.33:3:
irº:3&c. ºr
#kºº.º.
tº:
Science
É.
-
i
Er--H:
.
w
-
:
:
:
:
.
:
f
º
f



SN ----
:
4; *H
- -
UN
C-
C-
tººl
C
º
C
-
D-
*
C-
º
Tº
ſº
wº
C-
D-
C
C-
c
mº
C-
C-
Tº
ſº-
C-
C-
-
C-
C-l
L-
C-
|-l
º
[-]
wº
º
C
C
c-l
C-
D-
[-
ſº-
[-]
º
L-
ſº
--
º
C
º
s = *Tº
Mº',\,...ſº wº. Nº Mº ...º.º.A: \º, Jº

Mathematºs
QA
| |
, N 9-772.
} % w 7
. . A 2^ 2.-->
altibergipe ſtatbematical ſtionograpbg
EDITED BY JOHN WESLEY YOUNG
cHENEy PROFESSOR OF MATHEMATICS, DARTMOUTH COLLEGE
THE REORGANIZATION
OF MATHEM ATICS IN
SECONDARY EDUCATION
- (PART I)
A REPORT BY THE NATIONAL COMMITTEE ON MATH-
EMATICAL REQUIREMENTS |UNDER THE AUSPICES OF
THE MATHEMATICAL ASSOCIATION OF AMERICA, INC.
HOUGHTON MIFFLIN COMPANY
BOSTON - NEW YORK CHICAGO DALLAS SAN FRANCISCO
@Ibe ſiiuergite pregg Cambridge

COPYRIGHT, 1927
BY HOUGHTON MIFFLIN COMPANY
COPYRIGHT, IQ23, BY THE MATHEMATICAL ASSOCIATION OF AMERICA, INC.
ALL RIGHTS RESERVED INCLUDING THE RIGHT TO REPRODUCE
THIS BOOK OR PARTS THEREOF IN ANY FORM
Oſije 38tbergíbe 33regg
CAMBRIDGE . MASSACHUSETTS
PRINTED IN THE U.S.A.,
(%. 44
r&/26
* Cº.
--,
gº tº 4. 2."
2.
3.ſeR.
º
-
g
\
--"
)
ÖO
{\ſy
& - 3. &
**
t; 2
EDITOR'S INTRODUCTION
THE report of the National Committee on Mathematical
Requirements, entitled the Reorganization of Mathematics
£n Secondary Education, was published as a volume of
over 600 pages in 1923. It is generally recognized that
this report has had and continues to have a profound in-
fluence on the teaching of secondary school mathematics
in this country. The large edition of 25,000 copies of
this report was exhausted in the early spring of 1926. De-
mands for the report have continued since that time. It
is therefore a source of gratification that Part I of this
report, containing all the recommendations of the Com-
mittee, is now issued as one of the Riverside Mathemati-
cal Monographs.
The text of the report is a verbatim reprint of the orig-
inal text (except for the correction of obvious misprints).
The National Committee was discharged in December,
1923, and there is therefore no one in existence who has
authority to revise the report. It has seemed desirable,
however, to add some comments in the form of footnotes.
These additions have all been enclosed in square brackets.
Some supplementary material has been added in an ap-
pendix. This consists in part of significant extracts from
Part II of the report, and in part of brief statements pre-
pared especially for this edition.
I should like to express my gratitude to all those who
have helped in the publication of the present Monograph.
My special thanks are due to the former members of the
Ill
EDITOR'S INTRODUCTION
National Committee, and particularly to Professor David
Eugene Smith and Miss Vevia Blair; to Professor Thomas
S. Fiske, Secretary of the College Entrance Examination
Board, for his permission to reprint the Board's definitions
of college entrance requirements; and to Professor C. B.
Upton for his statement on standardized tests which will
be found in the Appendix.
J. W. YoUNG
THE NATIONAL COMMITTEE ON
MATHEMATICAL REQUIREMENTS
(Under the auspices of
The Mathematical Association of America, Inc.)
OFFICERS
J. W. Young, chairman, Dartmouth College, Hanover,
New Hampshire.
J. A. Foberg, vice chairman, State Normal School, Cali-
fornia, Pennsylvania.
MEMBERS
A. R. Crathorne, University of Illinois.
C. N. Moore, University of Cincinnati."
E. H. Moore, University of Chicago.
David Eugene Smith, Columbia University.
H. W. Tyler, Massachusetts Institute of Technology.
J. W. Young, Dartmouth College.
W. F. Downey, English High School, Boston, Massachu-
setts, representing the Association of Teachers of
Mathematics in New England.”
* Professor Moore took the place vacated in 1918 by the resignation of
Oswald Veblen, Princeton University,
* Mr. Downey took the place vacated in 1919 by the resignation of
G. W. Evans, Charlestown High School, Boston, Massachusetts.
V
THE NATIONAL COMMITTEE
Wevia Blair, Horace Mann School, New York City, repre-
senting the Association of Teachers of Mathematics in
the Middle States and Maryland.
J. A. Foberg, State Normal School, California, Penn-
Sylvania,” representing the Central Association of
Science and Mathematics Teachers.
A. C. Olney, Commissioner of Secondary Education,
Sacramento, California.
Raleigh Schorling, University of Michigan, Ann Arbor,
Michigan.
P. H. Underwood, Ball High School, Galveston, Texas.
Eula A. Weeks, Cleveland High School, St. Louis, Mis-
Souri.
3 Until July, 1921, of the Crane Technical High School, Chicago, Illi-
nois. &
PREFACE TO THE FIRST EDITION
THE National Committee on Mathematical Require-
ments was organized in the late summer of IQI6 under
the auspices of The Mathematical Association of America
for the purpose of giving national expression to the move-
ment for reform in the teaching of mathematics, which
had gained considerable headway in various parts of the
country, but which lacked the power that coördination
and united effort alone could give.
The original nucleus of the committee, appointed by
E. R. Hedrick, then president of the Association, con-
sisted of the following: A. R. Crathorne, University of
Illinois; E. H. Moore, University of Chicago; D. E.
Smith, Columbia University; H. W. Tyler, Massachu-
setts Institute of Technology; Oswald Veblen, Princeton
University; and J. W. Young, Dartmouth College, chair-
man. This committee was instructed to add to its
membership so as to secure adequate representation of
Secondary school interests, and then to undertake a
comprehensive study of the whole problem concerned
with the improvement of mathematical education and to
cover the field of secondary and collegiate mathematics.
This group held its first meeting in September, 1916, at
Cambridge, Massachusetts. At that meeting it was de-
cided to ask each of the three large associations of sec-
ondary school teachers of mathematics (The Association
of Teachers of Mathematics in New England, The Asso-
ciation of Teachers of Mathematics in the Middle States
Vll
PREFACE TO THE FIRST EDITION
and Maryland, and The Central Association of Science
and Mathematics Teachers) to appoint an official repre-
sentative on the committee. At this time also a general
plan for the work of the committee was outlined and
agreed upon.
In response to the request above referred to the follow-
ing were appointed by the respective associations: Miss
Wevia Blair, Horace Mann School, New York City, repre-
senting the Middle States and Maryland Association;
G. W. Evans, Charlestown High School, Boston, Massa-
chusetts, representing the New England Association; *
and J. A. Foberg, Crane Technical High School, Chicago,
Illinois, representing the Central Association.
At later dates the following members were appointed:
A. C. Olney, Commissioner of Secondary Education,
Sacramento, California; Raleigh Schorling, The Lincoln
School, New York City; P. H. Underwood, Ball High
School, Galveston, Texas; and Miss Eula A. Weeks,
Cleveland High School, Saint Louis, Missouri.
From the very beginning of its deliberations the com-
mittee felt that the work assigned to it could not be done
effectively without adequate financial support. The wide
geographical distribution of its membership made a full
attendance at meetings of the committee difficult if not
impossible without financial resources sufficient to defray
the traveling expenses of members, the expenses of clerical
assistance, etc. Above all, it was felt that, in order to
give to the ultimate recommendations of the committee
the authority and effectiveness which they should have,
* Mr. Evans resigned in the summer of 1919, owing to an extended trip
abroad; his place was taken by W. F. Downey, Inglish High School.
Boston, Massachusetts.
viii
PREFACE TO THE FIRST EDITION
it was necessary to arouse the interest and secure the
active coöperation of teachers, administrators, and or-
ganizations throughout the country — that the work of
the committee should represent a coöperative effort on a
truly national scale.
For over two years, owing in large part to the World
War, attempts to secure adequate financial support
proved unsuccessful. Inevitably also the war interfered
with the committee's work. Several members were en-
gaged in war work and the others were carrying extra
burdens on account of such work undertaken by their
Colleagues.
In the Spring of 1919, however, and again in 1920, the
Committee was fortunate in Securing generous appropri-
ations from the General Education Board of New York
City for the prosecution of its work.”
This made it possible greatly to extend the commit-
tee's activities. The work was planned on a large scale
for the purpose of organizing a truly nation-wide discus-
sion of the problems facing the committee, and J. W.
Young and J. A. Foberg were selected to devote their
whole time to the work of the committee. Suitable office
Space was Secured and adequate Stenographic and clerical
help was employed.
The results of the committee's work and deliberations
are presented in the following report. A word as to the
* Professor Veblen resigned in 1917 on account of the pressure of his war
duties. His place was taken on the committee by Professor C. N. Moore,
University of Cincinnati.
3 Again in November, 1921, the General Education Board made ap-
propriations to cover the expense of publishing and distributing the pre-
sent report and to enable the committee to carry on certain phases of its
work during the years 1922–23.
fx
PREFACE TO THE FIRST EDITION
methods employed may, however, be of interest at this
point. The committee attempted to establish working
contact with all organizations of teachers and others in-
terested in its problems and to secure their active assist-
ance. Nearly Ioo such organizations have taken part in
this work. A list of these organizations will be found in
the appendix to this report (p. 632).” Provisional reports
on various phases of the problem were submitted to these
coöperating Organizations in advance of publication, and
Criticisms, comments, and suggestions for improvement
were invited from individuals and special coöperating
committees. The reports previously published for the
committee by the United States Bureau of Education 5
and in The Mathematics Teacher" and designated as “pre-
liminary” are the result of this kind of coöperation. The
value of such assistance can hardly be overestimated and
the committee desires to express to all individuals, organ-
izations, and educational journals that have taken part
its hearty appreciation and thanks. The Committee be-
lieves it is safe to say, in view of the methods used in for-
mulating them, that the recommendations of this final
report have the approval of the great majority of pro-
gressive teachers throughout the Country.
4 [Not contained in the present edition.]
5 The Reorganization of the First Courses in Secondary School Mathe-
matics, U.S. Bureau of Education, Secondary School Circular, no. 5, Feb-
ruary, 1920. II pp. Junior High School Mathematics, U.S. Bureau of
Education, Secondary School Circular, no. 6, July, 1920. Io pp. The
Function Concept in Secondary School Mathematics, Secondary School Cir-
cular, no. 8, June, 1921. Io pp.
* “Terms and Symbols in Elementary Mathematics,” The Mathematics
Teacher, vol. I4 (March, 1921), pp. 107–18. “Elective Courses in
Mathematics for Secondary Schools,” The Mathemalics Teacher, vol. 14
(April, 1921), pp. 161-70. “College Entrance Requirements in Mathe-
matics,” The Malhematics Teacher, vol. 14 (May, 1921), pp. 224-45.
X
PREFACE TO THE FIRST EDITION
No attempt has been made in this report to trace the
origin and history of the various proposals and move-
ments for reform nor to give credit either to individuals or
organizations for initiating them. A convenient starting
point for the history of the modern movement in this
country may be found in E. H. Moore's presidential ad-
dress before the American Mathematical Society in 1902.
But the movement here is only one manifestation of a
movement that is world-wide and in which very many in-
dividuals and organizations have played a prominent
part. The student interested in this phase of the subject
is referred to the extensive publications of the Interna-
tional Commission on the Teaching of Mathematics, to
the Bibliography of the Teaching of Mathematics, 1900–12,
by D. E. Smith and C. Goldziher (U.S. Bureau of Edu-
cation, Bulletin, 1912, no. 29) and to the bibliography
(since 1912) to be found in this report (chap. XVI, p. 539f).”
The National Committee expects to maintain its office,
with a certain amount of clerical help, during the year
1922–23 and perhaps for a longer period. It is hoped
that in this way it may continue to serve as a clearing
house for all activities looking to the improvement of the
teaching of mathematics in this country, and to assist in
bringing about the effective adoption in practice of the
recommendations made in the following report, with such
modifications of them as continued study and experimen-
tation may show to be desirable.
7 E. H. Moore: On the Foundations of Mathematics, Bulletin of the Amer-
ican Mathematical Society, vol. 9 (1902–03), p. 4oz; Science, vol. 17,
D. 4OI.
* Not contained in the present edition.]
CONTENTS
I. A BRIEF OUTLINE OF THE REPORT
>II. AIMS OF MATHEMATICAL INSTRUCTION – GENERAL
PRINCIPLES
III. MATHEMATICs For YEARS SEVEN, EIGHT, AND NINE .
IV. MATHEMATICS FOR YEARS TEN, ELEVEN, AND TWELVE
V. COLLEGE ENTRANCE REQUIREMENTS .
VI. LIST OF PROPOSITIONS IN PLANE AND SOLID GEOMETRY
VII. THE FUNCTION CONCEPT IN SECONDARY SCHOOL MATHE-
MATICS
VIII. TERMS AND SYMBOLS IN ELEMENTARY MATHEMATICS
APPENDIX
I. The Present Status of Disciplinary Values in Education,
by Vevia Blair. (Extract.) e
2. The Training of Teachers of Mathematics, by R. C.
Archibald. (Extracts.) . . . . . . . . . .
3. The Testing Movement in Mathematics, by C. B. Upton
4. High School Courses in the Calculus, by Vevia Blair
5. College Entrance Examination Board, Document Io'ſ.
Definition of the requirements in Elementary Algebra,
Advanced Algebra, Trigonometry .
6. College Entrance Examination Board, Document IoS.
Definition of the requirements in Plane Geometry, Solid
Geometry, Plane and Solid Geometry (Major Require-
ment), Plane and Solid Geometry (Minor Requirement)
INDEX . . . . . . .
26
45
6I
78
92
. Ioff
. I23
I26
I33
. I37
. I39
I6I
• I'73
THE REORGANIZATION OF
MATHEMATICS IN SECONDARY
EDUCATION
I
A BRIEF OUTLINE OF THE REPORT
THE present chapter gives a brief general outline of the
contents of this report for the purpose of orienting the
reader and making it possible for him to gain quickly an
understanding of its scope and the problems which it
considers.
The valid aims and purposes of instruction in mathe-
matics are considered in Chapter II. A formulation of
such aims and a statement of general principles governing
the committee's work is necessary as a basis for the later
specific recommendations. Here will be found the reasons
for including mathematics in the course of study for all
secondary school pupils.
To the end that all pupils in the period of secondary
education shall gain early a broad view of the whole field
of elementary mathematics, and, in particular, in order
to insure contact with this important element in second-
ary education on the part of the very large number of
pupils who, for one reason or another, drop out of school
by the end of the ninth year, the National Committee
recommends emphatically that the course of study in
mathematics during the seventh, eighth, and ninth years
contain the fundamental notions of arithmetic, of algebra,
of intuitive geometry, of numerical trigonometry, and at
I
OUTLINE OF THE REPORT
least an introduction to demonstrative geometry, and
that this body of material be required of all secondary
School pupils.
A detailed account of this material is given in Chapter
III. Careful study of the later years of our elementary
schools, and comparison with European schools, have
shown the vital need of reorganization of mathematical
instruction, especially in the seventh and eighth years.
The very strong tendency now evident to consider ele-
mentary education as ceasing at the end of the sixth
school year, and to consider the years from the seventh
to the twelfth inclusive as comprising years of secondary
education, gives impetus to the movement for reform of
the teaching of mathematics at this stage.
While Chapter III is devoted to a consideration of the
body of materials of instruction in mathematics that is
regarded as of sufficient importance to form part of the
course of study for all secondary school pupils, Chapter
IV is devoted to a consideration of the types of material
that properly enter into courses of study for pupils who
continue their study of mathematics beyond the mini-
mum regarded as essential for all pupils. Here will be
found recommendations concerning the traditional sub-
ject-matter of the tenth, eleventh, and twelfth school
years, and also certain material that heretofore has been
looked upon in this country as belonging rather to college
courses of study; as, for instance, the elementary ideas
and processes of the Calculus. *
Chapter V is devoted to a study of the types of sec-
ondary school instruction in mathematics that may be
looked upon as furnishing the best preparation for suc-
cessful work in college. This study leads to the conclu-
2
OUTLINE OF THE REPORT
sion that there is no conflict between the needs of those
pupils who ultimately go to college and those who do not.
Certain very definite recommendations are made as to
changes that appear desirable in the statement of college
entrance requirements and in the type of College entrance
examination.* *
Chapter VI contains lists of propositions and construc-
tions in plane and in solid geometry. The propositions
are classified in such a way as to separate from others of
less importance those which are regarded as so funda-
mental that they should form the common minimum of
any standard course in the subject.”
The statement previously made in preliminary reports
of the National Committee and repeated in Chapter II,
that the function concept should serve as a unifying ele-
ment running throughout the instruction in the mathe-
matics of the secondary school, has brought many re-
quests for a more precise definition of the rôle of the
function Concept in Secondary School mathematics.
Chapter VII is intended to meet this demand.
| Such changes have, since the publication of this report, been incor-
porated in the requirements formulated by the College Entrance Exam-
ination Board. In order to facilitate comparison with the recommenda-
tions of the National Committee the definitions of the C.E.E.B. now in
force will be found in this monograph. (See Chapter VI and Appendix.)
They have been taken with permission from Document Io'7 (Definition
of the Requirements in Elementary Algebra, Advanced Algebra, Trig-
onometry) and Document IoS (Definition of the Requirements in Plane
and Solid Geometry), published by the College Entrance Examination
Board, 431 West 117th Street, New York City. Copies of these pam-
phlets, which contain much valuable information, will be mailed by the
Board free of charge to any teacher.]
[* The College Entrance Examination Board has also published such
lists (see Document 108 referred to in the previous note). The lists in
Chapter VI (pp. 79–91) indicate the differences between the National
Committee and the C.E.E.B. lists.]
3.
OUTLINE OF THE REPORT
Recommendations as to the adoption and use of terms
and symbols in elementary mathematics are contained
in Chapter VIII. It is intended to present a norm em-
bodying agreement as to best current practice.
The remaining chapters 3 give for the most part the
results of special investigations conducted for the Na-
tional Committee. The contents of these chapters are
indicated sufficiently in the general table of contents and
in the tables of contents preceding many of the chapters
in question. *
[ With the exception of certain extracts (see Appendix), these chapters
are not included in the present edition.]
II
AIMS OF MATHEMATICAL INSTRUCTION —
GENERAL PRINCIPLES
I. INTRODUCTION
A DISCUSSION of mathematical education, and of ways
and means of enhancing its value, must be approached
first of all on the basis of a precise and comprehensive
formulation of the valid aims and purposes of such edu-
cation." Only on such a basis can we approach intelli-
gently the problems relating to the selection and organ-
ization of material, the methods of teaching and the
point of view which should govern the instruction, and
the qualifications and training of the teachers who im-
part it. Such aims and purposes of the teaching of
mathematics, moreover, must be sought in the nature of
the subject, the rôle it plays in the practical, intellectual,
and spiritual life of the world, and in the interests and
capacities of the students.
Before proceeding with the formulation of these aims,
however, we may properly limit to some extent the field
of our enquiry. We are concerned primarily with the pe-
riod of secondary education — comprising, in the modern
* Reference may here be made to the formulation of the principal aims
in education to be found in the Cardinal Principles of Secondary Educa-
tion, published by the U.S. Bureau of Education as Bulletin no. 55, 1918.
The main objectives of education are there stated to be: (1) health;
(2) command of fundamental processes; (3) Worthy home membership;
(l) vocation; (5) citizenship; (6) worthy use of leisure; (7) ethical char-
acter. These objectives are held to apply to all education — clemcntary,
secondary, and higher — and all subjects of instruction are to contribute
to their achievement.
5
AIMS OF MATHEMATICAL INSTRUCTION
junior and senior high schools, the period beginning with
the seventh and ending with the twelfth school year, and
concerning itself with pupils ranging in age normally from
12 to 18 years. References to the mathematics of the
grades below the seventh (mainly arithmetic) and beyond
the senior high school will be only incidental.
Furthermore, we are primarily concerned at this point
with what may be described as “general” aims, that is to
say, aims which are valid for large sections of the school
population and which may properly be thought of as
contributing to a general education as distinguished from
the specific needs of vocational, technical, or professional
education.
II. THE AIMS OF MATHEMATICAL INSTRUCTION
With these limitations in mind we may now approach
the problem of formulating the more important aims that
the teaching of mathematics should serve. It has been
customary to distinguish three classes of aims: (1) Prac-
tical or utilitarian, (2) disciplinary, (3) cultural; and such
a classification is indeed a convenient one. It should be
kept clearly in mind, however, that the three classes men-
tioned are not mutually exclusive and that convenience
of discussion rather than logical necessity often assigns a
given aim to one or the other of these classes. Indeed,
any truly disciplinary aim is practical and, in a broad
sense, the same is true of cultural aims.
Practical aims. By a practical or utilitarian aim, in
the narrower sense, we mean then the immediate or direct
usefulness in life of a fact, method, or process in mathe-
matics.
1. The immediate and undisputed utility of the funda-
6
AIMs of MATHEMATICAL INSTRUCTION
mental processes of arithmetic in the life of every individual
demands our first attention. The first instruction in
these processes, it is true, falls outside the period of in-
struction which we are considering. By the end of the
sixth grade the child should be able to carry out the four
fundamental operations with integers and with Common
and decimal fractions accurately and with a fair degree
of speed. This goal can be reached in all schools — as it
is being reached in many – if the work is done under
properly qualified teachers and if drill is confined to the
simpler cases which alone are of importance in the prac-
tical life of the great majority. (See more specifically,
Chapter III, p. 29.) Accuracy and facility in numerical
computation are of such vital importance, however, to
every individual that effective drill in this subject should
be continued throughout the secondary school period,
not in general as a separate topic, but in connection with
the numerical problems arising in other work. In this
numerical work, besides accuracy and speed, the following
aims are of the greatest importance:
(a) A progressive increase in the pupil's understanding
of the nature of the fundamental operations and power
to apply them in new situations. The fundamental laws
of algebra are a potent influence in this direction. (See
3, below.)
(b) Exercise of common sense and judgment in com-
puting from approximate data, familiarity with the effect
of small errors in measurements, the determination of the
number of figures to be used in computing and to be re-
tained in the result, and the like.”
[* The extent to which this recommendation has found its way into
recent textbooks and the emphasis these topics have received constitute
7
AIMS OF MATHEMATICAL INSTRUCTION
~!
(c) The development of self-reliance in the handling of
numerical problems through the consistent use of checks
on all numerical work.
2. Of almost equal importance to every educated per-
son is an understanding of the language of algebra and the
ability to use this language intelligently and readily in
the expression of such simple quantitative relations as
occur in every-day life and in the normal reading of the
educated person.
Appreciation of the significance of formulas 3 and
ability to work out simple problems by setting up and
solving the necessary equations must nowadays be in-
cluded among the minimum requirements of any pro-
gram of universal education.
3. The development of the ability to understand and
to use such elementary algebraic methods involves a
study of the fundamental laws of algebra and at least a
certain minimum of drill in algebraic technique, which,
when properly taught, will furnish the foundation for an
understanding of the significance of the processes of arith-
metic already referred to. The essence of algebra as dis-
tinguished from arithmetic lies in the fact that algebra
concerns itself with the operations upon numbers in
general, while arithmetic confines itself to operations on
particular numbers.
4. The ability to understand and interpret correctly
graphic representalions of various kinds, such as nowadays
abound in popular discussions of current scientific, social,
one of the notable changes in the teaching of mathematics of the last few
years.]
[ The importance of the formula has received recognition by explicit
inclusion in the C.E.E.B. requirements in Elementary Algebra.] Sce
9. I49.
8
AHMS OF MATHEMATHCAHL INSTRUCTION
industrial, and political problems, will also be recognized
as one of the necessary aims in the education of every
individual. This applies to the representation of sta-
tistical data which are becoming increasingly important
in the consideration of our daily problems, as well as to
the representation and understanding of various Sorts of
dependence of one variable quantity upon another.
5. Finally, among the practical aims to be served by
the study of mathematics should be listed familiarity with
the geometric forms common in nature, industry, and life;
the elementary properties and relations of these forms,
including their mensuration; the development of space-
perception; and the exercise of spatial imagination. This
involves acquaintance with such fundamental ideas as
congruence and similarity and with such fundamental
facts as those concerning the sum of the angles of a tri-
angle, the pythagorean proposition, and the areas and
volumes of the common geometric forms.
Among directly practical aims should also be included
the acquisition of the ideas and concepts in terms of
which the quantitative thinking of the world is done, and
of ability to think clearly in terms of those concepts. It
scems more convenient, however, to discuss this aim in
connection with the disciplinary aims.
Disciplinary aims. We would include here those aims
which relate to mental training, as distinguished from the
acquisition of certain specific skills discussed in the pre-
ceding section. Such training involves the development
of certain more or less general characteristics and the
formation of certain mental habits which, besides being
directly applicable in the setting in which they are de-
veloped or formed, are expected to operate also in more
9
AIMS OF MATHEMATICAL INSTRUCTION
or less closely related fields — that is, to “transfer” to
other situations.
The subject of the transfer of training has for a number
of years been a very controversial one. Only recently
has there been any evidence of agreement among the
body of educational psychologists. We need not at this
point go into detail as to the present status of disciplinary
values since this forms the subject of a separate chapter
(Chapter IX; see also Chapter X).4 It is sufficient for
our present purpose to call attention to the fact that most
psychologists have abandoned two extreme positions as
to transfer of training. The first asserted that a pupil
trained to reason well in geometry would thereby be
trained to reason equally well in any other subject; the
second denied the possibility of any transfer and hence
the possibility of any general mental training. That the
effects of training do transfer from one field of learning to
another is now, however, recognized. The amount of
[4 Not included in the present edition. See, however, the extract from
Chapter IX on p. 123.
In connection with the latter part of this paragraph, the following opin-
ion expressed by Professor W. C. Bagley, and quotcd from pp. 97 and 98
of the complete report is significant:
The evidence from the experiments is clearly in favor of a certain
amount of transfer.
In my opinion, the possibilities of transfer are increased by the kind of
teaching that makes the student conscious of the procedure as such, and
keenly appreciative of its value as a general procedure.
The theory of “identical elements.” I regard as sound — but it is not
rich in pedagogical suggestiveness. The theory of transfer through
“concepts of method,” and “ideals of procedure” furnishes a definite
suggestion for teaching. The two theories are not inconsistent with one
another: a person who has gained an understanding and an appreciation
of a procedure will not be limited to the “identical elements” which come
by accident; he will search for identities — for places at which the pro-
cedure may be applied.]
IC)
AIMS OF MATHEMATICAL INSTRUCTION
•.
x,
~
*
\
transfer in any given case depends upon a number of con-
ditions. If these conditions are favorable, there may
be considerable transfer, but in any case the amount of
transfer is difficult to measure. Training in connection
with certain attitudes, ideals, and ideas is now almost
universally admitted by psychologists to have general
value. It may, therefore, be said that, with proper re-
strictions, general mental discipline is a valid aim in
education.
The aims which we are discussing are so important in
the restricted domain of quantitative and spatial (i.e.,
mathematical or partly mathematical) thinking which
every educated individual is called upon to perform that
we do not need for the sake of our argument to raise the
question as to the extent of transfer to less mathematical
situations. . *. -
In formulating the disciplinary aims of the study of
mathematics the following should be mentioned: *
(1) The acquisition, in precise form, of those ideas or
concepts in terms of which the quantitative thinking of the
world is done. Among these ideas and concepts may be
mentioned ratio and measurement (lengths, areas, vol-
umes, weights, velocities, and rates in general, etc.), pro-
portionality and similarity, positive and negative num-
bers, and the dependence of one quantity upon another.
(2) The development of ability to think clearly in terms
N of such ideas and concepts. This ability involves train-
1ng In
º, (a) Analysis of a complex situation into simpler parts,
This includes the recognition of essential factors and the
rejection of the irrelevant.
N (b) The recognition of logical relations between inter-
. II
AIMS OF MATHEMATICAL INSTRUCTION
dependent factors and the understanding and, if possible,
the expression of such relations in precise form.
(c) Generalization; that is, the discovery and formula-
tion of a general law and an understanding of its pro-
perties and applications.
(3) The acquisition of mental habits and attitudes which
will make the above training effective in the life of the in-
dividual. Among such habitual reactions are the follow-
ing: a seeking for relations and their precise expression;
an attitude of inquiry; a desire to understand, to get to
the bottom of a situation; concentration and persistence;
a love for precision, accuracy, thoroughness, and clear-
ness, and a distaste for vagueness and incompleteness; a
desire for orderly and logical organization as an aid to
understanding and memory.
(4) Many of these disciplinary aims are included in the
broad sense of the idea of relationship or dependence — in
what the mathematician in his technical vocabulary refers
to as a “function” of one or more variables. Training in
“functional thinking,” that is, thinking in terms of and
about relationships, is one of the most fundamental dis-
ciplinary aims of the teaching of mathematics.
Cultural aims. By cultural aims we mean those some-
what less tangible but none the less real and important
intellectual, ethical, esthetic or spiritual aims that are
involved in the development of appreciation and insight
and the formation of ideals of perfection. As will be at
once apparent the realization of Some of these aims must
await the later stages of instruction, but some of them
may and should operate at the very beginning.
More specifically we may mention the development or
acquisition of -
I2
AIMs of MATHEMATICAL INSTRUCTION
(1) Appreciation of beauty in the geometrical forms of
nature, art, and industry. -
(2) Ideals of perfection as to logical structure, precision
of statement and of thought, logical reasoning (as ex-
emplified in the geometric demonstration), discrimina-
tion between the true and the false, etc.
(3) Appreciation of the power of mathematics – of what
Byron expressively called “the power of thought, the
magic of the mind” 5 — and the rôle that mathematics
and abstract thinking, in general, have played in the de-
velopment of civilization; in particular in science, in in-
dustry, and in philosophy. In this connection mention
should be made of the religious effect, in the broad sense,
which the study of the infinite and of the permanence of
laws in mathematics tends to establish."
III. THE POINT OF VIEW GOVERNING INSTRUCTION
The practical aims enumerated above, in spite of their
vital importance, may without danger be given a second-
ary position in seeking to formulate the general point of
view which should govern the teacher, provided only that
they receive due recognition in the selection of material
and that the necessary minimum of technical drill is
insisted upon.
The primary purposes of the teaching of mathematics
should be to develop those powers of understanding and of
s D. E. Smith: “Mathematics in the Training for Citizenship,” Teach-
ers College Record, vol. 18 (May, 1917), p. 6.
* For an elaboration of the ideas here presented in the barest outline,
the reader is referred to the article by D. E. Smith already mentioned and
to his presidential address before the Mathematical Association of Amer-
ica, “Religio Matematici,” American Malhematical Monthly, vol. 28
(October, 1921), pp. 339-49, also published in The Mathematics Teacher,
vol. I4 (December, 1921), pp. 413–26.
H3
AIMS OF MATHEMATICAL INSTRUCTION
analyzing relations of quantity and of space which are
necessary to an insight into and control over our environ-
ment and to an appreciation of the progress of civilization
in its various aspects, and to develop those habits of thought
and of action which will make these powers effective in the
life of the individual. .
All topics, processes, and drill in technique which do not
directly contribute to the development of the powers men-
tioned should be eliminated from the curriculum. It is
recognized that in the earlier periods of instruction the
strictly logical organization of subject-matter 7 is of less
importance than the acquisition, on the part of the pupil,
of experience as to facts and methods of attack on sig-
nificant problems, of the power to see relations, and of
training in accurate thinking in terms of such relations.
Care must be taken, however, through the dominance of
the course by certain general ideas that it does not become
a collection of isolated and unrelated details.
Continued emphasis throughout the course must be
placed on the development of ability to grasp and to
utilize ideas, processes, and principles in the solution of
concrete problems rather than on the acquisition of mere
facility or skill in manipulation. The excessive emphasis
now commonly placed on manipulation is one of the main
obstacles to intelligent progress. On the side of algebra,
the ability to understand its language and to use it intelli-
gently, the ability to analyze a problem, to formulate it
mathematically, and to interpret the result must be
dominant aims. Drill in algebraic manipulation should
7 Dewey, How We Think, p. 62. “The logical from the standpoint of
subject-matter represents the goal, the last term of training, not the
point of departure.”
I4
AIMS OF MATHEMATICAL INSTRUCTION
^ſ
be limited to those processes and to the degree of complexity
required for a thorough understanding of principles and for
probable applications either in common life or in subse-
quent courses which a substantial proportion of the pupils
will take. It must be conceived throughout as a means
to an end, not as an end in itself. Within these limits,
skill in algebraic manipulation is important, and drill in
this subject should be extended far enough to enable
students to carry out the essential processes accurately
and expeditiously.”
On the side of geometry the formal demonstrative
work should be preceded by a reasonable amount of
J
informal work of an intuitive, experimental, and con-
structive character. Such work is of great value in
itself; it is needed also to provide the necessary familiar-
ity with geometric ideas, forms, and relations, on the
basis of which alone intelligent appreciation of formal
demonstrative work is possible.
The one great idea which is best adapted to unify the
course is that of the functional relation. The concept of
a variable and of the dependence of one variable upon
another is of fundamental importance to every one. It is
true that the general and abstract forms of these concepts
can become significant to the pupil only as a result of
very considerable mathematical experience and training.
There is nothing in either concept, however, which pre-
vents the presentation of Specific concrete examples and
[* The experience of the last few years has fully confirmed this recom-
mendation regarding the desirability of eliminating the former excessive
emphasis on complicated algebraic technique. The authors of some
recent texts have perhaps gone to the opposite extreme, by not taking
sufficiently to heart the caution contained in the last Section of the above
paragraph.)
I5
AIMS OF MATHEMATICAL INSTRUCTION
illustrations of dependence even in the early parts of the
course. Means to this end will be found in connection
with the tabulation of data and the study of the formula
and of the graph and of their uses.
The primary and underlying principle of the course
should be the idea of relationship between variables, in-
cluding the methods of determining and expressing such
relationship. The teacher should have this idea con-
stantly in mind, and the pupil’s advancement should be
consciously directed along the lines which will present
first one and then another of the ideas upon which finally
the formation of the general concept of functionality de-
pends. (For a more detailed discussion of these ideas
see Chapter VII.)
The general ideas which appear more explicitly in the
course, and under the dominance of one or another of
which all topics should be brought, are: (1) The formula,
(2) graphic representation, (3) the equation, (4) measure-
ment and computation, (5) congruence and similarity,
(6) demonstration. These are considered in more de-
tail in a later section of the report (Chapters III and
IV).
IV. THE ORGANIZATION OF SUBJECT-MATTER
“General ” courses. We have already called atten-
tion to the fact that, in the earlier periods of instruction
especially, logical principles of organization are of less
importance than psychological and pedagogical prin-
ciples. In recent years there has developed among many
progressive teachers a very significant movement away
from the older rigid division into “subjects” such as
arithmetic, algebra, and geometry, each of which shall be
IO
AIMS OF MATHEMATICAL INSTRUCTION
N
“completed” before another is begun, and toward a ra-
tional breaking down of the barriers separating these
Subjects, in the interest of an organization of subject-
matter that will offer a psychologically and pedago-
gically more effective approach to the study of mathe-
matics.
There has thus developed the movement toward what
are variously called “composite,” “correlated,” “uni-
fied,” or “general” courses. The advocates of this new
method of organization base their claims on the obvious
and important interrelations between arithmetic, algebra,
and geometry (mainly intuitive), which the student must
grasp before he can gain any real insight into mathemati-
cal methods and which are inevitably obscured by a
strict adherence to the conception of separate “subjects.”
The movement has gained considerable new impetus by
the growth of the junior high school, and there can be
little question that the results already achieved by those
who are experimenting with the new methods of organ- .
ization warrant the abandonment of the extreme “water-
tight compartment” method of presentation.
The newer method of organization enables the pupil to
gain a broad view of the whole field of elementary mathe-
matics early in his high-school course. In view of the
very large number of pupils who drop out of school at the
end of the eighth or the ninth school year or who for other
reasons then cease their study of mathematics, this fact
offers a weighty advantage over the older type of organ-
ization under which the pupils studied algebra alone dur-
ing the ninth school year, to the complete exclusion of all
contact with geometry.
It should be noted, however, that the specific recom,
17
AIMS OF MATHEMATICAL INSTRUCTION
mendations as to content given in the next two chapters
do not necessarily imply the adoption of a different type
of Organization of the materials of instruction. A large
number of high schools will for some time continue to
find it desirable to organize their courses of study in
mathematics by subjects — algebra, plane geometry, etc.
Such schools are urged to adopt the recommendations
made with reference to the content of the separate sub-
jects. These, in the main, constitute an essential simpli-
fication as compared with present practice. The econ-
Omy of time that will result in courses in ninth-year al-
gebra, for instance, will permit of the introduction of the
newer type of material, including intuitive geometry and
numerical trigonometry, and thus the way will be pre-
pared for the gradual adoption in larger measure of the
recommendations of this report.
At the present time it is not possible to designate any
particular order of topics or any organization of the ma-
terials of instruction as being the best or as calculated
most effectively to realize the aims and purposes here set
forth. More extensive and careful experimental work
must be done by teachers and administrators before any
such designation can be made that shall avoid undesirable
extremes and that shall bear the stamp of general ap-
proval. This experimental work will prove successful in
proportion to the skill and insight exercised in adapting
the aims and purposes of instruction to the interests and
capacities of the pupils. One of the greatest weaknesses
of the traditional courses is the fact that both the interests
and the capacities of pupils have received insufficient con-
sideration and study. For a detailed account of courses
in mathematics at a number of the most successful experi-
I8
AIMS OF MATHEMATICAL INSTRUCTION
mental schools the reader is referred to Chapter XII° of
this report. -
Required courses. The National Committee believes
that the material described in the next chapter should be
required of all pupils and that under favorable conditions
this minimum of work can be completed by the end of the
ninth school year. In the junior high school, comprising
grades seven, eight, and nine, the course for these three
years should be planned as a unit 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.
In particular, college entrance requirements should, dur-
ing these years, receive no specific consideration. For-
tunately there appears to be no conflict of interest during
this period between those pupils who ultimately go to
college and those who do not; a course planned in accord-
ance with the principle just enunciated will form a de-
sirable foundation for college preparation.” (See Chapter
V; also, the experience of the schools described in Chap-
ter XII9). º
Similarly, in case of the at present more prevalent 8–4
school organization, the mathematical material of the
seventh and eighth grades should be selected and organ-
ized as a unit with the same purpose; the same applies to
the work of the first year (ninth grade) of the standard
four-year high School, and to later years in which mathe-
matics may be a required subject.
[9 Not included in the present edition.]
[* It may be added that the present requirements of the College En-
trance Examination Board are entirely in harmony with the spirit of the
present report, so that the suggestions above made are now in no way in
conflict with the demands of the colleges.]
I9
AIMS OF MATHEMATICAL INSTRUCTION
In the case of some elective courses the principle needs
to be modified so as to meet whatever specific vocational
or technical purposes the courses may have. (See
Chapter IV.)
The movement toward correlation of the work in
mathematics with other courses in the curriculum, not-
ably those in science, is as yet in its infancy. The results
of such efforts will be watched with the keenest interest.
The junior high-school movement. Reference has
several times been made to the junior high school. The
National Committee adopted the following resolution on
April 24, 1920: **
The National Committee approves the junior high school
form of Organization, and urges its general adoption in the
conviction that it will secure greater efficiency in the teaching
of mathematics.
The committee on the reorganization of secondary edu-
cation, appointed by the National Education Association,
in its pamphlet on the Cardinal Principles of Secondary
School Education, issued in 1918 by the Bureau of Edu-
cation, advocates an organization of the school system
whereby the first six years shall be devoted to elementary
education and the following six years to secondary edu-
cation to be divided into two periods which may be desig-
nated as junior and senior periods.
To those interested in the study of the questions relat-
ing to the history and present status of the junior high
school movement, the following books are recommended:
Principles of Secondary Education, by Inglis (Houghton
Mifflin Company, 1918); “The Junior High School,” The
Fifteenth Yearbook (Part III) of the National Society for
the Study of Education (Public School Publishing Com-
2O
AIMS OF MATHEMATICAL INSTRUCTION
pany, 1919); The Junior High School, by Bennett (War-
wick & York, 1919); The Junior High School, by Briggs
(Houghton Mifflin Company, 1920); and The Junior
High School, by Koos (Harcourt, Brace & Co., 1920).”
V. THE TRAINING OF TEACHERS
While the greater part of this report concerns itself with
the content of courses in mathematics, their organization
and the point of view which should govern the instruc-
tion, and investigations relating thereto, the National
Committee must emphasize strongly its conviction that
even more fundamental is the problem of the teacher —
his qualifications and training, his personality, skill, and
enthusiasm.
The greater part of the failure of mathematics is due
to poor teaching. Good teachers have in the past suc-
ceeded, and will continue to succeed, in achieving highly
satisfactory results with the traditional material; poor
teachers will not succeed even with the newer and better
material. f** .
The United States is far behind Europe in the scientific
and professional training required of its secondary school
teachers (see Chapter XIV*). The equivalent of two or
three years of graduate and professional training in addi-
tion to a general College Course is the normal requirement
[* Other books on the junior high School movement, published since the
publication of this report, are: Junior High School Procedure, by Touton
and Struthers (Ginn & Co.); the monograph on Junior High School Prac-
tices, by Glass (University of Chicago Press); Junior High School Curri-
cula, by Hines (The Macmillan Company); Junior High School Mathe-
matics, by Barber (Houghton Mifflin Company); Psychology of the Junior
IIigh School, by Pechstein and McGregor (Houghton Mifflin Company);
The Junior High School, by Smith (The Macmillan Company).]
|* Not included in the present edition. But see extract on p. 126.]
2I
AIMS OF MATHEMATICAL INSTRUCTION
for secondary school teachers in most European countries.
Moreover, the recognized position of the teacher in the
Community must be such as to attract men and women of
the highest ability into the profession. This means not
Only higher salaries but smaller classes and more leisure
for continued study and professional advancement. It
will doubtless require a considerable time before the pub-
lic can be educated to realize the wisdom of taxing itself
sufficiently to bring about the desired result. But if this
ideal is continually advanced and supported by sound
argument there is every reason to hope that in time the
goal may be reached. \
In the meantime everything possible should be done to
improve the present situation. One of the most vicious
and widespread practices consists in assigning a class in
mathematics to a teacher who has had no special training
in the subject and whose interests lie elsewhere, because
in the construction of the time schedule he or she happens
to have a vacant period at the time. This is done on the
principle, apparently, that “anybody can teach mathe-
matics” by simply following a textbook and devoting
ninety per cent of the time to drill in algebraic manipula-
tion or to the recitation of the memorized demonstration
of theorems in geometry.
It will be apparent from the study of this report that a
successful teacher of mathematics must not only be highly
trained in his subject and have a genuine enthusiasm for
it but must have also peculiar attributes of personality
and above all insight of a high order into the psychology
of the learning process as related to the higher mental
activities. Administrators should never lose sight of the
fact that while mathematics if properly taught is one of
22
AIMS OF MATHEMATICAL INSTRUCTION
the most important, interesting, and valuable subjects of
the curriculum, it is also one of the most difficult to teach
successfully. -
Standards for teachers. It is necessary at the outset
to make a fundamental distinction between standards in
the sense of requirements for appointment to teaching
positions, and standards of scientific attainment which
shall determine the curricula of colleges and normal
schools aiming to give candidates the best practicable
preparation. The former requirements should be high
enough to insure competent teaching, but they must not
be so high as to form a serious obstacle to admission to
the profession even for candidates who have chosen it
relatively late. The main factors determining the level
of these requirements are the available facilities for pre-
paration, the needs of the pupils, and the economic or
salary conditions.
Relatively few young people deliberately choose before
entering college the teaching of Secondary mathematics
as a life work. In the more frequent or more typical case,
the college student who will ultimately become a teacher
of secondary mathematics makes the choice gradually,
perhaps unconsciously, late in the college course or even
after its completion, perhaps after Some trial of teaching
in other fields. The possible supply of young people who
have the real desire to become teachers of mathematics is
so meager in comparison with the almost unlimited needs
of the country that every effort should be made to develop
and maintain that desire and all possible encouragement
given those who manifest it. If, as will usually be the
case, the desire is associated with the necessary mathe-
matical capacity, it will not be wise to hamper the candi-
23.
AIMS OF MATHEMATICAL INSTRUCT ION
date by requiring too high attainments, though as a
matter of course he will need guidance in continuing his
preparation for a profession of exceptional difficulty and
exceptional opportunity.
Another factor which must tend to restrict require-
ments of high mathematical attainment is the importance
to the candidate of breadth of preparation. In college he
may be in doubt as to becoming a teacher of mathematics
or physics or some other subject. It is unwise to hasten
the choice. In many cases the secondary teacher must
be prepared in more than one field, and to the future
teacher of mathematics preparation in physics and draw-
ing, not to mention chemistry, engineering, etc., may be
at least as valuable as purely mathematical college elec-
tives beyond the calculus.
In the second sense — of standards of scientific attain-
ment to be held by the colleges and normal schools —
these institutions should make every effort
I. To awaken interest in the subject and the teaching
of it in as many young people of the right sort as
possible.
2. To give them the best possible opportunity for pro-
fessional preparation and improvement, both before
and after the beginning of teaching.
How the matter of requirements for appointment will
actually work out in a given community will inevitably
depend upon conditions of time and place, varying widely
in character and degree. In many communities it is
already practicable and customary to require not less than
two years of college work in mathematics, including ele-
mentary calculus, with provision for additional electives.
Such a requirement the committee would strongly recom-
24
AIMS OF MATHEMATICAL INSTRUCTION
mend, recognizing, however, that in some localities it
would be for the present too restrictive of the supply. In
Some cases preparation in the pedagogy, philosophy, and
history of mathematics could be reasonably demanded or
at least given weight; in other cases, any considerable
time spent upon them would be of doubtful value. In all
cases requirements should be carefully adjusted to local
conditions with a view to recognizing the value both of
broad and thorough training on the part of those entering
the profession and of continued preparation and improve-
ment by summer work and the like. Particular pains
should be taken that such preparation is made accessible
and attractive in the colleges and normal schools from
which teachers are drawn.
It is naturally important that entrance to the profes-
sion should not be much delayed by needlessly high or
, extended requirements, and the danger of creating a
teacher who may be too much a specialist for school
work and too little for college teaching must be guarded
against. There may naturally also be a wide difference
between requirements in a strong School offering many
electives and a weaker one or a junior high school. Prac-
tically, it may be fair to expect that the stronger schools
will maintain their standards not by arbitrary or general
requirements for entrance to the profession but often by
recruiting from other schools teachers who have both high
attainments and successful teaching experience.
Programs of courses for colleges and normal schools
preparing teachers in secondary mathematics will be
found in Chapter XIV, together with an account of exist-
ing conditions.”
[* Not included in the present edition. But see extract, pp. 126f, 130f.]
25
III
MATHEMATICS FOR YEARS SEVEN, EIGHT
. AND NINE
I. INTRODUCTION
THERE is a well-marked tendency among school adminis-
trators to consider grades one to six, inclusive, as consti-
tuting the elementary school and to consider the second-
ary school period as commencing with the seventh grade
and extending through the twelfth. Conforming to this
view, the content of the courses of study in mathematics
for grades seven, eight, and nine are considered together.
In the succeeding chapter the content for grades ten,
eleven, and twelve is considered. -
The committee is fully aware of the widespread desire
on the part of teachers throughout the country for a de-
tailed syllabus by years or half-years which shall give the
best order of topics with specific time allotments for each.
This desire cannot be met at the present time for the
simple reason that 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 feels that its recommen-
dations should be so formulated as to give every encour-
* “We therefore recommend a reorganization of the school system
whereby the first six years shall be devoted to elementary education de-
signed to meet the needs of pupils of approximately 6 to 12 years of age;
and the second six years to secondary education designed to meet the
needs of approximately 12 to 18 vears of age. . . . The six years to be
devoted to secondary education may well be divided into two periods,
which may be designated as the junior and senior periods.” Cardinal
Principles of Secondary Education, U.S. Bureau of Education, Bulletin
no. 55, 1918, p. 18.
26
YEARS SEVEN, EIGHT, AND NINE
agement to further experimentation rather than to restrict
the teacher's freedom by a standardized syllabus.
However, certain suggestions as to desirable arrange-
ments of the material are offered in a later section (Sec-
tion III) of this chapter, and in Chapter XII* there will
be found detailed outlines giving the order of presenta-
tion and time allotments in actual operation in schools
of various types. This material should be helpful to
teachers and administrators in planning Courses to fit
their individual needs and conditions.
It is the opinion of the committee that the material
included in this chapter should be required of all pupils.
It includes mathematical knowledge and training which
is likely to be needed by every citizen. Differentiation
due to special needs should be made after and not before
the completion of such a general minimum foundation.
Such portions of the recommended content as have not
been completed by the end of the ninth year should be re-
quired in the following year.
The general principles which have governed the selec-
tion of the material presented in the next section and
which should govern the point of view of the teaching
have already been stated (Chapter II). At this point it
seems desirable to recall specifically what was then said
concerning principles governing the organization of ma-
terial, the importance to be attached to the development
of insight and understanding and of ability to think
clearly in terms of relationships (dependence), and the
limitations imposed on drill in algebraic manipulation.
In addition we would call attention to the following
considerations:
tº Not included in the present edition.]
27
YEARS SEVEN, EIGHT, AND NINE
It is assumed that at the end of the sixth school year
the pupil will be able to perform with accuracy and with
a fair degree of speed the fundamental operations with
integers and with common and decimal fractions.” The
fractions here referred to are such simple ones in common
use as are set forth in detail under A(c) in the following
section. It may be pointed out that the standard of at-
tainment here implied is met in a large number of schools,
as is shown by various tests now in use (see Chapter
XIII"), and can easily be met generally if time is not
wasted on the relatively unimportant parts of the sub-
ject.
In adapting instruction in mathematics to the mental
traits of pupils care should be taken to maintain the men-
tal growth too often stunted by secondary school mate-
rials and methods, and an effort should be made to asso-
ciate with inquisitiveness, the desire to experiment, the
wish to know “how and why” and the like, the satisfac-
tion of these needs. t
In the years under consideration it is also especially
important to give the pupils as broad an outlook over
the various fields of mathematics as is consistent with
sound scholarship. These years especially are the ones
in which the pupil should have the opportunity to find
himself, to test his abilities and aptitudes, and to secure
information and experience which will help him choose
wisely his later courses and ultimately his life work.
[.. In most progressive schools the study of decimal fractions includes
an introduction to percentage, per cent being looked upon simply as
another type of decimal.]
[* Not included in the present edition. But see statement prepared by
Professor C. B. Upton on pp. 133-137.]
28
YEARS SEVEN, EIGHT, AND NINE
II. MATERIAL FOR GRADES SEVEN, EIGHT, AND NINE
In the material outlined in the following pages no
attempt is made to indicate the most desirable order of
presentation. Stated by topics rather than years the
mathematics of grades seven, eight, and nine may pro-
perly be expected to include the following:
A. Arithmetic. (a) The fundamental operations of
arithmetic.
(b) Tables of weights and measures in general practical
use, including the most common metric units (meter,
centimeter, millimeter, kilometer, gram, kilogram, liter).
The meaning of such foreign monetary units as pound,
franc, and mark.
(c) Such simple fractions as %, 34, 34, 34, 34, 34, 96; others
than these to have less attention.
(d) Facility and accuracy in the four fundamental
operations; time tests, taking care to avoid subordinating
the teaching to the tests or to use the tests as measures of
the teacher's efficiency. (See Chapter XIII.3)
(e) Such simple short cuts in multiplication and divi-
sion as that of replacing multiplication by 25 by multiply-
ing by Ioo and dividing by 4.
(f) Percentage. Interchanging common fractions and
per cents; finding any per cent of a number; finding what
per cent one number is of another; finding a number when
a certain per cent of it is known; and such applications of
percentage as come within the student's experience.
[5 Not included in the present edition. But see the statement by Pro-
fessor C. B. Upton especially prepared for this edition, on pp. 133–137.
It has been suggested that it would be a step in advance if the case of
division of one fraction by another received but little attention, repre-
senting as it does an operation rarely needed except in a limited number
of technical lines of work.]
29
YEARS SEVEN, EIGHT, AND NINE
(g) Line, bar, and circle graphs, wherever they can be
used to advantage.
(h) Arithmetic of the home: household accounts, thrift,
simple bookkeeping, methods of sending money, parcel
post. - -
Arithmetic of the community: property and personal
insurance, taxes. -
Arithmetic of banking: Savings accounts, checking
a CCOuntS.
Arithmetic of investment: real estate, elementary no-
tions of stocks and bonds, postal savings. -
(i) Statistics: fundamental concepts, statistical tables
and graphs; pictograms; graphs showing simple frequency
distributions.
It will be seen that the material listed above includes
some material of earlier instruction. This does not mean
that this material is to be made the direct object of
study but that drill in it shall be given in connection
with the new work. It is felt that this shift in emphasis
will make the arithmetic processes here involved much
more effective and will also result in a great saving of
time.
The amount of time devoted to arithmetic as a distinct
subject should be greatly reduced from what is at present
customary. This does not mean a lessening of emphasis
on drill in arithmetic processes for the purpose of securing
accuracy and speed. The need for continued arithmetic
work and numerical computation throughout the second-
ary school period is recognized elsewhere in this report.
(Chapter II.)
The applications of arithmetic to business should be
continued late enough in the course to bring to their study
3O
YEARS SEVEN, EIGHT, AND NINE
the pupil’s greatest maturity, experience, and mathe-
matical knowledge, and to insure real significance of this
study in the business and industrial life which many of
the pupils will enter at the close of the eighth or ninth
school year. (See I, below.) In this connection care
should be taken that the business practices taught in the
schools are in accord with the best actual usage. Arith-
metic should not be completed before the pupil has ac-
quired the power of using algebra as an aid.
B. Intuitive geometry. (a) The direct measurement
of distances and angles by means of a linear scale and pro-
tractor. The approximate character of measurement.
An understanding of what is meant by the degree of
precision as expressed by the number of “significant”
figures.
(b) Areas of the square, rectangle, parallelogram, tri-
angle, and trapezoid; circumference and area of a circle;
surfaces and volumes of solids of corresponding impor-
tance; the construction of the corresponding formulas.
(c) Practice in numerical computation with due regard
to the number of figures used or retained.
(d) Indirect measurement by means of drawings to
scale; use of Square ruled paper. -
(e) Geometry of appreciation; geometric forms in na-
ture, architecture, manufacture, and industry.
(f) Simple geometric constructions with ruler and com-
passes, T-square, and triangle, such as that of the perpen-
dicular bisector, the bisector of an angle, and parallel
lines. - -
(g) Familiarity with such forms as the equilateral tri-
angle, the 30°-60° right triangle, and the isosceles right
triangle; symmetry; a knowledge of such facts as those
3I.
YEARS SEVEN, EIGHT, AND NINE
concerning the sum of the angles of a triangle and the
pythagorean relation; simple cases of geometricloci in the
plane and in Space.
(h) Informal introduction to the idea of similarity.
The work in intuitive geometry should make the pupil
familiar with the elementary ideas concerning geometric
forms in the plane and in space with respect to shape,
size, and position. Much opportunity should be pro-
vided for exercising space perception and imagination.
The simpler geometric ideas and relations in the plane
may properly be extended to three dimensions. The
work should, moreover, be carefully planned so as to
bring out geometric relations and logical connections.
Before the end of this intuitive work the pupil should
have definitely begun to make inferences and to draw
valid conclusions from the relations discovered. In other
words, this informal work in geometry should be so or-
ganized as to make it a gradual approach to, and provide
a foundation for, the subsequent work in demonstrative
geometry.
C. Algebra. I. The formula – its construction, mean-
ing, and use (a) as a concise language; (b) as a shorthand
rule for computation; (c) as a general solution; (d) as
an expression of the dependence of One variable upon
another.
The pupil will already have met the formula in connec-
tion with intuitive geometry. The work should now
include translation from English into algebraic language,
and vice versa, and special care should be taken to make
sure that the new language is understood and used intel-
ligently. The nature of the dependence of one variable
in a formula upon another should be examined and an-
r)
32
YEARS SEVEN, EIGHT, AND NINE
alyzed, with a view to seeing “how the formula works.”
(See Chapter VII.)
2. Graphs and graphic representations in general —
their construction and interpretation in (a) representing
facts (statistical, etc.); (b) representing dependence;
(c) solving problems.
After the necessary technique has been adequately
presented graphic representation should not be consid-
ered as a separate topic but should be used throughout,
whenever helpful, as an illustrative and interpretative
instrument.
3. Positive and negative numbers — their meaning
and use: (a) as expressing both magnitude and one of two
opposite directions or senses; (b) their graphic represen-
tation; (c) the fundamental operations applied to them.
4. The equation — its use in solving problems:
(a) Linear equations in one unknown – their solution
and applications.
(b) Simple cases of quadratic equations when arising in
connection with formulas and problems.
(c) Equations in two unknowns, with numerous con-
Crete illustrations.
(d) Various simple applications of ratio and proportion
in cases in which they are generally used in problems of
similarity and in other problems of ordinary life. In view
of the usefulness of the ideas and training involved, this
subject may also properly include simple cases of variation.
5. Algebraic technique: (a) The fundamental opera-
tions.
Their connection with the rules of arithmetic should be
clearly brought out and made to illuminate numerical
processes. Drill in these operations should be limited
33
YEARS SEVEN, EIGHT, AND NINE
strictly in accordance with the principle mentioned in
Chapter II, page 141. In particular, “nests” of parenthe-
ses should be avoided, and multiplication and division
should not involve much beyond monomial and binomial
multipliers, divisors, and quotients.
(b) Factoring. The only cases that need be considered
are (i) common factors of the terms of a polynomial; (ii)
the difference of two squares; (iii) trinomials of the second
degree that can be easily factored by trial.
(c) Fractions. Here again the intimate connection
with the corresponding processes of arithmetic should be
made clear and should serve to illuminate such processes.
The four fundamental operations with fractions should
be considered only in connection with simple cases and
should be applied constantly throughout the course so as
to gain the necessary accuracy and facility.
(d) Exponents and radicals. The work done on ex-
ponents and radicals should be confined to the simplest
material required for the treatment of formulas. The
laws for positive integral exponents should be included.
The consideration of radicals should be confined to
transformations of the following types: Va’b = avb,
va/5 – val and Va/5 = Va/v/b, and to the nu-
merical evaluation of simple expressions involving the
radical sign. A process for finding the Square root of a
number should be included, but not for finding the square
root of a polynomial. - -
(e) Stress should be laid upon the need for checking
solutions.
D. Numerical trigonometry. (a) Definition of sine,
cosine, and tangent.
34
YEARS SEVEN, EIGHT, AND NINE
(b) Their elementary properties as functions.
(c) Their use in solving problems involving right
triangles.
(d) The use of tables of these functions (to three or
four places).
The introduction of the elementary notions of trig-
onometry into the earlier courses in mathematics has not
been as general in the United States as in foreign coun-
tries. (See Chapter XI.") Among the reasons for an
early introduction of this topic are these: Its practical
usefulness for many citizens; the insight it gives into the
nature of mathematical methods, particularly those con-
Cerned with indirect measurement, and into the rôle that
mathematics plays in the life of the world; the fact that
it is not difficult and that it offers wide opportunity for
concrete and significant application; and the interest it
arouses in the pupils. It should be based upon the work
in intuitive geometry, with which it has intimate con-
tacts (see B, (d), (h), above), and should be confined to
the simplest material needed for the numerical treatment
of the problems indicated. Relations between the trigo-
nometric functions need not be considered.
E. Demonstrative geometry. The demonstration of
a limited number of propositions, with no attempt to
limit the number of fundamental assumptions, the prin-
cipal purpose being to show to the pupil what “demon-
stration” means.
Many of the geometric facts previously inferred in-
tuitively may be used as the basis upon which the demon-
strative work is built. This is not intended to preclude
the possibility of giving at a later time rigorous proofs
[* Not included in the present edition.]
35
YEARS SEVEN, EIGHT, AND NINE
of some of the facts inferred intuitionally. It should be
noted that from the strictly logical point of view the at-
tempt to reduce to a minimum the list of axioms, postu-
lates, or assumptions is not at all necessary, and from a
pedagogical point of view such an attempt in an elemen-
tary course is very undesirable. It is necessary, however,
that those propositions which are to be used as the basis
of Subsequent formal proofs be explicitly listed and their
logical significance recognized. -
In regard to demonstrative geometry some teachers
have objected to the introduction of such work below the
tenth grade on the ground that with such immature pupils
as are found in the ninth grade nothing worth while could
be accomplished in the limited time available. These
teachers may be right with regard to conditions prevail-
ing or likely to prevail in the majority of schools in the
immediate future. The committee has therefore in a
later section of this chapter (Section III) made alter-
native provision for the omission of work in demonstra-
tive geometry.
On the other hand, it is proper to call attention to the
fact that certain teachers have successfully introduced a
limited amount of work in demonstrative geometry into
the ninth grade (see Chapter XII") and that it would seem
desirable that others should make the experiment when
conditions are favorable. Much of the opposition is
probably due to a failure to realize the extent to which
the work in intuitive geometry, if properly organized, will
prepare the way for the more formal treatment, and to a
misconception of the purposes and extent of the work in
demonstrative geometry that is proposed. In reaching a
ſ? Not included in the present edition.]
36
YEARS SEVEN, EIGHT, AND NINE
decision on this question teachers should keep in mind
that it is one of their important duties and obligations,
in the grades under consideration, to show their pupils
the nature, content, and possibilities of later courses in
their subject and to give to each pupil an opportunity to
determine his aptitudes and preferences therefor. The
omission in the earlier courses of all work of a demon-
strative nature in geometry would disregard one educa-
tionally important aspect of mathematics.
F. History and biography. Teachers are advised to
make themselves reasonably acquainted with the leading
events in the history of mathematics, and thus to know
that mathematics has developed in answer to human
needs, intellectual as well as technical. They should use
this material incidentally throughout their courses for the
purpose of adding to the interest of the pupils by means
of informal talks on the growth of mathematics and on
the lives of the great makers of the science. -
G. Optional topics. Certain schools have been able
to cover satisfactorily the work suggested in sections
A to F before the end of the ninth grade (see Chapter
XII°). The committee looks with favor on the efforts, in
such schools, to introduce earlier than is now customary
certain topics and processes which are closely related to
modern needs, such as the meaning and use of fractional
and negative exponents, the use of the slide-rule, the use
of logarithms and of other simple tables, and simple work
in arithmetic and geometric progressions, with modern
applications to such financial topics as interest and an-
nuities and to such scientific topics as falling bodies and
laws of growth.
[* Not included in the present edition.]
37
YEARS SEVEN, EIGHT, AND NINE
H. Topics to be omitted or postponed. In addition to
the large amount of drill in algebraic technique already
referred to, the following topics should, in accordance
with our basic principles, be excluded from the work of
grades seven, eight, and nine; some of them will properly
be included in later courses (see Chapter IV):
Highest common factor and lowest common multiple,
except the simplest cases involved in the addition of
simple fractions.
The theorems on proportion relating to alternation,
inversion, composition, and division.
Literal equations, except such as appear in common
formulas, including the derivation of formulas and of
geometric relations, or to show how needless computation
may be avoided.
Radicals, except as indicated in a previous section.
Square root of polynomials.
Cube root.
Theory of exponents.
Simultaneous equations in more than two unknowns.
The binomial theorem.
Imaginary and complex numbers.
Radical equations except such as arise in dealing with
elementary formulas.
I. Problems. As already indicated, much of the
emphasis now generally placed on the formal exercise
should be shifted to the “concrete” or “verbal” problem.
The selection of problem material is, therefore, of the
highest importance. -
The demand for “practical” problems should be fully
met in so far as the maturity and previous experience of
the pupil will permit. But above all, the problems must
38
YEARS SEVEN, EIGHT AND NINE
be “real” to the pupil, must connect with his ordinary
thought, and must be within the world of his experience
and interest. “The educational utility of problems is not
to be measured by their commercial or scientific value,
but by their degree of reality for the pupils. . . . They
must exemplify those leading ideas which it is desired to
impart, and they must do so through media which are real
to those under instruction. The reality is found in the
students, the utility in their acquisition of principles.” 9
There should be, moreover, a conscious effort through
the selection of problems to correlate the work in mathe-
matics with the other courses of the curriculum, especially
in connection with courses in science. The introduction
of courses in “general science” increases the opportuni-
ties in this direction.
J. Numerical computation, use of tables, etc. The
solution of problems should offer opportunity throughout
the grades under consideration for considerable arith-
metical and computational work. In this connection
attention should be called to the importance of exercising
common sense and judgment in the use of approximate
data, keeping in mind the fact that all data secured from
measurement are approximate. A pupil should be led to
see the absurdity of giving the area of a circle to a thou-
sandth of a square inch when the radius has been meas-
ured only to the nearest inch. He should understand the
Conception of “the number of significant figures” and
should not retain more figures in his result than are war-
ranted by the accuracy of his data.” The ideals of ac-
9 Carson: Mathematical Education, pp. 42–50. -
[to In view of this fact it is unfortunate that exercises in the addition and
subtraction of decimals should ever involve cases with varying numbers
39
YEARS SEVEN, EIGHT, AND NINE
curacy and of self-reliance and the necessity of checking
all numerical results should be emphasized. An insight
into the nature of tables, including some elementary no-
tions as to interpolation, is highly desirable. The use of
tables of various kinds (such as squares and square roots,
interest, and trigonometric functions) to facilitate com-
putation and to develop the idea of dependence should
be encouraged.
III. SUGGESTED ARRANGEMENTS OF MATERIAL
In approaching the problem of arranging or Organizing
this material it is necessary to consider the different situ-
ations that may have to be met.
I. The junior high school. In view of the fact that
under this form of school organization pupils may be ex-
pected to remain in school until the end of the junior
high School period instead of leaving in large numbers at
the end of the eighth school year, the mathematics of the
three years of the junior high school should be planned as
a unit, and should include the material recommended in
the preceding section. There remains the question as to
the order in which the various topics should be presented
and the amount of time to be devoted to each. The Com-
mittee has already stated its reasons for not attempting
to answer this question (see Section I of this chapter).
The following plans for the distribution of time are, how-
ever, suggested in the hope that they may be helpful; but
no one of them is recommended as superior to the others,
and only the large divisions of material are mentioned.
of decimal places. Since all such work results from measurements of
some kind, and since in any given problem those measurements should
always be made to same degree of accuracy, there is no reality in these
operations unless the numbers have the same number of decimal places.]
4O
YEARS SEVEN, EIGHT, AND NINE
TLAN A
First year: Applications of arithmetic, particularly in such
lines as relate to the home, to thrift, and the the various school
subjects; intuitive geometry. { *
Second year: Algebra; applied arithmetic, particularly in
such lines as relate to commercial, industrial and social needs.
Third year: Algebra, trigonometry, demonstrative ge-
ometry.”
By this plan the demonstrative geometry is introduced in
the third year, and arithmetic is practically completed in the
Second year.
PLAN B
First year: Applied arithmetic (as in plan A); intuitive
geometry. -
Second year: Algebra, intuitive geometry, trigonometry.
Third year: Applied arithmetic, algebra, trigonometry,
demonstrative geometry.*
By this plan trigonometry is taken up in two years, and the
arithmetic is transferred from the second year to the third
year.
PLAN C }
First year: Applied arithmetic (as in plan A), intuitive
geometry, algebra.
Second year: Algebra, intuitive geometry.
Third year: Trigonometry, demonstrative geometry,” ap-
plied arithmetic.
By this plan Algebra is confined chiefly to the first two
years.
PLAN ID
First year: Applied arithmetic (as in plan A), intuitive
geometry.
[* It is to be remembered that in all of these plans, “demonstrative
geometry” means merely a brief introduction, “the principal purpose be-
ing to show the pupil what ‘demonstration’ means.” See p. 35, E.]
4I
YEARS SEVEN, EIGHT, AND NINE
Second year: Intuitive geometry, algebra.
Third year: Algebra, trigonometry, applied arithmetic.
By this plan demonstrative geometry is omitted entirely.
PLAN E
First year: Intuitive geometry, simple formulas, elementary
principles of statistics, arithmetic (as in plan A).
Second year: Intuitive geometry, algebra, arithmetic.
Third year: Geometry, numerical trigonometry, arithmetic.
2. Schools organized on the 8–4 plan. It cannot be too
strongly emphasized that, in the case of the older and at
present more prevalent plan of the 8–4 school organiza-
tion, the work in mathematics of the seventh, eighth, and
ninth grades should also be organized to include the
material here suggested.
The prevailing practice of devoting the seventh and
eighth grades almost exclusively to the study of arith-
metic is generally recognized as a wasteful marking of
time. It is mainly in these years that American children
fall behind their European brothers and sisters. No es-
sentially new arithmetic principles are taught in these
years, and the attempt to apply the previously learned
principles to new situations in the more advanced busi-
ness and economic aspects of arithmetic is doomed to
failure on account of the fact that the situations in ques-
tion are not and cannot be made real and significant to
pupils of this age. We need only refer to what has
already been said in this chapter on the Subject of pro-
blems (I, Section II).
The same principles should govern the selection and ar-
rangement of material in mathematics for the seventh
and eighth grades of a grade school as govern the Selec-
42
YEARS SEVEN, EIGHT, AND NINE
tion for the corresponding grades of a junior high school,
with this exception: Under the 8–4 form of organization
many pupils will leave school at the end of the eighth
year. This fact must receive due consideration. The
work of the seventh and eighth years should be so planned
as to give the pupils in these grades the most valuable
mathematical information and training that they are
capable of receiving in those years, with little reference to
courses that they may take in later years. As to possi-
bilities for arrangement, reference may be made to the
plans given above for the first two years of the junior high
school. When the work in mathematics of the seventh
and eighth grades has been thus reorganized, the work of
the first year of a standard four-year high school should
complete the program suggested.
Finally, there must be considered the situation in those
four-year high schools in which the pupils have not had
the benefit of the reorganized instruction recommended
for grades seven and eight. It may be hoped that this
situation will be only temporary, although it must be
recognized that, owing to a variety of possible reasons
(lack of adequately prepared teachers in grades seven and
eight, +rck of suitable textbooks, administrative inertia,
and the like), the new plans will not be immediately
adopted and that, therefore, for some years many high
schools will have to face the situation implied.
In planning the work of the ninth grade under these
conditions teachers and administrative officers should
again be guided by the principle of giving the pupils
the most valuable mathematical information and training
which they are capable of receiving in this year, with
little reference to future courses which the pupil may or
43
YEARS SEVEN, EIGHT, AND NINE
may not take. It is to be assumed that the work of this
year is to be required of all pupils. Since for many this
will constitute the last of their mathematical instruction,
it should be so planned as to give them the widest outlook
consistent with sound scholarship. -
Under these conditions it would seem desirable that
the work of the ninth grade should contain both algebra
and geometry. It is, therefore, recommended that about
two thirds of the time be devoted to the most useful parts
of algebra, including the work on numerical trigonometry,
and that about one third of the time be devoted to geom-
etry, including the necessary informal introduction and,
if feasible, the first part of demonstrative geometry.
It should be clear that owing to the greater maturity of
the pupils much less time need be devoted in the ninth
grade to certain topics of intuitive geometry (such as
direct measurement, for example) than is desirable when
dealing with children in earlier grades. Even under the
conditions presupposed, pupils will be acquainted with
most of the fundamental geometric forms and with the
mensuration of the most important plane and solid fig-
ures. The work in geometry in the ninth grade can then
properly be made to center about indirect measurement
and the idea of similarity (leading to the processes of nu-
merical trigonometry), and such geometric relations as
the sum of the angles of a triangle, the pythagorean pro-
position, Congruence of triangles, parallel and perpen-
dicular lines, quadrilaterals, and the more important
simple constructions.
IV
MATHEMATICS FOR YEARS TEN, ELEVEN
AND TWELVE
I. INTRODUCTION
THE committee has in the preceding chapter expressed its
judgment that the material there recommended for the
seventh, eighth, and ninth years should be required of all
pupils. In the tenth, eleventh, and twelfth years, how-
ever, the extent to which election of subjects is permitted
will depend on so many factors of a general character
that it seems unnecessary and inexpedient for the present
committee to urge a positive requirement beyond the
minimum one already referred to. The subject must,
like others, stand or fall on its intrinsic merit or on the
estimate of such merit by the authorities responsible at a
given time and place. The committee believes never-
theless that every standard high school should not merely
offer courses in mathematics for the tenth, eleventh, and
twelfth years, but should encourage a large proportion of
its pupils to take them. Apart from the intrinsic interest
and great educational value of the study of mathematics,
it will in general be necessary for those preparing to enter
college or to engage in the numerous occupations involv-
ing the use of mathematics to extend their work beyond
the minimum requirement.
The present chapter is intended to suggest for students
in general courses the most valuable mathematical train-
ing that will appropriately follow the courses outlined in
the previous chapter. Under present conditions most
45
YEARS TEN, ELEVEN, AND TWELVE
of this work will normally fall in the last three years of the
high school; that is, in general, in the tenth, eleventh, and
twelfth years.”
The selection of material is based on the general prin-
ciples formulated in Chapter II. At this point attention
need be directed only to the following:
1. In the years under consideration it is proper that
some attention be paid to the students' vocational or
other later educational needs.
2. The material for these years should include as far
as possible those mathematical ideas and processes that
have the most important applications in the modern
world. As a result, certain material will naturally be in-
cluded that at present is not ordinarily given in secondary
school courses; as, for instance, the material concerning
the calculus. On the other hand, certain other material
that is now included in college entrance requirements will
be excluded.” The results of an investigation made by
the National Committee in connection with a study of
these requirements indicate that modifications to meet
these changes will be desirable from the standpoint of
both college and secondary school (see Chapter V).
3. During the years now under consideration an in-
creasing amount of attention should be paid to the logical
organization of the material, with the purpose of devel-
oping habits of logical memory, appreciation of logical
structure, and ability to organize material effectively.
It cannot be too strongly emphasized that the broaden-
ing of content of high School courses in mathematics sug.
gested in the present and in previous chapters will ma .
[* These years are coming to be known as the senior high school.]
[* “Will be made optional’’ more accurately expresses the meaning.]
46
YEARS TEN, ELEVEN, AND TWELVE
terially increase the usefulness of these courses to those
who pursue them. It is of prime importance that edu-
cational administrators and others charged with the ad-
vising of students should take careful account of this fact
in estimating the relative importance of mathematical
courses and their alternatives. The number of important
applications of mathematics in the activities of the world
is to-day very large and is increasing at a very rapid rate.
This aspect of the progress of civilization has been noted
by all observers who have combined a knowledge of math-
ematics with an alert interest in the newer developments
in other fields. It was revealed in very illuminating
fashion during the recent war by the insistent demand
for persons with varying degrees of mathematical training
for many war activities of the first moment. If the same
effort were made in time of peace to secure the highest
level of efficiency available for the specific tasks of modern
life, the demand for those trained in mathematics would
be no less insistent; for it is in no wise true that the appli-
cations of mathematics in modern warfare are relatively
more important or more numerous than its applications
in those fields of human endeavor which are of a con-
structive nature. -
There is another important point to be kept in mind in
considering the relative value to the average student of
mathematical and various alternative courses. If the
student who omits the mathematical courses has need of
them later, it is almost invariably more difficult, and it is
frequently impossible, for him to obtain the training in
which he is deficient. In the case of a considerable num-
ber of alternative subjects a proper amount of reading
in spare hours at a more mature age will ordinarily furnish
47
YEARS TEN, ELEVEN, AND TWELVE
him the approximate equivalent to that which he would
have obtained in the way of information in a high school
course in the same subject. It is not, however, possible
to make up deficiencies in mathematical training in so
simple a fashion. It requires systematic work under a
competent teacher to master properly the technique of
the subject, and any break in the continuity of the work
is a handicap for which increased maturity rarely com-
pensates. Moreover, when the individual discovers his
need for further mathematical training it is usually diffi-
cult for him to take the time from his other activities for
Systematic work in elementary mathematics.
II. RECOMMENDATIONS FOR ELECTIVE COURSES
The following topics are recommended for inclusion in
the mathematical electives open to pupils who have
satisfactorily completed the work outlined in the pre-
ceding chapter, comprising arithmetic, the elementary
notions of algebra, intuitive geometry, numerical trigo-
nometry, and a brief introduction to demonstrative
geometry.
I. Plane demonstrative geometry. The principal pur-
poses of the instruction in this subject are: To exercise
further the spatial imagination of the student, to make
him familiar with the great basal propositions and their
applications, to develop understanding and appreciation
of a deductive proof and the ability to use this method of
reasoning where it is applicable, and to form habits of
precise and succinct statement, of the logical organization
of ideas, and of logical memory. Enough time should be
spent on this subject to accomplish these purposes.
The following is a suggested list of topics under which
48
YEARS TEN, ELEVEN, AND TWELVE
the work in demonstrative geometry may be organized:*
(a) Congruent triangles, perpendicular bisectors, bisectors
of angles; (b) arcs, angles, and chords in circles; (c) par-
allel lines and related angles, parallelograms; (d) the sum
of the angles for triangle and polygon; (e) secants and
tangents to circles with related angles, regular polygons;
(f) similar triangles, similar figures; (g) areas; numerical
Computation of lengths and areas, based upon geometric
theorems already established.
Under these topics constructions, loci, originals and
other exercises are to be included.
It is recommended that the formal theory of limits and
of incommensurable cases be omitted, but that the ideas
of limit and of incommensurable magnitudes receive in-
formal treatment.
It is believed that a more frequent use of the idea of
motion in the demonstration of theorems is desirable,
both from the point of view of gaining greater insight and
of saving time."
If the great basal theorems are selected and effectively
organized into a logical system, a considerable reduction
(from thirty to forty per cent) can be made in the number
of theorems given either in the Harvard list or in the
report of the Committee of Fifteen. Such a reduction
is exhibited in the lists prepared by the committee and
priated later in this report (Chapter VI). In this con-
nection it may be suggested that more attention than is
* It is not intended that the order here given should imply anything as
to the order of presentation. (See also Chapter VI.)
4 Reference may here be made to the treatment given in recent French
texts such as those by Bourlet and Méray.
ſ: See also the College Entrance Examination Board lists referred to in
Note (2), p. 3.) . . . .'; … -
40
YEARS TEN, ELEVEN, AND TWELVE
now customary may profitably be given to those methods
of treatment which make consistent use of the idea of
motion (already referred to), continuity (the tangent as
the limit of a secant, etc.), symmetry, and the dependence
of one geometric magnitude upon another.
If the student has had a satisfactory course in intuitive
geometry and some work in demonstration before the
tenth grade, he may find it possible to cover a minimum
course in demonstrative geometry, giving the great basal
theorems and constructions, together with exercises, in
the ninety periods constituting a half-year’s work."
2. Algebra. (a) Simple functions of one variable:
Numerous illustrations and problems involving linear,
quadratic, and other simple functions including formulas
from science and common life. More difficult problems
in variation than those included in the earlier course.
(b) Equations in one unknown: Various methods for
Solving a quadratic equation (such as factoring, complet-
ing the square, use of formula) should be given. In Con-
nection with the treatment of the quadratic a very brief
discussion of complex numbers should be included. Sim-
ple cases of the graphic solution of equations of degree
higher than the Second should be discussed and ap-
plied.
[ The editor has not as yet heard of any school that has attempted to
cover such a minimum course in half a year. The experience of the last
few years has shown conclusively, however, that the work in intuitive
geometry of the junior high School makes it already possible to save, on
a conservative basis, from six to eight weeks on the traditional year course
in plane geometry. It may be expected that this situation will soon lead
to the organization of courses covering the fundamental topics of plane
and Solid geometry in one year, along the lines, perhaps, suggested by the
C.E.E.B. definition of “Plane and Solid Geometry, Minor Requirement.”
(See Document IoS, referred to in note (1) p. 3; see also below, p. 167.)]
5O
YEARS TEN, ELEVEN AND TWELVE
(c) Equations in two or three unknowns: The algebraic
solution of linear equations in two or three unknowns and
the graphic solution of linear equations in two unknowns
should be given. The graphic and algebraic Solution of a
linear and a quadratic equation and of two quadratics
that contain no first degree term and no acy term should be
included.
(d) Exponents, radicals, and logarithms: The definitions
of negative, zero, and fractional exponents should be
given, and it should be made clear that these definitions
must be adopted if we wish such exponents to conform to
the laws for positive integral exponents. Reduction of
radical expressions to those involving fractional exponents
should be given as well as the inverse transformation.
The rules for performing the fundamental operations on
expressions involving radicals, and Such transformations as
n /— ſº * 3. (l, emº
should be included. In close connection with the work
on exponents and radicals there should be given as
much of the theory of logarithms as is involved in their
application to Computation and sufficient practice in
their use in Computation to impart a fair degree of
facility. -
(e) Arithmetic and geometric progressions: The formulas
for the nth term and the sum of n terms should be derived
and applied to significant problems.
(f) Binomial theorem: A proof for positive integral ex-
ponents should be given; it may also be stated that the
formula applies to the case of negative and fractional
exponents under suitable restrictions, and the problems
SI
YEARS TEN, ELEVEN, AND TWELVE
may include the use of the formula in these cases as well
as in the case of positive integral exponents.
3. Solid geometry. The aim of the work in solid
geometry should be to exercise further the spatial imagi-
nation of the student and to give him both a knowledge
of the fundamental spatial relationships and the power
to work with them. It is felt that the work in plane
geometry gives enough training in logical demonstration
to warrant a shifting of emphasis in the work on solid
geometry away from this aspect of the subject and in the
direction of developing greater facility in visualizing
Spatial relations and figures, in representing such figures
on paper, and in solving problems in mensuration."
For many of the practical applications of mathematics
it is of fundamental importance to have accurate space
perceptions. Hence it would seem wise to have at least
some of the work in solid geometry come as early as pos-
sible in the mathematical courses, preferably not later
than the beginning of the eleventh school year. Some
schools will find it possible and desirable to introduce
the more elementary notions of Solid geometry in connec-
tion with related ideas of plane geometry.
The work in solid geometry should include numerous
exercises in computation based on the formulas estab-
lished. This will serve to correlate the work with arith-
metic and algebra and to furnish practice in Compu-
tation.
The following provisional outline of Subject-matter is
submitted:
(a) Propositions relating to lines and planes, and to
dihedral and trihedral angles.
[7 See Note 6, p. 5o.]
52
YEARS TEN, ELEVEN, AND TWELVE
(b) Mensuration of the prism, pyramid, and frustum;
the (right circular) cylinder, cone and frustum,
based on an informal treatment of limits; the sphere
and the spherical triangle.
(c) Spherical geometry.
(d) Similar solids.
Such theorems as are necessary as a basis for the topics
here outlined should be studied in immediate connection
with them. -
Desirable simplification and generalization may be
introduced into the treatment, of mensuration theorems
by employing such theorems as Cavalieri's and Simp-
son's, and the Prismoid Formula; but rigorous proofs or
derivations of these need not be included.
Beyond the range of the mensuration topics indicated
above, it seems preferable to employ the methods of the
elementary calculus (see Section 6, below).
It should be possible to complete a minimum course
covering the topics outlined above in not more than one
third of a year.”
The list of propositions in solid geometry given in
Chapter VI should be considered in connection with the
general principles stated at the beginning of this section.
By requiring formal proofs to a more limited extent than
has been customary, time will be gained to attain the
aims indicated and to extend the range of geometric
information of the pupil. Care must be exercised to
make sure that the pupil is thoroughly familiar with the
facts, with the associated terminology, with all the neces-
* See Note 6, p. 5o. It may be added that the practice of teaching
plane and solid geometry together has for years been in successful opera-
tion in many European Schools.
53
YEARS TEN, ELEVEN, AND TWELVE
sary formulas, and that he secures the necessary practice
in working with and applying the information acquired
to concrete problems.
4. Trigonometry. The work in elementary trigonom-
etry begun in the earlier years should be completed by
including the logarithmic solution of right and oblique
triangles, radian measure, graphs of trigonometric func-
tions, the derivation of the fundamental relations be-
tween the functions and their use in proving identities
and in solving easy trigonometric equations. The use of
the transit in connection with the simpler operations of
surveying and of the sextant for some of the simpler as-
tronomical observations, such as those involved in finding
local time, is of value; but when no transit or sextant is
available, simple apparatus for measuring angles roughly
may and should be improvised. Drawings to scale should
form an essential part of the numerical work in trigonom-
etry. The use of the slide-rule in computations requiring
only three-place accuracy and in checking other compu-
tations is also recommended.
5. Elementary statistics. Continuation of the earlier
work to include the meaning and use of fundamental con-
cepts and simple frequency distributions with graphic
representations of various kinds and measures of central
tendency (average, mode, and median).
6. Elementary calculus. The work should include:
(a) The general notion of a derivative as a limit indis-
pensable for the accurate expression of such fundamental
quantities as velocity of a moving body or slope of a
CUITVC.
(b) Applications of derivatives to easy problems in
rates and in maxima and minima.
54
YEARS TEN, ELEVEN, AND TWELVE
(c) Simple cases of inverse problems; e.g., finding dis-
tance from velocity, etc.
(d) Approximate methods of summation leading up to
integration as a powerful method of Summation.
(e) Applications to simple cases of motion, area, vol-
ume, and pressure.
Work in the calculus should be largely graphic and may
be closely related to that in physics; the necessary tech-
nique should be reduced to a minimum by basing it
wholly or mainly on algebraic polynomials. No formal
study of analytic geometry need be presupposed beyond
the plotting of simple graphs.
It is important to bear in mind that, while the elemen-
tary calculus is sufficiently easy, interesting, and valuable
to justify its introduction, special pains should be taken
to guard against any lack of thoroughness in the fund-
amentals of algebra and geometry; no possible gain
could compensate for a real Sacrifice of such thorough-
Il CSS.
It should also be borne in mind that the suggestion of
including elementary calculus is not intended for all
schools nor for all teachers or all pupils in any school.
It is not intended to connect in any direct way with col-
lege entrance requirements. The future college student
will have ample opportunity for calculus later. The
capable boy or girl who is not to have the college work
ought not on that account to be prevented from learning
something of the use of this powerful tool. The applica-
tions of elementary calculus to simple concrete problems
are far more abundant and more interesting than those of
algebra. The necessary technique is extremely simple.
The Subject is commonly taught in Secondary schools in
55
YEARS TEN, ELEVEN , AND TWELVE
England, France, and Germany, and appropriate English
texts are available.” "
7. History and biography. Historical and biographical
material should be used throughout to make the work
more interesting and significant.
8. Additional Electives. Additional electives such as
mathematics of investment, shop mathematics, Surveying
and navigation, descriptive or projective geometry will ap-
propriately be offered by schools which have special needs
or conditions, but it seems unwise for the National Com-
mittee to attempt to define them pending the results of
further experience on the part of these schools.
III. PLANS FOR ARRANGEMENT OF THE MATERIAL
In the majority of high schools at the present time the
topics suggested can probably be given most advantage-
ously as separate units of a three-year program. How-
ever, the National Committee is of the opinion that
methods of organization are being experimentally per-
fected whereby teachers will be enabled to present much
of this material more effectively in combined courses
unified by one or more of such central ideas as function-
ality and graphic representation.
As to the arrangement of the material the committee
gives below four plans which may be suggestive and
helpful to teachers in arranging their courses. No one
of them is, however, recommended as Superior to the
others.
9 Quotations and typical problems from one of these texts will be found
in a supplementary note appended to this chapter.
[* See statement by Miss Blair, p. 137.]
56
YEARS TEN, ELEVEN, AND TWELVE
PLAN A
Tenth vear: Plane demonstrative geometry, algebra.
Eleventh year: Statistics, trigonometry, solid geometry.
Twelfth year: The calculus, other elective.
PLAN B
Tenth year: Plane demonstrative geometry, solid geometry.
Eleventh year: Algebra, trigonometry, statistics.
Twelfth year: The calculus, other elective.
PLAN C
Tenth year: Plane demonstrative geometry, trigonometry.
Eleventh year: Solid geometry, algebra, statistics.
Twelfth year: The calculus, other elective.
PLAN ID
Tenth year: Algebra, statistics, trigonometry.
Eleventh year: Plane and solid geometry.
Twelfth year: The calculus, other elective.
Additional information on ways of organizing this ma-
terial will be found in Chapter XII."
SUPPLEMENTARY NOTE ON THE CALCULUS As A HIGH
SCHOOL SUBJECT
In connection with the recommendations concerning the
calculus, such questions as the following may arise: Why
should a college subject like this be added to a high school
program? How can it be expected that high School teachers
will have the necessary training and attainments for teaching
it? Will not the attempt to teach such a subject result in
loss of thoroughness in earlier work? Will anything begained
beyond a mere Smattering of the theory? Will the boy or
[* Not included in the present edition.]
57
YEARS TEN, ELEVEN, AND TWELVE
girl ever use the information or training secured? The sub-
sequent remarks are intended to answer such objections as
these and to develop more fully the point of view of the com-
mittee in recommending the inclusion of elementary work in
the calculus in the high school program.
By the calculus we mean for the present purpose a study of
rates of change. In nature all things change. How much do
they change in a given time? How fast do they change? Do
they increase or decrease? When does a changing quantity
become largest or smallest? How can rates of changing
quantities be compared?
These are some of the questions which lead us to study the
elementary calculus. Without its essential principles these
questions cannot be answered with definiteness.
The following are a few of the specific replies that might be
given in answer to the questions listed at the beginning of this
note: The difficulties of the college calculus lie mainly outside
the boundaries of the proposed work. The elements of the
subject present less difficulty than many topics now offered in
advanced algebra. It is not implied that in the near future
many secondary school teachers will have any occasion toº
teach the elementary calculus. It is the culminating subject
in a series which only relatively strong schools will complete
and only then for a selected group of students. In such schools
there should always be teachers competent to teach the ele-
mentary calculus here intended. No superficial study of
calculus should be regarded as justifying any substantial sac-
rifice of thoroughness. In the judgment of the committee the
introduction of elementary calculus necessarily includes suffi-
cient algebra and geometry to compensate for whatever di-
version of time from these subjects would be implied.
The calculus of the algebraic polynomial is so simple that a
boy or girl who is capable of grasping the idea of limit, of slope,
and of velocity, may in a brief time gain an outlook upon the
field of mechanics and other exact Sciences, and acquire a fair
degree of facility in using one of the most powerful tools of
58
YEARS TEN, ELEVEN, AND TWELVE
mathematics, together with the capacity for solving a number
of interesting problems. Moreover, the fundamental ideas
involved, quite aside from their technical applications, will
provide valuable training in understanding and analyzing
quantitative relations – and such training is of value to
every one.
The following typical extracts from an English text in-
tended for use in secondary Schools may be quoted:
“It has been said that the calculus is that branch of mathe-
matics which schoolboys understand and senior wranglers
fail to comprehend. . . . So long as the graphic treatment and
practical applications of the calculus are kept in view, the
subject is an extremely easy and attractive one. Boys can be
taught the subject early in their mathematical career, and
there is no part of their mathematical training that they enjoy
better or which opens up to them wider fields of useful ex-
ploration. . . . The phenomena must first be known practi.
cally and then studied philosophically. To reverse the order
of these processes is impossible.”
The text in question, after an interesting historical sketch,
deals with such problems as the following:
A train is going at the rate of 40 miles an hour. Represent
this graphically.
At what rate is the length of the daylight increasing or de-
creasing on December 31, March 26, etc.? (From tabular
data.)
A cart going at the rate of 5 miles per hour passes a mile-
stone, and 14 minutes afterwards a bicycle, going in the same
direction at I 2 miles an hour, passes the same milestone.
Find when and where the bicycle will overtake the cart.
A man has 4 miles of fencing wire and wishes to fence in a
rectangular piece of prairie land through which a straight
river flows, the bank of the stream being utilized as one side of
the enclosure. How can he do this so as to inclose as much
land as possible?
A circular tin canister closed at both ends has a surface area
50
YEARS TEN, ELEVEN, AND TWELVE
of roo square centimeters. Find the greatest volume it can
contain.
Post-office regulations prescribe that the combined length
and girth of a parcel must not exceed 6 feet. Find the maxi-
mum volume of a parcel whose shape is a prism with the ends
Square. m -
A pulley is fixed 15 feet above the ground, over which passes
a rope 30 feet long with one end attached to a weight which
can hang freely, and the other end is held by a man at a height
of 3 feet from the ground. The man walks horizontally away
from beneath the pulley at the rate of 3 feet per second. Find
the rate at which the weight rises when it is Io feet above the
ground.
The pressure on the surface of a lake due to the atmosphere
is known to be 14 pounds per Square inch. The pressure in
the liquid & inches below the surface is known to be given by
the law dp/dx = o.o.36. Find the pressure in the liquid at a
depth of Io feet.
The arch of a bridge is parabolic in form. It is 5 feet wide
at the base and 5 feet high. Find the volume of water that
passes through per second in a flood when the water is rushing
at the rate of Io feet per second.
A force of 20 tons compresses the spring buffer of a railway
stop through I inch, and the force is always proportional to
the compression produced. Find the work done by a train
which compresses a pair of such stops through 6 inches.
These may illustrate the aims and point of view of the pro-
posed work. It will be noted that not all of them involve cal-
culus, but those that do not lead up to it.
V
COLLEGE ENTRANCE REQUIREMENTS
THE present chapter is concerned with a study of topics
and training in elementary mathematics that will have
most value as preparation for College work, and with
recommendations of definitions of College entrance re-
quirements in elementary algebra and plane geometry.
General considerations. The primary purpose of
college entrance requirements is to test the candidate's
ability to benefit by college instruction. This ability de-
pends, so far as Our present inquiry is concerned, upon
(1) general intelligence, intellectual maturity, and mental
power; (2) specific knowledge and training required as
preparation for the various courses of the College cul-
riculum.
Mathematical ability appears to be a sufficient but not
a necessary condition for general intelligence." For this,
as well as for other reasons, it would appear that college
entrance requirements in mathematics should be formulated
primarily on the basis of the special knowledge and training
required for the successful study of courses which the
student will take in college.
The separation of prospective college students from the
others in the early years of the secondary school is neither
feasible nor desirable. It is therefore obvious that sec-
ondary School courses in mathematics cannot be planned
* A recent investigation made by the department of psychology at
})artmouth College showed that all students of high rank in mathematics
had a high rating on general intelligence; the converse was not true,
bowever.
6I
COLLEGE ENTRANCE REQUIREMENTS
with specific reference to college entrance requirements.
Fortunately there appears to be no real conflict of interest
between those students who ultimately go to college and
those who do not, so far as mathematics is concerned. It
will be made clear in what follows that a course in this
subject, covering from two to two and one half years in a
standard four-year high school, and so planned as to give
the most valuable mathematical training which the stu-
dent is capable of receiving, will provide adequate pre-
paration for college work.
Topics to be included in high-school courses. In the
selection of material of instruction for high-school courses
in mathematics, its value as preparation for college courses
in mathematics need not be specifically considered. Not
all college students study mathematics; it is therefore
reasonable to expect college departments in this subject
to adjust themselves to the previous preparation of their
students. Nearly all college students do, however, study
one or more of the physical sciences (astronomy, physics,
chemistry) and one or more of the Social sciences (history,
economics, political science, sociology). Entrance re-
quirements must therefore insure adequate mathematical
preparation in these subjects. Moreover, it may be
assumed that adequate preparation for these two groups
of subjects will be sufficient for all other subjects for
which the secondary schools may be expected to furnish
the mathematical prerequisites.
The National Committee recently conducted an inves-
tigation for the purpose of Securing information as to the
content of high-school courses of instruction most desir-
able from the point of view of preparation for college
work. A number of College teachers, prominent in their
- 62
COLLEGE ENTRANCE REQUIREMENTS
respective fields, were asked to assign to each of the topics
in the following table an estimate of its value as prepara-
tion for the elementary courses in their respective sub-
jects. Table I gives a summary of the replies, arranged
in two groups – “Physical sciences,” including astro-
nomy, physics, and chemistry; and “Social sciences,”
including history, economics, sociology, and political
science.
The high value attached to the following topics is sig-
nificant: Simple formulas — their meaning and use; the
linear and quadratic functions and variation; numerical
trigonometry; the use of logarithms and other topics re-
lating to numerical computation; statistics. These all
stand well above such standard requirements as arith-
metic and geometric progression, binomial theorem,
theory of exponents, simultaneous equations involving
one or two quadratic equations, and literal equations.
These results would seem to indicate that a modifica-
tion of present College entrance requirements in mathe-
matics is desirable from the point of view of college teach-
ers in departments other than mathematics. It is inter-
esting to note how closely the modifications suggested by
this inquiry Correspond to the modifications in secondary
school mathematics foreshadowed by the study of needs
of the high school pupil irrespective of his possible future
college attendance. The recommendations made in
Chapter II that functional relationship be made the
“underlying principle of the course,” that the meaning
and use of simple formulas be emphasized, that more at-
tention be given to numerical Computation (especially to
the methods relating to approximate data), and that
work on numerical trigonometry and statistics be in-
03
COLLEGE ENTRANCE REQUIREMENTS
TABLE I. - VALUE OF Top ICS As PREPARATION FOR ELEMENTARY
COLLEGE COURSES
[In the headings of the table, E = essential, C = of considerable value, S = of some value,
O = of little or no value, N = number of replies received. The figures in the first four
columns of each group are perceptages of the number of replies received.]
PHYSICAL Social
SCIENCES SCIENCES
E| C S O|N| E| C S O| N
Negative numbers — their meaning and use...... . . . . . . . |79| 5 |Io 5 39|45|I7|22|I7|18
Imaginary numbers — their meaning and use. . . . . . . . . . . . .23|21 ||25 |31 |39||13||1337|37|16
Simple formulas – their meaning and use. . . . . . . . . . . . . . . .93| 5 || 2 |-|41|47|26|2|I||5||19
Graphic representation of statistical data. . . . . . . . . . . . . . . [57]25|I5| 3|4ols?|24|14| 5|21
Graphs (mathematical and empirical):
(a) As a method of representing dependence........... 62|I6|22|—|37|15|54|15|15||13
(b) As a method of solving problems. . . . . . . . . . . . . . . . . .45|2O28; 6|25||18|18|46|18|II
The linear function, y = ma + b . . . . . . . . . e e º e s º s s e e s e s s 78! I 4| 8 |-|37|29|29|I4]29; 14
The quadratic function, y = ax + ba + c. . . . . . . . . . . . . . . [59]21 |17|3|34 8|33 |So! I 2
Equations: Problems leading to —
Linear equations in one unknown. . . . . . . . . . . . . . . . . . . . .98| 2 |-|-|4|I|4ol 7|20|33 |15
Quadratic equations in one unknown. . . . . . . . . . . . . . . . . 78|IS 5| 2 |4o 31|| 8 || 8||54|13
Simultaneous linear equations in 2 unknowns. . . . . . . . . . 7 I 24|| 3 || 3 ||38|33 || 8 |—|58|12
Simultaneous linear equations in more than 2 unknowns...|43|29|23| 6|35| 8| 8|17|67|12
One quadratic and one linear equation in 2 unknowns...|4o 24|27 9,33|-| 9| 9|82|II
Two quadratic equations in 2 unknowns. . . . . . . . . . . . . 31|I9 |28|2232|—| 9 |—|91|II
Equations of higher degree than the second. . . . . . . . . . . Io 32|32 |2631 –—| 9 |91|II
Literal equations (other than formulas) . . . . . . . . . . . . . . . . 43|I8|32 || 7 |28|—|Io!4o 5oiro
Ratio and proportion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |84. 8] 3| 5 |39|37|26|32 || 5 |19
Variation. . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . So I3|2O. I7|3|O I 7|33 ||25 ||25 |I2
Numerical computation:
With approximate data — rational use of significant
figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6v 36|—| 3 ||39|40|27|2013||12
Short-cut methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27|38|24|Io!37|29.35|23|12|17
Use of logarithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |62|29| 7 || 2 |42|I2]29|29|29; 17
Use of other tables to facilitate computation. . . . . . . . . . 24|45|26|| 5 ||38||1829|41|12|17
Use of slide-rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2439|26|12|38|II |39|28|22|18
Theory of exponents. . . . . tº ſº e º ſº tº tº g º º . . . . . . . . . . . . . . . . . . .3631 ||25 | 8 |36|—|2I]21 |57|14
Theory of logarithms. . . . . . . . . . . . . . . . . . . . e is tº e º ſº . . . . . . . .34|26|21 |18|38|| 7|13|20|6o 15
Arithmetic progression. . . . . . . . . . . . . § {º $ tº 9 & © e º ſº tº º . . . . . . I632/38||1337|23|29|I2|35||17
Geometric progression. . . . . tº gº tº e º e º e º e º e º e s e º ºs e e tº e º º e s tº I9; 27.40; I4|37 |23|25|I8|35||17
Binomial theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35|32 |I8|13|37 |I3|20/27 |4o;15
Probability. . . . . . . tº gº g º e º is tº º e e º 'º e º ºs º º ºs º ºs e º is tº º ºs tº e º sº e º e Q|32 |4|I|I9|32 |2O |35|35|IO |2O
Statistics:
Meaning and use of elementary concepts. . . . . . . . . . . . . |23|2831||17|29|55|36|| 5 || 5 |22
Frequency distributions and frequency curves. . . . . . . . . [I5] Ioſ35|32 |26|4733|Ioſ Io.21
Correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ‘. . . . . . . . . . [II]18|39|32 |2833|47|14| 512I
Numerical trigonometry:
Use of sine, cosine, and tangent in the solution of simple
problems involving right triangles. . . . . . . . . . . . . . . . . 68.21 3| 8|38||—|-|25||75||12
Demonstrative geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68||15||12| 6||34|—|21|4336/14
Plane trigonometry (usual course). . . . . . . . tº es e º ſº e º 'º is e e º ºs 57|27|II | 5 ||37 || 8 |23|31 ||38||13
Analytic geometry:
I'undamental conceptions and methods in the plane....|32 |45||19|| 3 |31 ||—|15|38|46|13
, Systematic treatment of — -
Straight line. . . . . tº e º ºs e º e º e s is s º e º e º e s e º e º e . . . . . . . .3437|2O 9|35| 9| 9|18|64|Ir
ircle. . . . . . . . . . . & & e º e º e º is tº e º g º e º e º e e . . . . . . . . . . . .294329| 9 |35| –|I8] Q|73 II
Conic sections. . . . . . . . tº g g º 'º e º 'º tº º s º e e s ∈ & s e s tº º e º e & I841 |26|1534 – 9|1873; II
Polar coördinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826|41 | 1534––1882|II
Empirical curves and fitting curves to observations... [I2;38||38||12|34} 8–|25 |67|12
64
COLLEGE ENTRANCE REQUIREMENTS
TABLE 2. — ToPICS IN ORDER OF VALUE As PREPARATION FOR
ELEMENTARY COLLEGE COURSES
[The figures in the column headed “E” are taken from Table I, taking in each case the
higher of the two “E” ratings there given. The column headed “E + C" gives in each
case the sum of the two ratings for “E” and “C.” An asterisk indicates that the topic
in question is now included in the definitions of the College Entrance Examination
Board.”]
E |E+C
*Linear equations in one unknown. . . . . . . . . . . tº e º 'º e º ºs & e g º ºs e s 9 e º e s sº * : * * * * 98 IOO
Simple formulas – their meaning and use. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q3 98
*Ratio and proportion. . . . . . . . . . . . . . . . . . . . e e º ſe g g º e º 'º e º 'º e s e is e º e < * * * * * * 84 92
*Negative numbers – their meaning and use. . . . . . . . . . . . . . . . . . . . . tº e º ºs & s ſº 79 84
*Quadratic equations in one unknown . . . . . . . . . tº gº ºs e is e e g º ei e is a s º is e e tº a tº is sº e s 78 93
The linear function: y = mx + b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Qº
*Simultaneous linear equations in two unknowns. . . . . . . . . . . . . . . . . . . . . . . . 7 I 95
Numerical trigonometry — the use of the sine, cosine, and tangent in the solu-
tion of simple problems involving right triangles. . . . . . . . . . . . . . . . . . . . . . . 68 89
*Demonstrative geometry. . . . . . . . . . . . . . . . . . . . . . . . . . º e º 'º e º 'º e º e < * * * * * * * 68 83
Use of logarithms in computation . . . . . . . . . . . . . . . . . . tº tº $ tº º & e º 'º a tº s º º a e e g is 62 9I
*Graphs as a method of representing dependence . . . . . . . . . . . . . . . . . . . . . . . . 62 78
Computation with approximate data — rational use of significant figures....| 61 97
The quadratic function: y = ax + ba + c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SQ 8o
Plane trigonometry — usual course . . . . . . tº e º we is º is s is º & & tº tº tº º tº gº tº gº e º ſº gº tº e º 'º s 2 57 84
Graphic representation of statistical data . . . . . . . . . . . . . . tº e º 'º e º ºs e e g º 'º e º e º e S7 82
Statistics — meaning and use of elementary concepts. . . . . . . . . . . . . . . . . . . . . SS QI
Variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tº º tº º º $ tº e º te tº tº tº e º 'º $ $ tº º SO 63
Statistics — frequency distributions and curves. . . . . . . . . . . . . . . . . . . . • . . . . . . 47 8o
Graphic solution of problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 65
*Literal equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ë e º e º 'º • •'. . . . . 43 6I
*Simultaneous linear equations in more than 2 unknowns. . . . . . . . . . . * & 6 tº e a 43 72
*Simultaneous equations, one quadratic, one linear . . . . . . . . . . . . . . . . . . . . . . 4o | . 64
*Theory of exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 36 67
*Binomial theorem. . . . . . . . . . . . . . . . . . . . e is tº $ tº dº º ſº tº e º ºs tº e º tº $ tº gº e º e º is e s is tº e º a 35 67
Analytic geometry of the straight line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 7 I
Theory of logarithms. . . . . . . . . . . . . . . . . . . . . . . . . . tº ſº e º 'º e º 'º e s e º e a s is e e º ſº e s 34 6o
Statistics — correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8o
Analytic geometry — Fundamental Conceptions. . . . . . . . . . . . . . . . . . . . . . . . . 32 77
*Simultaneous quadratic equations. . . . . . . . . . . . . . e e º e s tº e º 'º e º e º e s tº e º s e e º e 3 I SO
Analytic geometry of the circle. . . . . . . . . . . . . . . . . . . . . . . . . © tº e º $ tº t e º 'º e º 'º º s 29 72
Short-cut methods of computation . . . . . . . . . . . . . . . . . . . . . . . tº e e is s e s tº • . . . . . 29 65
Use of tables in computation (other than logarithms). . . . . & ſº tº g g g º º . . . . . . . . 24 69
Use of slide rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tº . . º. & tº e º e º a tº a $ 4 & 8 & © tº $ tº sp 24 63
*Arithmetic progression. . . . . . . . . . . . . . . . . . . . . . . . * = * * * * * * * * * * * * * * * * * * * * 23 52
*Geometric progression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tº e º 'º e º e º is * * * * * * * * 23 48
Imaginary numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . tº º 4 & & sº e º 'º a º e º * & s is tº e º 'º 8 e 23 44
Probability. . . . . . . . . & © tº gº e º e º O © e º & e s is is a tº s # 8 s tº º e º 'º e & e º e tº a e º ºs e tº tº º ſº tº * * * * 2O SS
Conic sections....... & ſº e º 'º e º 'º e º 'º º * e º g º t e º ºs e e º 'º e º is tº e g is e º e s . . . . . . . . . . . . . . 18 SQ
Polar coördinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * * * * * * * g º º º º is tº º ºs I8 44
I’mpirical curves and fitting curves to observations. . . . . . * & e º ºs e º ºs e s e e º 'º tº e I 2 SO
Equations of higher degree than the second. . . . . . . . . tº e s tº e º is tº sº e s & s tº e • . . . . . IO 42
• The list includes all the requirements of the college entrance examination board except
those relating to algebraic technique. The º: of “Negative numbers” has also been
T
given an asterisk, as it is clearly implied, thoug
definitions.
not explicitly mentioned in the C.E.E.B.
[It should be noted that the asterisk refers to topics included in the C.E.E.B. defini:
tions in force in 1923; – the present (1927) requirements include several topics not marked
with an asterisk in the above list. See the present requirements reprinted on p. 149.)
65
COLLEGE ENTRANCE REQUIREMENTS
cluded, have received widespread approval throughout
the country. That they should be in such close accord
with the desires of college teachers in the fields of physical
and social sciences as to entrance requirements is striking.
We find here the justification for the belief expressed
earlier in this report that there is no real conflict between
the needs of students who ultimately go to college and
those who do not. -
The attitude of the colleges. Mathematical instruc-
tion in this country is at present in a period of transition.
While a considerable number of our most progressive
schools have for severaſ years given Courses embodying
most of the recommendations contained in Chapters II,
III, and IV of the present report, the large majority of
schools are still continuing the older types of courses or
are only just beginning to introduce modifications. The
movement toward reorganization is strong, however,
throughout the country, not only in the standard four-
year high schools but also in the newer junior high schools.
During this period of transition it should be the policy
of the colleges, while exerting a desirable steadying influ-
ence, to help the movement toward a Sane reorganization.
In particular, they should take care not to place obstacles
in the way of changes which are clearly in the interest of
more effective college preparation, as well as of better
general education. t
College entrance requirements will continue to exert a
powerful influence on secondary school teaching. Unless
they reflect the spirit of Sound progressive tendencies,
they will constitute a serious obstacle.
In the present chapter revised definitions of college-
entrance requirements in plane geometry and elementary
66
COLLEGE ENTRANCE REQUIREMENTS
algebra are presented. So far as plane geometry is con-
Cerned, the problem of definition is comparatively simple.
The proposed definition of the requirement in plane ge-
ometry does not differ from the one now in effect under
the College Entrance Examination Board. A list of pro-
positions and constructions has, however, been prepared
and is given in the next chapter for the guidance of teach-
ers and examiners.
In elementary algebra a certain amount of flexibility is
obviously necessary both on account of the quantitative
differences among colleges and of the special conditions
attending a period of transition. The former differences
are recognized by the proposal of a minor and a major re-
quirement in elementary algebra. The second of these
includes the first and is intended to correspond with the
two-unit rating of the C.E.E.B.
In connection with this matter of units, the committee
wishes particularly to disclaim any emphasis upon a
special number of years or hours. The unit termino-
logy is doubtless too well established to be entirely ig-
nored in formulating college entrance requirements, but
the standard definition of unit 3 has never been precise,
and will now become much less so with the inclusion of
the newer six-year program. A time allotment of 4 or
5 hours per week in the seventh year can certainly not
have the same weight as the same number of hours in the
twelfth year, and the disparity will vary with different
3 The following definition, formulated by the National Committee on
Standards of Colleges and Secondary Schools, has been given the ap-
proval of the C.E.E.B.: “A unit represents a year's study in any subject
in a secondary school, constituting approximately a quarter of a full
year's work. A four-year secondary school curriculum should be re-
garded as represeating not more than sixteen units of work.”
67
COLLEGE ENTRANCE REQUIREMENTS
subjects. What is really important is the amount of sub-
ject-matter and the quality of work done in it. The “unit.”
cannot be anything but a crude approximation to this.
The distribution of time in the school program should not
be determined by any arbitrary unit scale.
As a further means of securing reasonable flexibility,
the committee recommends that for a limited time – say
five years — the option be offered between examinations
based on the old and on the new definitions, so far as dif-
ferences between them may make this desirable.
In view of the changes taking place at the present time
in mathematical courses in secondary schools, and the fact
that college entrance requirements should so soon as pos-
sible reflect desirable changes and assist in their adoption,
the National Committee recommends that either the
American Mathematical Society or the Mathematical As-
sociation of America (or both) maintain a permanent com-
mittee on College entrance requirements in mathematics,
such a committee to work in close coöperation with other
agencies which are now or may in the future be concerned
in a responsible way with the relations between colleges
and secondary Schools.”
PROPOSED DEFINITION OF COLLEGE ENTRANCE
REQUIREMENTSS
ELEMENTARY ALGEBRA
Minor requirement. The meaning, use, and evalua-
tion (including the necessary transformations) of simple
[.. Such a committee has not been appointed. It seems to be the gen-
eral opinion that such a committee is at present unnecessary.]
[5 For comparison see the definitions of the C.E.E.B. now in force,
which will be found on p. 140.]
68
COLLEGE ENTRANCE REQUIREMENTS
formulas involving ideas with which the student is fa-
miliar and the derivation of such formulas from rules ex-
pressed in words.
The dependence of one variable upon another. Nu-
merous illustrations and problems involving the linear
function y = mac + b. Illustrations and problems in-
volving the quadratic function y = ka”.
Graphs and graphic representations in general; their
construction and interpretation, including the representa-
tion of statistical data and the use of the graph to exhibit
d; pendence.
Positive and negative numbers; their meaning and use.
Linear equations in one unknown quantity; their use
in solving problems.
Sets of linear equations involving two unknown quan-
tities; their use in solving problems.
Ratio, as a case of simple fractions; proportion without
the theorems on alternation, etc.; and simple cases of
variation. -
The essentials of algebraic technique. This should
include —
(a) The four fundamental operations.
(b) Factoring of the following types: Common factors
of the terms of a polynomial; the difference of two squares;
trinomials of the second degree (including the square of a
binomial) that can be easily factored by trial. -
(c) Fractions, including complex fractions of a simple
typº.
(d) ſixponents and radicals. The laws for positive
integral exponents; the meaning and use of fractional
exponents, but not the formal theory. The considera-
tion of radicals may be confined to the simplification of
- 69
COLLEGE ENTRANCE REQUIREMENTS
expressions of the form Va’b and Va/5 and to the evalu-
ation of simple expressions involving the radical sign.
A process for extracting the Square root of a number
should be included but not the process for extracting
the square root of a polynomial.
Numerical trigonometry. The use of the sine, cosine,
and tangent in solving right triangles. The use of three
or four place tables of natural functions.
Major requirement. In addition to the minor require-
ment as specified above, the following should be included:
Illustrations and problems involving the quadratic
function y = ax + bac + c.
Quadratic equations in one unknown; their use in
solving problems.
Exponents and radicals. Zero and negative exponents,
and more extended treatment of fractional exponents.
Rationalizing denominators. Solution of simple types
of radical equations.
The use of logarithmic tables in computation without
the formal theory.
Elementary statistics, including a knowledge of the
fundamental concepts and simple frequency distributions,
with graphic representations of various kinds.
The binomial theorem for positive integral exponents
less than 8; with Such applications as Compound interest.
The formula for the nth term, and the sum of n terms,
of arithmetic and geometric progressions, with applica-
tions.
Simultaneous linear equations in three unknown quan-
tities and simple cases of simultaneous equations involv-
ing one or two quadratic equations; their use in Solving
problems.
7o
COLLEGE ENTRANCE REQUIREMENTS
Drill in algebraic manipulation should be limited, par-
ticularly in the minor requirement, by the purpose of
Securing a thorough understanding of important prin-
ciples and facility in carrying out those processes which
are fundamental and of frequent occurrence either in
Common life or in the subsequent courses that a sub-
stantial proportion of the pupils will study. Skill in
manipulation must be conceived of throughout as a means
to an end, not as an end in itself. Within these limits,
skill and accuracy in algebraic technique are of prime
importance, and drill in this subject should be extended
far enough to enable students to carry out the funda-
mentally essential processes accurately and with reason-
able speed.
The consideration of literal equations, when they
Serve a significant purpose, such as the transformation of
formulas, the derivation of a general solution (as of the
quadratic equation), or the proof of a theorem, is im-
portant. As a means for drill in algebraic technique they
should be used sparingly.
The solution of problems should offer opportunity
throughout the course for considerable arithmetical and
computational work. The Conception of algebra as an
extension of arithmetic should be made significant both
in numerical applications and in elucidating algebraic
principles. Emphasis should be placed upon the use of
common Sense and judgment in Computing from approxi-
mate data, especially with regard to the number of fig-
ures retained, and on the necessity for checking the re-
sults. The use of tables to facilitate computation (such
as tables of squares and Square roots, of interest, and of
trigonometric functions) should be encouraged.
7I
CoLLEGE ENTRANCE REQUIREMENTS
PLANE GEOMETRY
The usual theorems and constructions of good text-
books, including the general properties of plane rectilinear
figures; the circle and the measurement of angles; similar
polygons; areas; regular polygons and the measurement
of the circle. The solution of numerous original exer-
cises, including locus problems. Applications to the
mensuration of lines and plane Surfaces.
The scope of the required work in plane geometry is
indicated by the List of Fundamental Propositions and
Constructions, which is given in the next chapter. This
list indicates in Section I the type of proposition which,
in the opinion of the committee, may be assumed without
proof or given informal treatment. Section II contains
52 propositions and IQ Constructions which are regarded
as so fundamental that they should constitute the com-
mon minimum of all standard Courses in plane geometry.
^*ction III gives a list of subsidiary theorems which sug-
gests the type of additional propositions that should be
included in such courses.
College entrance examinations. College entrance ex-
aminations exert in many schools, and especially through-
out the eastern section of the Country, an influence on
secondary school teaching which is very far-reaching. It
is, therefore, well within the province of the National
Committee to inquire whether the prevailing type of
examination in mathematics serves the best interests of
mathematical education and of college preparation.
The reason for the almost controlling influence of
entrance examinations in the schools referred to is readily
recognized. Schools sending students to such colleges
for men as Harvard, Yale, and Princeton, to the larger
72
COLLEGE ENTRANCE REQUIREMENTS
colleges for women, or to any institution where examina-
tions form the only or prevailing mode of admission,
inevitably direct their instruction toward the entrance
examination. This remains true even if only a small
percentage of the class intends to take these examinations,
the point being that the success of a teacher is often meas-
ured by the success of his or her students in these exam-
inations.
In the judgment of the committee, the prevailing type
of entrance examination in algebra is primarily a test of
the candidate's skill in formal manipulation, and not an
adequate test of his understanding or of his ability to
apply the principles of the subject. Moreover, it is quite
generally felt that the difficulty and complexity of the
formal manipulative questions, which have appeared on
recent papers set by colleges and by such agencies as
the College Entrance Examination Board, has often been
excessive. As a result, teachers preparing pupils for these
examinations have inevitably been led to devote an
excessive amount of time to drill in algebraic technique,
without insuring an adequate understanding of the prin-
ciples involved. Far from providing the desired facility,
this practice has tended to impair it. For “practical
skill, modes of effective technique, can be intelligently,
non-mechanically used only when intelligence has played
a part in their acquisition.” (Dewey: How We Think,
p. 52.)
Moreover, it must be noted that authors and publish-
ers of textbooks are under strong pressure to make their
content and distribution of emphasis conform to the pre-
vailing type of entrance examination. Teachers in turn
are too often unable to rise above the textbook. An
73
COLLEGE ENTRANCE REQUIREMENTS
improvement in the examinations in this respect will
cause a Corresponding improvement in textbooks and in
teaching.
On the other hand, the makers of entrance examina-
tions in algebra cannot be held solely responsible for the
Condition described. Theirs is a most difficult problem.
Not only can they reply that as long as algebra is taught
as it is, examinations must be largely on technique,” but
they can also claim with considerable force that tech-
nical facility is the only phase of algebra that can be fairly
tested by an examination; that a candidate can rarely do
himself justice amid unfamiliar surroundings and subject
to a time limit on questions involving real thinking in
applying principles to concrete situations; and that we
must face here a real limitation on the power of an exam-
ination to test attainment. Many, and perhaps most,
teachers will agree with this claim. Past experience is
on their side; no generally accepted and effective “power
test” in mathematics has as yet been devised and, if
devised, it might not be suitable for use under conditions
prevailing during an entrance examination.
But if it is true that the power of an examination is thus
inevitably limited, the wisdom and fairness of using it as
the sole means of admission to college is surely open to
grave doubt. That many unqualified candidates are
admitted under this system is not open to question. Is
it not probable that many qualified candidates are at the
same time excluded? If the entrance examination is a
fair test of manipulative skill only, should not the colleges
* The vicious circle is now complete. Algebra is taught mechanically
because of the character of the entrance examination; the cKamination,
in order to be fair, must Conform to the character of the teaching.
74
COLLEGE ENTRANCE REQUIREMENTS
use additional means for learning something about the
candidate's other abilities and qualifications?
Some teachers believe that an effective “power test ''
in mathematics is possible. Efforts to devise such a test
should receive every encouragement. g
In the meantime, certain desirable modifications of
the prevailing type of entrance examination are possible.
The College Entrance Examination Board recently ap-
pointed a committee to consider this question and a con-
ference 7 on this subject was held by representatives of
the College Entrance Examination Board, members of
the National Committee, and others. The following
recommendations are taken from the report of the com-
mittee just referred to:
Fully one third of the questions should be based on topics of
such fundamental importance that they will have been thor-
oughly taught, carefully reviewed, and deeply impressed by
effective drill. . . . They should be of such a degree of difficulty
that any pupil of regular attendance, faithful application, and
even moderate ability may be expected to answer them satis-
factorily. -
There should be both simple and difficult questions testing
the candidate's ability to apply the principles of the subject.
The early ones of the easy questions should be really easy for
the candidate of good average ability who can do a little
thinking under the stress of an examination; but even these
questions should have genuine scientific content.
There should be a substantial question which will put the
7 At this conference the following vote was unanimously passed:
“Voted, that the results of examinations (of the College Entrance Ex-
amination Board), be reported by letters A, B, C, D, E, and that the defi-
nition of the groups represented by these letters should be determined in
each year by the distribution of ability in a standard group of papers
representing widely both public and private Schools.”
75
CoLLEGE ENTRANCE REQUIREMENTS
best candidates on their mettle, but which is not beyond the
reach of a fair proportion of the really good candidates. This
question should test the normal workings of a well-trained
mind. It should be capable of being thought out in the
limited time of the examination. It should be a test of the
Candidate's grasp and insight – not a catch question or a
question of unfamiliar character making extraordinary de-
mands on the critical powers of the candidate or one the solu-
tion of which depends on an inspiration. Above all, this
question should lie near to the heart of the subject as all well-
prepared candidates understand the subject.
As a rule, a question should consist of a single part and be
framed to test one thing — not pieced together out of several
unrelated and perhaps unequally important parts.
Each question should be a substantial test on the topic or
topics which it represents. It is, however, in the nature of the
case impossible that all questions be of equal value.
Care should be used that the examination be not too long.
. . . The examiner should be content to ask questions on the
important topics, so chosen that their answers will be fair to
the candidate and instructive to the readers; and beyond this
merely to Sample the candidate's knowledge of the minor
topics.
The National Committee suggests the following addi-
tional principles: The examination as a whole should, as
far as practicable, reflect the principle that algebraic
technique is a means to an end, and not an end in itself.
Questions that require of the candidate skill in alge-
braic manipulation beyond the needs of actual applica-
tion should be used very sparingly.
An effort should be made to devise questions which
will fairly test the candidate's understanding of principles
and his ability to apply them, while involving a minimum
of manipulative complexity. •
76
COLLEGE ENTRANCE REQUIREMENTS
The examinations in geometry should be definitely
constructed to test the candidate's ability to draw valid
conclusions rather than his ability to memorize an argu-
ment.
A chapter on mathematical terms and symbols is
included in this report.” It is hoped that examining
bodies will be guided by the recommendations there
made relative to the use of terms and symbols in ele-
mentary mathematics.
The College Entrance Examination Board, early in
1921, appointed a commission to recommend such re-
visions as might seem necessary in the definitions of the
requirements in the various subjects of elementary math-
ematics. The recommendations contained in the present
chapter have been laid before this commission. It is
hoped that the commission's report, when it is finally
made effective by action of the College Entrance Exam-
ination Board and the various colleges concerned, will
give impetus to the reorganization of the teaching of ele-
mentary mathematics along the lines recommended in
the report of the National Committee.”
8 See Chapter VIII.
[9 This hope has been fully realized. See the present C.E.E.B. requires
ments on p. 140.] -
VI
LIST OF PROPOSITIONS IN PLANE AND
SOLID GEOMETRY
General basis of the selection of material. The sub-
Committee appointed to prepare a list of basal proposi-
tions made a careful study of a number of widely used
textbooks on geometry. The bases of selection of the
propositions were two: (1) The extent to which the pro-
positions and corollaries were used in subsequent proofs
of important propositions and exercises; (2) the value of
propositions in completing important pieces of theory.
Although the list of theorems and problems is substan-
tially the same in nearly all textbooks in general use in
this country, the wording, the sequence, and the methods
of proof vary to such an extent as to render difficult a
definite statement as to the number of times a proposi-
tion is used in the several books examined. A tentative
table showed, however, less variation than might have
been anticipated. -
Classification of propositions. The classification of
propositions is not the same in plane geometry as in Solid
geometry. This is partly due to the fact that it is gen-
erally felt that the student should limit his construction
work to figures in a plane and in which the Compasses and
straight edge are sufficient. The propositions have been
divided as follows:
Plane geometry: I. Assumptions and theorems for in-
formal treatment; II. Fundamental theorems and con-
structions: A. Theorems, B. Constructions; III. Sub-
sidiary theorems.
78
LIST of PROPOSITIONS IN GEOMETRY
Solid geometry: I. Fundamental theorems; II. Funda-
mental propositions in mensuration; III. Subsidiary
theorems; IV. Subsidiary propositions in mensuration."
PLANE GEOMETRY
I. Assumptions and theorems for informal treatment.
This list contains propositions which may be assumed
without proof (postulates) and theorems which it is per-
missible to treat informally. Some of these propositions
will appear as definitions in certain methods of treatment.
Moreover, teachers should feel free to require formal
proofs in certain cases, if they desire to do so. The pre-
cise wording given is not essential, nor is the Order in
which the propositions are here listed. The list should
be taken as representative of the type of propositions
which may be assumed, or treated informally, rather than
as exhaustive.
I. Through two distinct points it is possible to draw one
straight line, and only one.
2. A line segment may be produced to any desired length.
3. The shortest path between two points is the line segment
joining them.
4. One and only one perpendicular can be drawn through a
given point to a given Straight line. [6*, not more than one
perpendicular.]
5. The shortest distance from a point to a line is the per-
pendicular distance from the point to the line.
6. From a given Center and with a given radius one and only
one circle can be described in a plane.
[.. In the lists below the numbers in [ ] give the numbers of the corre-
sponding propositions in the C.E.E.B. lists. The letters “cq” after a
proposition indicates that it is included in the C.E.E.B. requirement for
mathematics, cq, Plane and Solid Geometry, Minor Requirement. See
pp. 162, 168, for meaning of asterisks following some of the numbers.]
79
LIST OF PROPOSITIONS IN GEOMETRY
7. A straight line intersects a circle in at most two points.
8. Any figure may be moved from one place to another with-
Dut changing its shape or size.
9. All right angles are equal. |
Io. If the sum of two adjacent angles equals a straight
angle, their exterior sides form a straight line.
II. Equal angles have equal Complements and equal Sup-
plements. -
12. Vertical angles are equal.
13. Two lines perpendicular to the same line are parallel.
[7]
I4. Through a given point not on a given straight line, one
straight line, and only one, can be drawn parallel to the given
line.
I5. Two lines parallel to the same line are parallel to each
other.
16. The area of a rectangle is equal to its base times its
altitude.
II. Fundamental theorems and constructions. It is
recommended that theorems and constructions (other
than originals) to be proved on college entrance exami-
nations be chosen from the following list. Originals and
other exercises should be capable of solution by direct
reference to one or more of these propositions and con-
structions. It should be obvious that any course in
geometry that is capable of giving adequate training
must include considerable additional material. The
order here given is not intended to signify anything as to
the order of presentation. It should be clearly under-
stood that certain of the statements contain two or more
theorems, and that the precise wording is not essential.
The committee favors entire freedom in statement and
Sequence.
80
LIST OF PROPOSITIONS IN GEOMETRY
A. THEOREMS
1. Two triangles are congruent if * (a) two sides and the in-
cluded angle of one are equal, respectively, to two sides and
the included angle of the other; (b) two angles and a side of
one are equal, respectively, to two angles and the correspond-
ing side of the other; (c) the three sides of one are equal
respectively, to the three sides of the other. It", 2*, 5*,
cdº] .
2. Two right triangles are congruent if the hypotenuse and
one other side of one are equal, respectively, to the hypote-
nuse and another side of the other. Iro", cd".
3. If two sides of a triangle are equal, the angles opposite
these sides are equal; and conversely. [3*, 4*, cd”
4. The locus of a point (in a plane) equidistant from two
given points is the perpendicular bisector of the line joining
them. [30°, cdº]
5. The locus of a point equidistant from two given intersect-
ing lines is the pair of lines bisecting the angles formed by these
lines. [31*, cd”
6. When a transversal cuts two parallel lines, the alternate
interior angles are equal; and conversely. [II*, ca”, 14, cq]
7. The sum of the angles of a triangle is two right angles.
[23*, cdº]
8. A parallelogram is divided into congruent triangles by
either diagonal.
9. Any (convex) quadrilateral is a parallelogram (a) if the
* Teachers should feel free to separate this theorem into three distinct
theorems and to use other phraseology for any such proposition. For
example, in I, “Two triangles are equalif. . . ,” “a triangle is determined
by . . . ,” etc. Similarly in 2, the statement might read: “Two right
triangles are congruent if, besides the right angles, any two parts (not
both angles) in the one are equal to corresponding parts of the other.”
3 It should be understood that the converse of a theorem need not be
treated in connection with the theorem itself, it being sometimes better
to treat it later. Furthermore, a converse may occasionally be accepted
as true in an elementary Course, if the necessity for proof is made clear.
The proof may then be given later.
8T
LIST OF PROPOSITIONS IN GEOMETRY
opposite sides are equal; (b) if two sides are equal and parallel.
[20°, 21*, cdº
Io. If a Series of parallel lines cut off equal segments on One
transversal they cut off equal segments on any transversai.
[22*, cdº
II. (a) The area of a parallelogram is equal to the base
times the altitude. [72*, cq*]
(b) The area of a triangle is equal to one half the base times
the altitude. [73, cq]
(c) The area of a trapezoid is equal to half the sum of its
bases times its altitude. [75*, cd)
(d) The area of a regular polygon is equal to half the product
of its apothem and perimeter. [81*] .
12. (a) If a straight line is drawn through two sides of a
triangle parallel to the third side it divides these sides pro-
portionally. [57, Cd]
(b) If a line divides two sides of a triangle proportionally it
is parallel to the third side. (Proofs for commensurable cases
only.) [58*, cdº]
(c) The segments cut off on two transversals by a series of
parallels are proportional. [64, cq)
13. Two triangles are similar if (a) they have two angles of
One equal, respectively, to two angles of the other; (b) they
have an angle of one equal to an angle of the other and the in-
cluding sides are proportional; (c) their sides are respectively
proportional. [59°, 60°, 61*, cd”)
14. If two chords intersect in a circle, the product of the
segments of one is equal to the product of the segments of the
other. [67*.
15. The perimeters of two similar polygons have the same
ratio as any two corresponding sides. [69, Cd]
I6. Polygons are similar, if they can be decomposed into
triangles which are similar and similarly placed [71]; and
conversely. [76*
17. The bisector of an (interior or exterior) angle of a tri-
angle divides the opposite side (produced if necessary) into
82
LIST OF PROPOSITIONS IN GEOMETRY
segments proportional to the adjacent sides. [62, cd; 63]
18. The areas of two similar triangles (or polygons) are to
each other as the Squares of any two corresponding sides.
[76°, 77*, cd]
I9. In any right triangle the perpendicular from the vertex
of the right angle on the hypotenuse divides the triangle into
two triangles each similar to the given triangle. [65, Cd]
20. In a right triangle the square on the hypotenuse is
equal to the sum of the squares on the other two sides. (66*,
cd”
21. In the same circle or in equal circles, if two arcs are
equal, their central angles are equal; and conversely. [37,
cq] -
22. In any circle angles at the center are proportional to
their intercepted arcs. (Proof for commensurable case only.)
|38, col) |
23. In the same circle or in equal circles, if two chords are
equal their corresponding arcs are equal; and conversely.
[39, Cd]
24. (a) A diameter perpendicular to a chord bisects the
chord and the arcs of the chord. [41*, cd” (b) A diameter
which bisects a chord (that is not a diameter) is perpendicular
to it. [42, Cal
25. The tangent to a circle at a given point is perpendicular
to the radius at that point; and conversely. [43, 44, Cd]
26. In the same circle or in equal circles, equal chords are
equally distant from the center; and conversely. [40°, cd”
27. An angle inscribed in a circle is equal to half the central
angle having the same arc. [47”, cd”
28. Angles inscribed in the same segment are equal. [48, cq]
29. If a circle is divided into equal arcs, the chords of these
arcs form a regular inscribed polygon and tangents at the
points of division form a regular circumscribed polygon.
[80]
3o. The circumference of a circle is equal to 27tr. (Informal
proof only.) [87, Cd]
83
LIST OF PROPOSITIONS IN GEOMETRY
31.3 The area of a circle is equal to trº. (Informal proof
only.) [88, cq]
The treatment of the mensuration of the circle should be
based upon related theorems concerning regular polygons, but
it should be informal as to the limiting processes involved.
The aim should be an understanding of the concepts involved,
So far as the capacity of the pupil permits. -
B. CONSTRUCTIONS
I. Bisect a line segment [I, Cd] and draw the perpendicular
bisector.
2. Bisect an angle. [2, col] -
3. Construct a perpendicular to a given line through a given
point. (3, 4, Cd]
4. Construct an angle equal to a given angle. [5, cq]
5. Through a given point draw a straight line parallel to a
given straight line. [6, Cd]
6. Construct a triangle, given (a) the three sides; (b) two
sides and the included angle; (c) two angles and the included
side. [8, 9, Io, Cd)
7. Divide a line segment into parts proportional to given
Segments. [7, Cd (equal parts), 15, Cd)
8. Given an arc of a circle, find its center.
9. Circumscribe a circle about a triangle. [II, cq]
Io. Inscribe a circle in a triangle. [I2, Cd
II. Construct a tangent to a circle through a given point
[I3, 14, cq
12. Construct the fourth proportional to three given line
segments. [16, Cd]
13. Construct the mean proportional between two given
line segments. [I7, Cd]
14. Construct a triangle (polygon) similar to a given triangle
(polygon). [18]
3 The total number of theorems given in this list when separated, as
will probably be found advantageous in teaching, including the converses
indicated, is 52.
84
LIST OF PROPositions IN GEOMETRY
15. Construct a triangle equal to a given polygon.
I6. Inscribe a square in a circle. [19, Cd]
17. Inscribe a regular hexagon in a circle. [20, cq]
III. Subsidiary list of propositions. The following list
of propositions is intended to suggest some of the addi-
tional material referred to in the introductory paragraph
of Section II. It is not intended, however, to be exhaus-
tive; indeed, the committee feels that teachers should be
allowed considerable freedom in the selection of such
additional material, theorems, corollaries, originals, exer-
cises, etc., in the hope that opportunity will thus be
afforded for constructive work in the development of
courses in geometry. -
I. When two lines are cut by a transversal, if the cor-
responding angles are equal [I5, Cd) or if the interior angles on
the same side of the transversal are supplementary [16], the
lines are parallel.
2. When a transversal cuts two parallel lines, the cor-
responding angles are equal [12, cq], and the interior angles on
the same side of the transversal are supplementary. [I3]
3. A line perpendicular to one of two parallels is perpendicu-
lar to the other also. [8, ca]
4. If two angles have their sides respectively parallel or
respectively perpendicular to each other, they are either equal
or supplementary. [I7, Cd
5. Any exterior angle of a triangle is equal to the sum of the
two opposite interior angles. [24, Cd]
6. The sum of the angles of a convex polygon of n sides is
2(n) − 2) right angles. [25]
7. In any parallelogram (a) the opposite sides are equal;
(b) the opposite angles are equal; (c) the diagonals bisect each
other. [18, 19, Cd]
8. Any (convex) quadrilateral is a parallelogram, if (a) the
opposite angles are equal; (b) the diagonals bisect each other.
85
LIST OF PROPOSITIONS IN GEOMETRY
9. The medians of a triangle intersect in a point which is
two thirds of the distance from a vertex to the mid-point of
the opposite side. [35]
Io. The altitudes of a triangle meet in a point. [34]
II. The perpendicular bisectors of the sides of a triangle
meet in a point. [32]
I2. The bisectors of the angles of a triangle meet in a point.
[33] *
I3. The tangents to a circle from an external point are
equal. [45, Cd, part .
14.4 (a) If two sides of a triangle are unequal, the greater
side has the greater angle opposite it, and conversely. [26,
27, Cd]
(b) If two sides of one triangle are equal respectively to two
sides of another triangle, but the included angle of the first is
greater than the included angle of the second, then the third
side of the first is greater than the third side of the second, and
conversely. [28, 29]
(c) If two chords are unequal, the greater is at the less dis-
tance from the center, and conversely. [55, 56]
(d) The greater of two minor arcs has the greater chord, and
conversely. [53, 54]
I5. An angle inscribed in a semicircle is a right angle.
[49, Cd]
I6. Parallel lines cutting a circle intercept equal arcs on the
circle. [46, cd]
I7. An angle formed by a tangent of a circle and a chord
drawn through the point of contact is measured by half the
intercepted arc. [50, cq]
18. An angle formed by two intersecting chords is measured
by half the sum of the intercepted arcs. [51]
4. Such inequality theorems as these are of importance in developing the
notion of dependence or functionality in geometry. The fact that they
are placed in the “Subsidiary list of propositions” should not imply that
they are considered of less educational value than those in List II. They
are placed here because they are not “fundamental” in the same sense
that the theorems of List II are fundamental.
86
List of PROPOSITIONS IN GEOMETRY
19. An angle formed by two secants or by two tangents to a
circle is measured by half the difference between the inter-
cepted arcs. [52, part
20. If from a point without a circle a secant and a tangent
are drawn, the tangent is the mean proportional between the
whole secant and its external segment. [68]
2I. Parallelograms or triangles of equal bases and equal
altitudes are equal. [74, Cd] §
22. The perimeters of two regular polygons of the same
number of sides are to each other as their radii and also as
their apothems. [83] 5
SOLID GEOMETRY
In the following list the precise wording and the se-
quence are not considered:
I. FUNDAMENTAL THEOREMS
I. If two planes meet, they intersect in a straight line. [1,
cq]
2. If a line is perpendicular to each of two intersecting lines
at their point of intersection it is perpendicular to the plane
of the two lines. [2*, cd".
3. Every perpendicular to a given line at a given point lies
in a plane perpendicular to the given line at the given point.
[3, Cd)
4. Through a given point (internal or external) there can
pº one and only one line perpendicular to a plane. [6°, 7”,
cdº]
5. Two lines perpendicular to the same plane are parallel.
[toº, cd” º
6. If two lines are parallel, every plane containing one of the
lines and only one is parallel to the other. [13, coll
7. Two planes perpendicular to the same line are parallel.
[16, Că]
[5 For additional theorems in Plane Geometry contained in the C.E.E.B
List see p. 169.]
87
LIST OF PROPOSITIONS IN GEOMETRY
8. Ef two parallel planes are cut by a third plane, the lines
of intersection are parallel. [I7, cq]
9. If two angles not in the same plane have their sides re-
Spectively parallel in the same sense, they are equal and their
planes are parallel. [21*, cdº].
Io. If two planes are perpendicular to each other, a line
drawn in one of them perpendicular to their intersection is
perpendicular to the other. [23°, cd".
II. If a line is perpendicular to a given plane, every plane
which contains this line is perpendicular to the given plane.
[26*, cdº]
12. If two intersecting planes are each perpendicular to a
third plane, their intersection is also perpendicular to that
plane. [27*, cd)
13. The sections of a prism made by parallel planes cutting
all the lateral edges are congruent polygons. [37*.
I4. An oblique prism is equal to a right prism whose base
is equal to a right section of the oblique prism and whose
altitude is equal to a lateral edge of the oblique prism.
[40°] f .
I5. The opposite faces of a parallelepiped are Congruent.
[42] .
I6. The plane passed through two diagonally opposite
edges of a parallelepiped divides the parallelepiped into two
equal triangular prisms. [43*) -
17. If a pyramid (or a cone) is cut by a plane parallel to the
base:
(a) The lateral edges (or elements) and the altitude are
divided proportionally; [47", cd” (pyramid)]
(b) The section is a figure similar to the base; [47", cd”
(pyramid), 56° (cone)]
(c) The area of the section is to the area of the base as the
square of the distance from the vertex is to the square of the
altitude of the pyramid (or cone).
18. Two triangular pyramids having equal bases and equal
altitudes are equal. [50]
88
LIST OF PROPOSITIONS IN GEOMETRY
19. All points on a circle of a sphere are equidistant from
either pole of the circle. [67]
20. On any sphere a point which is at a quadrant’s distance
from each of two other points not the extremities of a diameter
is apole of the great circle passing through these two points.
[68*.
2I. If a plane is perpendicular to a radius at its extremity
on a sphere, it is tangent to the Sphere. [69; and Conversely,
7o
22. A sphere can be inscribed in [71] or circumscribed about
[72*] any tetrahedron.
23. If one spherical triangle is the polar of another, then
reciprocally the second is the polar triangle of the first. [76*
24. In two polar triangles each angle of either is the supple.
ment of the opposite side of the other. [77]
25. Two symmetric spherical triangles are equal. [79]
II. FUNDAMENTAL PROPOSITIONS IN MENSURATION
26. The lateral area of a prism (41*] or a circular cylinder
[54, coll is equal to the product of a lateral edge or element,
respectively, by the perimeter (circumference) of a right sec-
tion.
27. The volume of a prism [46, cd; triangular, 45° (in-
cluding any parallelepiped [44, Cd) or of a circular cylinder
[55, ca) is equal to the product of its base by its altitude.
28. The lateral area of a regular pyramid [48*] or a right
circular cone [57, coll is equal to half the product of its slant
height by the perimeter (circumference) of its base.
29. The volume of a pyramid 152, cd; triangular, 51* or a
cone [59, cq] is equal to one third the product of its base by its
altitude.
30. The area of a sphere. [91, cd]
31. The area of a spherical polygon.
32. The volume of a sphere. [92, cd]
89
LIST OF PROPOSITIONS IN GEOMETRY
III. SUBSIDIARY THEOREMS
33. If from an external point a perpendicular and obliques
are drawn to a plane, (a) the perpendicular is shorter than any
oblique [8, Cd]; (b) obliques meeting the plane at equal dis-
tances from the foot of the perpendicular are equal [9]; (c) of
two obliques meeting the plane at unequal distances from the
foot of the perpendicular, the more remote is the longer. [9]
34. If two lines are cut by three parallel planes, their cor-
responding segments are proportional. [22]
35. Between two lines not in the same plane there is one
common perpendicular, and only one. [29, Cd]
36. The bases of a cylinder are Congruent. [53]
37. If a plane intersects a sphere, the line of intersection is
a circle. [64*]
38. The volume of two tetrahedrons that have a trihedral
angle of one equal to a trihedral angle of the other are to each
other as the products of the three edges of these trihedral angles.
39. In any polyhedron the number of edges increased by
two is equal to the number of vertices increased by the num-
ber of faces.
40. Two similar polyhedrons can be separated into the same
number of tetrahedrons similar each to each and similarly
placed. -
41. The volumes of two similar tetrahedrons are to each
other as the cubes of any two corresponding edges.
42. The volumes of two similar polyhedrons are to each
other as the cubes of any two corresponding edges. [61]
43. If three face angles of one trihedral angle are equal, re-
spectively, to the three face angles of another the trihedral
angles are either congruent or symmetric. [36]
44. Two spherical triangles on the same sphere are either
congruent or symmetric if (a) two sides and the included angle
of one are equal to the corresponding parts of the other; (b)
two angles and the included side of one are equal to the cor-
responding parts of the other; (c) they are mutually equilat-
eral; (d) they are mutually equiangular. |80, 81, 82, 83]
90
LIST OF PROPOSITIONS IN GEOMETRY
45. The sum of any two face angles of a trihedral angle is
greater than the third face angle. [33]
46. The sum of the face angles of any convex polyhedral
angle is less than four right angles. [34]
47. Each side of a spherical triangle is less than the sum of
the other two sides. [74]
48. The sum of the sides of a spherical polygon is less than
360°. [75] -
49. The sum of the angles of a spherical triangle is greater
than 180° and less than 540°. [78*]
50. There cannot be more than five regular convex poly-
hedrons. [35]
51. The locus of points equidistant (a) from two given
points; (b) from two given planes which intersect. [30, 32]
IV. SUBSIDIARY DROPOSITIONS IN MENSURATION
52. The volume of a frustum of (a) a pyramid or (b) a cone.
53. The lateral area of a frustum of (a) a pyramid [regular,
49] or (b) a cone of revolution. [58]."
54. The volume of a prismoid (without formal proof).
* For additional theorems in Solid Geometry contained in the C.E.E.B.
List see p. 169. -
VII
THE FUNCTION CONCEPT IN SECONDARY
SCHOOL MATHEMATICS I
IN Chapter II, and incidentally in later chapters, consid-
erable emphasis has been placed on the function concept
or, better, on the idea of relationship between variable
quantities as one of the general ideas that should domi-
nate instruction in elementary mathematics. Since this
recommendation is peculiarly open to misunderstanding
on the part of teachers, it seems desirable to devote a
Separate chapter to a rather detailed discussion of what
the recommendation means and implies. t
It will be seen in what follows that there is no disposi
tion to advocate the teaching of any sort of function
theory. A prime danger of misconception that should be
removed at the very outset is that teachers may think it
is the notation and the definitions of such a theory that
are to be taught. Nothing could be further from the
intention of the committee. Indeed, it seems entirely
safe to say that the word “function” had best not be used
at all in the early courses.
What is desired is that the idea of relationship or de-
pendence between variable quantities be imparted to the
* The first draft of this chapter was prepared for the National Com-
mittee by E. R. Hedrick, of the University of Missouri [at present, of
the University of California, Southern Branch]. It was discussed at the
meeting of the Committee, September 2–4, 1920; revised by the author,
and again discussed December 29, 30, 1920, and is now issued as part of
the committee's report.
92
THE FUNCTION CONCEPT
pupil by the examination of numerous concrete instances
of such relationship. He must be shown the workings
of relationships in a large number of cases before the
abstract idea of relationship will have any meaning for
him. Furthermore, the pupil should be led to form the
habit of thinking about the connections that exist be-
tween related quantities, not merely because such a habit
forms the best foundation for a real appreciation of the
theory that may follow later, but chiefly because this
habit will enable him to think more clearly about the
quantities with which he will have to deal in real life,
whether or not he takes any further work in mathe-
matics. -
Indeed, the reason for insisting so strongly upon atten
tion to the idea of relationships between quantities is that
such relationships do occur in real life in connection with
practically all of the quantities with which we are called
upon to deal in practice. Whereas there can be little
doubt about the small value to the student who does not
go on to higher studies of some of the manipulative pro-
Cesses criticized by the National Committee, there can
be no doubt at all of the value to all persons of any in-
crease in their ability to see and to foresee the manner in
which related quantities affect each other.
To attain what has been suggested, the teacher should
have in mind constantly not any definition to be recited
by the pupil, not any automatic response to a given cue,
not any memory exercise at all, but rather a determina-
tion not to pass any instance in which one quantity is
related to another, or in which one quantity is determined
by one or more others, without calling attention to the
fact, and trying to have the student “see how it works.”
93
THE FUNCTION CONCEPT
These instances occur in literally thousands of cases in
both algebra and geometry. It is the purpose of this
chapter to outline in some detail a few typical instances
of this character. -
I. RELATIONSHIPS IN ALGEBRA
In algebra the instance of the function idea which usu,
ally occurs to one first is in connection with the study of
graphs. While this is natural enough, and while it is
true that the graph is fundamentally functional in char-
acter, the supposition that it furnishes the first oppor-
tunity for observing functional relations between quan-
tities betrays a misconception that ought to be cor-
rected.
1. Use of letters for numbers. The very first illus-
trations given in algebra to show the use of letters in the
place of numbers are essentially functional in character.
Thus, such relations as I = prl and A = Tr”, as well as
others that are frequently used, are statements of general
relationships. These should be used to accustom the
student not only to the use of letters in the place of num-
bers and to the solution of simple numerical problems, but
also to the idea, for example, that changes in r affect the
value of A. Such questions as the following should be
considered: If r is doubled, what will happen to A 2 If
p is doubled, what will happen to I? Appreciation of the
meaning of such relationships will tend to clarify the en-
tire subject under consideration. Without such an appre-
ciation, it may be doubted whether the student has any
real grasp of the matter. -
2. Equations. Every simple problem leading to an
equation in the first part of algebra would be better
94
THE FUNCTION CONCEPT
understood for just such a discussion as that mentioned
above. Thus, if two dozen eggs are weighed in a basket
which weighs 2 pounds, and if the total weight is found to
be 5 pounds, what is the average weight of an egg’ If a
is the weight in ounces of one egg, the total weight with
the 2-pound basket would be 24x + 32 ounces. If the
student will first try the effect of an average weight of
I ounce, of 194 ounces, 2 ounces, 2% ounces, the meaning
of the problem will stand out clearly. In every such
problem, preliminary trials really amount to a discussion
of the properties of a linear function.
3. Formulas of pure Science and of practical affairs.
The study of formulas as such, aside from their numerical
evaluation, is becoming of more and more importance.
The actual uses of algebra are not to be found solely nor
even principally in the solution of numerical problems
for numerical answers. In such formulas as those for
falling bodies, levers, etc., the manner in which changes
in one quantity cause (or correspond to) changes in an-
other, are of prime importance, and their discussion need
cause no difficulty whatever. The formulas under dis-
cussion here include those formulas of pure science and of
practical affairs which are being introduced more and
more into our texts on algebra. Whenever such a for-
mula is encountered, the teacher should be sure that the
students have some comprehension of the effects of
changes in one of the quantities upon the other quantity
or quantities in the formula.
As a specific instance of such scientific formulas, con-
sider, for example, the force F, in pounds, with which a
weight W, in pounds, pulls outward on a string (centri-
fugal force) if the weight is revolved rapidly at a speed º,
95
THE FUNCTION CONCEPT
in feet per second, at the end of a string of length r feet.
This force is given by the formula
F – W. "
32 r
When such a formula is used the teacher should not be
Contented with the mere insertion of numerical values for
W, v, and r to obtain a numerical value for F.
The advantage obtained from the study of such a for-
mula lies quite as much in the recognition of the behavior
of the force when one of the other quantities varies. Thus
the student should be able to answer intelligently such
questions as the following: If the weight is assumed to be
twice as heavy, what is the effect upon the force? If the
speed is taken twice as great, what is the effect upon the
force? If the radius becomes twice as large, what is the
effect upon the force? If the speed is doubled, what
change in the weight would result in the same force?
Will an increase in the speed cause an increase or decrease
in the force? Will an increase in the radius r cause an
increase or a decrease in the force?
As another instance (of a more advanced character)
consider the formula for the amount of a sum of money P,
at Compound interest at r per cent, at the end of n years.
This amount may be denoted by An. Then we shall have
An = P (I + r.)". Will doubling P result in doubling A, 2
Will doubling n result in doubling A, 3 Since the com-
pound interest that has accumulated is equal to the dif-
ference between P and An, will the doubling of r double
the interest? Compare the correct answers to these
questions with the answers to the similar questions in the
case of simple interest, in which the formula reads
96
THE FUNCTION CONCEPT
A, - P + Prm and in which the accumulated interest is
simply Prm.
The difference between such a study of the effect pro-
duced upon one quantity by changes in another and the
mere substitution of numerical values will be apparent
from these examples.
4. Formulas of pure algebra. Formulas of pure alge-
bra, such as that for (x + h)*, will be better understood
and appreciated if accompanied by a discussion of the
manner in which changes in h cause changes in the total
result. This can be accomplished by discussing such
concrete realities as the error made in computing the area
of a square field or of a square room when an error has
been made in measuring the side of the square. If w is the
true length of the side, and if the student assumes various
possible values for the error h made in measuring x, he
will have a situation that involves some comprehension
of the functional workings of the formula mentioned.
The same formula relates to such problems as the change
from one entry to the next entry in a table of squares.
A similar situation, and a very important one, occurs
with the pure algebraic formula for (x + a) (y -- b). This
formula may be said to govern the question of the keeping
of significant figures in finding the product acy. For, if a
and b represent the uncertainty in 3 and y, respectively,
the uncertainty in the product is given by this formula.
The student has much to learn on this score, for the re-
tention of meaningless figures in a product is one of the
commonest mistakes of both student and teacher in com-
putational work.
Such formulas occur throughout algebra, and each of
them will be illuminated by such a discussion. The for-
Q7
THE FUNCTION CONCEPT
mulas for arithmetic and geometric progression, for
example, should be studied from a functional stand-
point.
5. Tables. The uses of the functional idea in connec-
tion with numerical computation have already been men-
tioned in connection with the formula for a product.
Work which appears on the surface to be wholly numeri-
cal may be of a distinctly functional character. Thus, any
table, e.g., a table of Squares, corresponds to or is con-
structed from a functional relation, e.g., for a table of
squares, the relation y = x*. The differences in such a
table are the differences caused by changes in the values
of the independent variable. Thus, the differences in a
table of squares are precisely the differences between acº
and (x + h)” for various values of ac.
6. Graphs. The functional character of graphic repre-
sentations was mentioned at the beginning of this section.
Every graph is obviously aspepresentation of a functional
relationship between two or more quantities. What is
needed is only to draw attention to this fact and to study
each graph from this standpoint. In addition to this,
however, it is desirable to point out that functional rela-
tions may be studied directly by means of graphs without
the intervention of any algebraic formula. Thus, such a
graph as a population curve, or a curve representing wind
pressure, obviously represents a relationship between two
quantities, but there is no known formula in either case.
The idea that the three concepts, tables, graphs, alge-
braic formulas, are all representations of the same kind of
connection between quantities, and that we may start in
some instances with any of the three, is a most valuable
addition to the student's mental equipment, and to his
98
THE FUNCTION CONCEPT
control over the quantities with which he will deal in his
daily life. --
II. RELATIONSHIPS IN GEOMETRY
Thus far the instances mentioned have been largely
algebraic, though certain mensuration formulas of geo-
metry have been mentioned. While the mensuration
formulas may occur to one first as an illustration of
functional concepts in geometry, they are by no means
the earliest relationships that occur in that study.
I. Congruence. Among the earliest theorems are those
on the congruence of triangles. In any such theorem, the
parts necessary to establish Congruence evidently deter-
mine completely the size of each other part. Thus, two
sides and the included angle of a triangle evidently de-
termine the length of the third side. If the student
clearly grasps this fact, the meaning of this case of con-
gruence will be more vivid to him, and he will be prepared
for its important applications in Surveying and in trig-
onometry. Even if he never studies these subjects, he
will nevertheless be able to use his understanding of the
situation in any practical cases in which the angle be-
tween two fixed rods or beams is to be fixed or is to be
determined, in a practical situation such as house build-
ing. Other congruence theorems throughout geometry
may well be treated in a similar manner.
2. Inequalities. In the theorems regarding inequali-
ties, the functional quality is even more pronounced.
Thus, if two triangles have two sides of one equal re-
spectively to two sides of the other, but if the included
angle between these sides in the one triangle is greater
than the corresponding angle in the other, then the third
99
THE FUNCTION CONCEPT
sides of the triangles are unequal in the same sense. This
theorem shows that as one angle grows, the side opposite
it grows, if the other sides remain unchanged. A full
realization of the fact here mentioned would involve a
real grasp of the functional relation between the angle
and the side opposite it. Thus, if the angle is doubled,
will the side opposite it be doubled? Such questions arise
in connection with all theorems on inequalities.
3. Variations in figures. A great assistance to the im-
agination is gained in certain figures by imagining varia-
tions of the figure through all intermediate stages from
one case to another. Thus, the angle between two lines
that cut a circle is measured by a proper combination of
the two arcs cut out of the circle by the two lines. As the
vertex of the angle passes from the center of the circle to
the circumference and thence to the outside of the circle,
the rule changes, but these changes may be borne in mind,
and the entire scheme may be grasped, by imagining a
continuous change from the one position to the other, fol-
lowing all the time the changes in the intercepted arcs.
The angle between a secant and a tangent is measured in
a manner that can best be grasped by another such con-
tinuous motion, watching the changes in the measuring
arcs as the motion occurs. Such observations are essen-
tially functional in character, for they consist in careful
observations of the relationships between the angle to be
measured and the arcs that measure it.
4. Motion. The preceding discussion of variable fig-
ures leads naturally to a discussion of actual motion. As
figures move, either in whole or in part, the relationships
between the quantities involved may change. To note
these changes is to study the functional relationships be-
IOO
THE FUNCTION CONCEPT
tween the parts of the figures. Without the functional
idea, geometry would be wholly static. The study of
fixed figures should not be the sole purpose of a Course
in geometry, for the uses of geometry are not wholly on
static figures. Indeed, in all machinery, the geometric
figures formed are in continual motion, and the shapes of
the figures formed by the moving parts change. The
study of motion and of moving forms, the dynamic aspects
of geometry, should be given at least some consideration.
Whenever this is done, the functional relations between
the parts become of prime importance. Thus a linkage
of the form of a parallelogram can be made more nearly
rectangular by making the diagonals more nearly equal,
and the linkage becomes a rectangle if the diagonals are
made exactly equal. This principle is used in practice
in making a rectangular framework precisely true.
5. Proportionality theorems. All theorems which
assert that certain quantities are in proportion to certain
others, are obviously functional in character. Thus even
the simplest theorems on rectangles assert that the area
of a rectangle is directly proportional to its height, if the
base is fixed. When more serious theorems are reached,
such as the theorems on similar triangles, the functional
ideas involved are worthy of considerable attention. That
this is eminently true will be realized by all to whom trig-
onometry is familiar, for the trigonometric functions are
nothing but the ratios of the sides of right triangles. But
even in the field of elementary geometry a clear under-
standing of the relation between the areas (and volumes)
of similar figures and the corresponding linear dimensions
is of prime importance.
IOI
THE FUNCTION CONCEPT
III. RELATIONSHIPS IN TRIGONOMETRY
The existence of functional relationships in trig-
onometry is evidenced by the common use of the words
“trigonometric functions” to describe the trigonometric
ratios. Thus the sine of an angle is a definite ratio whose
value depends upon and is determined by the size of the
angle to which it refers. The student should be made
conscious of this relationship and he should be asked such
questions as the following: Does the sine of an angle in-
crease or decrease as the angle changes from zero to 90°P
If the angle is doubled, does the sine of the angle double?
If not, is the sine of double the angle more or less than
twice the sine of the original angle? How does the value
of the sine behave as the angles increase from 90° to 180°P
From 180° to 270°P From 270° to 360°P Similar ques-
tions may be asked for the cosine and for the tangent of an
angle. -
Such questions may be reinforced by the use of figures
that illustrate the points in question. Thus an angle
twice a given angle should be drawn, and its sine should
be estimated from the figure. A central angle and an
inscribed angle on the same arc may be drawn in any
circle. If they have one side in common, the relations
between their sines will be more apparent. Finally, the
relationships that exist may be made vivid by actual
comparison of the numerical values found from the
trigonometric tables. h
Not only in these first functional definitions, however,
but in a variety of geometric figures throughout trig-
onometry do functional relations appear. Thus the law
of cosines states a definite relationship between the three
sides of a triangle and any one of the angles. How will
I O2
THE FUNCTION CONCEPT
the angle be affected by an increase or decrease of the side
opposite it, if the other two sides remain fixed? How will
the angle be affected by an increase or a decrease of one
of the adjacent sides, if the other two sides remain fixed?
Are these statements still true, if the angle in question is
obtuse?
As another example, the height of a tree, or the height
of a building, may be determined by measuring the two
angles of elevation from two points on the level plain in a
straight line with its base. A formula for the height (h)
in terms of these two angles (A, B) and the distance (d)
between the points of observation, may be easily written
down [h = d sin A sin B / sin (A – B). Then the effect
upon the height of changes in One of these angles may be
discussed.
In a similar manner, every formula that is given or
derived in a course on trigonometry may be discussed
with profit from the functional standpoint.
IV. CONCLUSION
In conclusion, mention should be made of the great rôle
which the idea of functions plays in the life of the world
about us. Even when no calculation is to be carried out,
the problems of real life frequently involve the ability to
think correctly about the nature of the relationships
which exist between related quantities. Specific men-
tion has been made already of this type of problem in con-
nection with interest on money. In everyday affairs,
such as the filling-out of formulas for fertilizers or for
feeds or for spraying mixtures on the farm, the similar
filling-out of recipes for cooking (on different scales from
that of the book of recipes), or the proper balancing of the
IO3
THE FUNCTION CONCEPT
ration in the preparation of food, many persons are at a
loss on account of their lack of training in thinking about
the relations between quantities. Another such instance
of very common occurrence in real life is in insurance.
Very few men or women attempt intelligently to under-
Stand the meaning and the fairness of premiums on life
insurance and on other forms of insurance, chiefly be-
cause they cannot readily grasp the relations of interest
and of chance that are involved. These relations are not
particularly complicated, and they do not involve any
great amount of calculation for the comprehension of the
meaning and of the fairness of the rates. Mechanics,
farmers, merchants, housewives, as well as scientists and
engineers have to do constantly with quantities of things,
and the quantities with which they deal are related to
other quantities in ways that require clear thinking for
maximum efficiency.
One element that should not be neglected is the occur-
rence of such problems in public questions which must be
decided by the votes of the whole people. The tariff,
rates of postage and express, freight rates, regulation of
insurance rates, income taxes, inheritance taxes, and
many other public questions involve relationships be-
tween quantities – for example, between the rate of in-
come taxation and the amount of the income – that
require habits of functional thinking for intelligent de-
cisions. The training in such habits of thinking is there-
fore a vital element toward the creation of good citizen-
ship.
It is believed that transſer of training does operate be-
tween such topics as those suggested in the body of this
chapter and those just mentioned, because of the existence
IO4
THE FUNCTION CONCEPT
of such identical or common elements, whereas the trans-
fer of the training given by courses in mathematics that
do not emphasize functional relationships might be
questionable.
While this account of the functional character of cer-
tain topics in geometry and in algebra makes no claim to
being exhaustive, the topics mentioned will suggest others
of like character to the thoughtful teacher. It is hoped
that sufficient variety has been mentioned to demonstrate
the existence of functional ideas throughout elementary
algebra and geometry. The committee feels that if this
is recognized, algebra and geometry can be given new
meaning to many children, and indeed to many educa-
tors, and that all students will be better able to control
the actual relations which they meet in their own lives.
VIII
TERMS AND SYMBOLS IN ELEMENTARY
MATHEMATICS:
A. Limitations imposed by the committee upon its work.
The Committee feels that in dealing with this subject it
should explicitly recognize certain general limitations, as
follows:
I. No attempt should be made to impose the phrase-
ology of any definition, although the committee should
state clearly its general views as to the meaning of dis-
puted terms. 9
2. No effort should be made to change any well-defined
Current usage unless there is a strong reason for doing so,
which reason is supported by the best authority, and,
other things being substantially equal, the terms used
should be international. This principle excludes the use
of all individual efforts at coining new terms except under
circumstances of great urgency. The individual opinions
of the members, as indeed of any teacher or body of
teachers, should have little weight in comparison with
general usage if this usage is definite. If an idea has to be
expressed so often in elementary mathematics that it
becomes necessary to invent a single term or symbol for
the purpose, this invention is necessarily the work of an
individual; but it is highly desirable, even in this case,
* The first draft of this chapter was prepared by a subcommittee con-
sisting of David Eugene Smith (chairman), W. W. Hart, H. E. Hawkes,
E. R. Hedrick, and H. E. Slaught. .
Roč
ELEMENTARY TERMS AND syMBOLs
that it should receive the sanction of wide use before it is
adopted in any system of examinations.
3. On account of the large number of terms and Sym-
bols now in use, the recommendations to be made will
necessarily be typical rather than exhaustive.
I. GEOMETRY
B. Undefined terms. The committee recommends
that no attempt be made to define, with any approach to
precision, terms whose definitions are not needed as parts
of a proof.
Especially is it recommended that no attempt be made
to define precisely such terms as Space, magnitude, point,
straight line, surface, plane, direction, distance, and Solid,
although the significance of such terms should be made
clear by informal explanations and discussions.
C. Definite usage recommended. It is the opinion of
the committee that the following general usage is desir-
able:
I. Circle should be considered as the curve; but where
no ambiguity arises, the word “circle” may be used to
refer either to the curve or to the part of the plane in-
closed by it.
2. Polygon (including triangle, square, parallelogram,
and the like) should be considered, by analogy to a circle,
as a closed broken line; but where no ambiguity arises, the
word polygon may be used to refer either to the broken
line or to the part of the plane inclosed by it. Similarly,
segment of a circle should be defined as the figure formed
by a chord and either of its arcs.
3. Area of a circle should be defined as the area (numer-
ical measure) of the portion of the plane inclosed by the
IO7
EV,EMENTARY TERMS AND SYMBOLS
circle. Area of a polygon should be treated in the same
way. - -
4. Solids. The usage above recommended with re-
Spect to plane figures is also recommended with respect
to Solids. For example, sphere should be regarded as a
Surface, its volume should be defined in a manner similar
to the area of a circle, and the double use of the word
should be allowed where no ambiguity arises. A similar
usage should obtain with respect to such terms as poly-
hedron, cone, and cylinder.
5. Circumference should be considered as the length
º measure) of the circle (line). Similarly, peri-
meter should be defined as the length of the broken line
which forms a polygon; that is, as the sum of the lengths
of the sides.
6. Obtuse angle should be defined as an angle greater
than a right angle and less than a straight angle, and
should therefore not be defined merely as an angle greater
than a right angle. *
7. The term right triangle should be preferred to “right-
angled triangle,” this usage being now so standardized in
this country that it may properly be continued in spite of
the fact that it is not international. Similarly for acute
triangle, obtuse triangle, and oblique triangle.
8. Such English plurals as formulas and polyhedrons
should be used in place of the Latin and Greek plurals.
Such unnecessary Latin abbreviations as Q.E.D. and
Q.E.F. should be dropped.
9. The definitions of axiom and postulate vary so much
that the committee does not undertake to distinguish
between them.
D. Terms made general. It is the recommendation
s Io8
ELEMENTARY TERMS AND syMBOLS
of the committee that the modern tendency of having
terms made as general as possible should be followed.
For example:
I. Isosceles triangle should be defined as a triangle
having two equal sides. There should be no limitation to
two and only two equal sides.
2. Rectangle should be considered as including a Square
as a special case. -
3. Parallelogram should be considered as including a
rectangle, and hence a square, as a special case.
4. Segment should be used to designate the part of a
straight line included between two of its points as well
as the figure formed by an arc of a circle and its chord,
this being the usage generally recognized by modern
writers.
E. Terms to be abandoned. It is the opinion of the
committee that the following terms are not of enough
consequence in elementary mathematics at the present
time to make their recognition desirable in examinations,
and that they serve chiefly to increase the technical
vocabulary to the point of being burdensome and
unnecessary:
I. Antecedent and consequent.
2. Third proportional and fourth proportional.
3. Equivalent. An unnecessary substitute for the more
precise expressions “equal in area” and “equal in vol-
ume,” or (where no confusion is likely to arise) for the
single word “equal.”
4. Trapezium.
5. Scholium, lemma, oblong, scalene triangle, sect, peri-
gon, rhomboid (the term “oblique parallelogram” being
sufficient), and reflex angle (in elementary geometry).
IOQ
ELEMENTARY TERMS AND SYMBOLS
6. Terms like flat angle, whole angle, and conjugate
angle are not of enough value in an elementary course to
make it desirable to recommend them.
7. Subtend, a word which has no longer any etymologi-
Cal meaning to most students and teachers of geometry.
While its use will naturally continue for some time to
Come, teachers may safely incline to such forms as the
following: “In the same circle equal arcs have equal
chords.” -
8. Homologous, the less technical term “correspond-
ing” being preferable.
9. Guided by principle A 2 and its interpretation,
the committee advises against the use of such terms as
the following: Angle-bisector, angle-sum, consecutive in-
terior angles, supplementary consecutive exterior angles,
quader (for rectangular solid), sect, explement, transverse
angles.
Io. It is unfortunate that it still seems to be necessary
to use such a term as parallelepiped, but we seem to
have no satisfactory substitute. For rectangular paral-
ielepiped, however, the use of rectangular solid is rec-
ommended. If the terms were more generally used in
elementary geometry it would be desirable to consider
carefully whether better ones could not be found for the
purposes than isoperimetric, apolhem, icosahedron, and
dodecahedron.
F. Symbols in elementary geometry. It should be
recognized that a symbol like L is merely a piece of
Shorthand designed to afford an easy grasp of a written
or printed statement. Many teachers and a few writers
make an extreme use of symbols, Coining new ones to
meet their own views as to usefulness, and this practice
I IO
ELEMENTARY TERMS AND SYMBOLS
is naturally open to objection.” There are, however, cer-
tain symbols that are international and certain others of
which the meaning is at once apparent and which are
sufficiently useful and generally enough recognized to be
recommended.
For example, the symbols for triangle, A, and circle,
G), are international, although used more extensively in
the United States than in other countries. Their use,
with their customary plurals, is recommended.
The symbol L, generally read as representing the single
word “perpendicular” but sometimes as standing for the
phrase “is perpendicular to,” is fairly international and
the meaning is apparent. Its use is therefore recom-
mended. On account of such a phrase as “the LAB,”
the first of the above readings is likely to be the more
widely used, but in either case there is no chance for
confusion.
The symbol | for “parallel” or “is parallel to ” is fairly
international and is recommended.
The symbol • for “similar” or “is similar to” is inter-
national and is recommended.
The symbols s and = for “congruent” or “is con-
gruent to ” both have a considerable use in this country.
The committee feels that the former, which is fairly inter-
national, is to be preferred because it is the more dis-
tinctive and suggestive.
The symbol Z for “angle” is, because of its simplicity,
coming to be generally preferred to any other and is
therefore recommended.
2 This is not intended to discourage the use of algebraic methods in
the solution of such geometric problems as lend themselves readily to
algebraic treatment.
Jſ iſ
ELEMENTARY TERMS AND SYMBOLs
Since the following terms are not used frequently
enough to render special symbols of any particular value
the world has not developed any that have general ac-
ceptance and there seems to be no necessity for making
the attempt: Square, rectangle, parallelogram, trapezoid,
Quadrilateral, semicircle.
The symbol AB for “arc AB" cannot be called inter-
national. While the value of the symbol — in place of
the short word arc is doubtful, the committee sees no
objection to its use. r
The symbol . . . for “therefore” has a value that is gen-
erally recognized, but the sumbol " . " for “since” is used
so seldom that it should be abandoned.
With respect to the lettering of figures, the committee
calls attention for purposes of general information to a
convenient method, found in certain European and in
some American textbooks, of lettering triangles: Capitals
represent the vertices, corresponding small letters repre-
sent opposite sides, Corresponding small Greek letters
represent angles, and the primed letters represent the
corresponding parts of a congruent or similar triangle.
This permits of speaking of a (alpha) instead of “angle
A,” and of “small a’ instead of BC. The plan is by no
means international with respect to the Greek letters.
The committee is prepared, however, to recommend it
with the optional use of the Greek forms.
In general, it is recommended that a single letter be
used to designate any geometric magnitude whenever
there is no danger of ambiguity. The use of numbers
alone to designate magnitudes should be avoided by the
use of such forms A, A, ... -
With respect to the symbolism for limits, the committee
II 2
ELEMENTARY TERMS AND syMBOLs
calls attention to the fact that the symbol = is a local
one, and that the symbol – (for “tends to”) is both in-
ternational and expressive and has constantly grown in
favor in recent years. Although the subject of limits is
not generally treated scientifically in the secondary
school, the idea is mentioned in geometry and a symbol
may occasionally be needed.
While the teacher should be allowed freedom in the
matter, the committee feels that it is desirable to dis-
courage the use of such purely local symbols as the
following:
= for “equal in degrees,”
ass for “two sides and an angle adjacent to one of
them,” and Q
sas for “two sides and the included angle.”
G. Terms not standardized. At the present time
there is not sufficient agreement upon which to base
recommendations as to the use of the term ray and as to
the value of terms like Coplanar, collinear, and concurrent
in elementary work. Many terms, similar to these, will
gradually become standardized or else will naturally drop
out of use.
II. ALGEBRA AND ARITHMETIC
H. Terms in algebra. I. With respect to equations
the committee calls attention to the fact that the classi-
fication according to degree is comparatively recent and
that this probably accounts for the fact that the termin-
ology is so unsettled. The Anglo-American custom of
designating an equation of the first degree as a simple
equation has never been satisfactory, because the term
has no real significance. The most nearly international
II?
ELEMENTARY TERMS AND SYMBOLS
terms are equation of the first degree (or “first degree equa-
tion”) and linear equation. The latter is so brief and
suggestive that it should be generally adopted.
2. The term quadratic equation (for which the longer
term “second degree equation” is an unnecessary Syno-
nym, although occasionally a convenient one) is well
established. The terms pure quadratic and affected quad-
ratic signify nothing to the pupil except as he learns the
meaning from a book, and the committee recommends
that they be dropped. Terms more nearly in general use
are complete quadratic and incomplete quadratic. The
committee feels, however, that the distinction thus de-
noted is not of much importance and believes that it can
well be disperosed with in elementary instruction.
3. As to other special terms, the committee recom-
mends abandoning, so far as possible, the use of the fol-
lowing: Aggregation for grouping; Vinculum for bar;
evolution for finding roots, as a general topic; involution
for finding powers; extract for find (a root); absolute term
for constant term; multiply an equation, clear of fractions,
cancel and transpose, at least until the significance of the
terms is entirely clear; aliquot part (except in commercial
work).
4. The committee also advises the use of either system
of equations or set of equations instead of “simultaneous
equations,” in such an expression as “solve the following
set of equations,” in view of the fact that at present no
well established definite meaning attaches to the term
“simultaneous.”
5. The term simplify should not be used in cases where
there is possibility of misunderstanding. For purposes of
computation, for example, the form V8 may be simpler
II4
ELEMENTARY TERMS AND SYMBOLS
than the form 2 V2, and in some cases it may be better to
express V% as Vo.75 instead of 4 V3. In such cases, it is
better to give more explicit instructions than to use the
misleading term “simplify.”
6. The committee regrets the general uncertainty in
the use of the word surd, but it sees no reasonable chance
at present of replacing it by a more definite term. It
recognizes the difficulty generally met by young pupils
in distinguishing between coefficient and exponent, but it
feels that it is undesirable to attempt to change terms
which have come to have a standardized meaning and
which are reasonably simple. These considerations will
probably lead to the retention of such terms as rationalize,
extraneous root, characteristic, and mantissa, although in
the case of the last two terms “integral part” and “frac-
tional part” (of a logarithm) would seem to be desirable
substitutes.
7. While recognizing the motive that has prompted a
few teachers to speak of “positive ac’’ instead of “plus ac,”
and “negative y” instead of “minus y” the committee
feels that attempts to change general usage should not be
made when based upon trivial grounds and when not
sanctioned by mathematicians generally.
I. Symbols in algebra. The symbols in elementary
algebra are now so well standardized as to require but few
comments in a report of this kind. The committee feels
that it is desirable, however, to call attention to the fol-
lowing details:
I. Owing to the frequent use of the letter 3, it is pre-
ferable to use the center dot (a raised period) for multipli-
cation in the few cases in which any symbol is necessary.
For example, in a case like I 2: 3: ... (3 – 1). a, the center
II.5
ELEMENTARY TERMS AND SYMBOLS
dot is preferable to the syſmbol × ; but in cases like
2a(a – a) no symbol is necessary. The committee recog-
nizes that the period (as in a.b) is more nearly inter-
national than the center dot as in (a b); but inasmuch
as the period will continue to be used in this country as
a decimal point, it is likely to cause confusion, to ele-
mentary pupils at least, to attempt to use it as a symbol
for multiplication.
2. With respect to division, the symbol + is purely
Anglo-American, the symbol : Serving in most countries
for division as well as ratio. Since neither symbol plays
any part in business life, it seems proper to consider only
the needs of algebra, and to make more use of the frac-
tional form and (where the meaning is clear) of the sym-
bol /, and to drop the symbol + in writing algebraic
expressions.
3. With respect to the distinction between the use of
+ and — as symbols of Operation and as symbols, of
direction, the committee sees no reason for attempting
to use Smaller signs for the latter purpose, such an attempt
never having received international recognition, and the
need of two sets of symbols not being sufficient to warrant
violating international usage and burdening the pupil
with this additional symbolism.
4. With respect to the distinction between the symbols
= and = as representing respectively identity and equal-
ity, the committee calls attention to the fact that, while
the distinction is generally recognized, the consistent use
of the symbols is rarely seen in practice. The committee
recommends that the symbol = be not employed in ex-
aminations for the purpose of indicating identity. The
teacher, however, should use both symbols if desired.
II6
ELEMENTARY TERMS AND SYMBOLS
5. With respect to the root sign, V, the committee
recognizes that convenience of writing assures its contin-
ued use in many cases instead of the fractional exponent.
It is recommended, however, that in algebraic work in-
volving complicated cases the fractional exponent be
preferred. Attention is also called to the fact that the
convention is quite generally accepted that the symbol
Va (a representing a positive number) means only the
positive square root and that the symbol Va means only
the principal nth root, and similarly for a”, ai. The
reason for this convention is apparent when we come to
consider the value of V4 + Vo-E V16+ V25. This
convention being agreed to, it is improper to write a = V4,
as the Complete Solution of acº — 4 = o, but the result
should appear as a = + V4. Similarly, it is not in ac-
cord with the convention to write V4 = + 2, the conven.
tional form being + V4 = + 2; and for the same reason
it is impossible to have V(–1) = + 1 since the symbol
refers only to a positive root. These distinctions are not
matters to be settled by the individual opinion of a teacher
or a local group of teachers; they are purely matters of
convention as to notation, and the Committee simply sets
forth, for the benefit of teachers, this statement as to
what the convention seems to be among the leading writ-
ers of the world at the present time.
6. When imaginaries are used, the symbol i should be
employed instead of V- I except possibly in the first
presentation of the subject.
7. As to the factorial symbols 5 ! and |5, to represent
5-4-3-2-1, the tendency is very general to abandon the
second one, probably on account of the difficulty of print-
II 7
ELEMENTARY TERMS AND SYMBOLS
ing it, and the committee so recommends. This question
is not, however, of much importance in the general courses
in the high school.
8. With respect to symbols for an unknown quantity
there has been a noteworthy change within a few years.
While the Cartesian use of ac and y will doubtless continue
for two general unknowns, the recognition that the for-
mula is, in the broad use of the term, a central feature of
algebra has led to the extended use of the initial letter.
This is simply illustrated in the direction to solve for r the
equation A = Tr". This custom is now international and
should be fully recognized in the schools.
9. The committee advises abandoning the double
colon (::) in proportion, and the symbol oc in variation,
both of these symbols being now practically obsolete.
J. Terms and symbols in arithmetic. I. While it is
rarely wise to attempt to abandon suddenly the use of
words that are well established in our language, the Com-
mittee feels called upon to express regret that we still
require very young pupils, often in the primary grades,
to use such terms as sublrahend, addend, minuend and
multiplicand. Teachers themselves rarely understand
the real significance of these words, nor do they recognize
that they are comparatively modern additions to what
used to be a much simpler vocabulary in arithmetic. The
committee recommends that such terms be used, if at all,
only after the sixth grade.
2. Owing to the uncertainty attached to such expres-
sions as “to three decimal places,” “to thousandths,”
“correct to three decimal places,” “correct to the nearest
thousandth,” the following usage is recommended: When
used to specify accuracy in computation, the four expres-
II.8
ELEMENTARY TERMS AND SYMBOLS
sions should be regarded as identical. The expression
“to three decimal places” or “to thousandths” may
be used in giving directions as to the extent of a compu-
tation. It then refers to a result carried only to thou-
sandths, without considering the figure of ten-thou-
sandths; but it should be avoided as far as possible be-
cause it is open to misunderstanding. As to the term,
“significant figure,” it should be noted that o is always
significant except when used before a decimal fraction
to indicate the absence of integers or, in general, when
used merely to locate the decimal point. For example
the zeros italicized in the following are “significant,”
while the others are not: O.5, 9.50, IO2, 30,200. Further,
the number 2396, if expressed correct to three significant
figures, would be written 24oo.” It should be noted that
the context or the way in which a number has been ob-
tained is sometimes the determining factor as to the
significance of a o.
3. The pupil in arithmetic needs to see the work in the
form in which he will use it in practical life outside the
schoolroom. His visualization of the process should
therefore not include such symbols as +, −, X, +,
which are helpful only in writing out the analysis of a
problem or in the printed statement of the operation to
be performed. Because of these facts the committee
recommends that only slight use be made of these sym-
bols in the written work of the pupil, except in the ana-
lysis of problems. It recognizes, however, the value of
such symbols in printed directions and in these analyses.
3 The italicizing of significant zeros is here used merely to make clear the
committee's meaning: The device is not recommended for general
adoption.
II9
ELEMENTARY TERMS AND SYMBOLS
III. GENERAL OBSERVATIONS AND RECOMMENDATIONS
K. General Observations. The committee desires
also to record its belief in two or three general observa-
tions.
I. It is very desirable to bring mathematical writing
into closer touch with good usage in English writing in
general. That we have failed in this particular has been
the subject of frequent comment by teachers of mathe-
matics as well as by teachers of English. This is all the
more unfortunate because mathematics may be and
should be a genuine help toward the acquisition of good
habits in the speaking and writing of English. Under
present conditions, with a style that is often stilted and
in which undue compression is evident, we do not offer to
the student the good models of English writing of which
mathematics is capable, nor indeed do we always offer
good models of thought processes. It is to be feared that
many Leachers encourage the use of a kind of vulgar
mathematical slang when they allow such words as “tan.”
and “cos,” for tangent and cosine, and habitually call
their subject by the title “math.”
2. In the same general spirit the committee wishes to
observe that teachers of mathematics and writers of text-
books seem often to have gone to an extreme in searching
for technical terms and for new symbols. The committee
expresses the hope that mathematics may retain, as far as
possible, a literary flavor. It seems perfectly feasible
that a printed discussion should strike the pupil as an
expression of reasonable ideas in terms of reasonable
English forms. The fewer technical terms we introduce,
the less is the subject likely to give the impression of
being difficult and a mere juggling of words and symbols.
I 2G)
ELEMENTARY TERMS AND syMBOLs
3. While recognizing the claims of euphony, the fact
that a word like “historic” has a different meaning from
“historical” and that confusion may occasionally arise
if “arithmetic” is used as an adjective with a different
pronunciation from the noun, the committee advises that
such forms as geometric be preferred to geometrical.
This is already done in such terms as analytic geometry
and elliptic functions, and it seems proper to extend the
custom to include arithmetic, geometric, graphic, and the
like.
L. General recommendations. I. In view of the desira-
bility of a simplification of terms used in elementary in-
struction, and of establishing international usage so far
as is reasonable, the committee recommends that the
subject of this report be considered by a committee to be
appointed by Section IV of the next International Con-
gress of Mathematicians, such committee to contain
representatives of at least the recognized international
languages admitted to the meetings.
2. The committee suggests that examining bodies,
contributors to mathematical journals, and authors of
textbooks endeavor to follow the general principles
formulated in this report.
APPENDIX
I. THE PRESENT STATUS OF DISCIPLINARY VALUES
IN EDUCATION (Extract)
By VEVIA BLAIR, HoRACE MANN SCHOOL FOR GIRLs
[EDITOR’s NoTE: The following is an extract from Chapter IX of the com-
plete report. Miss Blair made a very thorough study of the publications
of psychologists on the subject of “formal discipline” and on the basis of
this study formulated the inferences with which the following extract be-
gins. These inferences were then submitted to forty leading psychologists
of this country for their consideration and comment. Of the twenty-
seven who replied three did not give permission to use their names. The
names of the remaining twenty-four together with their replies in full were
published in the complete report. Here the space available permits the
publication only of Sections 3 and 4 of Miss Blair's investigation, giving
respectively the “Formulation of Inferences” and “The Judgment of
Psychologists and General Conclusions.”
3. A FORMULATION OF INFERENCES
From the general considerations listed above the following
seven statements were formulated and submitted to psycholo-
gists for their consideration. The psychologists were asked
for any further information they would be willing to give, and
it was hoped that a free expression of opinion on their part in
regard to any debatable point would tend to obviate some of
the objectionable features of a questionnaire.
I. Transfer of training is an established fact, and may be
positive, negative, or zero.
II. The true amount of transfer from one field to another has
not yet been found by experiment, on account of one or more of
the following handicaps: -
(a) The maturity or previous training of the subjects
tested. -
(b) The absence from the training of the factors most
favorable to transfer.
(c) The inadequacy of the tests to measure the traits sought.
III. It is a reasonable inference that a substantial amount of
transfer to some related field would be ſound by an adequale lest, if
M23
APPENDIx
Y.
in teaching children, emphasis were placed upon the trait which
the selected subject was most capable of developing, and if the
factors controlling transfer were present in the training.
IV. Even if no great amount of transfer of training to any one
field should ever be found by experiment, it would still be true that
if small amounts of transfer of a valuable trait extended to a large
number of fields, the sum total of all these small amounts would
be a very valuable educational asset.
V. Negative transfer or interference may take place when in the
training of a certain trait, auxiliary habits are cultivated, which
have to be broken down before the trait can function in a new situa-
'tion.
VI. Zero transfer may occur when the habits tending to inter-
fere, and those tending to transfer, just offset each other.
VII. There are elements of situations so fundamental in their
- nature that they occur again and again in connection with almost
anything else. Special training with these elements has general
value.
4. THE JUDGMENT OF PSYCHOLOGISTS AND GENERAL
CONCLUSIONS
In the accompanying table an attempt has been made to
show roughly the nature of the responses to each statement.
The figures in the body of the table indicate numbers of men;
totals are given both in numbers and in percentages. Ques-
tions misunderstood are listed as omitted.
The general conclusions drawn from this table and from the
comments which some of the psychologists added to their
answers may be stated as follows: -
I. The two extreme views for and against disciplinary values
practically no longer exist. As the question now stands, as
transfer of training, the psychologists quoted here almost
unanimously agree that transfer does exist.
2. A large majority agree that there is a possibility of
negative transfer, and of zero transfer, caused by interference
effects. Professor Thorndike is of the opinion that negative
I 24
APPENDIX
STATEMENT YES owº, Doubt FUL No OMITTED TOTAL
I. . . . . . . 2I 2 * I 24.
II. . . . . . . I9 2 I 2 24
II a... . . . . I3 2 2 2 5 24.
II b. . . . . I7 I 2 2 2 24.
II c. . . . . I5 2 2 2 3 24.
III. . . . . . . II 3 2 I 7 24
IV. . . . . . . II 5 I 2 5 24
.V. . . . . . . 2O I 3 24.
VI. . . . . . . 17 S 2 24
VII. . . . . . . 2I I 2 24
Total. . . . . I65 24 I 2 IO 29 24O
Per cents... 69 IO 5 4. I 2 IOO
transfer is comparatively rare and can be avoided by proper
methods of training.
3. Very few if any experiments have shown the full amount
of transfer between the fields chosen for investigation. The
reason for this is to be found in the imperfections of the ex-
perimental setting. Of the defects mentioned in Statement
II, absence from the training of the factors most favorable to
transfer is the one most generally considered important,
though strong emphasis is placed in several instances upon
the inadequacy of the tests.
4. The amount of transfer in any case where transfer is ad-
mitted at all, is very largely dependent upon methods of
teaching. This is probably the strongest note struck by the
psychologists in their comments. Twelve of them out of
twenty-four make some explicit reference to the matter.
5. A majority of the psychologists, judging from their re-
marks in connection with Statements III, IV, and VII seem
to believe that, with certain restrictions, transfer of training
is a valid aim in teaching.
6. Transfer is most evident with respect to general ele-
ments — ideas, attitudes, and ideals. These act in many
instances as the carriers in transfer. Often they form the
common element so generally held to be the sine qua non of
transfer. Twenty-two of the twenty-four psychologists ex-
I 25
APPENDIX
press the opinion that special training in connection with these
elements has general value; and one of them, Professor San-
ford, adds: “This I believe to be true, and to be the basis of
the generally held belief among practical teachers of the ex-
istence and value of ‘formal training.’”
[The names of the psychologists, whose replies are published
in full in the complete report are: J. R. Angell, W. C. Bagley,
H. V. Bingham, E. G. Boring, S. S. Colvin, G. O. Ferguson,
V. A. C. Henmon, Joseph Jastrow, C. H. Judd, G. T. Ladd,
R. M. Ogden, W. P. Pilsbury, E. C. Sanford, W. D. Scott,
C. E. Seashore, Daniel Starch, E. K. Strong, L. M. Terman,
E. L. Thorndike, H. C. Warren, M. F. Washburn, J. B.
Watson, R. S. Woodworth, R. M. Yerkes.]
2. THE TRAINING OF TEACHERS OF MATHEMATICS
(Extracts)
By RAYMOND CLARE ARCHIBALD, BROWN UNIVERSITY
[EDITOR’s NOTE: The following are two brief extracts from the extensive
report by Professor Archibald published as Chapter XIV of the complete
report. The first is taken from Section I of this chapter and is intended
to give a summary of conditions in foreign countries. The second is
Section IV in full and relates to a tentative standard and program for the
training of teachers of mathematics for secondary Schools in this country.]
I. THE TRAINING OF TEACHERS OF MATHEMATICS FOR
SECONDARY SCHOOLS IN FOREIGN COUNTRIES
In 1912 about 150 reports * regarding the teaching of
mathematics in seventeen foreign countries were presented to
the International Congress of Mathematicians at Cambridge,
England. With these as a basis the writer prepared an ex-
tensive report, published by the Bureau of Education, Wash-
ington, D.C., in 1918, on The Training of Teachers of Mathe-
matics for the Secondary Schools of the Countries represented in
* The International Commission on the Teaching of Mathematics
stated in July, 1921 (Enseignement Mathématique), that it had published
187 volumes or fascicules which it listed, containing 3 Io reports of
13,505 pages.
126
º, APPENDIX
the International Commission on the Teaching of Mathematics.
For each country considered, rather detailed information and
full bibliographies were given with reference to the training of
teachers to serve in better secondary schools for boys.
A summary of the material thus set forth is here adapted
for an introductory chapter which may be suggestive to those
who thoughtfully consider in what way desirable reforms in
Connection with our secondary Schools may be brought about.
That the contrast in standards and ideals may be more marked,
this summary is placed in immediate Contiguity to Some in-
dications of corresponding conditions in the United States.
In instituting such a comparison it should be borne in mind
that there is no consensus of opinion throughout the world as
to the periods to be devoted to primary, or to secondary, edu-
cation. Most foreign countries agree that twelve or thirteen
years normally lead to a university. But while the United
States and Australia hold that only four years should be de-
voted to secondary education, seven countries (Austria,
Finland, Hungary, Italy, Japan, Roumania, and Russia) set
aside eight years, and three countries (France, with her two
extra years for the mathematical or military or naval or en-
gineering specialist, Germany, and Sweden) nine years.
In contrast to eight years of elementary education * in the
United States, France and Germany consider that the best
results are obtained when even the three or four years allotted
to primary instruction are given in connection with secondary
Schools. It is then not surprising to find that there is great
difference between the scholastic equipment of students com-
ing from these two general types of school. The graduates of
the classe de mathématiques spéciales in France, or of the
German Gymnasium are about on a par with the youth who
has finished his junior year in one of the better American col-
!eges. And in other countries also, like Denmark, Japan, and
* The recent movement towards the establishment of junior high schools
tends to reduce the elementary period to six years and to extend the
secondary period to six years.
127
APPENDIX
Sweden, the graduate of a higher secondary school has covered
considerable of the work of which the equivalent is offered in
colleges in the United States. It is only in the light of such
considerations that the full force of, say, Sweden's requirement
of ten or eleven years of preparation before a graduate of a
Gymnasium may return as a regularly appointed professor can
be adequately appreciated.
tº e ſº e © © © O , C © O
SUMMARY
These very brief sketches of the conditions in foreign coun-
tries can suggest only in imperfect outline some outstanding
features as to the manner in which prospective teachers of
mathematics in the secondary schools are trained. After
leaving the secondary Schools the training is derived, broadly
speaking, from two Sources: I. Courses in a university or a
similar higher institution; and II. Professional training.
I. All the countries require some university training on the
part of candidates for appointment as secondary school
teachers.3 The maximum requirements are in Denmark and
Netherlands, each 6 years, and in Sweden, about 8 years. On
the other hand, for minor positions in the athénées of Belgium
and regular positions in Canton Vaud, Switzerland (where
most Cantons require 4 or 5 years), only 2 or 2% years of
attendance at a university are compulsory. The complete
record is as follows: Australia (Victoria and New South
Wales), 3 years; Austria, 3% to 4; Belgium, 4 to 5, 2 for minor
positions in athénées; Denmark, 6; England, 3 to 4; Finland,
4 to 5; France, not less than 3, in addition to 2 years in classes
de mathématiques Spéciales; Germany, 3 to 4, but rarely 3 and
often 5 are taken; Hungary, 4; Italy, 4 to 5; Spain, 5, for lower
positions, 4; Sweden, usually 8; Switzerland, 4 to 5 years for
the most part, in one or two cases 2 to 3.
Let us consider an example to bring out more clearly the
3 So far as this statement concerns Japan, reference is made to the
higher middle Schools.
I 28
APPENDIX.
implications of these statements. Since the future mathe-
matical teacher entering a German university is about on a
scholastic par with the student who has finished the junior
year at a college in the United States, we may state, roughly,
that the German teacher has generally had at least three
years more of scientific training than the prospective teacher
going out as a college graduate in the United States. More-
over, in contrast, his training has, as a rule, been based upon a
secondary education relentlessly thorough, and conducted by
those who are masters of their professions; in the university
his line of work has been very specialized. Such specialization
is characteristic in the case of most of the foreign countries
considered above in connection with the preparation of their
teachers for secondary schools.
The lists of courses taken by such students indicate how the
majority of the universities emphasize instruction in such sub-
jects as plane and Solid analytic geometry, Calculus, functions
of a real variable, differential equations, differential geometry,
descriptive geometry, mechanics, and physics. Projective geo-
metry is prominent in such countries as Belgium, Denmark,
France, Italy, and Spain; various topics of higher algebra
in Hungary, Italy, the Netherlands, Roumania and Russia;
foundations of mathematics in Austria, Belgium, Japan, and
Spain; and history of mathematics in Belgium and Denmark.
It is to be borne in mind that mere names of courses men-
tioned in connection with different countries do not, without
much elaboration, convey a very definite idea. Calculus as a
requirement for the prospective teacher in Japan is vastly
different from that in Belgium where the equivalent of de la
Vallée Poussin's treatise on analysis may be the basis for part
of the first two years of work in a university. What is given
in our first and second courses of Calculus in American col-
leges is covered in special Secondary Courses of France, so that
the student at a university starts in with courses in analysis
and differential geometry such as are outlined in the first vol-
ume and part of the second of GourSat’s treatise on analysis.
I29
APPENDIX
II. In addition to attendance at universities, some coun-
tries require professional training. Australia (Victoria and
New South Wales), England (generally), Finland, Roumania,
and some states of Germany each require one year (it is only
in theóry that Austria requires a year); other states of Ger-
many and Denmark require two years each; in addition to a
year in a seminary, Sweden requires two years of probation as
teacher before regular appointment and in a similar way
Hungary requires three; and in Italy after four years of trial
a teacher may be dropped. In France the professional train-
ing may possibly be estimated at half a year. In six coun-
tries no professional training is made compulsory. These
countries are: Belgium, Japan (in higher middle Schools),
the Netherlands, Russia, Spain, and Switzerland. Details
regarding the nature of professional training in different
countries, and as to its comparative efficiency, may be found
in the report referred to in the first paragraph of this section,
page I26.
IV. A TENTATIVE STANDARD FOR THE TRAINING OF
TEACHERS OF MATHEMATICS, AND COURSES PRIMARILY
INTENDED FOR SUCH TEACHERS
We have noted the very wide divergence in the United
States of standards regarding the certification of teachers of
mathematics in senior high Schools. Is it possible at the pre
sent time to set up any standard that would be generally ac-
ceptable? The answer must be unqualifiedly in the negative.
Indeed, in States where the standards are legally very high, it
has recently been found extremely difficult to secure a suffi-
cient number of teachers with the desired qualifications. (See
the remarks of National Committee, pp. 23 ff.)
There is, however, a general belief that, with public realiza.
tion of the fundamental importance of secondary education,
the economic condition of the -secondary school teacher will
materially improve and the attractions of his work essentially
I3O
APPENDIX
increase. Looking forward then to a happier period, it seems
decidedly advisable to formulate a standard for certification
which would be generally desirable if it could be put into
effect — a formulation suggestive to colleges and universities
throughout the land in raising standards to a reasonable
status, even though this be far below that of several European
countries. It will be observed that the standard here Sug-
gested is practically equivalent to that demanded by one
State of the Union acting in coöperation with its state uni-
versity. It is hoped, therefore, that other States will be the
more ready to consider the adoption of a policy approxi-
mating the following tentative ideals:
To receive permanent appointment as a teacher of mathematics
in a senior high school a candidate should satisfy the following re-
quirements, or their equivalent:
I. Graduation from a standard four year college, or university,
or from an institution offering courses of at least equal diffi-
culty and educational value.
2. Credit for at least the following mathematical Courses (given
by teachers of mathematics in colleges or universities):
(a) Plane and spherical trigonometry;
(b) Plane analytic geometry and the elements of analytic
geometry of three dimensions;
(c) College algebra (I semester 4);
(d) Differential and integral calculus, with applications to
geometry and mechanics (3 semesters);
(e) Synthetic projective geometry (I semester);
(f) Scientific training in geometry (2 semesters) — (i) first
semester: Text, J. Petersen's Methods and Theories for
the Solution of Problems of Geometrical Constructions §
and accompanying lectures to present the history of the
famous problems, and the history of elementary geo-
4 Here, and in what follows, it shall be supposed that the term “semes-
ter” implies about 45 lectures or recitations — 3 hours a week for I5
weeks.
s Specific books are mentioned, here and later, in order to give definite-
ness to the suggestions. It is to be understood, however, that other
texts are not necessarily regarded as less desirable.
I3 I
APPENDIx
metry; (ii) second semester: Texts, F. Klein's Famous Pro-
blems of Elementary Geometry (except chapters on tran-
scendence of e and T), J. W. Young's Lectures on Funda–
mental Concepts of Algebra and Geometry (selected chap-
ters), and, for half the semester, J. Hadamard's Leçons
de Géométrie Elémentaire, vol. I, Géométrie plane, vol. 2,
Géométrie dans l'espace, especially the chapters on pro-
portional lines, areas, regular polygons, dihedral and
polyhedral angles, polyhedra, Cylinders, Cones, and
spheres, and the notes on Euclid's postulate, notions of
area, definitions of volumes, regular polyhedra, and
groups of rotations.
(g) Scientific training in algebra (2 semesters) — Lectures
on the history of elementary algebra and topics from the
following texts: J. W. Young's Lectures on Fundamental
Concepts of Algebra and Geometry (selected chapters),
Fine's College Algebra (especially the parts on numbers,
fundamental theorem of algebra, h.c.f. and l.c.m., sym-
metric functions, convergence and divergence of series),
F. Klein's Famous Problems of Elementary Geometry
(chapters on transcendence of e and T).
3. Credit for at least the following scientific courses: Theoretical
and practical physics (3 semesters), chemistry (2 Semesters).
4. Credit for at least the following theoretical professional courses
(4 semesters; given by teachers of education): History of edu-
cation, Principles of education, methods of teaching (includ-
ing the teaching of elementary algebra and geometry), educa-
tional psychology, organization and function of secondary
education.
5. Satisfactory performance of the duties of a teacher of mathe-
matics in a secondary school for a period of not less than Io
year, or 20 semester, hours. It is considered by many com-
petent authorities that the most satisfactory Conditions for
this practical professional training are in Connection with a
year of postgraduate work in a School of education organized
so that continuous directed teaching of classes in public
schools throughout the year is available to students.
It is believed that college semester-courses in rational solid geo-
metry, descriptive geometry, analytic projective geometry, theory
I32
APPENDIX
of statistics, mathematics of investment, surveying, practical and
descriptive astronomy, and in as many other mathematical topics
as possible, are also desirable.
It is generally conceded as further desirable that the prospective
teacher should have studied during the college course the following
subjects: History, economics, Sociology, political science, general
psychology, philosophy, and ethics.
Finally, it is suggested to state departments of education
that the best interests of its teachers, and their charges, would
be conserved, if during the first twenty years of service, an
appreciable portion of periodic salary increases of the teacher
were made dependent upon continued scientific development
(for example, by means of summer-school courses), and upon
active coöperation with colleagues in promoting the interests
of mathematics and the ideals and purposes of mathematical
organizations.
3. THE TESTING MOVEMENT IN MATHEMATICS
By C. B. UPTON, TEACHERS COLLEGE, COLUMBIA UNIVERSITY
The testing movement in mathematics began about twenty
years ago when Courtis first published his Test in Arithmetic.
Since that time numerous other tests in arithmetic, algebra,
and geometry have been developed. A study of these tests
indicates that the following types are now available.
Speed tests. This type of test, best illustrated by the
Courtis Research Test in Arithmetic, or the original
Rugg-Clark Test in Algebra, aims to see how many ex- 837
amples of equal difficulty a pupil can do in a given time. 882
For example, he may be asked to add as many ex– 959
amples, each similar to the one at the right, as he can in 003
8 minutes. II.8
Scales or power tests. Mathematical scales, illus- 781
trated by the Woody Arithmetic Scale, or the Hotz Alge- 750
bra Scale, are so constructed that they begin with most 222
sinple exercises and gradually progress to those that are 525
I33
APPENDIX
more difficult. Such tests, also called “power” tests, serve
in certain ways for diagnostic purposes. For example, if a
pupil works nine examples in such a test and fails on the tenth,
it is usually because some new difficulty is presented in the
tenth which he has not yet mastered. Hence the teacher has
a way of knowing where the pupil’s weakness lies.
Diagnostic tests. The mathematical scales just men-
tioned are one form of diagnostic test. Another form, illus-
trated by one of the diagnostic tests in arithmetic, pre-
sents in subtraction, for example, problems of various types
of difficulty in the subtraction of whole numbers and frac-
tions, though the problems are not all arranged in order of
difficulty, as in the Woody Scale. The aim of such tests is
to discover the types of examples with which the pupil is not
familiar.
Achievement tests. Each of the types of tests mentioned
above provides a certain measure of achievement. To these
should be added certain tests which are really a new type of
examination in mathematics, such as the Schorling-Sanford
Achievement Test in Plane Geometry.
Reasoning or problem tests. Such tests aim to examine
a pupil’s ability to work the usual concrete problems in arith-
metic or algebra, whereas the tests described above have been
confined to the abstract operations.
Tests of mathematical ability. Tests of this type are in
some respects similar to an intelligence examination. The
Rogers Test of Mathematical Ability, for example, aims to
determine in advance a pupil’s innate mathematical ability
in order to guide him in his future study of mathematics in
the high School.
Tests such as those described above have contributed con-
siderably to the efficiency of the teaching of mathematics in
both elementary and secondary Schools by fixing standards of
accomplishment; by giving a basis for Comparing the work of
One class or of one school with another; by directing the at-
tention of teachers to certain large essentials of the Subject,
I34
APPENDIX
thus keeping them from neglecting certain topics and over.
emphasizing others; by giving definite evidence concern-
ing the relative difficulty of certain topics; and by pro-
viding teachers with a measure of the efficiency of their in-
struction. *
On the other hand, these same tests have not been free from
harmful influence. They have overemphasized abstract work
and the mechanical elements in arithmetic and algebra at the
expense of the applications of the subject; they have also
tended to give too much attention to the speed factor, en-
couraging students to plunge ahead mechanically without
thinking through a solution. Perhaps their most detrimental
influence has been that they have helped to perpetuate in the
course of study certain undesirable types of subject-matter
which more progressive teachers have long considered it de-
sirable to eliminate. For example, in a certain algebra test
one finds the first example shown below, while a well-known
achievement test in arithmetic stresses such atrocities as that
shown in the second example. The result has been that many
3 — 23; * + I_ _ + =
(1) (x – I)3 "(…) QC - I (?)
(2) #1-#434-5 4-H
I8 3 9 6 2
teachers, knowing that their pupils are to be examined by some
of these tests, have continued to drill upon topics which for
some time have been obsolete. Perhaps the chief criticism of
most of our tests is that they are too far removed from what is
usually being taught in our best modern schools. This is due
to the fact that many of these tests were prepared by those
who were not specialists in the teaching of mathematics and
who had given no study to the modern tendencies in this sub-
ject. In Constructing a test it is first necessary to determine
what subject-matter it is desirable to teach, and then to so
I35
APPENDIX
construct the test that it will measure the pupils' mastery of
that particular subject-matter.
Another defect of the testing movement has been that it has
not covered all of the instruction in mathematics. In arith-
metic and algebra Stress, has been given to abstract work.
While certain tests in problem-solving have also been avail-
able, in general they are not satisfactory. The reason for
this is that the makers of the problem-solving tests are not
quite sure what they want to test. The most neglected field
is that of geometry, where thinking and problem-solving pre-
dominate and where abstract skills are relatively few. Thus
we see that if a modern school should be tested in mathematics
with all the tests now available, we would still fall far short of
having an adequate measure of all of that school's instruction
in mathematics.
One important development growing out of the testing
movement has been the creation of practice exercises which
aim to remedy the weaknesses disclosed by the standardized
tests. Such exercises are now available in both arithmetic and
algebra and are serving as an efficient aid to instruction. These
practice exercises are far more detailed than any of our
standardized tests in their gradation of the difficulties of each
topic and hence serve as one of the best means of diagnosing
a pupil’s difficulties that we now have. In fact, the most
promising development in the future seems to be that of mak-
ing a more effective combination of practice elements and
diagnosis in a carefully graded series of exercises, which can
easily be corrected and scored by untrained assistants or by
the pupils themselves, and which will also provide some
flexible standard of accomplishment. Such a type of test will
naturally test only the topics that are actually being taught;
in fact it is possible for teachers to develop such tests them-
selves as a measure of the efficiency of their instruction and as
an aid in finding the individual weaknesses of the pupils. In
such tests the time element should be subordinated to the
instructional feature. In fact, as testing programs become
136
APPENDIX
more and more common, speed is being stressed less,
while accuracy and understanding are receiving more at-
tention. Even in arithmetic, where the speed element is more
important than in algebra or geometry, the rapid develop-
ment of modern calculating machinery has made a high de-
gree of speed in pencil calculation less necessary than it was
formerly. -
The future for the testing movement seems bright, and
promises to profit by the mistakes of the past. The tests of
the next decade will have less of the aspect of the standard-
ized examination and more of the element of being an aid to
and a measure of daily classroom instruction.
4. HIGH SCHOOL COURSES IN THE CALCULUS
By VEVIA BLAIR, HoRACE MANN SCHOOL FOR GIRLS
Elective courses in the calculus are gradually finding their
way into the high school. These courses differ in content,
arrangement of subject-matter, and method of presentation,
in accordance with differing aims of the schools in offering
them. Outlines are given below of a few typical courses of
this kind.
Wadleigh High School, New York, has for some years been
developing a course in the calculus in connection with ad-
vanced algebra, trigonometry, and Solid geometry. In the
second semester of the eleventh year, differentiation and inte-
gration of algebraic polynomials and their applications to
problems in maxima and minima, and areas are taught in
connection with the advanced algebra. Equations of tan-
gents and normals of conic sections are obtained by differen-
tiation, and points of inflection are determined. In the
twelfth year, the treatment of trigonometric functions is
taken up in connection with the formal study of trigonometry,
and algebraic expressions are integrated by trigonometric
substitution. The application of integration to volumes and
surfaces is made in connection with Solid geometry; in particu-
137
APPENDIX
lar, the prismoid is defined as a solid whose cross-section can
be expressed as a function of 2 of degree not higher than
three, and the volume is then obtained by integration and
applied to the usual elementary solids and to other solids
satisfying the definition of the prismoid.
The University High School of the University of California,
Oakland, has recently given a combination course in the
twelfth year in Solid geometry and the calculus. This course
follows the recommendation of the National Committee
in simplifying the treatment of the mensuration theorems
by the use of Cavalieri's and Simpson’s theorems and by
the elementary methods of the calculus. This course is
fully described in the University High School Journal, June,
I926.
The Choate School, Wallingford, Connecticut, in 1926–27
introduced a course in the calculus in the twelfth year, using
as a text LOngley and Wilson, Introduction to the Calculus
(Ginn & Co.).
The Lincoln School, Wew York, in 1921 introduced a short
unit of the calculus in the eleventh year using as a text Young
and Morgan, Elementary Mathematical Analysis (The Mac-
millan Company). This work was discontinued in 1924,
when an elective course in the calculus was introduced in the
twelfth year. The subject-matter of this course is to be found
in Chapters III and IV of Griffin, Introduction to Mathe-
matical Analysis (Houghton Mifflin Company), in Chapters
VI and VII of Gale and Watkeys, Elementary Functions and
Applications (Henry Holt & Co.), and in Griffin's Supple-
mentary Exercises (Houghton Mifflin Company). The time
given to the course is five fifty-minute periods per week for one
SemeSter.
Horace Mann School for Girls, Wew York, offers a course in
the tenth year in general mathematics, in which has been in-
cluded since 1921 a unit of differential calculus. No text is
used. The derivative is applied to rates and to problems m
maxima and minima. The extent of difficulty is the applica-
I38
APPENDIX
tion of the derivative of f(x) = u" to a problem involving the
v/3' + 81 + I5 – 3:
function y = . Since 1923 an elective course
in the calculus has been offered in the twelfth year. Deriv-
atives are derived for f(x) = uv, f(x) = u/v, f(x) = sin x
and f(x) = cos x. The extent of difficulty is to be found in the
development of the formula sin & = x - * +: *º º + . . .
Integration is used to find areas, volumes of revolution, liquid
pressure, and work done by a variable force. The limit of
difficulty is to be found in the calculation of the volume of a
ring made by revolving the circle acº + (y – 7)* = 9 about the
axis of ac, the integral of y = Va” – 3:4 being assumed. As
the aim of the course is largely cultural, emphasis is placed
upon the historical development of the calculus and upon the
methods by which its results are obtained. The time given
to the course is three forty-minute periods per week for about
twenty weeks. - º
Besides the textbooks mentioned above certain texts pub-
lished in England may be found available for high school use
in this country, such as, for example, First Steps in the Cal-
culus, by C. Godfrey and A. W. Siddons (Cambridge Uni-
versity Press), The Calculus for Beginners, by J. W. Mercer
(Cambridge University Press), Elements of Differential and
Integral Calculus, by A. E. H. Love (Cambridge U.P.), and
A First Course in the Calculus, by W. P. Milne, and G. J. B.
Westcott (London). #
5. COLLEGE ENTRANCE EXAMINATION BOARD
Document Io'ſ
Definition of the requirements in Elementary Algebra,
Advanced Algebra, and Trigonometry, adopted by the
College Entrance Examination Board April 21, 1923:
I39
APPENDIX
ELEMENTARY ALGEBRA (PART I)
ALGEBRA TO QUADRATICS
1. The meaning, use, evaluation, and necessary transfor-
mations of simple formulas involving ideas with which the
pupil is familiar, and the derivation of such formulas from
rules expressed in words.”
The following are types of the formulas that may be con-
sidered:
W = $tr; (the sphere)
A = }h(b + b') (the trapezoid)
s = $gtº (falling bodies)
A = p(I + ri) (amount at simple interest)
A = p(I + r)* (amount at Compound interest)
2. The graph, and graphical representation in general.
The construction and interpretation of graphs.”
The following are types of the material adapted to this
purpose: Statistical data; formulas involving two variables,
such as
A = Tr",
and y = x* + 33 – 2;
formulas involving three variables, but considered for the
case in which an arbitrary value is assigned to One of them,
as V = Trºh for a fixed value, Say 4, of h.
3. Negative numbers; their meaning and use.”
This requirement includes the fundamental operations with
negative numbers and the interpretation of a negative result
in a problem, provided such an interpretation is germane to
the problem. J
4. Linear equations in one unknown quantity, and simul-
taneous linear equations involving two unknown quantities,
with verification of results.4 Problems.5
The coefficients of a single linear equation in one unknown
* The numbers refer to the Notes to Teachers, p. 140.
I4O
APPENDIX
quantity may be literal fractions. In the case of simulta-
neous equations, literal coefficients are restricted to simple
integral expressions, and to cases readily reducible to such
expressions.
5. Ratio, as a case of simple fractions; proportion, as a case
of an equation between two ratios; variation." Problems.
6. The essentials of algebraic technique, including:
a) The four fundamental operations."
b) Factoring of the following types:*
(1) Monomial factors;
, (2) The difference of two squares;
(3) Trinomials of the type 3% + pac + q.
c) Fractions, including complex fractions of simple type.”
The requirement includes complex fractions of
about the following degree of difficulty:
0. a + 35 (l, . C
p4; C – 54 i".
d a -35 m p
q T a c + 5d 11 q
d) Numerical verification of the results secured under
a), b), and c).
7. Exponents and radicals.
a) The proof of the laws for positive integral exponents.
b) The reduction of radicals, confined to transformations
of the following types:
ab– avb, V;-
3 8/–
I ^/02 I ^/ a
3/– = TT = —?
Wa Taº v/a, a
and to the evaluation of simple expressions involving
the radical sign.”
c) The meaning and use of fractional exponents, limited
Vº, Yº-Yº
b Vi, T * b.'
I4 I
APPENDIX
to the treatment of the radicals that occur under b)
above.
d) A process for finding the square root of a number, but
no process for finding the Square root of a polynomial.
8. Numerical trigonometry.
The use of the sine, cosine, and tangent in Solving right
triangles.
The use of four-place tables of natural trigonometric
functions is assumed, but the teacher may find it useful to
include some preliminary work with three-place tables.
It is important that the pupil should acquire facility in
simple interpolation; in general, emphasis should be laid
On carrying the computation to the limit of accuracy per-
mitted by the table. -
ELEMENTARY ALGEBRA (PART II)
QUADRATICS AND BEYOND
1. Numerical and literal quadratic equations in one
unknown quantity.” Problems.
The requirement includes the solution of the general
quadratic equation
ax” + ba + c = o,
the conditions for the reality and for the distinctness of
the roots, and the formulas for the sum and the product of
the roots. Simple cases in which w is replaced by 2” or by a
linear binomial, and problems leading to quadratics, are also
included; furthermore:
The interpretation of the graph of such an expression as
* – 32 + 5, meaning thereby the graph of the Corresponding
equation,
y = x* – 33 -H 5.
2. The binomial theorem for positive integral exponents,
with applications.”
It is not intended under this topic to include problems
I42
APPENDIX
involving irrational numbers, or surds, or the expansion of
the powers of a binomial having more than one fractional
coefficient. Such simple applications as that to compound
interest are included.
3. Arithmetic and geometric series.
The requirement is limited to the formulas for the nth
term, the sum of the first n terms, the value of such an infi-
nite decreasing geometric series as 1 + 2 + 34 + 38 + . . . ,
and to simple applications.”
4. Simultaneous linear equations in three unknown
quantities.
The coefficients may be integers, numerical fractions, or
algebraic monomials.
5. Simultaneous equations, consisting of one quadratic
and One linear equation, or of two quadratic equations of
the following types:
ſº + by = c, ſº + bºy” = cl,
acy = h; a2wº + bºy” = ca.
These may be expressed in other forms, such as
a’ = (r-H y) (r — y), acy = r^.
The coefficients may be integers, numerical fractions, or
algebraic monomials.
Graphical treatment is expected in the cases of equations
of the types
z -- y4 = a”, w” — y' = a”, acy = a, y' = ax
6. Exponents and radicals. -
a) The theory and use of fractional, negative, and zero
exponents.”
b) The rationalization of the denominator in such
fractions as
a + v/5
c – Vd
I43
APPENDIX
c) The solution of such equations as
^/3 – 23 — 3 = 30.
7. Logarithms.”
a) The fundamental formulas;
b) Computation by four-place tables;
c) Applications to the trigonometry of the right tri-
angle.
ADVANCED ALGEBRA
a) THEORY OF EQUATIONS
The theorem that an equation of the nth degree has n
roots, if every such equation has one root. Factoring of poly-
nomials in one variable, and the remainder theorem. The
coefficients as symmetric functions of the roots. . Simple
transformations of equations, limited to the removal of the
second term, and increase of the roots by a given number and
multiplication of the roots by a given factor. Conjugate
complex roots of equations with real coefficients.
Equations with whole numbers or fractions as coefficients.
Condition for a rational root.”
Approximate solution of numerical equations. Descartes's
Rule of Signs. Preliminary location of the roots by the
graph. Determination of the roots to two or three signifi-
cant figures.”
b) DETERMINANTS
Definition of determinants of the second and third orders
by the explicit polynomial formulas.” Evaluation of such
determinants by the familiar rules.” The simple transfor-
mations,” proved directly from the definition, and illustrated
by examples in which the elements of the determinant are small
integers.
Determinants of the fourth order; their evaluation and
transformations.”
Application to linear equations: (i) non-homogeneous
I44
APPENDIX
equations in two, three, and four unknowns; (ii) homoge-
neous equations in two and three unknowns; the treatment to
include all cases in which the number of equations does not
exceed the number of unknowns.” The case of compatibility
of three non-homogeneous equations in two unknowns is also
included.”3
c) BRIEF TOPICS
Complex numbers, numerical and geometric treatment.”
Simultaneous Quadratics.”
Scales of Notation.”
Mathematical Induction.”
Permutations and Combinations. Probability.”
TRIGONOMETRY
I. Definition of the six trigonometric functions of angles
of any magnitude, as ratios. The computation of five of
these ratios from any given one. Functions of oº, 30°, 45°,
60°, 90°, and of angles differing from these by multiples of
90°.”
2. Determination, by means of a diagram, of such func-
tions as sin (A + 90°) in terms of the trigonometric functions
of A.
3. Circular measure of angles; length of an arc in terms
of the Central angle in radians.
4. Proofs of the following fundamental formulas,” and of
simple identities derived from them.
a) The Ratio Formulas:
sin & cos x T I
tan & = —, = —, = —, CSC 3: = −,
1In 33
COt 3 = ge
tan c'
b) The Pythagorean Formulas: , ,
sin’ & H- cos’ & = 1, 1 + tanº & = Secº 2, 1 + cot” w = cscº ac;
I45
APPENDIX
c) The Addition Theorems:
sin (2 + y) = sin a cos y + cos x sin y,
cos (a + y) = COs & Cos y – sin a sin y,
tan & -- tan y
I — tan & tan y
tan (x + y) = ;
d) The Double-Angle Formulas, for sin 23, cos 2x, tan 2x.
5. Solution of simple trigonometric equations of the gen-
eral order of difficulty of the following:3: -
6 sin x + cos x = 2; cos 2x = sin 2x; tan (x + 30°) = cota.
6. Theory and use of logarithms, without the introduction
of work involving infinite series. Use of trigonometric tables,
with interpolation.3%
7. Derivation of the Law of Sines and the Law of Cosines.
8. Solution of right and oblique triangles (both with and
without 33 logarithms) with special reference to the applica-
tions. Value will be attached to the systematic arrangement
of the work.
NOTES TO TEACHERS
It is the purpose of the College Entrance Examination
Board to set its requirements in Mathematics in such a way
that the work of preparation for them shall be, at the same
time, of the greatest intrinsic value to the pupil in connection
with his subsequent work in mathematics, physics and engi-
neering, Science, economics, and other subjects which make
use of mathematics. It appears that this purpose will best
be served by specifying the minimum requirement in each
subject and leaving to the teacher a free hand for developing
in the pupil power, and mastery of the subject. To prepare
pupils satisfactorily for the examinations, teachers will,
therefore, find it necessary to extend the work somewhat
beyond the limits set in each topic.
It is not practicable to specify in advance all the non-
mathematical terms that may be used in problems on the
I46
APPENDIX
applications, of mathematics, and it will never be possible to
escape altogether the difficulties arising from the fact that
words and expressions which form a part of the language of
common life for one group of students may be strange to
another group; but it seems worth while to specify that in
connection with problems in physics the candidate in Trigo-
nometry will be expected to know the principle of the paral-
lelogram of forces, and for problems involving the points of
the Compass, although he will not be called upon to interpret
such expressions as “northeast by east a quarter east,” he
will be expected to know the names of the eight principal
directions – north, northeast, east, and so on to northwest.
The order in which the topics are listed is not intended
to imply any recommendation as to the order of presentation
in teaching. For example, the teacher may prefer, in algebra,
to begin with work in oral algebra and with problems leading
to simple equations.
The formal statement of the requirements is supplemented
by the following running commentary.
ELEMENTARY ALGEBRA
In conformity with the general principles above laid down,
a less extensive treatment of certain purely algebraic topics
than has hitherto been the practice in Elementary Algebra is
herewith specified, in order that time may be saved for the
introduction of intuitive geometry and numerical trigonome-
try, as well as for better emphasis on the great basal prin-
ciples of algebra. The examinations are expected to be more
searching with respect to the parts of algebra that will be
used by the pupil in his later work, and particularly with re-
spect to his ability to solve applied problems, to handle
formulas, to interpret graphs, and to solve the types of equa-
tions which he will subsequently need to use.
The amount of time and the degree of maturity needed in
preparing for these examinations are expected to correspond
to the present requirements.
I47
APPENDIX
The requirements in algebra here formulated represent in
each case the minimum requirement in that topic. Just as
the power of the candidate in geometry is tested by originals
not included in any standard list of propositions, so in algebra
questions on any topic, which go somewhat beyond the stated
requirement, but which are within the power of a candidate
properly trained to meet that requirement, may form a part
of the examination.
ELEMENTARY ALGEBRA (PART I)
ALGEBRA TO QUADRATICS
1. In the work done with formulas, the general idea of
the dependence of one variable upon another should be repeat-
edly emphasized. The illustrations should include formulas
from science, mensuration, and the affairs of everyday life.
Throughout the course, there should be opportunity for a
reasonable amount of numerical work and for the clarification
of arithmetical processes.
2. The pupil should be required to construct the graph
of such an expression as i
3.” – 23 + 3,
i.e., to plot the curve
y = x* – 23 + 3,
and to interpret the meaning of any graph constructed.
3. The relation of real numbers to points on a line should
be made clear, the cardinal principle being that to every such
number corresponds a point on the scale, and conversely. In
this way, the relation of positive to negative numbers, of
integers to fractional and irrational numbers, and of approxi-
mations to the values of both rational and irrational numbers
will be rendered intelligible. -
4. Beside numerical linear equations in one unknown,
involving numerical or algebraic fractions, the pupil will be
expected to solve such literal equations as contribute to an
I48
APPENDIX
understanding of the elementary theory of algebra. For
example, he should be able to solve the equation
ar" — a
S = —- } for (!.
ºr — I
In the case of simultaneous linear equations, he should be
able to Solve such a set of equations as
ax + by = k,
ca, -i- dy = l,
in order to establish general formulas. But the instruction
should include a somewhat wider range of cases, as for ex-
ample:
ax + (a + b)y = ab, ax + by = ab,
Or b
ax + (a – b)y = —ab; -(-)-.
The work in equations will include cases of fractional
equations of reasonable difficulty; but, in general, cases will
be excluded in which long and unusual denominators appear
and in which the highest Common factor of the denominators,
or the lowest common denominator, cannot be found by
inspection.
5. Problems in linear equations, as in ratio, proportion,
and variation, will whenever practicable be so framed as to
express conditions that the pupil will meet in his later studies.
Nevertheless, such studies are generally so technical as to
render it impossible to use the phraseology and laws that
represent real situations, particularly in connection with
proportion and variation. On this account it must be expected
that the problems will frequently involve situations that are
manifestly fictitious; but even such problems will nevertheless
serve the purpose of showing the pupil how to translate from
algebraic symbols to ordinary written language, and vice
versa, and to use algebraic forms to aid in the Solution of
problems.
I49
APPENDIX
6. Tt is not considered necessary to treat ratio and pro-
portion as a distinct topic. The pupil is expected to look
upon .
:
à
as a fractional equation, or as a formula involving fractions,
and to transform it accordingly. Similarly, he is expected to
look upon the equation y = Cx; as representing simple variation
without rendering it obscure by the use of such a form as y oc ac.
Such terms as “alternation,” “composition,” etc., should not
be introduced. The corresponding theorems, the substance
of which can be presented without the use of the technical
terms, are of value for Some applications in geometry and
other branches, but are not included in the requirements in
algebra.
7. It is not expected that pupils will be called upon to
perform long and elaborate multiplications or divisions of
polynomials, but that they will have complete mastery of
those types that are essential in the subsequent work with
ordinary fractional equations, and with Such other topics as
are found in elementary algebra. In other words, these opera-
tions should be looked upon chiefly as a means to an end.
8. Factoring has been reduced to the cases needed in
later work. Under (3), only simple cases, like
* - 5x + 6 and 3:” – 3 – 2
are contemplated. A problem may, however, combine two or
more of the cases here suggested, as in the polynomial
* + 43 – y” + 4, Or 2aw” - 4ax + 2a,
2(a – b) – 33:(b — a).
The teacher may wish to use such examples as
a” — b” + a - b Or zy + 2x + 3 y + 6
I5o
APPENDIX
for purposes of instruction; but these cases are not included
in the requirement.
9. Complex fractions may often be treated advantageously
as cases in the division of simple fractions, but usually it is
more convenient, in practice, to multiply both numerator
and denominator of the complex fraction by a factor that will
reduce it to a simple fraction.
The meaning of the operations with fractions should
be made clear by numerical illustrations, and the results
of algebraic calculations should be frequently checked by
numerical substitution as a means to the attainment of
accuracy in arithmetical work with fractions.
Io. In all work involving radicals, such theorems as
vab = Vavö and * = va may be assumed. Proofs of
b Vb
these theorems should be given only in so far as they make
clearer the reasonableness of the theorems; and the reproduc-
tion of such proofs is explicitly excepted from the require-
ments here formulated.
ELEMENTARY ALGEBRA (PART II)
QUADRATICS AND BEYOND
II. The coefficients of the equation may be common
fractions or simple literal fractions. In such cases it is
usually best to begin by freeing the equation of fractions.
The method of solution by completing the square should be
the one primarily used, with emphasis on the systematic
arrangement of the necessary steps and Computations.
Equations of the form
aw” -- bac = o
should be solved by factoring; beyond this, the method of
solution by factoring should be restricted to simple cases,
Such as
w” – 53 + 6 = o, and w” + x – 2 = o.
#5 I
APPENDIX
The teacher may wish to use such examples as the factoring
of 23." -- 53 – 7 for purposes of instruction; but these cases are
not included in the requirement. It should be understood
that the recommendations here made do not exclude the
method of solution by the formula.
12. The pupil should be able to write the first few terms
of the expansion of (a + b)" for any positive integral value of
n, although in general it will be sufficient to take n = 8.
13. In connection with the infinite decreasing geometric
series, such illustrations as that of the repeating decimal
should be used, but the subject of repeating decimals will not
be required on the examinations. One important reason for
the inclusion of this topic is that it serves as an introduction
to the idea of the limit.
At some point in the course, the expression of a” — b" as
the product of a - b and a second factor should be taken up,
and this may be done in connection with the formula for the
sum of n terms of a geometric progression. It is sufficient to
restrict m to values not exceeding 8. The corresponding
factoring of a” + b" when n is odd should be included.
I4. Fractional, negative, and Zero exponents should be
presented as cases in which a symbol is defined in such a way
as to give universality to a law already established for certain
special cases. That is, because
a"a.” E a"+"; (a")" s: a"; a"b" -: (ab)",
when m and n are positive integers,
p
a 9, a", a”
are so defined as to render these laws permanent.
15. The graph of y = Io" may profitably be used for the
purpose of explaining the theory. Only the base Io need be
considered in presenting the subject, but other bases may be
used for purpose of illustration.
I52
APPENDIX
Proofs of the theorems
log PQ = log P-H log Q;
log P’ = n log P;
may be required on examinations; but proofs of the derived
theorems, such as
P º, I
log O = log P – log Q, and log VP = n log P,
although these can readily be deduced from the first two,
will not be required in examinations.
The requirement includes the computation of such an ex-
pression as

Nº. + O.9834.
3. I42
the calculation of s when the formula vº = 2gs is given, and
values of v and g are given to three or four significant figures;
and the complete solution of a right triangle in which the
given parts have been measured to four significant figures.
It excludes the solution of exponential equations, like 2* = 3,
and the computation of such expressions as 6.821*.
ADVANCED ALGEBRA
I6. The requirement relating to a rational root is restricted
to the case in which the coefficients are integers and the
coefficient of the highest power of the unknown is unity.
17. After a root has been shown to lie between two values,
ac, and ac, which are fairly near together, the points of the
graph whose ac-Coördinates are a, and 3:2 are computed,
and the chord determined by these points is drawn. The
intersection of this chord with the axis of a yields the next
approximation. A graphical Solution of the problem may
often be used with profit. For closer approximations to the
root, this or other methods may be used. But the elaborate
developments suggested by the name of “Horner's Method,”
I53
APPENDIX
which involve multiplying the roots by powers of Io, con-
structing a special array for the numerical work (over and
above the scheme for reducing the roots by a given number),
and finally “contracting” the work are not a part of the re-
quirement.
18. That is,
for determinants of the third order.
I9. Namely,
A.
º } = ad – be, and the six-term formula
/
|
\f
\
^
i
\!
20. Such as adding columns, etc. -
21. It is important that the pupil should know that a
determinant of the fourth order cannot be evaluated by such
a scheme as that mentioned under Note 19, but is evaluated
by expansion according to the elements of a row or a column.
On the other hand, by transformations such as those of
Note 20, some elements can be reduced to o and the numerical
work thus abbreviated.
22. Thus, under (i), the pupil will be expected to know in
what the solution of the equations
a;3 + bºy + c, - O,
a23 -H by + C2 = O,
consists, both when the determinant of the coefficients,
a, b,
a2 b,
j
has a value different from o and when it equals o.


I54
APPENDIX
In order to avoid too great detail, the requirement is
restricted to the case, when the determinant of the equations
vanishes, in which at least one (first) minor is different from o.
But the instruction should make clear to the pupil how he
can proceed in any numerical case and find the Complete
Solution. Thus, under (ii), the solution of the equations
33 -H 4y – 22 = o,
2x + 3 y + 42 = o
53 + 7) + 22 = o,
consists in giving to z an arbitrary value and then determining
3, and y in terms of z from the first two equations.
23. It is desirable that the extension of all the foregoing
definitions and theorems to more general cases be pointed
out; but all such extensions, as also the rule for the multiplica-
tion of determinants, are explicitly excluded from the re-
quirement.
24. The reduction of the sum, difference, product, and
quotient of two complex numbers to the form a + bi, The
geometrical construction of the sum and difference, but not
of the product and quotient, of two complex numbers.
25. Two equations of the type
a,3:4-H bºxy + c, y' = d.,
a.a.” + b,<y + c, y' = d.
26. The point of this requirement is the fact that the
base Io is accidental, that any other base greater than I could
be used, and that computation would be performed by the aid
of the addition table and the multiplication table, as is cus-
tomary with the base Io. -
27. A clear exposition of this important method of reason-
ing. Applications to such problems as Summing the series
I* + 2* + 3 + . . . --n’,
and
13 + 23 + 33 + . . . -- nº;
I55
APPENDIX
the derivation of the formula for
(a, + a, + . . . -- aw)”;
and, possibly, the proof of the binomial theorem for a posi-
tive integral exponent; the simpler applications are the more
illuminating, and the application to the proof of the binomial
theorem will not be required in examinations. -
28. Permutations and combinations, restricted to the
case of n objects, all of which are different. Probability,
restricted to problems of moderate difficulty.
The Brief Topics should really, as the name implies, be
treated briefly in the course, and their importance should not
be overrated on the question paper by the setting of difficult
questions. Their value lies in their mathematical content,
rather than in their being a means of further developing
technique. h
It is not contemplated that the total requirement should
be greater than that in Trigonometry or Solid Geometry.
TRIGONOMETRY
29. The values of these functions should be read from
diagrams, and not made merely a matter of memorizing.
30. This is a minimum list to be memorized. With respect
to formulas c), the requirement is restricted to the case that
a; and y áre both acute; but the instruction should include
the generalization of the formulas to the case of any angles
by the addition of 90°.
In restricting explicitly the formulas to be memorized,
the object has been to give the pupil perspective and to
aid him in an efficient study of the subject. The Board
does not wish to discourage the pupil from memorizing
certain other formulas when it is convenient for him to do so.
But these formulas should hold distinctly a secondary place in
the pupil’s mind, and they should not be allowed to compete
with the fundamental formulas.
I56
APPENDIX
31. The instruction should include an explanation of the
notation sin Tºx (or arc sin x), cos−3, tan Tºx, cot-ºx on the
basis of the definition:
3 = sin-ºx, if x = sin y,
etc., y being measured in terms of radians, and the further
definition of the principal value of each of these functions
should be pointed out. But this topic is not included in the
requirement. -
32. The use of four-place logarithmic and trigonometric
tables is recommended, but both four-place and five-place
tables will be furnished the candidates in the examinations
of the Board.
The computation of such expressions as 6.821**9, and
the solution of exponential equations, like 2* = 3, are included
in this requirement.
33. This implies the case in which the sides of the triangle
are represented by Small integers, or in which the data re-
specting the sides are correct only to two significant figures.
This case would naturally be considered at the beginning,
before logarithms are introduced, for it contributes to an ap-
preciation of the actual geometrical magnitudes and the re-
lations between them.
Graphical solutions by means of scale and protractor
should be freely used as checks.
Both the representation of the angle by degrees, minutes
and seconds, and also the decimal Subdivision of the degree
should be taught, but only one method of dividing the degree
is required.
The use of slide rules in the instruction is to be encouraged,
but they will not be permitted in the examination for the
reason that, when some candidates use slide rules and some
do not, it is impossible to mark the papers justly.
157
APPENDIX
ForMULAs of TRIGONOMETRY
r. The Ratio Formulas:
tan ºc = — cot & = — Sec 3: = cSc w = −
COS X, sin x.
tan &
COt 3: =
2. Formulas read off from a figure; e.g., cos (90° -- a),
sin ( – 3), etc., in terms of the trigonometric functions of z.
3. The Pythagorean Formulas:
sin” ac-H cos’ ac = I,
I + tan” & = secº 3,
I + Cot” c = Cscº 3.
4. The Addition Theorems:
sin (x + y) = sin & cos y + cos x sin y,
cos (x + y) = cos & Cos y – sin & sin y,
tan a + tan y
tan (x + y) = I – tan & tan y
5. The Double-Angle Formulas:
sin 23 = 2 sin x cos x,
cos 23 = cos’ & – sin” &c,
= 2 cos” a - I,
= 1 – 2 sin” ac,
2 tan &
I — tan? 3:
:-
tan 23;
6. The Half-Angle Formulas, a) in Implicit Form:
f . . 20
I - COS 3; - 2 Sin” —
2
9,
I + cos w = 2 cos” –
2
* . 30 %
§In 30 - 2 SIIl T COST
2 2
I58
APPENDIX
b) In Explicit Form:
. 30 I – cos x
Sln — = + V — *
2 2
3C I + COS 3.
COS T = + *sº
2 2
t 3C NHº:
an - = + *-****
2 I + COS ac
7. a) Law of Sines:
- (! b C

sin A T sin B T sin C
b) Law of Cosines:
a* = b” + c2 – 2bc COSA.
8. Sums and Products.
sin & -- sin y = 2 sin 94 (x + y) cos 34 (3 – y),
sin & – sin y = 2 sin 94 (3 – y) cos 34 (3 + y),
cos x + cos y = 2 cos 34 (a + y) cos 34 (3 – y),
COs a - Cos y = - 2 sin Vá (a + y) sin 94 (3–y).
9. Special Formulas for Triangles:
a) Law of Tangents:
a – b tan 3% (A - B)
a + b tan 94 (A + B)
b) For the case in which the three sides are given:

anº-Vº (s — c)
s(s – a)
Area = Vs(s – a) (s – b)(s–c)
s = % (a + b + c).
Formulas 1, 3, 4, and 5 (and also 7) should be memorized,
like the multiplication table. The proofs of I and 3 are im-
mediate. In the case of the addition theorems, the proof for
I59
APPENDIX
the sine and the cosine of the sum of two acute angles should
be given and the formulas extended to the case of any angles
by adding 90° to a or y. As already noted, the proof will be
required on examinations only for the case in which ac and y
are both acute. The third formula, 4, and the formulas, 5,
follow at once from these. The formulas for sin (x - y), etc.,
are obtained by replacing y by — y and making the obvious
reductions. These formulas need not be memorized, but
should be deduced when required.
Formulas 2 include sin (x + 90°), sin (90° — «), sin (x +
180°), sin (180° – 3), sin ( — ac), etc., and the corresponding
cases for the cosine, tangent, and cotangent. None of the
formulas should be memorized, but each should be read off
from the appropriate figure, which the pupil Soon comes to
visualize without pencil and paper.
The half-angle formulas, 6, should be deduced when needed
from the double-angle formulas, 5. The double signs in 6 b)
depend on the magnitude of the angle and cannot be grouped
automatically. s
Formulas 7 should be memorized, as already noted, and the
pupil should be familiar with the proofs.
Formulas 8 are not to be memorized, although they are oc-
casionally used in practice, and it is well for the pupil to know
that such formulas exist. Their proof affords a useful exercise
in the manipulation of the trigonometric identities.
Any triangle can be solved by the law of sines and the law
of cosines, and these formulas are to be preferred when the
sides are represented by simple numbers. Formulas 9 are well
adapted to logarithmic computation in the cases to which they
apply. The pupil should be able to use them, or formulas
equivalent to them; but he is not required to memorize them,
or to learn their proof. The formula for the area of a triangle
when two sides and the included angle are given, namely,
Area = % ac sin B,
is not included, since it should be read off, whenever needed,
immediately from the figure.
16o
APPENDIX
6. COLLEGE ENTRANCE EXAMINATION BOARD
Document Io8 (Extract)
Definition of the requirements in Plane Geometry, Solid
Geometry, Plane and Solid Geometry, Major Requirement,
and Plane and Solid Geometry, Minor Requirement:
SYLLABI IN GEOMETRY
The following syllabi have been prepared to indicate the
scope of the requirements in geometry. It will be seen that
the content of the standard requirements in plane geometry
and in Solid geometry is essentially the same as heretofore.
A redistribution of emphasis on the various parts of the work,
designed to lighten the demands on the candidate's memory
and to give increased opportunity for attention to the de-
velopment of geometrical understanding, is explained in sub-
sequent paragraphs of this introductory Statement.
To meet the situation created by the gradual disappearance
of solid geometry from the Schools, the present requirements
(Mathematics C, D, and CD) have been so modified as to per-
mit teachers greater freedom in the treatment of the material,
and a new requirement (Mathematics cq), covering selected
parts of plane and solid geometry and calling for about as
much preparation as is represented by the present requirement
in plane geometry, has been added.
As in the past, the examinations will consist partly of
questions on book propositions and partly of originals. In
the former type of question, the candidate will be asked to
give demonstrations of standard theorems which are assumed
to have been presented to him in his course of study, or to re-
produce standard constructions. Under the latter type are
included the demonstration of theorems which are not as-
sumed to be familiar to the candidate, problems of measure-
ment and calculation, and problems in the working out of un-
familiar constructions and the identification of unfamiliar loci.
Questions calling for simple geometrical knowledge and under-
standing may fall under either type.
I61
APPENDIX
An important novelty in the present statement of require-
ments relates to the treatment of book propositions. These
will be chosen, not from the entire syllabus at large, but from
a restricted list of propositions, designated in each syllabus by
asterisks. The starred propositions have been chosen partly
with regard to their intrinsic importance, but partly also with
regard to their availability for examination purposes. A
theorem may be of the highest importance, and yet may have
a proof which, for One reason or another, is of such a nature
that little information as to the candidate's ability is ob-
tained by asking him to reproduce it. On this account, some
of the most important propositions in the course, such as
those on the circumference and area of the circle, are omitted
from the starred list.
In the case of the unstarred propositions, the candidate will
be expected to be familiar with their content, so as to be able
to answer questions about their substance or use them as a
basis for solving originals, but he will not be assumed to have
their demonstrations in mind. A few propositions have been
included on account of their general importance and interest
without reference to direct or indirect use for examination
purposes. The demonstration of an unstarred proposition
may sometimes be called for on an examination, but only if it
is of such a nature that the candidate can reasonably be ex-
pected to work it out as an original. It will undoubtedly be
desirable to present the proofs of most of the unstarred propo-
sitions in regular sequence in the course, formally or inform-
ally, as one of the best supplements to the instruction directly
needed for the satisfaction of the minimum requirements. It
is intended, however, that it shall be most advantageous for
the pupil, in preparing for the examinations, to spend his time
in developing facility in independent thinking on geometrical
topics, rather than in memorizing such proofs.
About one third of the examination will be devoted to book
propositions from the starred list, except in the case of Solid
geometry as explained later. The remainder will be divided
162
APPENDIX
between easy originals which ought to be within the reach of
any properly trained candidate, and originals which, though
still not of excessive difficulty, are designed to test the powers
of the better candidates. The easy as well as the more
difficult originals may be unfamiliar theorems, or problems in
mensuration, Construction, or loci. Whether papers so planned
are found to be intrinsically easier or harder than those
that have been set in the past, it is intended that the marking
shall be so administered that the difficulty of obtaining a pass-
ing grade is not materially changed.
The questions constituting the examination (that is, the
main Subdivisions of the paper, numbered from 1 to 6, more
or less) will as a rule be weighed equally by the readers; but a
bonus may be given for exceptional work on a given question,
and, on the other hand, the general character of a candidate's
paper may indicate serious deficiencies, even though a con-
siderable total be scraped together from his fragmentary
answers to many questions. The readers will appraise a book
as a whole, and not cling to any mechanical method of mark-
ing when occasion demands the exercise of human judgment.
The questions may be answered in any order, and it is ad-
visable that the candidate make sure of those which he can
answer best. When a single question is composed of two or
more parts, the relative weights to be assigned to the parts
will be determined by the readers according to circumstances.
Each of the following syllabi has been arranged in what is
regarded as a practicable teaching order, but this order is in
no wise prescribed. The candidate may base his answers on
any sequence of propositions which is logically sound, and
may even quote propositions not listed in the syllabus at all,
if they have been presented as standard propositions in his
course. The readers occasionally have to use their discretion
in deciding whether an answer made unduly easy by the cita-
tion of an unusual reason deserves full Credit. For example,
iſ a candidate proves the starred theorem *23 in Plane Geo-
metry merely by deducing it from theorem 24 or 25, he may
I03
APPENDIX
reasonably be expected to make up for a considerable saving
of time here by a higher standard of performance on the rest
of the paper than would otherwise be required.
For convenience of reference, formulas have been ap-
pended to Some of the theorems in mensuration. The nota-
tion used in these formulas is not prescribed, but if the candi-
date uses a different notation, he should be sure that its mean-
ing is unmistakable. :*
It is to be emphasized once more that the syllabi have been
prepared as a basis for the Conduct of examinations, not as a
recommendation for the content and arrangement of ideal
Courses. Many of their features, especially in connection with
the distribution of asterisks, have been determined by the
peculiar exigencies of examination technique. .
In particular, it will be noticed that proofs involving the
theory of limits are not required for examination purposes.
This is for the reason that the definition and maintenance of
a satisfactory standard of performance is difficult for ex-
aminers and readers, as well as for teachers and candidates.
Apart from the examination requirement, it is in the interest
of Sound mathematical training that each class of pupils re-
ceive the best and most thorough instruction in the theory of
limits that they can accept. An immeasurably improved
understanding of the meaning of limits has been perhaps the
most important achievement of mathematical Science in the
past hundred years. As this improved knowledge becomes
generally available, the importance of the subject for elemen-
tary instruction will increase rather than diminish. In the
light of developments already in progress, there is reason for
the belief that the present period of transition, in which the
Setting of any kind of uniform standards appears impractica-
ble, will not be of long duration.
- PLANE GEOMETRY
The significance of the starred and the unstarred proposi-
tions has been explained above.
I64
APPENDIX
The originals on the examination will in general depend for
their solution on propositions mentioned in the syllabus. But
occasionally an original will be so framed that a solution will
occur more readily to the candidate who is familiar with such
important geometrical facts as the properties of the 30° and
the 45° right triangles.
With regard to constructions, the candidate is expected to
be able to perform and to describe accurately those listed at the
end of the syllabus, and also, as originals, others based on
these. He is not required to give proofs of constructions, un-
less a proof is specifically called for by the question, and such
proofs will not be regarded as constituting a part of the book-
work requirement, but will have the status of originals. The
candidate is expected to be provided with ruler and com-
passes, and to use them. In default of these instruments,
however, he will receive credit for a satisfactory free-hand
sketch showing all construction lines.
SOLID GEOMETRY
The work in solid geometry may well be supplemented by a
considerable amount of material which remains outside the
scope of the examination requirement. The importance of
limit proofs has already been emphasized. Other valuable
material is contained in the theorems of Cavalieri and Euler,
and the so-called prismoid or prismatoid formula.
A few theorems on similar solids have been included in the
syllabus, unstarred. The candidate should be familiar with
their substance, as a basis for problems in measurement and
calculation, but they should not be made the occasion of a
large amount of demonstrational work.
The candidate should know what is meant by the per-
pendicularity of skew lines, but will not be required to deal
with the measure of an angle between two skew lines which are
not perpendicular to each other.
Attention is again called to the fact that the order in which
the propositions are listed is in no wise to be regarded as a
I65
APPENDIX
prescription of the order in which the proofs shall be taken
up. Thus, for example, the sphere may be treated early in
the course, and the treatment of dihedral angles based on the
geometry thus developed.
The Board wishes to accord all due latitude in the treat-
ment of the subject of solid geometry. It recognizes the value
of the further training in logical demonstration which supple-
ments the study of plane geometry and is given in standard
courses at the present time. It recognizes also that the intui-
tive geometry of the early school course may well be carried
further as regards both a firmer grasp on space relations aſid
the visualization of Space figures, and the mensuration of
surfaces and Solids in space.
The examinations will be constructed with reference to this
larger interpretation of the requirement. In the past, the
candidate has been expected to answer six questions, and this
will be assumed for convenience in defining the nature of the
new examinations. These papers will consist of seven ques-
tions, of which the candidate will be expected to answer six.
Two of these questions will call for demonstrations of propo-
sitions from the starred list, but not both of these proposi-
tions will be chosen from Book VI. Many teachers have felt
that the amount of formal demonstration demanded by this
Book has been excessive and has obscured the subject-matter.
The purpose of the new requirement is to give the teacher
freer hand, enabling him, if he so desires, to teach the facts
Concerning the relations of lines and planes in Space by means
of problems and constructions.
In connection with the drawing and measurement of geo-
metrical solids, the candidate should be thoroughly familiar
with the terminology of the subject, including such terms as
the following: polyhedron; regular polyhedron; diagonal; face
diagonal; prism, right prism; oblique prism; truncated prism;
parallelepiped; right parallelepiped; rectangular parallele-
piped; pyramid; frustum of a pyramid; vertex; lateral edge;
Slant height; altitude; cylinder and cone, and the customary
166
APPENDIX
terms relating to them, defined with complete generality,
though restricted in problems of mensuration to the right
circular cylinder and cone: zone; lune, great circle, Small circle,
segment of one base and sector of a sphere.
Problems of mensuration, stated in words or in terms of a
figure, form an important topic of the course. The candidate
is expected to be familiar with the fundamental formulas of
mensuration in plane geometry, with the use of natural sines,
Cosines, and tangents in Solving right triangles, as taught in
numerical trigonometry, and with the formulas of Solid geo-
metry enumerated in the Appendix to the Syllabus.
Another topic explicitly included is the solution of locus pro-
blems, to the extent of visualizing, describing, and representing
the figure on paper; formal proofs are not in general required.
With the considerable reduction in the formal proofs de-
manded by the present requirement, it is expected that candi-
dates will show greater facility in visualizing space figures and
in representing such figures on paper. Moreover, a firmer
grasp of the facts of solid geometry is contemplated, and pro-
blems in mensuration which test all of these things will form a
part of the examinations. The type of training expected of a
candidate, so far as this new material is concerned, is indi-
cated by a list of problems in the Appendix to the Syllabus.
PLANE AND SOLID GEOMETRY
MAJOR REQUIREMENT
This requirement is based on the syllabi for the two pre-
ceding requirements, taken together, and corresponds to the
present requirement, Mathematics CD, I}% units.
PLANE AND SOLID GEOMETRY
MINOR REQUIREMENT
This requirement is designed to cover the most important
parts of plane and Solid geometry, in such a way that the
preparation for it can be completed in the time usually de-
167
APPENDIX
voted to the standard requirement in plane geometry. It is
designated as Mathematics ca, and counts I unit.
The notes bearing on the syllabi for the preceding require-
ments, with regard to originals and constructions and the
matter of skew lines, are applicable here.
A number of theorems have been omitted from the present
syllabus which would be used in proving some of the un-
starred theorems that are included, so that the latter theorems
will necessarily be treated somewhat informally, if the course
is not to be made unduly long. The preliminary propositions
that are needed for proving the starred theorems have been
retained.
An omitted theorem may be used as an original on an ex-
amination, if it is of suitable difficulty. And an original may
involve a simple omitted theorem incidentally — for example,
the corollary that, if two parallel lines are cut by a transversal,
the interior angles on the same side of the transversal are sup-
plementary — but then the fact that the candidate has to re-
cognize and prove the point in question will be taken into
account in estimating the difficulty of the problem.
[Here follow lists of theorems and constructions in Plane
Geometry (89 theorems, 20 constructions), 92 theorems in
Solid Geometry, theorems and constructions in Plane and
Solid Geometry, Minor Requirements (96 theorems, 19 con-
structions). The theorems and constructions of these lists
are indicated in the National Committee’s lists, pp. 79-91,
by placing after each theorem or construction the number in
square brackets [ ] of the corresponding theorem or con-
struction of the C.E.E.B. list. The starred propositions are
also indicated by placing an asterisk after the Corresponding
number. The designation cd placed after a theorem or con-
struction on pp. 79-91 indicates that it is included in the
C.E.E.B. requirement for Plane and Solid Geometry, Minor
Requirement (Mathematics cal). Ali propositions of the
C.E.E.B. lists not contained in the National Committee's
lists are given below, with their respective numbers.]
I68
APPENDIX
PLANE GEOMETRY
Two right triangles are congruent if the hypotenuse and an ad-
jacent angle of one are equal respectively to the hypotenuse and an
adjacent angle of the other. [9, cq]
Through any three given points not lying in a straight line one
circle, and only one, can be drawn. [36*]
If tangents to a circle from an external point are drawn, they
make equal angles with the line joining the given point to the
Center. [45, Cd, part -
An angle formed by a secant and a tangent, intersecting at a
point outside the circle, is measured by half the difference between
the intercepted arcs. [52, part]
A circle can be circumscribed about any regular polygon. [78*]
A circle can be inscribed in any regular polygon. [79]
Regular polygons of the same number of sides are similar. [82]
If the number of sides of a regular polygon inscribed in a circle
be increased indefinitely, the apothem of the polygon will approach
the radius of the circle as its limit. [84]
The circumference of a circle is the limit which the perimeters of
regular inscribed and circumscribed polygons approach when the
number of their sides is increased indefinitely; and the area of the
circle is the limit of the areas of these polygons. [85]
The areas of two circles are to each other as the squares of their
radii. [89)
SOLID GEOMETRY
Through a given point in a given line one plane, and only one,
can be passed perpendicular to the line. [4, coll
Through a given external point one plane, and only one, can be
passed perpendicular to a given line, [5, coll
If one of two parallel lines is perpendicular to a plane, the other is
also perpendicular to the plane. [II, cq]
If two lines are parallel to a third line, they are parallel to each
other. [I2, coll
Through either of two skew lines one plane, and only one, can be
passed parallel to the other line. [I4, cq) *
Through a given point in Space one plane, and only one, can be
passed parallel to each of two skew lines, or else parallel to one line
and containing the other, [I5, cq]
I69
APPENDIX
A line perpendicular to one of two parallel planes is perpendicular
to the other also. [18, cq] . t
Through a given point outside a plane one plane, and only one,
can be passed parallel to the given plane. [19, caj
If two intersecting lines are each parallel to a plane, the plane of
these lines is parallel to that plane. [2O, Cd]
If two planes are perpendicular to each other, a line perpendicu-
lar to one of them at any point of their intersection will lie in the
other. [24, cq]
If two planes are perpendicular to each other, a line drawn per-
pendicular to one of them through any point of the other will lie in
the second plane. [25, Cd]
Through a given line not perpendicular to a given plane, one
plane and only one can be passed perpendicular to the given plane.
i28, coll -
The locus of points equidistant from the vertices of a triangle is
the line through the center of the circumscribed circle, perpendicu-
lar to the plane of the triangle. [31]
Two prisms are congruent if three faces which include a tri-
hedral angle of one are respectively congruent to three faces which
include a trihedral angle of the other, and are similarly placed. [38]
Two right prisms having congruent bases and equal altitudes are
congruent. [39]
The areas of two similar polyhedrons are to each other as the
squares of any two corresponding edges. [60)
The areas of two similar cylinders, or of two similar cones, are to
each other as the squares of any two corresponding lines. [62]
The volumes of two similar cylinders, or of two similar cones, are
to each other as the cubes of any two corresponding lines. [63]
Through any two given points on the surface of a sphere an arc
of a great circle can be drawn. [65]
Through any three given points on the surface of a sphere one
circle and only one can be drawn. [66]
An angle formed by arcs of two great circles is measured by the
arc of a great circle described from its vertex as a pole and included
between its sides, produced if necessary. [73*)
If two sides of a spherical triangle are equal, the angles opposite
these sides are equal. [84] -
If two angles of a spherical triangle are equal, the sides opposite
thºse angles are equal. [85] -
I7o
APPENDIX
If two angles of a spherical triangle are unequal, the sides oppo
site these angles are unequal, and the side opposite the greater
angle is the greater. [86]
If two sides of a spherical triangle are unequal, the angles oppo-
site these sides are unequal, and the angle opposite the greater side
is the greater. [87]
A spherical triangle is equivalent to a lune whose angle is half
the spherical excess of the triangle. [88]
A Convex spherical polygon is equivalent to a lune whose angle
is half the spherical excess of the polygon. [89)
The shortest line that can be drawn on the surface of a sphere,
connecting two given points of the sphere, is an arc of a great circle.
[90)
[Document IoS also contains an Appendix in which will be
found valuable suggestions to teachers on “Formulas of
Mensuration for Solids,” “Drawing,” and a list of “Pro-
blems.”
INDEX
Accounts, household, 3o
Accuracy, I 2, 39; in computing, 39,
II.8
Acute triangle, Io&
Addend, II8
Affected quadratic, II4
Aggregation, II4
Aims, of mathematical instruction,
6 f., 13; practical, 6 f.; disciplin-
ary, 6, 9 f., 123 f.; cultural, 6,
I2 f.; of secondary education, 5
Algebra, fundamental laws of, 7, 8,
33, 14o; language of, 8, 32; tech-
nique of, 8, 33 f., 69, 71, I41;
topics in, for grades seven, eight,
and nine, 34; topics in, for grades
ten, eleven, and twelve, 5o f.;
college entrance requirements in,
68 f., 140 f.; relationships in, 94 f.;
terms in, I 13 f.; symbols in,
II5 f.; and arithmetic, 8, 34, 71;
referred to, 8, 16, 41, 42, 55, 61, 67
Aliquot part, I I4
Alternation, 38, 69, ISO
American Mathematical
xi, 68
Analysis, of a complex situation, II
Analytic geometry, 55, 64, 65
Angell, J. R., 126
Angle (s), measurement of, 31, 54,
72; of a triangle, 32, 49; bisectors
of, 49; in circles, 49, Ioo; of paral-
lel lines, 49; of a polygon, 49; in
solid geometry, 52; obtuse, IoS;
reflex, flat, whole, Io9, IIo; Con-
jugate, IIo; symbol for, III
Angle-bisector, I lo
Angle-sum, I Io
Annuities, 37
Antecedent, Io9
Apothem, I To
Appreciation, of beauty, I 3; Of
logical structure, 46; of a de-
ductive proof, 48
Society,
Approximate characteristics of
measurement, 31
Approximate data, computing from,
7, 31, 39, 65, 71
Approximate methods of summa-
tion, 55
Archibald, R. C., 126
Architecture, 31
Arcs, in circles, 49; Symbol for, 112
Area (s), of geometric figures, 31,
49, 72, IoI, Io'7, IoS; problems
in, 55; of a circle, Io?; of a
polygon, IoS
Arithmetic, utility of, 7; and alge-
bra, 7, 33, 71; list of topics in,
for grades seven, eight, and nine,
29; time devoted to, 30; applica-
tions of to business, 30; terms
and symbols in, II8; referred to,
I6, 41, 42
Arithmetic progression, 37, 51, 64,
65, 70, 143
Association of Teachers of Mathe-
matics in New England, vii; in
the Middle States and Maryland,
VII
Astronomical observations, 54
Attitudes, mental, 12
Average, 54
Axioms, 30, IoS
Bagley, W. C., Io, I26
Barber, H. C., 2I
Bennett, H. E., 21
Bingham, H. V., I 26
Binomial theorem, 38, 51, 64, 65,
7O, I42
Biography, 37, 56
Bisector, 31, 49
Blair, Vevia, iv, viii, 56, 137
Bonds, 3o
Bookkeeping, 3o
Boring, l'. G., 126
Bourlet, 49
I73
INDEX
Briggs, T. H., 21
Broken line, Io'7
Business aspects of arithmetic, 42
Byron, 13
Calculus, as preparation for teach-
ers, 24; as a high-school subject,
46, 53, 54, 57 f., 137 f.
Cancel, II4
Carson, G. St. L., 39
Cavalieri's theorem, 53, 138
Centimeter, 29
Centrifugal force, formula for, 96
Central Association of Science and
Mathematics Teachers, viii
Chance, Io4
Characteristic, 115
Checks, on Solutions, 34; in compu-
tation, 40, 54, 71
Chords, in a circle, 49
Circle, circumference of, 31, 72, Io8;
area of, 31, Io'7; arcs, angles and
chords in, 49, Ioo; secants and
tangents of, 49, Ioo; analytic
geometry of, 64, 65; as a topic in
geometry, 72; meaning of the
term, Io'7; Segment of, Io'7; sym-
bol for, I I I
Circumference, of circle, IoS
Citizenship, 13
Civilization, progress of, 14
Clearing of fractions, I 14
Coefficient, I 15
College entrance requirements, iv,
61 f., 72 f.; no consideration given
to, in junior high school, 19; pur-
pose of, 61; units of, 67; definition
of, 68 f., 139 f., 161 f.; referred to,
6
4 -
College entrance examinations,
72 f.; limitations on, 73, 74
College Entrance Examination
Board, iv, 3, 8, 19, 49, 50, 65, 66,
67, 68, 73, 75, 77, 79, 87, 91, 139,
I () I.
Collinear, 113
Colvin, S. S., 126
Combinations, 145
Commission, on college entrance
requirements in mathematics, 77
Committee of Fifteen, 49
Complete equation, I 14
Complex fractions, 69, 141
Complex numbers, 38, 5o, 145
Composite courses, 17 f.
Composition, 38, 15o
Compound interest, 70, 96, 14o
Computation from approximate
data, 7, 64, 65, 71; numerical,
importance of, 30, 64, 65; by
logarithms, 51; checks, 54; ac-
Curacy in, 39, 40, II8; referred
to, 16, 37, 54, 71
Concentration, 12
Concurrent, II3
Cone, 53, IoS
Congruence, of geometric figures, 9,
I6; of triangles, 49, 99; Symbol
for, III
Conic sections, 64, 65
Conjugate angle, IIo
Consecutive interior angles, IIo
Consequent, Io9
Constructions, in geometry, 31, 44.
5O
Constructive geometry, 15, 31 (see
also Intuitive geometry)
Continuity, in geometry, 5o
Cooking, recipes in, Io 3
Coördinates, polar, 64, 65
Coplanar, I 13
Correlated courses in mathematics,
17 f. .
Correlation, of courses, 17 f.
Correlation (in statistics), theory of,
64, 65
Corresponding, I To
Cosine (s), 34, 70, Io2, 142; law of,
Io2, 146
Crathorne, A. R., vii
Cube root, 38
Cultural aims, 6, 12 f.
Curve, slope of, 54; empirical, 64,
05
Cylinder, 53, IoS
Dartmouth College, 61
Decimal places, I 18, 110
Deductive geometry, see Demon-
Strative geometry
I 74
INDEX
Deductive proof, appreciation of,
48 \
Degree, of equation, II.3, II4
Demonstration, 16, 52
Demonstrative geometry, I 5, 44,
48, 64, 65, 72; intuitive geometry
a foundation for, 32; as a topic in
the junior high School, 35 ſ., 41,
42; purposes of, 48; of space |
(three dimensions), 32, 165; basal
propositions of, 49, 79 f., 161; re-
duction of number of proposi-
tions in, 49; list of topics in, 49 f.;
relationships in, 99 f.; terms in,
Io'7 f.; symbols in, IIo f.
Dependence, graphic representa-
tion of, 9, 64, 65; idea of, I I, I5,
27, 40, 69, 92 f.; expressed by
formulas, 32; in geometry, 5o,
86
Derivative, 54
Descartes’ Rule, 144
Descriptive geometry, 56
Determinants, 144 f., 154
Dewey, John, I4
JDifferential calculus, see Calculus
Direction, Io'7
Disciplinary aims, 6, 9 f., 123 f.
Disciplinary values, 123 f.
T)istance, measurement of,
definition of, Io'7
Division, limitation of, in algebra,
34; in proportion, 38; Symbol for,
II9
Dodecahedron, IIo
Downey, W. F., viii
Drawing, to Scale, 31, 54
T)rill, in arithmetic, 7, 13, 3o; in
algebraic technique, 8, 13; im-
portance of, 14, 15, 71; limitation
On, I4, I5, 28, 7I
Dynamic geometry, IoI
3 I;
Economic aspects of arithmetic, 42
I'lection of subjects, 45 f.
Elective courses in mathematics,
20, 48 f., 56
Empirical curves, 64, 65
lºngland, 56
Equation (s), as topic in algebra,
33; literal, 38, 64, 65, 71; simul-
taneous, 38, 5 I, 64, 65, 69, 7O,
II4, 14o; radical, 38; in one un-
known, 5o, 64, 65, 94; linear, 33,
64, 65, 69, 7o, 94, II4, I4o; quad-
ratic, 50, 64, 65, 7o, I I4, I42; pure
and affected, I I4; complete and
incomplete, II.4; of higher degree
than the second, 5o, 64, 65, I44;
trigonometric, 54; degree of, II3;
simple, I 13; system or set of,
I 14; theory of, 144
Equilateral triangle, 31
Equivalent, Io9
Errors, effect of, 7, 97
Esthetic aims, 12, 13
Ethical aims, I 2, 13
Europe, conditions in, 42, 56
Evans, G. W., viii
Evolution, II4
Examinations, college entrance,
72 f., 8o; limitations on, 73, 74,
I46 f.
Experimental geometry,
also Intuitive geometry)
Experimental work in organization
of material, 56
Explement, I Io
Exponents, 34, 69, 7o, II5, I4I;
fractional, 37, 51, 69, 141, I43;
negative, 37, 51, I43; theory of,
37, 64, 65, 143
Extract, I 14
Extraneous root, II 5
I5 (see
Factor, highest common, 38
Factorial, 117
Tactoring, cases to be included, 34,
69, 141; in quadratic equations,
5O
Falling bodies, formulas for, 95, 14o
Ferguson, G. O., I 26
Fifteen, Committee of, 49
Figures (geometry), congruent, 9,
16; similar, 49, IoI ; Variations
in, Ioo; motion in, Ioo, IoI;
Static and dynamic, IoI
Finance, 37
Fiske, T. S., iv
I'itting curves, 64, 65
I 75
INDEX
Flat angle, Iro
Foberg, J. A., viii., ix
Force, centrifugal, 96
Formula (s), significance of, 8; in
geometry, 31; in algebra, 32, 5o,
69, 95, 14o; for solving a quadratic
equation, 5o; of Science and
practical affairs, 95, Io9; re-
ferred to, I6, 42, 64, 65
Fourth proportional, Io9
Fractional exponents, 37, I4I
Fractions, in arithmetic, 28, 29, 34;
in algebra, 34, 69, I41
Franc, 29
France, 56
Frequency distributions, 30, 54, 64,
65
Frustum, 53
Function (s), idea of, I2, 15, 16, 56,
65, 86, 92 f.; of one variable, 5o;
linear, 64, 65, 69, 95; quadratic,
64, 65; theory of, 92; and graphs,
94; trigonometric, 40, Io2; in life,
Io3 f.
Functional character of graphs, 94,
98
Functional thinking, 12
Fundamental laws of algebra, 7, 8,
33, I4 I
Fundamental operations, in arith-
metic, 6, 7, 28, 29; applied to
negative numbers, 33; in algebra,
33, 34, 69, I41; applied to frac-
tions, 34; applied to radicals, 51
General courses in , mathematics,
I6 f.
General Folucation Board, ix
General science, 39
Generalization, 12
Geometric forms, familiarity with,
9, 15, 44; congruence of, 9, 16, 44;
similarity of, 9, 44, 49, IoI; in
nature, 3 I
Geometrical progression, 37, 51, 64,
35, 7o, I43
Geometrical relations, 15, 32, 99 f.
Geometry, demonstrative (see De-
monstrative geometry); intui-
tive (see Intuitive geometry);
of appreciation, 31; of three di-
mensions, 32; basal propolitions
of, 49, 5o, 79 f.; list of topics
in plane, 49; analytic, 55, 64,
65; descriptive (see Descriptive
geometry); projective (see Pro-
jective geometry); relationships
in, 99 f.; terms in, Io'7 f.; Symbols
in, IIo f.; referred to, I5, 16, 44,
56, 61, 66 .
Germany, 56
Glass, 2I
Godfrey, C., 139
Goldziher, C., xi
Gram, 29
Graphic representation, importance
of, 8, 64, 65; in statistics, 30, 33;
representing dependence, 33; of
negative numbers, 33; in College
entrance requirements, 69, I4o
(see also Graphs)
Graphic solution of equations, 51
Graphs, line, bar, and circle, 30; as
topic in algebra, 33; as a method
of solving problems, 33; of the
trigonometric functions, 54; in
college entrance requirements,
69, 14o; functional character of,
94, 98 (see also Graphic repre-
sentation)
Griffin, F. L., 138
Habits, mental, acquisition of, 12,
I4; of precise statement, 48
Hadamard, J., I32
Hart, W. W., Ioë
Harvard list, 49
Harvard University, 72
Hawkes, H. E., Ioé
Hedrick, I. R., vii, 92, Ioë
Henmon, V. A. C., 126
Highest common factor, 38
Hines, 21
History of mathematics, 37, 56
Homologous, IIo
Icosahedron, I To
Ideals of perfection, 13
Identities, trigonometric, 54; Sym-
bol for, II6
z76
INDEX
Imaginary numbers, 38, So, 64, 65;
symbol for, II 7
Imagination, spatial, exercise of,
9, 32, 48
Incommensurable cases, 49
Incomplete equation, I 14
Indirect measurement, 31, 35, 44
Industry, 13; geometric forms in, 31
Inequalities, in geometry, 99
Infinite, study of, I3
Insurance, 30
Integral calculus, see Calculus
Intelligence, 61
Interest, 37, Io4; tables, 40, 71;
compound, 70, 96, 14o; simple,
96, I4o
International Commission on the
Teaching of Mathematics, xi
International Congress of Mathe-
maticians, I2I
Interpolation, 40
Intuitive geometry, I5, 18, 31, 35,
4I, 42, 5o; topics in, 31 f.; as an
approach to demonstrative ge-
Ometry 35, 5o
Inversion, 38
Investment, arithmetic of,
mathematics of, 56
Involution, II4
Isoperimetric, IIo
Isosceles triangle, 31, Io9
3O;
Jastrow, Joseph, 126
Judgment, exercise of, 7
Judd, C. H., 126
Junior high school, 6, 17; principles
of organization for, 19; books on,
20, 21; approval of, 20; mathe-
matics for, 26 f.; arrangement of
material for, 40 f.
Kilogram, 29
Kilometer, 29
Klein, F., 132
Ladd, G. T., 126
Law of sines and cosines, I46
Laws, fundamental, of algebra, 7,
8, 33, 34; general formulation of,
I2; permanence of, I3
Lemma, Io9
Letters, use of, for numbers, 94
Levers, formulas for, 95 .
Limits, theory of, 49; referred to,
.54, 58
Line, straight, 64, 65
Linear equations, 33, 64, 65, 69,
I4O
Linear functions, 50, 64, 65, 69
Liter, 29
Literal equations, 38, 64, 65, 71
Loci, geometric, 32, 49, 72
Logarithms, 37, 51, 64, 65, 70, II5,
I44; theory of, 64, 65; tables of,
7O, I44.
Logarithmic solution of triangles,
54
Logical demonstration, 52
Logical organization, I2, I4, 46, 48
Logical relations, II, 32
Love, A. E. H., 139
Lowest common multiple, 38
McGregor, 21
Magnitude, Io'7
Manipulation, skill in, 14, 73;
limitation of drillin, I5, 28, 33, 71
Mantissa, IIS
| Manufacture, geometric forms in,
; 3I
Mark, 29
Material, selection of, 13; organiza-
tion of, I6 f.; arrangement of, 40,
4T, e
Mathematical Association of Amer-
ica, vii, 68
Mathematical induction, 145
Mathematical instruction, aims of,
5 f., 13; value of, in times of
peace, 47; value of, compared
with other subjects, 47 f.
Mathematics, history of, see His-
tory of mathematics
Maxima and minima, 54, 6o
Measurement, errors in, 7; direct,
31, 44; indirect, 31, 35, 44; ap-
proximate character of, 31; re-
ferred to, I6 (see also Mensura-
tion)
Mechanics, 58
I77
INDEX
Median, 54
Memory, 12, 46, 48
Mensuration, of geometric forms,
9, 44, 72; in Space, 53 (see also
Measurement)
Mental attitudes, 12
Mental habits, I2
Méray, 49
Mercer, J. W., 139
Meter, 29
Metric units, 29
Millimeter, 29
Milne, W. P., 139
Minima, problems in, 54, 6o
Minuend, I 18
Moore, C. N., ix
Moore, E. H., vii, x,
Morgan, F. M., 138
Motion, idea of, 49; problems in, 55;
in figures, Ioo
Multiplicand, II8
Multiplication, short cuts in, 29;
limitation on, in algebra, 34; Sym-
bol for, II5
Multiply, an equation, II4
Navigation, 56
Negative exponents, 37, 70
Negative numbers, 33, 64, 65, I4o
Notation, Scales of, 145
Notes to teachers, 146 f.
Numbers, positive and negative,
33, 64, 65, I40; imaginary and
complex, 38, 5o, 64, 65, I45
Numerical Computation, 39, 54, 64,
65
Numerical trigonometry, see Trig-
onometry, numerical
Oblique parallelogram, Io9
Oblique, triangle, Io&
Oblong, Io9
Obtuse, angle, Io8
Obtuse triangle, Io&
Ogden, R. M., 126
Olney, A. C., viii
Optional topics, 37
organization, logical, 12, 14, 46, 48;
of subject-matter, 16 f., 56
Originals, in geometry, 49
{
Parallel lines, 31, 44; angles of, 49;
symbol for, III
Parallelepiped, IIo
Parallelogram, 49, Io?, Io9; area of,
31; oblique, Io9 -
Parcel post, 3o
Parentheses, nests of, 34
Pechstein, 2I
Percentage, 29
Perigon, Io9
Perimeter, Io8
Permanence, of laws, 13
Permutations, I45
Perpendicular bisectors, 49; symbol
for, I Io
Perpendicular lines, 44
Persistence, I2
Petersen, J., 131
Philosophy, I3
Physics, correlation with, 55
Pictograms, 30
Pilsbury, W. P., 126
Plane, Io'7
Plane geometry, list of basal propo-
sitions, 78 f. (see also Geometry)
Plurals, Io&
Point, Io'7
Polar coördinates, 64, 65
Polygon (s), angles of, 49; regular,
49, 72; similar, 72; meaning of
term, Io'7
Polyhedron, Io&
Polynomials, 55, 7o
Postal savings, 30
Postulates, 36, Io9
Pound, 29
Practical aims, 6 f.
Precision, love for, I2; of measure-
ment, 3 I
Pressure, problems in, 55
Princeton University, 72
Prism, 53
Prismoid formula, 53
Probability, 64, 65, 145
Problems, concrete, 14, 38; practi-
cal, 38; real to the student, 39;
utility of, 39; in public questions,
IO4.
Progressions, arithmetic, and geo-
metric, 37, 5 I, 64, 65, 79
t?8
INDEX
Projective geometry, 56
Proportion, 33, D4, 65, 69, IoI, I4o;
theorems on, 38, 69
Proportional, third and fourth, Io9
Propositions, basal, in geometry,
49; reduction of, 49; list of, 79 f.
Protractor, 31 -
Psychologists, on the transfer of
training, Io, I 23 f.
Pure equation, II4
Purposes, primary,
mathematics, I.3, 14
Pyramid, 53
Pythagorean theorem, 32
Q.E.D., Q.E.F., Io&
Quader, IIo -
Quadratic equation, 33, 5o, 64, 65,
7O, II4, I42
Quadratic function, 5o, 64, 65, 70,
of teaching
I42
Quadrilaterals, 44
Quantitative thinking, II
Questionnaire investigations, 62 f.
Radian, 54
Radical equations, 38, 7o, I43
Radicals, 34, 38, 51, 69, I4I, I43
Rates, 54, 58
Ratio, 33, 64, 65, 69, 140; of the
sides of a triangle, IoI; trig-
onometric, IoI
Rationalize. II.5, I43
Ray, II3
Real estate, 30
Rectangle, area of, 31, IoI; mean-
ing of term, Io9
Rectangular solid, IIo
Reflex angle, Io9
Regular polygon, 49, 72
Relations, logical, II, 32; thinking
in terms of, 27; spatial, 5* be-
tween trigonometric functions,
4.
Riitionship (s), idea of, I 2, 92 f.;
in real life, 93; in algebra, 94 f.;
in geometry, 99 f.; in trigonome-
try, Io2 f.
Religious effect of the study of
mathematics, 13
Required courses, 19 f., 26 f.
Rhomboid, Io9
Right triangle, 31, IoI, Io&; solu-
tion of, 35
Root, square, 34, 38; Cube, 38; ex-
traneous, II 5; symbol for, and
meaning of, II 7
Sanford, E. C., 126
Scale, drawing to, 31, 54
Scalene, Io9
Scholium, Io9
Schorling, Raleigh, viii
Science, 13, 29, 50, 58
Seashore, C. E., 126
Scott, W. D., 126
Secant, 49
Secondary education, aims of, 5
Sect, Io9
Segment, of a circle, Io'7, Io9; of a
line, Io9
Self-reliance, in numerical work, 8,
4O -
Senior high school, 5; mathematics
in, 45 f.
Set of equations, II4
Sextant, 54
Shop mathematics, 56
Short cuts, in arithmetic, 29, 64,
65
Siddons, A. W., 139
Significant figures, 31, 39, 97, II9
Similar figures, 49, IoI
Similar polygons, 72
Similar solids, 53
Similar triangles, 49, IoI
Similarity, 9, 16, 44; introduction
of the idea of, 32; symbol for,
III
Simple equation, II3
Simplify, I 14
Simpson’s theorem, 53, 138
Simultaneous equations, 38, 51, 64,
65, 69, 7O, II 4, I4O, I43
Sine, 35, 70, Ioz, 142; law of, I46
Slaught, H. E., IoG
Slide rule, 37, 54, 64, 65
Slope of a curve, 54, 58
Smith, T). E., iv., vii, xi, T3, 21, ſcó
Solid geometry, 32, 52 f.; relation
I 79
INDEX
to plane geometry, 52; correla-
tion with arithmetic and algebra,
52; time devoted to, 53; list of
propositions, 87 f., 169 f.
Solid (s), mensuration of, 52; simi-
lar, 52; meaning of term, Io&;
rectangular, IIo
Space, Io'7
Space perception, development of,
9, 32, 52 . . e
Spatial imagination, exercise of, 9,
32, 48, 52
Sphere, 53, IoS, 14o
Spherical geometry, 53
Spherical triangle, 53
Square, 31, Io? -
Squares, tables of, 40, 71, 97 -
Square root, of a number, 34, 7o,
I42; of a polynomial, 34, 38, 142;
tables of, 40, 71; completing the,
5o
Square ruled paper, 31
Starch, Daniel, 126
Static geometry, IoI
Statistical data, graphic representa-
tion of, 9, 30, 69
Statistics, 30, 42, 54, 64, 65, 70, 140;
frequency distributions, 54, 64,
65, 7o; graphic representation of,
54, 64, 65, 7o
Straight line, 64, 65, Io'7
Strong, E. K., 126
, Struthers, 21
Subtend, I Io
Subtrahend, 118
Summation, methods of, 55
Surd, II 5
Surface, 31, IoS
Surveying, 54, 56
Symbols, Ioë f.; use of, IIo, III;
in geometry, IIo f.; in algebra,
II5 f.
Symmetric functions, 144
Symmetry, 31; use of, in geometry,
5o
System of equations, II4
Tables, 16; of weights and measures,
29; statistical, 30; of trigono-
metric functions, 35, 40, 70, 71,
Io2, 142; of logarithms, 37; use
of, 39, 71, 97; interpolation in,
4o; of Squares and Square roots,
40, 71, 97; interest, 40, 71
Tangent (geometry), to a circle, 49;
as limit of a secant, 5o
Tangent (trigonometry), 34, 7o, Ioz,
I42
Tariff, IoA
Taxes, 30, IO4
Teachers, training of, 21 f.; 126 f.;
in Europe and the U.S., 21, 22,
I 26 f.; standards for, 23 f., 130 f.;
notes to, 146 f.
Technical vocabulary, Io9
Technique, in algebra, 8, 33 f., 69 f.,
7I, I4I; in calculus, 55
Terman, L. M., 126
Terms, meaning of, Io'7 f.; unde-
fined, Io'7; to be abandoned, Io9;
in geometry, Io; f.; in algebra,
II3 f.
Testing movement in mathematics,
I33 f.
Tests, in arithmetic, 29, 133; power
test in algebra, 74, 134
Textbooks, influenced by examina-
tions, 73
Theorems in geometry, reduction of
number of, 49; list of, 78 f.
Therefore, symbol for, II 2
Thinking, quantitative, II; func-
tional, I2
Thorndike, E. L.,
Thoroughness, I2
Thrift, 30
Touton, F. C., 21
Training, of teachers (see Teachers,
training of); transfer of, Io, I 23
Transfer of training, Io, I 23 f.
Transit, 54
Transpose, II4
Transverse angles, IIo
Trapezium, Io9
Trapezoid, I4o
Triangles (s), angfes of, 32, 49;
area of, 31; equilateral, 31;
isosceles, 31, Io9; right, 31, 1or;
Io8; Solution of right, 35, 70,
congruent, 49, 99; similar, 49,
I26 -
18O
INDEX
rot: meaning of, Io'7; acute and
obtuse, Io&; oblique, Io&; scalene,
Io9; symbol for, III; notation
for, II 2
Trigonometric functions, 4o, IoI,
I45; graphs of, 54; relations be-
tween, 54; tables of, 35, 40, 70, 71,
I O2
Trigonometric identities and equa-
tions, 54
Trigonometry, numerical, 18, 34,
4I, 42. 64, 65, 7o, I42; arguments
for early introduction of, 35, 64;
plane, 54, 64, 65, IoI, I45 f., 156 f.
Tyler, H. W., vii
Undefined terms, Io'7
Underwood, P. H., viii
Unified courses, 16 f.
United States, Bureau of Educa-
tion, x, 20
Units, metric, 29; monetary, 29
Upton, C. B., iv, 28, 29, 133
Utilitarian aims, 6 f.
Utility, of arithmetic, 6 f.
Value of the study of mathematics,
45 f.
Variable, idea of, 15, 16; function of,
I5, 16, 50, 92 f.
Variation, 33, 5o, 64, 65, 69, I4o;
in figures, Ioo
Veblen, Oswald, vii, ix
Velocity, 54, 58
Vinculum, I 14
Vocabulary, technical, Io9
Volume (s), 32; problems in, 55; of
similar figures, IoI -
Warren, H. C., 126
Washburn, M. F., 126
Watson, J. B., 126
Weeks, Eula A., viii
Weights, tables of, 29
Westcott, G. J. B., 139
Whole angle, IIo
Woodworth, R. S., 126
Yale University, 72
Yerkes, R. M., 126
Young, J. W., vii, ix, 132, 138

UNIVERSITY OF wº * ~ - i
|||||||||||||||
gº
cy)7 », a fºr 22, (2. #za, (2
f/erººn to cº, itsºs


* Fº
ºº::
º: ºº is
ºf a º
º, a
a 2.2
* Ex
&
º . .
* * * *-
* * *-
* : * ~ * º
ºr wº B.º.º.
a tº º
º.
• *.* :
º
ºs- E.
* ~ * * * * * * * * * * * * * * *
*** * * * *-*.*.*.** *** -
º
sº - - - ºr sº-º-º:
§:...º.º.º.
º
* * * : * : * : * *
* * * * *
** r * * * * * * *
º wº
... . º sº v----ºn-
ºfº.º.º. "...º.º. º.
º:
*** *-ºx º
*** .." sº r, º
º
º: …..
... :::::
; : ºº sº.
º
º
Tº sº ºr º- Cº-
Kºi...º.º.º.º.º.º.º.º.
†-º-º-º:
…sº r
Eº-
-
º
º
* * : * r * * * * * *
* * * * * * * * * *
* - C - a - * : *
ºwº - ºr º &
º
;:::::::
-** *** * * *
º:
º
*::::::::::::::::
º-º-º-º-º-º-esº- sº
********
*-º ºrº 2-8 x * ..." º
• * * * *
Cº-ºw ºf ºw º
rºzºr-º-º: &
#
X.
s =
i
;
5.º.
* * *
º:
w" N - * * *r
•- A - a *
º
--> * * *-ºs º-sº
sº..…. .
: º -- -
* -- *-*.
cº-º-º-º-
º
:
* - sº tº ºr ºv
ºf s -r * *
g º
* * * * a- -
- we t-tº-ex” sº s- a - a -
- sº re--sº-ves - a --- ºr-->
* * *-* *- : * ~ * : * *- : * > *
. . . . . . . . . . ." . . . . . . .
Riº Hºrºs.:
* * * * *** * *** *** * *
B - - - - - > * * * * * * * * * **** *
• * ~ * : * ~ * ---
- sº
; :
º, tºº. .
E.-
i
:
tº
** * *-** wº
ºr ºr ar.
sº sº tº c.
..º.º.
* - Mº i º 'º a
fºr tº:********* *** 22 ºf
º “s sº.º.w…sºs...} := sº-ºº-ºº-ºººººººº tº
** : * *- : * * ** ::::::::::: Łº-º-º: sº Fº tº sº;
*** -ººrºº tº it"-º-º-º-º-º-, * * -
a ºr - º -- a -- -
*** * * * * *. ºw *
º
* * *
*** * *
- sº-
E-º-º-º:
tº-3
*~
ºs- ºf
º “.
* : * * * * *-* * * *** ****- : * :
*** ****** *
prºſºrºrº sº.
º
º
a cº-stº
* * **, *º-Ajº.
• ** -&-ºº-ºº º
-º-º-º-º:
lº q v' " A. P.
* * *
*# * * * *
s — 3 s º – a « I - e º z
º
º:5.
sº
º
ºº::
ºs- ºr
Eº
#
* *** *
* * * x
:.
;
& "...º.º.º.º. ººº-ºº:
& º
º
* * * *
-º-º-º-º:
Fºº
:
:
º
:
#
#
§:
ºfºº
=sººrºº:
rº-ººººº. §
§§§
::::::::::::::::::::::::::::
ºś
::::::: ſº §º ºº::::::::
**** : º tºº º: º º
º º ::::::::::::: #º:
- #º
º Fº
º # ºr sº
ºº: ºw, ºt
ºf:
Fº º:
tº:
gº
Fº
*…*
--
zerº ſº
º:
º t
º ſº -
º sº
::::::::::::::: :
£º
º ... º. f*:
#########
* -
sº
§:
º:
ºº:: wº
ſº
*:::::::::::::::
-- ~ * ~ *********
º * *
***** **t
** ***** **
*::: * * • * *
:
#
:
:
#
* * * * * > .
* -- * * * * * * * : *
tºº º
** **, wº ºr º- *** * sº-º-º-º-ºrt
* º-ºº-ººs & 2 tº
º: º ºr { *
Prº
º
º
* ºr
* * *
r &
&
ºº: "º
sº-sº sº. ºs º ºs- *** *t
cºst.º.º.º.º. *** . . .º.º.
º º
º
º, º 'º', ;
* * * * * * * * *
£ºº. ºº
fºº
ºº::
ºº: º, . Rº:
sº
§º
º:::tº: ºº: º
º & º ºº::::::
ºw tºº.
*Cº-ºº-ºººº- :::::::::::::::::::::
rº-ºº: s - ?: s ºf ºrº-Lº. at wºn
******
sººn
º
-º ºs
tº tº a ºne ºr --º
*** * * ºryzººs ºr ºst
** * * **** *** **-w: * * *
§§::::::::::::
º-- * **s ºvºvº-º-ºrº tº wº
ºº:::::::::::
zºr-sº ******
****** if:
£º.º.º.
*:::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::
º º *** -->> x. c. * *- : *-*.s ſº -
º º-
tº 28 e.
ºf º-sºº's 5 º
º º * : *-
ſº º
*** * * *
º *:::
º º, “º sº ºf £3
****** ***********
vs rººrººººººº sº
- * * * * º
- §º º -
- :*:
arrºrs tº º
§§§ #
º;
§:
#:
-
º º
sº
- º, tºº: I , ºr . º.º.º.
#
Prº- º
::::::::
º, . .
º
º sº sº
º: §::::::::::
ºº: ºº: "º §º
* *
Er
º
* * * : * *
ºº: º3
º
*ºrº, º lºº
º:
ºº: ºº iº ** * *
º-ººººººº sº
ºw sº
º:
tº ºr
º