundamental principlesand laws.

The high school curriculum can contain many practical

problems in mathematics, but there are some things that

the curriculum must contain that are not intimately related

to the detail of daily affairs. We have not yet found a

way to vitalize the things in which pupils can not detect

close relation to life. If we can get the pupil to realize

the value of efficiency and get him to go about the task

of realizing it, then he will soon sense a need for mathe-

matics as a means of testing himself as to the power he is

developing and the efficiency he is attaining. Teachers

must train general faculties; there is the call of the

world for efficiency. The teachers must help the pupils

to know the difference between efficiency and inefficiency.

Mathematics is an art; it has moral virtue in law and

order of the universe. High school students may be given

7 6 Ibid., 110.

32

a foreview of this. There are eternal verities in mathe-

matics. For instance, every algebraic equation has a root

now and forever; the square on the hypotenuse is equal to

the sum of the squares on the other two sides of the right

triangle. These eternal truths give the pupil something

with which to balance himself in this everchanging world

of other general scientific theory.77

W. T. Stratton.--Democracy in our mathematical teaching

does not consist in giving the same courses to all regard-

less of their tastes, capabilities, and the things for which

they are preparing. As long as such a small percentage of

our high school students go on to college we should not

adjust our courses entirely to suit the needs of those who

do.78

. .1 athematics should be treated as ascience, to be sure, but as a science to be ap-plied to practical living; it should be made tofunction in the lives of the pupils. . . . Thevalue of mathematical knowledge to the majorityof students ig9 our secondary schools lies in itsapplications.

Students cannot apply the mathematics they have learned.

It has been said that a student could apply his mathematics,

if he really learned it; but this does not work out in prac-

tice.

.. T. Stratton, "Mathematics and Life," (Discussionof preceding article), School Science and lathematjc , XV(1915), 115-120.

7 8 Ibi., 116. 7 Ibid., 117.

If he has been given a chance to apply hismathematics he will at least get the idea thatmathematics can be used, and he will come toregard it as a luable tool with which to doeffective work.

Teachers are prone to have different mathematical in-

terests from those of children, but of late it seems that

there is a tendency "toward a more careful consideration of

the interests and tastes of pupils."81

"After all, the success or failure of the subject

depends almost entirely upon the teacher."8 2 The teacher

must make himself, in a sense, unnecessary to the class.

Summary of Theories

We cannot have failed to note the diverse opinions of

educators concerning the theory underlying mathematical in-

struction. This indicates somewhat the transition through

which mathematics is going. Mathematics evolved in the

early ages as a result of a real need felt by man, but the

Greek philosophers raised the level for study even above

reality. For a long period mathematics and philosophy were

closely related. One of the authoritative statements of

the philosopher Kant8 3 was that the amount of real science

to be found in any subject was the amount of mathematics

contained therein.

8 0 Ibid., 118. 81Ibbid.., 119. 821bidd.., 120.

83E. W. Hobson, "Tendencies of Modern Mathematics,"Educational R eew, (1910), 524.

34

Until 1900, the theory upon which mathematical teaching

was based was that all students (and it was required of all)

were to become mathematicians who would likewise teach

others to become mathematicians. Beginning at approximately

that time, influences which we term "pragmatic" crept into

education. Although many educators still held to tradition

in ascertaining aims and methods of instruction, many others

advocated teaching mathematics for the practical value to

the individual. The most widely recognized educators

throughout the period under study were those who advocated

teaching the child, with a view to his individuality, for

worthy citizenship.

CHAPTER III

RESULTS OF SURVEYS OF 1923

Much investigation has been carried on in the nation

and state in the interest of improving mathematical in-

struction in the schools. The purpose of this chapter is

to report the findings of these investigations.

Report and Recommendationf ofthe National Committee

The Committee was appointed in 1916, but on account

of the World War it was unable to secure financial ap-

propriations for its work until 1919-1920. The activity

was conducted on a large scale in order to get nation-wide

discussion. Almost one hundred teacher organizations

assisted in making this report possible. The Committee

served "as a clearing house for all activities looking to

the improvement of the teaching of mathematics in this

country . . "

Comparison with European schools has shown the vital

need of reorganization of mathematical instruction,

o rganization of Mathematics in Secondary Education,A Report by The rational Committee on Mathematical Require-ments under the auspices of The Mathematical Association ofAmerica, Inc., (1923), pp. vii-527.

2Ibid.,!p. x.

35

36

especially in the seventh and eighth years. On account of

the new movement to consider the grades seventh to twelfth

as years of secondary education, impetus is given the move-

ment for reform.

On account of the fact that a large number of children

will drop out of school by the end of the ninth year, the

National Committee recommends that it be required of all

students of the seventh, eighth, and ninth grades that they

study the fundamental notions of arithmetic, algebra, in-

tuitive geometry, numerical trigonometry, and at least an

introduction to demonstrative geometry.

The Committee sets out the general principles or aims

of mathematical instruction to be these:

1. Practical or utilitarian aims.

2. Disciplinary aims.

3. Cultural aims.

By practical or utilitarian aims is meant the immediate

or direct usefulness of the subject. The child should have

learned already the four fundamental operations using in-

tegers, common and decimal fractions, and the use of them,

with fair speed and accuracy. The pupil must have an under-

standing of the language of algebra. He will need to study

the fundamental laws of algebra, Operations in algebra

furnish foundation for understanding of significance of

processes of arithmetic. He will learn to interpret graphic

37

representations of various kinds. He will need to be

familiar with geometric forms and mensuration and acquire

space-perception and spatial imagination.

Disciplinary aims relate to mental training. Mental

habits are to be formed which will operate in more or less

closely related fields--transfer--which is difficult to

measure.

Cultural aims "are involved in the development of

appreciation and insight and the formation of ideals of

perfection."3 There is beauty, perfection, and power in

mathematics.

The following statement summarizes aims:

The primary purposes of the teaching ofmathematics should be to develop those powersof understanding and analyzing relations ofquantity and of space which are necessary toan insight into and control over our environ-ment and to an appreciation of the progress ofcivilization in its various aspects, and todevelop those habits of thought and of actionwhich will make these powers effective in thelife of the individual.

Drill in algebraic manipulation should belimited to those processes and to the degreeof complexity required for a thorough under-standing of principles and for probable appli-cations either in common life or in subsequentcourses which a substantial proportion of thepupils will take.

Algebra must be conceived always as a means to an end and

not an end in itself.

p.%10.. ,pp. 10-11. _Ibi., p. 11_I _,P. 10.

The primary underlying principle in the study of geom-

etry should be the idea of relationship between variables.

It is now realized that "general,""unified," "corre-

lated," or "composite" courses are needed in mathematics.

It used to be that one "subject" had to be completed before

another was begun, but now it is known that there are

important interrelations. This broadens the view early

in the high school course. Not all schools can yet adopt

the new plan, and they are advised to organize their

courses in the subjects so as to follow the recommendations

for content of those separate subjects.

Much experimental work needs to be done to determine

the possibility of designating the order of any particular

topics.

The Committee believes no consideration need be given

during grades seven, eight, and nine, to college entrance

requirements, but the course should be planned "with the

purpose of giving each pupil the most valuable mathematical

training he is capable of receiving in those years, with

little reference to courses which he may or may not take

in succeeding years." 6

There is a movement to correlate mathematics with

other courses, as science, as was the case in the infancy

of mathematics.

6 bid.,vp.*14.

39

The Committee feels that the adoption of the junior

high school form of organization will secure greater ef-

ficiency in the teaching of mathematics.

Even more important than organization or content of

the mathematics course is the problem of the teacher--

his qualifications and training, his personality,skill, and enthusiasm. . . . The greater partof the failure of mathematics is due to poorteaching.

Administrators should never lose sight ofthe fact that while mathematics if properlytaught is one of the most important, interesting,and valuable subjects of the curriculum, it isalso one og the most difficult to teach suc-cessfully.

Mathematics for Year& Seven, Eight,and Nine

Teachers desire a detailed syllabus by years or half

years, with specific time allotment for each. This desire

cannot be met on account of the fact that at present no one

knows what is the best order of topics nor how much time

should be devoted to each in an ideal course. The Committee

recommends further experimentation rather than restriction

of teacher's freedom.

The Committee has outlined topics in each subject which

should constitute minimum essentials, and it recommends that

these be required of all pupils, because they include mathe-

matical knowledge and training which are likely to be needed

7lbid., p. 15. 8Ih1 ., p. 16.o 9iid., P. 19.

40

by every citizen. If there are special needs in particular

cases, differentiation should be made after and not before

completion of the minimum essentials.

In adapting instruction in mathematics tothe mental traits of pupils care should be takento maintain the mental growth too often stuntedby secondary school materials and methods, andan effort should be made to associate with in-quisitiveness, the desire to experiment, thewish to know 'how and why' and the like, thesatisfaction of these needs.

In the years under consideration it isalso especially important to give the pupilsas broad an outlook over the various fields ofmathej 8 tics as is consistent with sound scholar-ship.

The following is an abbreviated outline of the topics

to be covered in mathematics courses of grades seven, eight,

and nine, but no suggestion is made here for the order of

presentation.

A. Arithmetic.

1. Fundamental operations of arithmetic.

2. Tables of weights and measures.

3. Simple fractions, including eighths.

4. Short cuts in multiplication and division.

5. Easy percentage.

6. Line, bar, and circle graphs.

7. Arithmetic of the home, community, banking,

and investment.

8. Easy statistics.

1Tbid.,p. 20.

41

If there is repetition in that some of the above have

already been presented earlier, drill is to be given in

connection with new work.

The time devoted to arithmetic as a distinct subject

should be reduced. Drill need not be lessened, but more

work should be meaningfully related to other fields.

B. Intuitive geometry.

1. Measurement to "significant" figures.

2. Areas.

3. Drawings to scale.

4. Appreciation.

5. Simple geometric constructions.

6. Familiarity with geometric forms.

7. Introduction to similarity.

This work should provide an approach to and a founda-

tion for work in demonstrative geometry.

C. Algebra.

1. The formula.

2. Graphs--construction and interpretation.

3. Positive and negative numbers.

4. Equations.

5. Algebra technique.

D.I umerical trigonometry.

1. Definitions.

2. Properties.

42

3. Use of properties.

4. Use of tables.

Trigonometry is introduced earlier into courses in

mathematics in foreign countries than in the United States.

Relations of trigonometric functions need not be con-

sidered.

B. IDemonstrative geometr,.--Show the pupil what "dem-

onstration" means. Use facts previously intuitively in-

ferred, then later prove some of them. Basic propositions

should be explicitly listed and logical significance recog-

nized.

F. History and biography.--Teachers should know that

mathematics has developed in answer to human needs, need to

know leading events in history of mathematics in order to be

able to add to the interest of the pupils.

G. ptional topic.--If time permits, teachers may in-

troduce topics and processes not covered above, such as use

of fractional and negative exponents, slide rule, logarithms,

simple arithmetic and geometric progressions, interest and

annuities, and laws of falling bodies and of growth.

H. Topics to be omitted pr p ostconed.--These are better

for later courses:

1. Highest common factor and lowest common

multiple--outside simple fractions.

2. Theorems on proportion.

43

3. Literal equations.

4. Radicals.

5. Square root of polynomials.

6. Cube root.

7. Theory of exponents.

8. Simultaneous equations with more than two un-

knowns.

9. Binomial theorem.

10. Imaginary and complex numbers.

11. Radical equations above elementary type.

I. Problems.--Selection of problem material is of

greatest importance. Emphasis needs to be shifted from

formal exercise to concrete practical problems. Problems

must be real to the pupil. Relate them to other courses

in the curriculum. "General science" increases this op-

portunity.

J. Numerical coputati on , use of taml ,, ,,,.--There

should be opportunity for considerable arithmetical and

computational work. Measurement must be recognized as

approximate. Help the pupil understand the conception of

"the number of significant figures." An elementary notion

as to interpolation is desirable. Use of tables should be

encouraged.

Several plans are suggested for order of topics and

time to be consumed, but no one is recommended as superior

to the others.

44

Mathematics for Yearsjen, Eleven,and Twelve

The Committee does not recommend that the mathematics

of these years be required of all pupils. It does believe,

however, that every standard high school should offer courses

in mathematics for these grades and should encourage a large

proportion of its pupils to take them. Those pupils pre-

paring to enter college should extend their work in mathe-

matics beyond the minimum requirements.

Attention should be given the students' vocational or

other later educational needs. Material should include

"those mathematical ideals and processes that have the most

important applications in the modern world."1 2 Increasing

attention should be given "the logical organization of

material, with the purpose of developing habits of logical

memory, appreciation of logical structure, and ability to

organize material effectively."l3

"The number of important applications of mathematics

in the activities of the world is today very large and is

increasing at a very rapid rate."14 The pupil should be

impressed with the fact that need for mathematics which

arises later in life, cannot be easily met; it will be more

difficult to learn, and he will find it difficult to take

l1 d= .,p. 32. 121bid., p. 33.13lb d. 14 .

45

the time from other activities for systematic work in

elementary mathematics.

The following is a suggested outline of electives

which might be opened to pupils having completed the work

outlined for previous grades.

A. Plane demonstrative geometry.

B. Algebra.

1. Simple functions of one variable.

2. Equations in one unknown.

3. Equations in two or three unknowns.

4. Exponents, radicals, and logarithms.

5. Arithmetic and geometric progressions.

6. Binomial theorem.

C. Solid goetry.-including mensuration topics.

D. Trigonometry.

E. Elementary statistics.

F. Elementary calculus.--Calculus is not intended for

all schools, all teachers, nor all pupils in any school.

The subject is commonly taught in secondary schools in

England, France, and Germany.

G. Histor and biography.--"Historical and biographical

material should be used throughout to make the work more

interesting and significant."1 5

151bid. ,p. 38.

46

H. Additional electives.--These may be included if

desired:

1. Mathematics of investment.

2. Shop mechanics.

3. Surveying and navigation.

4. Descriptive geometry.

5. Projective geometry.

There is no suggestion as to definite order for pre-

senting these subjects, but some teachers have found it

effective to combine courses. Several arrangements are

listed, but no one is recommended as being superior to the

others.

College Entrance Requirements16

The primary purpose of college entrance re-quirements is to test the candidate's ability tobenefit by college instruction. This abilitydepends . . . upon (1) general intelligence, in-tellectual maturity, and mental power; (2) spe-cific knowledge and training required as prepara-tion for the various courses of the college cur-riculum.

Mathematical ability appears to be sufficient,but not a necessary condition for general intel-ligence. For this, as well as for other reasons,it would appear that college entrance requirementsin mathematics should be formulated primarily onthe basis of the special knowledge and trainingrequired for the successful study of courseswhich the student will take in college.

The separation of prospective college studentsfrom the others in the early years of sec daryschool is neither feasible nor desirable.

1iM.16Ibid., P. 43.

47

It is not necessary to consider, in selecting the

material for the high school course in mathematics, its

value as preparation for college courses, since all college

students do not study mathematics. Most college students,

however, study at least one of the physical sciences and

at least one of the social sciences; so entrance require-

ments must insure adequate mathematical preparation for

these subjects. It is assumed that adequate preparation

for these two groups will be sufficient mathematical pre-

requisites for other subjects.

Most progressive schools already offer mathematical

instruction as recommended by this committee, but "the

large majority of schools are still continuing the older

types of courses or are just beginning to introduce material

modifications." 1 8 Mathematical instruction is in a period

of transition.

College entrance requirements will continueto exert a powerful influence on secondary schoolteaching. Unless they reflect the spirit of soundprogressive tendeies, they will constitute aserious obstacle.

The Committee suggests:

The examination as a whole should, so faras practicable, reflect the principle that alge-braic technique is a means to an end, and not anend in itself.

20TIbid., p. 54.91% d18Ibid., p. 47.

48

The question of disciplinary value is still with us,

and the principal reason for keeping some of the subjects

in the curriculum has been that they discipline the mind.

Conclusions from various studies show that psycholo-

gists almost unanimously agree that transfer of training

exists. The amount of transfer is dependent largely upon

methods of teaching.

General standards for appointment of teachers of mathe-

matics in the better secondary schools are fairly high.

From studies it is found that public opinion is in

favor of teaching algebra and geometry in high schools.

Report of Texas Educg{ionalSurvey of 1923

The curriculum of a very large number of schools is

planned with preparation for college as the chief end in

view. The majority of teachers have little professional

experience and are influenced by the curriculum of the

college from which they have recently come.

The following table has been compiled to show the rank

of the mathematics courses taken by high school pupils, as

gathered from data of annual reports of superintendents of

counties and independent districts in 1922-1923.. rot all

21C. H. Judd, Secondary Education, Texas Educational

Survey Report, Texas Educational Survey Commission, III(1924), 89-97.

49

counties or independent districts are represented, but a

large per cent are included. Negro and white children are

included in the count. 2

TABLE 1

ENROLL }NT IN MATHEMATICS COURSES, 1922-23

Common IndependentSubjects School School Total

District District

Algebra............ 33,631 88,301 121,932Geometry, Plane.... 7,631 27,177 34,730Arithmetic......... 12,109 11,857 23,966Geometry, Solid.... 259 5,858 6,117Mental Arithmetic.. 3,125 2,143 5,268Trigonometry....... 57 3,163 3,220

Table 1 shows that algebra has a greater enrollment

than all the other mathematics courses offered. Let us

note how these subjects compare with other subjects of the

school curriculum. 2 3

TABLE 2

PUPIL ENROLLMENT IN HIGH SCHOOL SUBJECTS

Common Independent

Subjects School School TotalDistrict District

Algebra..................... 33,631 88,301 121,932

Composition................. 20,167 72,321 92,488

2 2 Taken from Table 26 of urgey Reprt, . cit., p. 90.

23 Ibid.

50

TABLE 2.--Continued

Common IndependentSubjects School School Total

District District

Ancient and Medieval History. 22,424 43,081 65,505Modern istory............... 8,597 27,292 35,889Plane Geometry............... 7,553 27,177 34,730Physical Geography........... 18,382 16,117 34,499

pani.................... 1,830 31,527 33,357Latin............ ...... 2,962 29,807 32,769

American Literature.......... 6,372 24,367 30,739American istory............. 7,796 19,363 27,159Civil Government............. 11,916 14,781 26,697English Grammar.............. 4,279 21,383 25,652Physiology and Hygiene....... 10,432 14,509 24,941Arithmetic.................. 12,109 11,857 23,966English Literature........... 1,691 20,631 22,322S pelling................... 5,071 15,669 20,731Home Economics............... 420 18,600 19,020Agriculture.................. 11,734 7,205 18,939General Science.............. 2,581 11,536 14,117Writing....................... 5,427 8,077 13,504Drawing...........e"....... 4,756 5,705 10,461usic.,.................... 2,361 7,464 9,825

Physics.....#............... 1,302 8,497 9,799ology#.. ............ 455 5,071 9,526

Public Speaking.............. 2,342 6,434 8,776Chemistry.................... 388 8,321 8,709English History.............. 618 7,062 7,680Descriptive Geography........ 4,176 3,047 7,223Manual Training.............. 177 6,553 6,730Reading......."............. 1,940 4,342 6,282Typewriting.................. 187 5,940 6,127Solid Geometry.9 .**........ 259 5,858 6,117History of the United States. 2,039 3,907 5,946Mental Arithmetic............ 3,125 2,143 5,268Economics....... .... ....... 207 4,847 5,054Bookkeeping.................. 364 4,061 4,425Stenography.................. 108 3,655 3,763Language Lessons.............. 1,351 2,217 3,568Commercial Law................ 797 2,728 3,525Trigonoretry............*... 57 3,163 3,220General History.............. 1,513 1,372 2,885NatureStudy................. 2,171 637 2,808renh., ..................... 26 2,090 2,116

51

TABLE 2--Continued

Common IndependentSubjects School School Total

District District

History of Texas............... 1,046 795 1,841German....................... 1,486 331 1,817Hygiene and Home-Making....... 188 1,625 1,813oo .c gy#............. 777 904 1,681

Botany....................... 28 1,613 1,641Cotton Classing.............. 426 472 898ardenng............ ....." s582 283 865

Zoology....................... 169 627 796

Trades and Industry........... ... 786 786Psychology.................... 432 320 752Methods of Teaching........... 619 65 684Schol Management............. 107 102 209Printing .".f. ..... .a........ ...... .. . 195 195History of Education......... 57 26 83

Algebra hasa registration far beyond any othersubject. It is a first-year requirement in everyhigh school in Texas . . . . The aggregate of regis-trations in all sciences is less than the registra-tion in the single subject algebra.

It is the judgment of the Survey Staff that thisexcessive emphasis on algebra is wrong. By puttingthis subject into the curriculum and making it anabsolute requirement in the first year, a great manypupils who fail are prevented from getting into coursesin history and science which from every point of viewwill be more useful in later life and are just as goodfor general mental training as algebra.

The Survey Staff finds that the powerful in-fluence of the State University is in large measureresponsible for the emphasis on algebra. That in-stitution insists on mathematics as one of the re-quirements in its own curriculum and gives on alloccasions as much encouragement as it can to the highschools to continue the present requirement. Someway ought to be found to remove this extravagant em-phasis. One suggestion which may be worth consideringif algebra cannot be eliminated, is to transfer it

52

from the first year to the third or fourth yearor to change it radically by combining it withgeometry.

A study of figures on the rate at which pupils drop

out of school just before and after the high schools try to

impose algebra, ancient history, and composition shows the

judgment of Texas young people concerning the highly aca-

demic and conservative offerings of Texas high schools.

The pupils do not want what is required of them; so they

leave school. There was an approximate drop of between

12,000 and 15,000 boys and girls from the high school en-

rollment between grades seven and nine during the school

year 1922-23. The small schools failed to an even greater

degree than the larger ones to supply the desires of their

pupils. In these respects, "Texas needs an awakening." 2 5

Summary of Theories UnderlyingRecommendations

The study of mathematics causes the pupil to realize

the necessity of stating facts with care.

The training received from working with mathematical

symbols will be valuable to the student after leaving school.

Activities of gigantic industries are directed by means of

symbols--the telephone girl, the man in the signal tower,

the president of a railroad or other great corporation.

Every business makes use of the graph in statistical reports.

2 51bi6., p. 97.24_I id, P. 91.

53

The department of mathematics is a field forexcellent service. There is no doubt but that ateacher's success is readily measured in exactterms by the skill of his pupils in the masteryof such definite tasks as2 gre presented in alge-bra and geometry courses.

Education aims to fit the student for his environment,

and

every man in the course of his life meets manyproblems which demand logical thinking for theirsolution, and consequently his education mustprovide some training in clear and logicalthought. The subject which above all othersis fitted to give this training is mathematics.. . . I am unable to see any essential dif-ference between mathematical reasoning and anyother kind of reasoning; the mind goes throughthe same kind of process in both cases.

Another reason for studgng mathematics isits immense practical value.

Of ultimate value to humanity are the applications of mathe-

matics to the various sciences.

. : . Mathematics furnishes a clear and con-cise symbolical language in Which the results ofscientific investigation can be expressed. Theimmense value of2g good symbolism can hardly beover-emphasi zed.

Algebra is indispensable as a tool for further mathe-

matical work. Geometry is the foundation of architecture

2 6 Anni e D. Durham, "'Mathematics as a Language," TheTexas Mathematics Teachers' Bulletin, IX, No. 3 (May 22,1924 , pp. 30-31.

2 7 Paul M. Batchelder, "The Place of Mathematics," TheTexas Mathematics Teachers' Bulletin, IX, Yo. 1, #2342(Tovember 8, 1923), pp. 28-40.

28Ibid., p. 35.

54

and decorative design. Both algebra and geometry "afford

the best possible means for developing ideas of form, ac-

curacy, logical sequence, and self-mastery."29

Foremost as a reason for studying geometryhas always stood, and will always stand, thepleasure and the mental uplift that comes fromthe contact with the great body of human learn-ing, and par+ cularly with the exact truth thatit contains.

One cannot be educated without knowing something about a

great many things. We learn some things for the pleasure

of knowing.

The student should be led to feel that hisaccomplishments are well worth his time andeffort regardless of practical applications.Culture is worth consideration.3

29S. I. Sewell, "Vitalizing the Teaching of Algebraand Geometry," The Texas Mathematics Teachers' Bulletin,IX, I o. 2, #2406, (February , 1924), p. 6.

3 0 Quoted from David Eugene Smith, Ibid., p. 6.

31Sewell, . cit., p. 7.

CHAPTER IV

CURRENT PRACTICE AND THEORYf

"Current Practice and Theory" shall be used herein to

pertain to the period beginning with 1923 and extending to

the present date.

Recommended Course of Studyin Bulletin 2431

When the bulletin was published a tentative program

was suggested. Algebra had just been reorganized into "a

one-year course including quadratics, to be followed later

by a half year advanced course for those pupils expecting

to attend college,"2 and the geometry course was under

revision with a new text expected to be adopted soon.

It is suggested that the courses of study result

from the combined effort of teachers and administrative

officers.

The authors tell of an experimental course that had

been conducted the year preceding, and its success caused

recommendation that the colleges accept one and one-half

1H. F. Alves and V. B. Brown, "Teaching of athematics,"Bulletin of the State Department of Education, Texas HighSchools, Austin, Texas, September, 1928.

21bid., p. 3.

55

56

units credit from the high schools. One year of algebra had

been given in one-half year. The experiment showed the

tendency to produce more efficient results in mechanics and

reasoning in both arithmetic and algebra.

Objectives set out in the bulletin are these: 3

1. To give pupils command of the mathematicalprocesses which are necessary for a complete ad-justment to the environment in which they live.

2. To give a closer contact with life andsociety, by use of material in itself useful.

3. To furnish and increase the incentive forstudying mathematics for the love of the subjectand the appreciation of its beauty and power, thusproviding a way for the worthy use of leisure time.

4. To give an introductory knowledge of thebroad fields of mathematics with materials thatform contacts with life, thus developing adapta-tions and interests or the lack of them.

5. Training in 'functional thinking.'

Teachers are to formulate specific objectives in each

case and make the classroom activity purposeful.

Recommended Course of4 Studyin Bulletin 254

The author recommends that the traditional two years

of high school mathematics be taught in one and one-half

years to students not purposing to go to college. It was

suggested that undue attention had in the past been given

least common multiple, highest common factor, fractions,

progressions, cube root, and a variety of methods of solving

3Ibid., p. 3.

M. B. Brown, "The Teaching of 1 athematics," Bulletinof State Department of Education, Austin, May, 1929.

57

quadratic equations. It is suggested that one method be

learned thoroughly, and that every problem in every set need

not be done. The mathematical training should be for reason-

ing and not merely for imparting information. It is sug-

gested that the solution of quadratics by factoring be taught

and that only one method of completing the square be given.

Geometric progression is to be omitted, and the geometric

phase of ratio and proportion is to be left out of algebra.

Recommended Course of5Studyin Bulletin 293~

One and one-half years of algebra are suggested for

those pupils who are not going to college. On account of

chiding on the part of college authorities for failure of

students just out of high school, it should be required

that those students who plan to go to college review algebra

during the last half of the last year in high school. Less

time should be given to drill, and more time should be given

to reasoning.

The author suggests board work and stress on good Eng-

lish. The teacher is warned not to give the pupil credit

merely for "trying," as mathematical work is either right or

wrong. The teacher is urged to teach the pupil how to study

and how to think,

5M. B. Brown, "Teaching of Mathematics," Bulletin ofState Department of Education, Austin, September, 1931.

58

Recommended Course of Studyin Bulletin 3256

Because the subject matter of second year al-gebra is little used by the average citizen, itis thought well to suggest that the classes inthat part of the work be limited to such as haveshown real mathematical ability in the work of thefirst year and who know that they are to attendcollege.

It is suggested that all new steps in mathematics be

discovered by the pupil, the teacher assisting. Drill is

not to be neglected, and new terms are to be explained as

they appear.

Recommended Course of8Studyin Bulletin 379

"In preparing this course, the unit of understanding

has been used as a basis of organization."9 The order is

only suggestive. No courses are set out as being required

at any level. The child's experience determines the ac-

tivity.

Arithmetic work can be motivated and made meaningful

through arithmetic of the home. Activities should be select-

ed, taking into consideration the interests, abilities, and

plane of the group. Drill must come only after meanings

6M. B. Brown, "Teaching of Mathematics," Bulletin ofState Department of Education, Austin, October, 1933.

7Ii..,p. 48.8 State Curriculum Executive Committee, "Teaching Mathe-

coati s, " Bulletn of State Depart .n o~f Education, Austin,December 20, 1937.

Ibid., p. 13.

59

have been developed. "We learn from insight and not from

repeated drill.1O

Mathematics should be conceived as a modeof thinking, as a tool for thinking, and as theability to think about the social situations oflife in quantitative terms. Emphasis should beplace n how to think rather than what tothink.

The child should be a part of the wholeprocess; he should participate in the planning,the purposing, and the executing. He should beencouraged to help choose the unit of work andselect the activities with which to develop it.He should understand the purposes of the entirecourse and of each unit and just how the programis contributing to his ability to meet lifesituations more effectively. }uch of the dif-ficulty with mathematics lies in the fact thata great deal of the time many of the pupils aremore or less following a blind form, not reallyKnowing the why and wherefore, not challenged towide-awake thorough thinking about the situation.

Conscious effort should be made to develophabits of accuracy, of neatness, and of estimatingand checking results.

The development of self-criticism and gglfevaluation should be stressed at all times.

The only justification any subject can offerfor its place in the curriculum is its contribu-tion to a life of meaningful experiences . . . .Any hope of transfer lies in the extent to whichthe classroom and future experiences have commonor identical elements so associated as to bgrecalled and recognized in the new setting.

There is more danger of having too littlethan of having too much informal work. Some ofthe pupils who do not mature enough to do ab-stract thinking now will do it later if suppliednow with the experiences upon which it must

1 0Ibid.., p. 13. 1 bi d., pp. 13-14.

1 2 bid., p. 14. 11b~id., p.60.

60

ultimately be based. The amount of informal workneeded will depend upon the amount that has alreadybeen done in general mathematics and the type ofindividuals composing the class. But at any rate,the informal is a very important and commonly usedtype of thinking in transacting the affairs ofordinary life.

The laboratory is a very essential part of the teaching

of intuitive geometry.

It is during the eighth and ninth years that many

pupils drop out of high school. In general these pupils

have had trouble with or failed to achieve profitable

results in algebra or geometry. A course in general mathe-

matics is recommended to be more profitable. Well-organized

and effectively taught courses in general mathematics might

do much to correct the problem. These courses should be

given in such a way that the student will acquire some

notion of algebra and geometry and their application in

the economic and industrial world. Such a course might

tend to influence the lengthening of the school life of

the child. Success rather than failure in the first years

of high school might lead to a desire for further study.

Arithmetic, algebra, and geometry should be included, but

it is difficult to say which, if any, should predominate

the others.

Algebra should be taught as a means, not an end. It

can be made a method of simplifying thought processes.

1 4 Ibid., p. 69.

61

Algebra and arithmetic are closely related, but the term

algebra is often terrifying. Reasoning is definitely a

part of our thought processes. Memory is required to a

certain extent in algebra, but no material progress can

be made in the subject without the use of reason. The

student must think through life problems to logical con-

clusions. Reflective thinking is important in algebra.

Intuitive geometry has been recommended for junior

hig h school and demonstrative geometry for placement in

the senior high school. One is not a preparation for the

other, however. Intuitive geometry allows greater activity.

Emphasis should be placed upon a definitionwith a meaning rather than upon a precise state-ment of words. A rather definite notion of themensuration formulas and their appliction shouldbe an outcome of intuitive geometry.

All definitions so learned should be consistent with those

used in demonstrative geometry. Intuitive geometry should

be a vital part of the child's educational experiences.

Demonstrative geometry tends to give thestudent some notion of space, a group of geo-metric facts which are useful in the socialand economic world, a notion of mensuration,and a list of mensuration formulas and someapplication of tlgm in the economic and in-dustrial fields.

The word geometry means to measure the earth, "but the key

to demonstrative geometry is demonstration."1 Industry

15bid., p. 80.17 l1.

62

today is interested in the ability of the prospective em-

ployee to think.

A prospective employee who is master of thefundamentals and who can start from there andthink through his problems to a satisfactory con-clusion can be placed in a company sponsoredtraining school and readily made to order forthe company's purpose. Nowhere else in theelementary or secondary school curriculum willthe student meet the question of logic quiteso forcefully as in demonstrative geometry.It is here that te first meets the notion of arigorous proof.

Recent investigations prove

that there is transfer of training from geometryto other fields provided the elements of simi-larity between the type of logical thinking usedin geometry and that used in the other fieldsare definitely shown. here is transfer whenwe teach for transfer.

Many children will not profit by study of geometry above the

intuitional course. It is not a very easy task to select

those students who could profit from further study. Progress

in intuitional geometry would not determine one's ability

for handling demonstrative geometry.

Success in algebra would be of very littleassistance, if any, in predicting progress ingeometry. If prognostic tests could be developedthat would show who could profit by demonstrativegeometry, we should well along toward a solu-tion of the problem.

Newton was said to have been a poor student of mathematics

until he took up the study of geometry. A student should

SIbia., p.1. 191bid. owsmW

not be denied the privilege of attempting geometry if he is

eager to do so.

The number of theorems used in demonstrative geometry

is decreasing and the number of exercises and originals is

increasing.

There is a strong movement away from theprinted proof of the theorem and to substitutea few suggestions or hints in order that thestudent may develop the proof himself. Theexercises have not only been increasing innumber, but a larger per cent of them are con-siderably easier as well. Since a real testof the student's success in geometry is hisability to solve originals, it is well2{'hatthe tendency is toward more exercises.

There is a contention in certain sectionsof the country to combine plane geometry andsolid geometry into a one-year course as asubstitute for the present year of plane ge-onetry and one-half year of solid geometry. 2 2

The problem confronting the teachers ofmathematics in Texas is not more geometry,but better geometry; not more mathematics,but improved teaching of mathematics to theeffect that the significance of the coursesmay be more thoroughly appreciated. Mathe-matics is essential to life and progress andvith less worry about what is going to happento mathematics in the curriculum and moreemphasis upon the proper presentation ofwell-organized, meaningful courses, the sub-ject of mathematics will sell itself to stu-dents and patrons.3

Materials available may be used in activities which

the children are allowed to choose and plan. Interest leads

to development of meanings. Laboratory and experimental

221bi .#- Iid., p. 82.o 23I_ *id.

64

methods are both good in plane and solid geometry. Proofs

should be brief. Students should learn how to accept,

reject, and evaluate parts of the proofs. Any simple ex-

ercise which develops geometric reasoning is commendable.

The second year or half-year algebra course is now

designed to be elected by students preparing to cork in

sciences that require mathematics for a foundation. It

is also recommended for students who have the ability to

master mathematics and to enjoy it.

General Statement

It is well to note that whereas in the bulletins for

years 1928-1933, the prescribed mathematics courses were

algebra, plane and solid geometry, trigonometry, and ad-

vanced arithmetic, the courses outlined in the 1937 bulletin

are termed general mathematics, algebra, and geometry.

Evident Procedures Resulting from Nationaland State Recommendations of 1923

Requirements in mathematics as set out by the National

Committee in 1923 became the basis for college entrance ex-

aminations, which the secondary schools now help compile.

This step is declared by David Eugene Smith2 4 to be the

most potent factor in the reform of the teaching of mathe-

matics of the present time.

24 irst Yearbook of the ational Council of Teachersof Mathematics, (1926), p. 9.

65

It was suggested both by the National Committee and

by the State Curriculum Executive Committee that algebra

be looked upon chiefly as a means to an end. This state-

ient "struck out a large amount of entirely useless and

uninteresting work that had cumbered up the inherited

course.

Due to the opportuneness of planning the junior high

school course of study at the time of the publication of

the work of the National Committee on reorganization, it

probably highly influenced that curriculum. 2 6

There has been some experimentation. Some outstanding

schools have "shown conclusively that it is possible to

give their pupils an interesting and modern course in mathe-

matics and at the same time prepare them to pass college

entrance examinations.",27

The Committee recommended change in textbooks, in order

that the child might meet new material on the ground of his

own language and experience. ". . . Textbook writers on

junior high school mathematics organized the work into units

and introduced considerable intuitive geometry and some

algebra in the seventh and eighth grades." There appeared

25Ibid., p. 10.

2 6 W. D. Reeve, Fourth Yearbook of the rational Councilof Teachers of Mathematics, (1929), p. 146.

27Ib . , p. 147. 2 8 1bi., p. 146.

66

some that "attempted to lead the child, step by step,

through the maze of difficulties attendant upon the first

study of any new operation."29 Several sample problems

were used. Explanations were detailed in child language.

The pupil was led to independent work.

Since 1930 the progressive education movement is

being felt in general, "and there is a growing tendency

to organize instruction in activity units instead of the

traditional formal classes." 3 0

In the present course of study arithmetic is taught

in the junior high school. It has one of its objectives

a broad outlook upon the whole field of mathematics, as

recommended by the National Committee. General mathe-

matics courses are offered in the eighth and ninth grades.

The ninth grade course consists of a combination of arith-

metic, algebra, and geometry. First year algebra is placed

in the tenth grade, while second year algebra is offered as

an elective and may be an entire year or a half-year course

in the eleventh grade. It is designed especially for those

students who are preparing for work in the sciences and

professions which require a mathematical foundation, and

for those students who have ability to master and enjoy

295. V. Studebaker, "New Trends in the Teaching of

Mathematics," 'athematics Teacher, XXXII, Yo. 5, p. 198.

Ibid., p. 198.

67

mathematics. This course includes geometry and trigonometry,

which may be given in any combination and amount the teacher

sees fit to use. There is evidence of teacher freedom seen

in the Texas course of study, and as recommended by the

National Committee.

The Society for the Promotion of Engineering Education

realizes that there is a movement in various parts of the

country to postpone or abbreviate the mathematics courses

usually given in secondary schools. This movement does not

recognize the fact that those courses are essential pre-

requisites for future training of scientific and engineering

students. The members of the Conference on Mathematics of

the Society for the Promotion of Engineering Education have

made record of a recommendation 3 1 that there be no post-

ponement in mathematical education in the secondary schools

for those who are to seek careers in science and engineer-

ing. They feel that a full four-year course of mathematics

should be made available for capable students, beginning in

the first of the last four years in the secondary schools.

This course should begin with college preparatory algebra

and include thorough work in trigonometry and solid geometry.

This, however, does not mean that university preparatory

courses be required of all students.

3 1Society for the Promotion of Engineering Education,"A Resolution by the S. P. E. E.,1" The American Mathematicalonhly, VII (October, 1940), p. 588.

68

Recommendations of TexasAuthorities

In this section, the views and recommendations of Texas

high school teachers and administrators and college officials

are presented. Some new trends in Texas schools are set out.

According to C. W. Webb of El Paso,32 the trend is now

away from arithmetic drill units which are usually small and

unrelated. The incidental theorists hold that the child

will get the arithmetic he needs if he learns it only in-

cidentally as he studies other subjects, and it will grow

out of the natural activities and interests of the child.

This theory is gaining ground. Wise teachers, however, do

not confine their activities wholly to incidental work.

They know that meaning is essential in learning arithmetic,

but at the same time social situations may be planned and

the pupils led to see the mathematics in many more situa-

tions. According to investigations made by Dr. Hanna and

others it has been shown that planned instruction is neces-

sary to supplement incidental learning.

There is a tendency to select the content matter on the

basis of social usefulness and the possibility of effective

use,

There is a decided trend to defer concepts and processes

until the need is evident and matches the child's maturation.

32C. A. Webb, "Significant Trends in the Teaching ofArithmetic," The Texas Outlook, (October, 1939), Texas StateTeachers Association, pp. 33-35.

69

It is poor judgment to use the same method of instruc-

tion at all times, but there is no best method for mathe-

matical instruction. Dr. Ettlinger of the State University

warns that "there is grave danger in building the course

entirely around unit projects, for that may make it merely

vocational with no depth at all and only a fair amount of

surface.133

Tests are being used in El Paso to determine the

readiness of the child for certain types of instruction.

These are diagnostic and remedial tests that are developed

in the locality for local use.

Emphasis on speed has shifted to accuracy, neatness,

and meaning.

The development of a mathematical vocabulary is a

noticeable trend in many communities.

There is a tendency toward planning units of work to

build certain concepts. This trend is seen in the work of

the Fort 'Worth teachers and in the Texas course of study.

Another trend that is illustrated in the El Paso system

is the remedial instruction that is provided both at the

elementary and secondary levels.

There is a tendency to postpone algebra and geometry.

Social mathematics courses are offered the non-college groups.

33H. ,. Ettlinger, "Numbers or Mathematics?" The TexasMathematics Teachers' Bulletin, XX (February, 19367 7 p. 9.

70

The following 'tendencies are also noted:3 4

1. The more regular appearance of consumermathematics courses.

2. The appearance and use of the unit methodof instruction.

3. Increased use of the laboratory by mathe-matics teachers.

4. The increase in quantitative experiences,5. An increase in descriptive experiences.6. A wide use of reports.7. More extensive reading in mathematics.8. Adjustment of materials to individual

differences, interests, needs, and capacities.9. The study of the processes involved rather

than the seeking of an answer.10. That of enriching arithmetic and providing

more attractive textbooks.11. The movement from subject matter to child-

hood experiences which deal with thinking.12. More and more attention is being devoted

to the larger concepts, which, after all, aremore important.

While there is no agreement as to the best method of

teaching mathematics, Warren A, Rees, of Houston Junior

College, asserts that a majority of the "students should

acquire a love or at least a wholesome respect for the

subject.,35

Rees continues by giving a few admonitions concerning

the high school mathematics course.

There should be sufficient time allowed in the elemen-

tary grades for the student to master the four fundamental

operations.

34Webb, o.. '"; 9.,pp. 34-35.

35"Suggestions for a Minimum High School MathematicsCourse," The Texas Mathematics Teachers' Bulletin, XVIII,No. 1, (February, 1934), 6.

71

Students should realize that mathematics is a logical

science.

Students must be careful to use good English in the

mathematics class. All statements should be complete sen-

tences.

It is better to do a few things well than to attempt

too many things.

Students should realize that mathematics is not static

and that there are many things about mathematics which no

one knows. Acquaintance with at least a few periodicals

devoted to new mathematics would be very helpful to the

student.36

That the secondary school does not prepare the student

for college mathematics has been shown by a number of

studies.37 The number of failures in mathematics is

greater than in the other subjects and almost twice the

number of failures in English, French, and history. Sherer

believes this is because students are not inspired in high

school; there is no urge or desire to explore. "The desire

to investigate is perhaps the most important condition for

36 Ii. , pp. 5-9.

37Charles R. Sherer, "Does the Present High SchoolCurriculum in Mathematics Prepare Students for College?"The Texas Mathematics Teachers' Bulletin, VWIII, I1o. 1,TTebruary, 1934), pp. 43-48.

72

preparation for college."3 8 Dr. Nathan A. Court believes

that the greater number of failures in mathematics is due

to the fact that it is easier to rate a student, at least

more exactly, in mathematics than in the other subjects.

The pupil either knows or he does not know mathematics.39

Dr. Reeve says that at its best, mathematics needs no

defense. mathematics is at its best when teaching is at

its best. Slow progress results from lack of preparation

of teachers. Texas does not require any semester hours of

mathematics for teachers, "Not one-third of the teachers

of mathematics in Texas have majored in mathematics." 4 0

Few Texas mathematics teachers belong to the national or-

ganization. In 1937 only 122 Texans were members of the

National Council of Teachers of mathematics.

Our State Department and State Board of Edu-cation have taken a step forward in requiringthat high school teachers must teach in theirfields of preparation and that all teachers mustattend summer school in order to keep up withthe more modern ideas of teaching . . . . Wefeel that this movement will lead to ggtterteaching of the high school subjects.

38Ii., p. 45.

E. E. Heimann, "Modern Tendencies in the Teaching ofMathematics," The Texas Mathematics Teachers' Bulletin,XIX, No. 1 (February, 1935), p. 75.

0Elizabeth Dice, "Present Trends and Their Significancein the Teaching of Mathematics in Texas," Texas mathematicsTeacher' ulletin, :XI (February, 1937), p. 54.

irs. Harry Brewton, "An Attempted Revision in HighSchool Algebra," The Texas Math ematics Teachers' Bulletin,XIX, No . 1, ( February , 1935), p . 12.

73

Mrs. Brewton carried on an experimental course at her

school in HemphillC' for those children who were not going

to college. It might be called community mathematics; it

was for the purpose of developing children for citizenship.

We are trying to show the children that theydo use mathematics in everyday life, to stimulatethe child's interest, and to cultivate an apprecia-tion of the fact that the men of ti streets coulduse algebra if they only knew how.

The spirit of every mathematics classroomshould be one of adventure and exploration, andwe should give the children opportunities ofassuming a more creative role. After all, theamount of algebra covered is of minor importanceas compared with the need of making the pupilsconscious of the quantita ve aspects of theworld in which they live. -

Mrs. W. E. Odom., head of the mathematics department of

Allan Junior High School, Austin, Texas, in 1923 developed

a course in intuitive geometry for the seventh grade.I45 t

was used in the seventh grade due to its being introductory

to algebra which was being offered in the eighth grade.

The geometry course was observational and carried on with

laboratory method. It was closely related with arithmetic.

Mrs. Odom is of the opinion that the mastery of the funda-

mental processes does not require a long period of time.

This course gives children opportunity to use the inquisi-

tive and investigating mind. The mathematical vocabulary

42Ibid., pp. 10-17. 4 3 Ibid., p. 13. 44Ibid., p. 17.

4 5 "Geometry in the Seventh Grade," The Texas Mathe-matics Teachers' Bulletin, VIII, No. 3, TIay, 1923T, 31-37.

74

and the mental imagery of geometrical forms are enlarged,

These are especially beneficial to those children who will

drop out of school before studying demonstrative geometry.

Pupils are led to discover geometric truths through the

use of geometric instruments.

It is the opinion of Genelle Bell of Beeville and

B. F. Holland of Austin that in general teachers of geom-

etry are making at Least three types of mistakes in teach-

ing.46 They are attempting to teach without adequate

training. They are placing too much emphasis upon informa-

tional and computational aspects of geometry and too little

upon methods or processes. They are failing to make the

types of thinking involved in geometry carry over into

other types of problems.

The new concept of geometry teaching that is now

evolving is a functional concept, which regards geometry

as a laboratory subject.

The functions of geometry have been set out by the

National Committee on the Reorganization of Mathematics and

the State Curriculum Executive Committee of Texas.

The method used in connection with the plan is such

that no textbook is used except as a reference or for com-

parison or illustration. The class is plunged into argument

46G. Bell and 3. F. Holland, "New Horizons in Geometry,"The Texas Outlook, (June, 1939), pp. 44-45.

75

over the meaning of some word. The need arises for accurate

definition, and it is agreed that definitions are essential.

Sentences containing assumptions and conclusions are an-

alyzed. The class may investigate assumptions of communism,

fascism, or some religious creed. Then it becomes neces-

sary to investigate what makes a proof and the types of

reasoning involved.

Results may be constantly evaluated by means of many

types of criteria,

Now let us note some of the outstanding general trends

in the teaching of mathematics and the significance of some

of them. 4 7

1, "Requiring blind memory work is decreasing."48 We

spend about one-seventh of the time on factoring that we did

ten years ago and memorize less than one-seventh as much.

2. "Crowding the work is decreasing."49 We are taking

time to think through a few fundamental processes.

3. "Some why instead of all how is a favorable trend

in the teaching of mathematics."50

4. "Informal work is increasing."5 1

5. "The growth of general mathematics is noteworthy."52

Texas adopted a textbook in that subject in 1936. General

4 7Elizabeth Dice, "Present Trends and Their Significancein the Teaching of Mathematics in Texas," The Texas -matics Teachers' Bulletin, XXI (February, 1937), 42-55.

48Ibid., 43. 4 9 Ibid., 44. 5 0 Ibid., 45.511bid., 46. 521bid., 47.

76

mathematics courses are endorsed by Texas Christian Uni-

versity and North Texas State Teachers College.

General mathematics can be easily introduced in those

schools having a complete junior high school program. Other

Texas cities can follow the plan supported by mathematical

leaders all over the United States. General mathematics is

suggested for the eighth grade, beginning algebra for the

ninth grade, plane geometry for the tenth grade, and in-

dependent half-year courses for the eleventh grade. These

half-year courses may be solid geometry, trigonometry, ad-

vanced arithmetic, and advanced algebra. Pupils who like

mathematics may elect as many as they like of the last

four. M{ost colleges offer trigonometry, but fewer offer

solid geometry; therefore, it seems that it is well to

elect solid geometry rather than trigonometry--if one

cannot get both. The Tyler High School has not offered

trigonometry since 1937 for that reason.

Eighth grade general mathematics should be required

of all students, but no more need be required for those

students not going to college. Tinth and tenth grade alge-

bra and geometry should be required of the college group.

There is a rapid trend toward no required mathematics.

6. "The mental age of the child is being considered." 5 3

Some pupils are immature when introduced to algebra and

5 3 Ibid., 49,

77

therefore fail first year algebra several times, but do

well in fourth year algebra. Capacity should be measured

by the jay a person works and not by the kind of work he

does.

.7. "New mathematics is being weighed." 5 4

8. "Less mathematics is being required for high school

graduation and for college entrance, and fewer pupils are

electing the non-required courses." (This is not a posi-

tive trend for mathematics.) Texas schools require as few

as no units and as many as four units of mathematics for

graduation. "The State Department of Education notes a

decrease in t he number of pupils who are electing mathe-

matics; so do Dallas, San Antonio, Fort Worth, and Houston." 5 6

Austin High School, in keeping with the decrease in entrance

requirement of the University of Texas has since 1930 de-

creased its requirements from three to two units. There are

no required mathematics courses in the San Antonio or Fort

Worth schools, and Houston's requirement has been decreased.

Most colleges have required for entrance fewer mathematics

courses in the last five to ten years. If the trend con-

tinues, by the year 2000 practically no mathematics will be

required for college entrance. In 1800 Harvard required no

mathematics, and Yale and Columbia required only arithmetic,

55I5i..2.54 Ibid.,54. 56 Ibid .

78

By 1900 most colleges required at least three years of

mathematics for entrance. The pendulum began swinging

back again in about 1925.

9. "The contribution of improved textbooks." 57

10. "Units."58

11. "OWjectives."59

12. "Junior high school influences."6 0

13. ', . . The poor teaching of mathematics is not a

trend but a standstill.",t61

There are still a few schools in Texas where the

traditional subjects are offered with traditional aims and

purposes, but in general Texas schools incorporate the ob-

jectives and aims of the progressive schools. It is

realized more than ever that the purpose of education is

training for citizenship. New education considers the

whole child.

57Ibid.,53. 58Iid., 54,

59Ibid., 54. 60Ibid., 53. 61Ibid.

CHAPTER V

SUU A Yr

Changes as Covered in This Study

Until about 1900 mathematics was taught as a formal

discipline on the theory that the study of it would create

reasoning power. Colleges required not less than three

years of mathematics for entrance. The subjects of mathe-

matics were water-tight compartments. Tradition blocked

progress. A study of enrollment in the secondary schools

of the United States reveals the following: 1

1900 -- 519,000

1910 -- 915,000

1920 -- 2,200,000

1930 -- 4,400,000

1935 -- 6,100,000

In 1900, 56.3% of the public high school pupils were studying

algebra, and 27.4% were studying geometry.

In 1935, only slightly more than 25% of the pupils were

studying algebra, and less than 15% were studying either

plane or solid geometry.

1H. R. Douglas, "Let's Face the Facts," Tie YathematicsTeacher, XXX (1937), 56-62.

79

80

Many universities and colleges no longer require en-

trance credits in mathematics. As a result the number of

high schools requiring mathematics for graduation is decreas-

ing, and fewer students are electing to study mathematics.

Changes have been influenced by the work of the Inter-

national Commission, the College Entrance Examination Board,

and the National Committee. Closer relations have been

brought into existence between the secondary schools and

the colleges. The junior high school movement has done much

to bring mathematics within the reach of all pupils. Much

improvement has been brought about by the schools of educa-

tion. Textbooks have been greatly improved. Aims that guide

in teaching each branch have been clarified. The right of

children to see the purpose of their studies has been recog-

nized. There has been notable advance in the testing of

pupils' abilities and achievements. More progress has been

made in the teaching of mathematics than in any other field

of elementary or secondary education.2

During the past twenty-five years we have been able to

detect greater appreciation of mathematics on the part of

the general public. t

2David Eugene Smith, "A General Survey of the Progressof Mathematics Courses in Our High Schools in the Last Twenty-Five Years," First Yearbook of the National Council ofTeachers of athematics, 1926)o ,pr. 30-31.

3Sl aught , =_. .st., p. 186.

81

General mathematics was introduced about twenty-five

years ago mainly for the benefit of the weak pupils. It is

like a layer cake--a little arithmetic, a little algebra,

a little geometry, arithmetic, algebra, etc.

Another type, termed integrated mathematics,4 has

recently been introduced which does away with compartments.

It is not necessarily intended for weak pupils; yet it

calls for no more ability than compartment mathematics.

It emphasizes the function of mathematics; it preserves

the unity of mathematics and its application by emphasizing

not only the relation of the various branches of mathematics

to each other, but also their application to related fields

in other subjects. Application is made to business, statis-

tics, life insurance, installment buying, small loans, music,

economics, and science.

In the future mathematics will be taught for its

genuine social significance. Leanings rather than drill

will be the object of instruction. 5

Preparation on the part of the teachers has been in-

adequate, but steps have been taken by boards of education

to raise requirements.

John A. Swenson, "The Newer Type of Mathematics Com-ared with the Old," School Science and Mathematics, XXVIII1938), 107-112.

5John A. Studebaker, "New Trends in the Teaching ofMathemati cs, " The Vathematic Teacher, XXXIX (1939), 201-202.

82

From the above, it is evident that the ten-dency in the teaching of mathematics during thepart of the present century already spent hasbeen somewhat away from the manipulative and themechanical side to a saner and more useful kindof mathematics which can help the pupil t9 abetter understanding of the modern world.0

Texas' Requirements in Relation toNational Recommendations

The Texas course of study for mathematics, published in

1937, parallels the suggestions of the Committee on reorgani-

zation as set out in 1923. The objectives, procedures, etc.

more than slightly incorporate the aims and standards as

outlined in the Fifteenth Yearbook of the National Council

of Teachers of Mathematics on The Place of Mathematics in

Secondary Education, published in 194, as a report of the

Joint Commission of the Mathematical Association of America

and the National Council of Teachers of 1Mathematics. Both

groups hold that the study of mathematics is useful and

helpful in intelligent adjustment of the individual to the

present-day world. Parallel suggestions are given in each

outline for work of years seven through twelve. The same

courses are recommended for the different levels. The same

views are held in regard to evaluation and testing, informal

class work, conceiving of mathematics as a mode of thinking

throughout the entire secondary level, and suggestions as to

method of reporting progress of the child. Both bodies

6 Swenson, p2. Cit., 112.

suggest the same courses for students who will follow

certain lines of :ork. either body favors segregating

those students who will or will not go to college.

herein We Are Failing: Recommendations

The Commission suggests that mathematics should be

required of all students through the ninth grade. Texas

requires that two credits in mathematics be presented for

graduation by students in high schools of less than

$1,000,000 taxable valuation and fewer than one hundred

pupils. Larger schools may not require any credits in

mathematics. The Commission recommends that the larger

schools offer advanced courses and college algebra in the

twelfth grade, thereby giving consideration to the student

who may take up the study of engineering. It suggests that

mathematical statistics and mathematical theory of finance

are important for the later secondary level. The Commission

emphasizes the importance of the history of mathematics and

its close association with the teaching of mathematics.

Texas should incorporate this suggestion.

The Commission advises that students be led to see that

acquaintance with mathematics helps one live more intelli-

gently in a scientific world. There is an increasing use of

mathematics in modern life, and a growing demand for persons

having adequate preparation in the subject.

The Commission warns against letting the provision for

84

slow pupils and those of low intelligence determine the

general educational pattern.

The most basic short-coming of our mathematical situa-

tion in Texas lies in the failure to conceive of the impor-

tance of the teacher, The welfare of the pupils and the

future of mathematics demands a deeper and broader study

of educational problems. The importance of aims and ob-

jectives is secondary to the place of the teacher. In

Texas, the prospective teacher need not have studied mathe-

matics in college at all. It was reported in 1935 that

"less than 50 per cent of the high school teachers of mathe-

matics have any special preparation in their field of teach-

ing." 7 A study made in the North shows that one-third of

the teachers of mathematics there who hold a college degree

never had any college mathematics. 8

A recent ruling of our State Department and State Board

of Education requiring that high school teachers must teach

in their fields of preparation is a forward step and should

be strictly enforced.

Mathematics must be taught for social value. The pupil

needs to know mathematics in order to protect himself against

'7 irs. Harry Brewton, "Attempted Revision in High SchoolAlgebra," The Texas Mathematics Teachers' Bulletin, (February,1935), p. 12.

8Charles R. Sherer, "Does the Present High School Cur-riculum in Mathematics Prepare Students for College?" TheTexas Mathematics Teachers' Bulletin, (February, 1934),pp. 43-48.

85

frauds and fallacies to which he may be subjected. Being

able to compare critically the sizes of containers, to

figure budgets, taxes, and installment buying are of vital

importance to every man, woman, and child. This will mean

teachers must have a different preparation from what they

now have. Teachers must be specialists in order to care

for college preparatory, vocational, and natural science

pupils together with the great mass of unspecialized pupils

who have need for mathematics of general appeal.

Mary E. Decherd in an article entitled "Service Mathe-

matics" cites a study the evaluation of which brings out

the conclusion that we do not need to teach more or dif-

ferent mathematics, but that we need to teach more effi-

cienty the elementary mathematics that we already try to

teach, 1 0

It is almost universally agreed that mathematics need

not be required of every person training for a profession,

but the cultured person should k:now certain facts concerning

the study of mathematics. The person who has not chosen a

profession and who does not study mathematics past elemen-

tary school will find the doors of numerous opportunities

Winston E. Romi, Thither Mathematics?" The Mathe-matics Teacher, QXI (1938), 293-296.

10 he Texas Mathematics Teachers' Bulletin, IX, No. 2,( ebruary, 1924), 37-38.

86

closed to him. The pupil should be made to realize that he

is living in a scientific age and that training in mathe-

matics will help him understand more of what is going on.

The students who are competent in mathematics do not regret

the time they have spent on it. They should know that

mathematics is an easy subject if well taught and that

there are many pupils who like it. Pupils should be in-

formed concerning mathematics even if they do not desire

to become mathematicians. Keyser considers it a tragedy

that many otherwise cultured people cannot read mathematical

literature in spite of the fact that "the best wisdom of

man is found in the literature of mathematics."12 Ayre says

that mathematics is "the one universal language; it is a

part of civilization." 1 3 Students and others who have

cultural ambition should be provided general knowledge con-

cerning mathematics through the medium of lectures, articles,

and kooks on the subject. For general reading purposes the

following books are suggested, and these may be supplemented

1 1 Raleigh Schorling, "The Place of Mathematics inGeneral Education," School Science and Iathematics, )L(1940), 14-26.

1 2 assius Jackson Keyser, "The Role of Mathematics inthe Tragedy of Our Modern Culture," Scripta Mathematica,VI (1939), 83.

131. G. Ayre, "Our fathemati cal Universe," The fathe-matics Teacher, XXXII (1939), 357.

87

at any general library by other books together with mathe-

matical articles published in monthly journals and magazines:

Bell, E. T., en _of mathematics.

Dantzig, Tobias, Tumber: The Lan gua e of Science.

Hogben, Launcelot, Mathematics for the illion.

A broad.view of mathematics as a science is needed.14

Changes are needed both in method and content of mathe-

matical teaching. Reorganization needs to take place and

should center around fundamental notions or concepts. Pro-

gressive education requires that mathematics be useful in

life. Life situations must be provided; however, when this

provision is made the problem is not yet solved. The

teacher-pupil relationship is of primary importance. 1 5

The teacher of mathematics must have a broad general edu-

cation; a professional attitude, by which is meant an en-

thusiastic interest in mathematics; an integrated philosophy

of education; a devotion to the teaching profession; and a

sense of responsibility for contributing to his professional

field. Scholarship is a fundamental qualification, but there

is a difference in learning to know and learning to teach.

The teacher must oe a seller of mathematics; he must secure

the customers. He must master the subject and yet be

14 aurice L. Hartung, mathematicss in Progressive Edu-cation," _The Mathematics Teacher, XXXII (1939), 265-269.

15 Ibid.

88

conscious of the processes by which he acquired mastery.

There must be mastery of fundamentals which constitute

minimum essentials, as well as certain technical equipment

that is necessary in order to do effective instruction.

The teacher must be able to recognize student difficulties

and weaknesses.16

Caution must be used in the elimination of any given

topics from the course of study. 1 7

Transfer, though possible, is limited in scope. The

reason that it does not take place is because teaching is

not done for that purpose. Teachers must call attention to

applications of formulas and give various situations in

which each mathematical item is applicable. 18

Incorporating some of the ideals of Wren, 1 9 we further

recommend:

That provision be made for significant mastery of sub-

ject matter fundamentals and specific preparation for teach-

ing at particular school levels. Degrees should then be

issued carrying the title "Teacher of Mathematics."

16p. L. Wren, "The Professional Preparation of athe-matics Teachers," Tie Mathematics Teacher, XXXII (1939),99-105.

17E. R. Hedrick, "The Contribution of Mathematics toGeneral Education," The Mathematics Teacher, ).XIII (1940),

1 8 Ibid.

1911 L. Wren, "The Professional Preparation of athe-

matics Teachers," The Mathematics Teacher, XXXII (1939),99-105.

That professional attitude of mind be built up on the

part of teachers at all levels.

That better training be provided in upper levels.

That all mathematical bodies combine in a program to

sell mathematics as a significant subject to the public

at large and the educationist in particular.

That there be a cooperative committee created, made up

of members of the different mathematical bodies and an

educational psychologist to study the mathematical program

of public education and also to formulate a program for

better preparation of teachers.

J. . Bledsoe, of the East Texas State Teachers College,

sums up not only Texas' need, but the need of the entire

nation when he says that

the greatest need in the mathematics teaching inour public schools today is a supply of teacherswho possess a sufficient knowledge and apprecia-tion of the subject to enable them to lead thepupils into a thorough mastery of the facts andprinciples, and render them skillful in applyingthese -rrinciples the solution of the everydayproblems of life.

Colleges and universities, as well as the high schools,

should be more interested in teaching students to do better

those desirable things they will do anyway.

2 0 "Laking Mathematics Interestin," 'ThTe Texas athe-matics Teachers' Bulletin, IX, No. L3 ay, 1924), 12.

UThomas B. Portwood, "The Place of High School Mathe-matics," The Clearing Heuse, XII (1938) , 140-143.

B IBLIOGRAPHY

Reports

Fifteenth Yearbook of the National Council of Teachers ofmathematics, Tew York, Bureau of Publications, TeachersCollege, Columbia University, 1940.

Prt Yearbook o the National Council of Teachers of ivathe-matics, New York, Bureau of Publications, TeachersCollege, Columbia University, 1926.

Fourth Yearbook o fthe National Council ofTeachers ofiathematics, New York, Bureau of Publications, TeachersCollege, Columbia University, 192,9.

Judd, C. i., _Seconder Education, Texas Educational 2rYexe , vol. III, Austin, Texas, Texas Educational

Survey Commission, 1924.

National Committee, Report of, The organization of {athe-matics in Secondary Education, Oberlin, Ohio, TheMathematical Association of America, Inc., 1923.

National Committee, Report of, The Reorganization of 'athe-matics in Secondary Education, Part I, Boston, NewYork, etc., Houghton Mifflin Company, 1927.

Perry, John, Discussion on the Teachn off Mathematics,British Associati on Jeeting in Glasgow, 1901, New York,The Macmillan Company , 1901.

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Alves, H. F., and Brown, M. B., Teaching of Mathematics,Bulletin of the State Department of Education, TexasHigh Schools, Austin, Texas, September, 1928.

Brown, M. B., The Teachn of Mathematics, Bulletin of theState Department of Education, Texas High Schools,Austin, Texas, May, 1929.

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90

91

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Durham, Annie D., "Mathematics as a Language," The TexasMathematics Teachers' Bulletin, IX, Io. I Povember,1923), tNo. 2342, 30-31.

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Hartung, Maurice L., "Mathematics in Progressive Education,"Te Mathematics Teacher, XXXII (1939), 265-269.

Hedrick, E. R., "The Contribution of Mathematics to GeneralEducation," The Mathematics Teacher, XGDIII (1940),

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Hobson, E. W., "Tendencies of Modern Kathematics," Educa-tional Review, XL (1910), 524-530.

Keyser, Cassius J., "The Role of Mathematics in the Tragedyof Our Modern Culture," Scrita Kathematica, VI (1939),81-87,

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V4

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