Front Cover
 Half Title
 Table of Contents
 Making general instructional...
 Getting an overview of mathematics...
 Improving mathematical instruc...

Group Title: Bulletin - Florida State Department of Education ; no. 50
Title: A Brief guide to teaching mathematics in the secondary schools
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00067266/00001
 Material Information
Title: A Brief guide to teaching mathematics in the secondary schools
Series Title: Bulletin State Dept. of Education
Physical Description: 60 p. : ill. ; 23 cm.
Language: English
Creator: Florida -- State Dept. of Education
Publisher: State Dept. of Education
Place of Publication: Tallahassee Fla
Publication Date: 1946
Subject: Mathematics -- Study and teaching -- Florida   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
non-fiction   ( marcgt )
Funding: Bulletin (Florida. State Dept. of Education) ;
 Record Information
Bibliographic ID: UF00067266
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 22198259

Table of Contents
    Front Cover
        Front Cover 1
        Front Cover 2
    Half Title
        Half Title
    Table of Contents
        Table of Contents 1
        Table of Contents 2
        Foreword 1
        Foreword 2
    Making general instructional plans
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
    Getting an overview of mathematics in the secondary school as an aid to planning
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
    Improving mathematical instruction
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
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Full Text





T75. 009759

BULLETIN No. 50 1 946



Wleacoin Mahemalica

Making General Instructional Plans................... ............................... 1
Trends in Secondary Education............................................................................ 1
Using the School to Develop Democratic Living....................... .- 2
Relating the School to Community Resources or Needs................ 3
Providing a Wide Range of Opportunities..................................... 4
Broadening the Program to Meet Needs of Youth Groups...... 6
Focusing Efforts of All Teachers Upon Common Goals............. 7
The Place of Mathematics in the Secondary School Program...... 8
Conditions in American Society Which Demand
T raining in M them atics.......................................................................................... 8
Necessity for Providing This Training in the
Secondary S chool............................................................................................................. 9
Mathematics Training for Laymen............................................ ........... 10
Training in Mathematics for Specialists;....... .................. 15

Getting an Overview of Mathematics in the Secondary School
As An Aid to Planning.................................................................................................18
Inadequacies of the Present Problem............ ................ ...................... 18
Inadequacies in C ontent................................................................................................... 18
Inadequacies in Methods............................. ........................................... 19
State A adopted T extbooks.......................................................................................................... 20
R equirem ents ...................................... ............................................... .......................... 22
Organization of Mathematics Offerings Within the School........... 22
Grade Placement of Mathematical Concepts........................................ 22
Preliminary Consideration ................................................................................. 23
D description of Courses............................................. ............ .......................... 23
Seventh and Eighth Grades...................................................................... ......... 23
N in th G rade................................................................................................................................ 30
Algebra and Geometry ................................ ............................ ....... 31
T w elfth G rade................................................................................................................. 32
Mathematics for the Small High School.................................................... 33
Mathematics in the Larger School............................. .... .......... 34

Improving Mathematical Instruction............... .................................................. 37
Materials of Instruction......................................................................................................... 37
T yp es of M aterial..................................... ..... .......... ............................... 37
Suggested Bibliography for Students' Library.......................................... 37
Materials Available in the Community.................................. ................ 38
Description of Materials That Can Be Made.......................... ............. 39
G row th in S ervice.................................... .......... ........................ .......................... 5
M again es .............................................................................................................................. ..... 57
Textbooks Which Will Aid in an Enriched Offering................... .... 57


At the request of many teachers, principals, supervisors, and
county superintendents, A Brief Guide to Teaching Mathematics in
Secondary Schools, Bulletin 51, has been prepared. Companion vol-
umes dealing with the teaching of English and social studies in the
secondary school are being issued simultaneously. In the near future,
a bulletin on the teaching of science in the secondary school will be

The four fields of English, mathematics, science, and social studies
represent a large portion of the secondary curriculum, particularly
of that part dealing with general education. The degree to which
these basic fields meet the needs of society and of the individual
will determine, in a large measure, whether or not as a nation we
shall meet our world responsibilities of the future.

The material in this bulletin is derived in part from the math-
ematics section of the State Department of Education, Bulletin 10,
A Guide to a Functional Program in the Secondary School. State
adopted textbooks, including 1946 adoptions, are included. Practical
techniques and procedures have been added in accord with recent ex-
periences in teaching mathematics.

It should be kept in mind that the purpose of this bulletin is to
give specific help to a large number of teachers who have entered
teaching in Florida high schools during the wartime emergency. The
material as presented is consistent with the purposes of education
in Florida and with sound educational principles. It is not necessary,
however, for one to have had extensive courses in the teaching of
mathematics to use the many practical suggestions contained

In short, this guide should be helpful at all levels of professional
development. Teachers will find it most helpful to read carefully
the suggestions pertaining to the state adopted texts.

The current secondary series of bulletins, including this one, were
prepared under the general direction of Dr. W. T. Edwards, Director
of the Division of Instruction, State Department of Education. Dr.
Clara M. Olson of the P. K. Yonge Laboratory School, University of

Florida served as general consultant and editor of the series. Mrs.
Dorothy L. Phipps, Head of the Department of Mathematics of
Gainesville High School, Gainesville, prepared the manuscript for
publication. Especial appreciation is accorded these educators as well
as other teachers and leaders who made valuable suggestions from
time to time. In presenting this bulletin, we are not unmindful of
the appreciation due the committee that prepared Bulletin 10 upon
the excellence of which much of this material rests.



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The effectiveness of any one teacher's work in a school depends,
in a large measure, upon the extent to which the program of the
whole school is well planned by the whole faculty. If secondary
teachers are to plan for effective instruction in their subject fields,
they must (1) know something of trends affecting the program of
the secondary school as a whole, (2) understand the desirability
of cooperative planning by the whole faculty, and (3) actually
participate in planning the general program of the whole school.


For the past quarter of a century, the percentage of boys and
girls entering the secondary school and continuing through it to
graduation has increased. Unfortunately, all youth entering the
secondary school have not completed, for one reason or another, the
program offered them. The startling fact is that 75 percent of the
youth who enter the secondary school never complete it. For those'
who have completed it, the educational values derived have not
always been satisfactory. These facts have caused educators, busi-
ness men, and other interested and competent adults to appraise
critically the purposes and the program of the school. Youth, too,
have looked critically at the opportunities and the limitations in
the program of education provided by the secondary school.2 The
net result has been the emergence of certain trends, or emphases;
in secondary education. Experiences during the war years and
plans for post-war education tend to confirm the trends. These
trends, both from the point of view of society and of the individual;
place emphasis on:

'Compare Planning for American Youth, an Educational Program for
Youth of Secondary School Age, Nattiial Association of Secondary School
Principals, N.E.A., 1944, p. 3.
'See Bulletin No. 2 Ways to Better Instruction in Florida Schools, pp.
29-32 and.p,: 35 and Bulletin No. 10 A Guide to a Functional Program in the
Secondary School, pp. 50-51 and all of Chapter Three.


1. Using the school to develop democratic living. Democracy
in the American heritage is both a personal way of life and a
system of social and political values. It includes those values which
sponsors of democracy from antiquity to modern times have deemed
essential to humane living and to effective self-government.3 Its
dynamic nature may be perceived in the evolving concepts of poli-
tical, economic, and social democracy; its force, in the moral or
ethical values which it creates. In its scheme the personal worth
and the dignity of every individual shape the motivating ideal.
Equally significant is the obligation of every individual to further
the general welfare of the group. This two-fold nature of democ-
racy places directly upon the school the responsibility for develop-
ing democratic living; that is, for clarifying and extending for
every boy and girl the meaning and the values of democracy. It
requires that the curriculum of both the elementary and the secon-
dary school shall be directed toward: (1) the development of an
individual who assumes increasing responsibility for self-direction
and for the development of his potentialities in such a way as to
bring about optimum satisfaction both to himself and to society;
and (2) the development of an individual who assumes increasing
responsibility for clarifying the meaning of democracy and for
solving personal-social problems in the light of this ideal.4
To develop democratic living in a school requires thoughtful
planning and careful guidance by individual teachers and by the
faculty as a whole. It requires a total program in which values
consistent with the democratic ideal are consciously sought both in
and out of the classroom. It requires methods, techniques, and con-
tent through which these values may become an integral part of
the behavior of all youth. Procedures, such as delegating respon-
sibility to committees, making decisions in accordance with the
preference of the majority, permitting the minority to be heard,
encouraging youth to decide what they will study, and attempting
to eliminate social cliques, in themselves, will not produce demo-

'Compare Beard The Unique Function of Education in American De-
mocracy as quoted on p. 21 of Bulletin No. 10.
'Compare discussion, especially footnote, Bulletin No. 2, pp. 97-98. For
stimulating analysis of assets of American democracy and implications for
the school, see Bulletin No. 10, p. 39-42 and p. 23 ff.


cratic living. However necessary such procedures are in effecting
democratic living, they may fall short of the mark unless they are
so planned and so used as to develop in all the boys and girls social
sensitivity and an ever-widening understanding of and concern for
humane living and efficient self-government. They are ineffective
also if they do not provide actual experience in intelligent social
action. Chaos in or out of the classroom carried on in the name of
individual "freedom and democracy," is as inimical to the develop-
ment of democratic living as mob rule is to the development of
orderly and effective government. Fascistic control by the teacher
with a sugar coating of democracy is equally inimical.

2. Relating the school to community resources and needs. A
school can, and should, raise the level of living in the community
it serves. To do so, school officials, the faculty as a group, and in-
dividual teachers must:

1. Know what the resources of the community are.
2. Appreciate the limitations (needs) and the potenti-
alities of the community.
3. Cooperate with available agencies in overcoming the
limitations and developing the potentialities of the
4. Project the program of the school into the life of the
community, especially in the areas of health, home
living, applied economics, and recreational oppor-
5. Devise ways of utilizing the resources of the community
realistically in developing the social intelligence and
the technical competence for which the individual
teacher in her own subject field must take responsi-
6. Allocate time for and gear direct and incidental in-
struction to improvement of community living.

The current emphasis on resources in the education of all chil-
dren and youth has set groups of educators and other civic-minded
adults to exploring ways of carrying out the foregoing sugges-


tions." The whole problem of resources-human, natural, and
cultural-is worthy of intensive study by small and large groups
of teachers. It is well for teachers to ponder the fact that the youth
of the local community, the state, the region, and the nation are
our most precious resource.

3. Providing a wide range of opportunities. The school is, and
should be regarded as, a state investment in democratic citizenship,
health, personal living, and vocational competence. Every com-
munity, therefore, should provide a well-balanced educational pro-
gram consisting of a wide range of opportunities based upon the
abilities and interests of all of its youth. A school that provides a
wide range of opportunities does the following things:

1. Safeguards the health of each pupil. In doing this it
provides opportunity for healthful living in the school
each day; provides for adequate health examinations
followed by immunization and correction of discovered
defects; maintains a nutritious lunch program and
makes best educational use of the lunchroom; provides
clinical facilities; bolsters health practices with adequate
health instruction;. relates instructional practices to
health; insures mental and emotional, as well as physi-
cal, health of all' pupils; initiates drives to rid
community of sources of disease and infection; and
cooperates with community agencies in improving and
maintaining the health of the community."

'Resources education is receiving considerable attention in Florida as a
result of the impetus coming from the emphasis on resources in the Southern
Region. Compare point of view expressed in Building a Better Southern
Region Through Education, Southern States Work-Conference on Adminis-
trative Problems, Tallahassee, Florida, pp. 1 and 2.
Bibliographies on Florida resources and on how to study resources may
be secured from the Curriculum Laboratories of the University of Florida
and the Florida State College for Women.
Workshops in Florida resources have been held at the Florida State
College for Women. County workshops, on.local and county resources have
been held. Examples of these -re the Madison County Workshop, reported
in the Journal of the Florida Education Association, April,' 1945, and .the
Pinellas County Workshop, whose published report Pinellas Resources,
1945,.may be secured.from the Pinellas County Board of Public Instruction,
Clearwater, Florida.
:'See also Bulletin No. 4," State. Department of Edtcaation, Tallahassee,
F]..ridl .. :" . .


2. Provides an adequate and appropriate program of
physical fitness and recreational opportunities and ex-
periences. The latter include an appreciation of and
experience in a variety of wholesome leisure activities.
3. Enables youth to progress noticeably in the direction
of becoming self-sustaining. Growth toward this goal
is dependent upon: an understanding of factors affect-
ing economic status; adequate vocational guidance;
acquisition of specific skills and understandings re-
lated to work; assistance to youth in finding employ-
ment after they leave school and in securing retraining
when necessary; and actual work experiences while in
4. Provides instruction in home and family relationships
which will enable youth to make an intelligent choice
of a marriage partner and to understand the basic prin-
ciples for establishing and maintaining a home. This
includes such areas as: the function of marriage and
the mutual obligations of each marriage partner; per-
sonal hygiene; feeding and care of infants; care of
children; budgeting on limited incomes; furnishing
homes on limited income; handling of family finance,
including insurance, savings, and loans; and the prob-
lem of further intellectual and social growth of each
marriage partner.
5. Develops skills, attitudes, and understandings necessary
to democratic citizenship. This includes: understanding
the historical background of our institutions; under-
standing the rights and duties of the citizen of a
democratic society; increasing participation in the life
of the school and the community. Activities which
facilitate growth as an intelligent citizen include:
group discussion; committees; forums; round table dis-
cussion; debates; community surveys; tournaments;
community development programs; student govern-
ment; trips to study government, a region, or the various
phases of community and regional life; and experiences
with varying cultural groups.


6. Develops youth as individuals. This includes develop-
ment of special interests and talents and help for youth
in such understanding of themselves as unique individ-
uals and as cooperating members of society as will lead
to the most satisfying self-integration.
Since youth differ in sex and race, in home background, in emo-
tional and physical health, in intelligence and aptitudes, in hobbies,
and in job interests, the school will have to develop a strong basic
program of general education, a vital and varied program of
specialized interests, and a cooperative program with many in-
dividuals and agencies in the community, county, state, and region,
if it meets the obligations set forth above.7 Ways to achieve these
desirable ends are worthy of the serious study of small and large
groups, especially in rural areas or in small communities Where
opportunities are limited. The foregoing program will, of necessity,
be developed gradually if it is to have deep roots. Faculties should
plan how best to begin and what steps to take progressively in
order to build soundly and wisely. To give up and say "it is beyond
our school" is to become defeatists. If necessary, faculties should
explore the possibilities of cooperating with other administrative
units in order to make the needed opportunities available to all
4. Broadening the program to meet needs of youth groups.
In every school there are groups of youth with varying interests
and abilities. For example, in a large rural consolidated school
there will be those who upon graduation expect to remain in the
local community and find their life work there, to go to the city
mainly to seek employment in commerce or in industry, to go to
college for additional general education, to go to trade school or
business college, or to go to the college or university for professional
education. The same groups will be found in a city school. It is
improbable, however, that.any appreciable number will be planning
to go to rural districts immediately upon graduation.

'Compare similar statements in Planning for American Youth, National
Association of Secondary School Principals, N.E.A. and Education for all
American Youth, Educational Policies Commission, N.E.A., 1944. Compare
also the four areas-work, citizenship, personal problems, and leadership--
pp. 53-55 in Bulletin No. 10, State Department of Education, Tallahassee,


From the point of view of society there will be need to see: (1)
that all youth receive sufficient general education to make them
socially competent both as individuals and as citizens of a de-
mocracy; (2) that a sufficient number be educated to perform the
work and the services needed by a complex, industrial, democratic
nation; and (3) that the supply of scientists, frontier thinkers,
mathematicians, statesmen, and creators in the field of the fine and
the applied arts be kept adequate for the maintenance and con-
tinuous growth of a great nation. The school must broaden its pro-
gram to provide the necessary educational opportunities for all of
the foregoing groups of youth. To do so may entail making ad-
ministrative changes in the school units of a local, county, or
regional system. It is a problem for all faculties to study, however.
5. Focusing effort of all teachers upon common goals. In order
to realize the goals set forth or implied in the foregoing discussion,
it is necessary that community of effort be emphasized in the nurs-
ery school, the elementary school, the secondary school, the junior
college, and the trade school. Community of effort is also neces-
sary among subject-field specialists. No longer can any one division
of the school or any one teacher accept responsibility for unrelated
or isolated areas of the youth's education. The truth of the matter
is that unrelatedness and isolation are a psychological impossibility
in the learning process. If the school or the teacher does not make
the adjustment, the youth will-in some way. The trouble with un-
planned education is that the adjustment may not be desirable.
Among the common responsibilities of all teachers are the de-
velopment of:
1. Skills and abilities in reading.8
2. Skills and abilities in oral and written expression.
3. Skills and abilities in utilizing and interpreting all
types of graphic and other visual aids.
4. Critical thinking.
5. Desirable work habits.

'For a detailed and helpful discussion, see A Guide to Teaching in the
Intermediate Grades, Bulletin No. 47, State Department of Education, 1944,
pp. 20-36.


6. Democratic living.
7. Intellectual curiosity.
8. Enjoyment of living.
9. Pride in clean and attractive surroundings.
10. Health.
11. Understanding of social relationships.
12. Wise use of all types of resources.
13. School-community relationships.
14. Standards and tastes in recreational activities.
15. Spiritual values.
16. Determination on part of all pupils to make the most
of their lives.

The quality of living that characterizes the school is the respon-
sibility of all the teachers. If it is conducive to the development of
democratic living, all the teachers are to be praised; if it falls short
of this mark, all the teachers share the blame for the failure. The
same is true of the contribution the school makes to the general
education of all of its pupils and to the quality of living in the


In the discharge of their duty to society, mathematics teachers
must accomplish two things: (1) give to every citizen sufficient
training in the subject so that he can exploit his abilities and oppor-
tunities for his own advantage and for the benefit of society as a
whole, and (2) discover, train, and inspire those students whose tal-
ents lie in mathematics or in related fields.

1. Conditions in American society which demand training in
mathematics. Appreciation of the place mathematics should occupy
in education is impossible without a knowledge of how mathematics
has contributed to our modern civilization. The conveniences which


make life so pleasant, the automobile and the airplane, the telephone
and the radio, highways, bridges, and great buildings would be
impossible without a vast background of mathematical knowledge.
The ships, planes, guns, and tanks which have defended our country
from its enemies could have been neither developed nor used with-
out the considerable mathematical knowledge of a great many
specialists. To insure the continuance and improvement of our way
of life, it will be necessary not only to find and develop an increas-
ing number of gifted persons capable of carrying on the present
progress, but also to develop in the mass of citizens an appreciation
of the contributions of these specialists.
Mathematical knowledge is essential to carrying on the affairs
of everyday life also. Nearly everyone has need for some mathe-
matics in his vocation, and every consumer needs to understand
taxes, budgeting, installment buying, profit and loss, interest, and
insurance. In order to be an intelligent member of society, one
must be able to comprehend large numbers; the vast sums involved
in the public debt, and statistics concerning public health are likely
to induce either an attitude of panic or one of indifference in an
uneducated individual. To the degree that individual citizens have
the mathematical attitudes of searching for true relationships,
recognizing assumptions in an argument, striving to separate cause
from effect, to that degree will society as a whole resist the attacks
of propaganda of competing ideologies.
Mathematics can contribute to the proper use of that ever in-
creasing leisure which results from continued technological ad-
vances. Every individual should have sufficient mathematical
knowledge to be able to follow reports of technical advances, to
read and interpret graphs, to suspect possible flaws in printed ma-
terial, and to check figures which may be given. There probably
are not many persons who would enjoy mathematical recreations,
but those few should have sufficient training to permit this diver-
sion. The best protection from the slot machine and the punch-
board is an acquaintance with the mathematical basis of probability
which underlies such machines and all games of chance.
2. Necessity for providing this training in the secondary
school. It is sometimes argued that instruction in a subject should


be delayed until a need is felt for the material in question. In the
case of mathematics at least, this is not always feasible, for its sub-
ject matter must be learned in sequence, the mastery of each part
being dependent upon the mastery of prerequisite material. For
example, if a person needs to solve a differential equation, for de-
signing a radio improvement, it would be too late for him to de-
velop the necessary skills in arithmetic, algebra, geometry, trigo-
nometry, analytic geometry, and differential and integral calculus,
all of which would be necessary before he could begin to work out
the desired process. To the time factor must be added the diffi-
culty of self-instruction; only an unusually gifted person with a
very persistent nature can have much hope for success in learning
mathematics unaided, and then only if he has a firm foundation
in early courses on which to build. For these reasons, and because
continuous practice and review are essential, the learning of mathe-
matics must be extended over several years. Administrators of
secondary schools should see that capable students have those fun-
damental mathematical experiences which will enable them to en-
gage in the occupations and pursuits best suited to their talents. If
these occupations and pursuits include all physical sciences, engi-
neering, economics, and finance, then the students must have ac-
quired the tools of advanced mathematics. The secondary school
should help him to acquire these tools.

Mathematics is a mode of thought, and habits of thinking must
be practiced over a period of years if they are to become second
nature. Learning to be exact, not to exaggerate, to require precise
definition of terms, and to seek the cause for every effect and the
result of every cause, can not be learned overnight. The ability
to abstract basic ideas from a given situation is developed grad-
ually. These features of mathematics, along with its sequential
nature, make it necessary to begin the training of future scientists
in the elementary and secondary school.
3. Mathematics training for laymen. The relationship of our
civilization to mathematical invention and progress indicates that
every citizen needs a working knowledge and appreciation of math-
ematics. Man's need for number concepts goes far back into the
dawn of pre-history and has expanded with the increasing complex-


ity of civilization. Recognition of the part that mathematics has
played and continues to play in the development of our culture
should be a part of the education of every member of society. The
problem of the school is to set up courses and provide experiences
which will do this; content and methods best suited to the purpose
must be determined.

The policy of determining the content of school courses solely on
the basis of items commonly used by adults is to be condemned.
The good of society and of the individual demands that the criterion
of choice be what could be used to advantage. The following list
of the needs of non-specialists is significant in that it points out the
mathematical materials that could be used by the ordinary citizen,
if training is such that transfer is effected. It is included in the
hope that it will aid in teaching for transfer and in answering the
questions of pupils.

Arithmetic. The paramount mathematical need of the average
citizen is for a greater knowledge of arithmetic than is now com-
mon. By arithmetic is meant more than computational facility and
understanding of principles. There is needed also familiarity with
applications to a wide variety of problems or situations that con-
front people, and ability to understand certain mathematical ideas
and procedures that may be encountered in ordinary reading.9

From among the many activities of ordinary life which require
some use of arithmetic, a few examples are listed here:
1. The home
(1) Budgeting income, keeping accounts, checking bills.
(2) Quantity and installment buying.
(3) Estimating depreciation on hom6 or car.
(4) Buying or settling fire or burglary insurance.
2. Personal finances
(1) Handling funds: depositing, checking, remitting.
(2) Borrowing: paying loans and interest.
(3) Saving and investing: choosing securities.
(4) Buying life and disability insurance, and annuities.
(5) Paying taxes: contributing to charities.

The material quoted herewith is from Mathematics in Secondary Education,
The Fifteenth Yearbook of the National Council of Teachers of Mathematics,
Bureau of Publications, Columbia University, (New York, 1940), pp. 207-211.


3. Recreational activities
(1) Buying season tickets for sports, plays, lectures, music.
(2) Planning trips-expenses, time schedules.
(3) Arranging social functions.
(4) Helping with neighborhood entertainments.

Among the numerous items often encountered in general reading
matter, the comprehension of which requires a considerable knowl-
edge of arithmetic, may be mentioned some relating to:
1. Civic and social life
(1) Taxes-property, income, inheritance, sales, imports.
(2) Use of public funds-for roads, public defense, social security,
health, interest on debt.
(3) Social statistics-population, vital statistics, distribution of
wealth, dependency.
2. Economic conditions
(1) Industrial activity, volume of output, dividends.
(2) Agricultural acreage, crops, prices.
(3) Labor conditions, wage levels, unemployment.
(4) Price indices, real wages, living standards.
(5) Financial conditions, interest rates, public debt, monetary stand-
(6) Exports, imports, international exchange.
3. General information
(1) Scientific and technical advances, involving use of very large
or very small numbers: e. g., relating to stellar distance, infra-
molecular physics, engineering feats, number theory.
(2) Statistical data involving averages-mean daily temperature,
or average annual rainfall; average crop per acre; average
(3) Further data involving ratio, proportion, or percentage, dietetic
and budgetary matters, probability of death or illness, "expecta-
tion" in games of chance, cost of essential elements in materials
containing large portions of waste.
(4) Diagrams, records, puzzles in reading material for recreations,
and in accounts of sports.

Graphic representation. To understand numerous mathematical
ideas which are commonly encountered in general reading, one
should be able to interpret graphs such as the following:
1. Line graphs
As used for market quotations, growth of population, progress of a
campaign, course of a fever temperature.
S2. Bar graphs
Showing the distribution of telephones, radios, automobiles.


3. Circle graphs
Portraying the percentages of land given to cultivation, grazing, for-
est, waste; or the distribution of the national income.
4. Scale drawings
As used in surveying, or to find forces or velocities.
Algebra. In studying investments, installment buying, statistical
aspects of one's business, solving mixture problems, or indeed in
dealing most effectively with various matters listed under arithmetic
above, one needs some command of formal algebra. For the aver-
age citizen, however, the chief need is for such algebraic knowledge
as will give him insight into and appreciation of mathematics as-
pects of the modern world, scientific achievements, economic ques-
tions, and the like. In particular he needs to be familiar with:
1. Positive and negative numbers
Their frequent use in describing such opposites as assets and liabili-
ties, forces or motions in opposite directions, temperatures above and
below zero, rates of increase and decrease.
2. Formulas and the function idea
Such related variables as the age and value of a house, amount of
use and annual cost of a car, driving speed and distance required for
stopping, length and period of a pendulum.
3. Equations
Their constant employment in technical and scientific work, as a
general way of finding unknown quantities.
.4. Coordinates
Their more common use in plotting curves, mapping, and as a basis
for engineering projects.
5. Indirect measurement
Finding elevations of inaccessible peaks, or constructing a long tun-
nel from both ends, understandable on the basis of numerical trig-
onometry using simple algebraic equations,
6. Modern methods of calculation
The power of generalized exponents as logarithms.
7. Permutations and combinations
Their frequent application in devising distinctive labels, etc.
Informal geometry. Some geometric knowledge is occasionally
employed as a working tool about the home, in such activities as:
1. Making mensurational calculations
2. Using a protractor and other instruments
But, as in the case of algebra, the most important uses that the
ordinary citizen can make of geometry are those by which he may
achieve insight. For instance, a good foundation in geometry will
contribute materially toward an informed appreciation of matters
relating to:


1. Architecture and decorative art
Recognition of the skillful use of geometric relations in patterns,
arches, vaults, columns, buttresses, apses, windows.
2. Engineering and manufacture
Realization of the importance of triangles for rigidity of structures;
of angles in surveying, geography, and navigation; of circles in
machinery and appliances; of polygons in furniture, jewelry, dia-
3. Natural forms
Awareness of geometric figures in natural objects, terrestrial, and
celestial, eclipse phenomena, stellar configurations, hexagonal cells,
By way of summary it is to be said that the important uses of
mathematics for the ordinary citizen are in large measure cultural.
Mathematics provides an outlook and a means of understanding.
There are important aspects of the world that only mathematics can
interpret to the citizen. Mathematics furnishes a mode of thinking
about many aspects of life, and a very general kind of language. A
liberal view of education regards such matters as genuine needs of
even the ordinary citizen. It rejects the thought of a person being
satisfied with such a minimum working equipment as would enable
him to exist as nothing more than a hewerr of wood and a drawer
of water."

Public Service and Leadership. Individuals who exercise influ-
ence on either organizations or communities should understand pub-
lic affairs thoroughly. Some mathematical aspects of such a many-
sided competence are these abilities:
1. To judge the significance of complicated numerical data
Here one needs a knowledge of statistical analysis, involving con-
siderable mathematics.
2. To follow quantitative studies of social phenomena where necessary
Studies of cyclic changes commonly employ trigonometric analysis;
those dealing with population growth and other trends often involve
3. To recognize fallacious conclusions
(1) By checking against known facts.
(2) By scrutinizing critically the authority for each step in the
argument and the cogency of each implication.
(3) By realizing that the validity of conclusions depends not only
on correct steps of deduction but also on the basic assumptions
or presuppositions.
SCultural satisfaction. The more highly educated group of per-
sons who wish to be familiar with the most significant modern
theories, the central problems, and the principal methods of inves-


tigation, in various fields, can make good use of mathematical
knowledge far beyond the elementary field:
1. To understand quantitative procedures
Through some familiarity with calculus and with group concepts.
2. To follow philosophical discussions:
(1) As to the nature of space: using ideas of non-Euclidean geom-
(2) As to the nature of number: complex numbers, series, theory of
(3) As to the problem of knowledge: foundations of mathematics.
(4) As to the nature of scientific law: functions of real variables
and of many variables: point sets.
Obviously laymen can scarcely devote to mathematics all the
time needed for high achievement along all the lines mentioned;
but, to the extent to which the indicated equipment is lacking,
insofar are an individual's potentialities circumscribed.

4. Training in mathematics for specialists. It is evident from
the discussion which precedes that the existence and progress of
civilization depend upon finding, inspiring, and training the special-
ists in the various fields of national life. The following needs for
specialists in fields dependent upon mathematics have been con-
densed from the Fifteenth Yearbook:

The mathematical needs of those who are to use this subject as
a working tool in their vocations or professions differ greatly, ac-
cording to the field.

The physical sciences. There is scarcely any limit to the amount
or variety of mathematics that can profitably be employed in some
branches of physics or in certain laboratory researches.
1. By physicists
Advanced mathematics.
2. By chemists
Calculus and differential equations.
3. By pharmacists
Percentage, proportion, formulation and solution of simultaneous
linear equations relating to mixtures.
Engineering and related activities. The mathematical require-
ments of engineering schools are doubtless reliable indications of a
suitable equipment in this field:


1. For engineers
At least calculus and differential equations, besides analytical geom-
etry and descriptive geometry.
2. For engineering employees
As a minimum, facility in handling algebraic formulas and graphs,
and mechanical drawing; a good knowledge of geometry; some fa-
miliarity with trigonometric functions and the solution of triangles.
Larger equipment is very helpful.
3. For skilled mechanics
Ability to apply arithmetic principles, to compute without making
mistakes, to determine the degree of precision required in instruments,
and to keep calculations correspondingly accurate; facility in dealing
with formulas and graphs that involve positive and negative num-
bers, square roots, quadratic and simultaneous equations, sines, log-
arithms, very large and very small numbers, occasional interpolation
to seconds; knowledge of many geometric facts needed in working
with triangles, and experience in analysis: an extensive mathemat-
ical vocabulary and ability to read technical books and articles.
The earth sciences
S1.. Physiography and cartography
A good command of trigonometry; some analytic geometry and cal-
2. Meteorology
Elementary mathematics through integral calculus.
3. Geology
(a) Locating oil deposits: graphic, algebraic, and geometric tech-
(b) Stratigraphic work: trigonometry,
(c) Seismology: elementary mathematics through integral calculus.
4. Navigation (marine or aerial)
Trigonometry at least.
5. Ballistics
Analytic geometry, calculus, differential equations.
6. Practical astronomy
Elementary mathematics including spherical trigonometry.
The life sciences. Statistical methods will not be explicitly men-
tioned since they occur so generally.
1. Biochemistry
Differential equations.
2. Other metabolic studies
These studies use involved irrational algebraic functions, and some
employ nomographic charts.
3. Heat and energy problems
4. Growth and senescence
Differential equations.
5. Population problems
Actuarial problems, requiring probability, calculus and higher


6. Genetics
Algebra and probability.
7. Forestry and agricultural experimentation
Using sampling theory and other mathematical analysis.
8. Biometry and anatomy
Trigonometry, frequency curves, and correlation.
9. Neurology
Psychology and education. Statistical methods, matrices, cal-
culus, trigonometry, polar coordinates, integral calculus, differen-
tial equations.
The social sciences. In the social sciences, economics makes the
largest demands upon mathematics, though political science and
sociology make considerable use of statistical methods and employ
growth curves in connection with cultural change and the develop-
ment of institutions. Sub-divisions of economics which utilize
mathematics beyond arithmetic, including calculus in some cases, are
in part the following:
1. Synthetic economics
2. Economic trends and cycles
3. Index numbers
4. General economic theory
5. Mathematics of finance and accounting
Industry and commerce. Statistics, sampling theory, mathe-
matics of finance.
Aesthetics and the fine arts. A broad familiarity with the prin-
ciples of geometry will be found helpful in this field. This is obvi-
ous as regards architecture and some decorative arts.
The need for mathematics on the part of both the ordinary citi-
zen and the specialist, pointed out in the material quoted from the
Fifteenth Yearbook, seems to indicate that there is need to provide
more suitable material. For those whose use of mathematics is
likely to be limited to the management of their personal affairs and
the reading of articles of general interest, the secondary school must
offer 'materials which will develop both insight into all the mathe-
matical phases of ordinary life and mastery of the skills involved.
For those who are by interest and ability suited to mathematical
pursuits, the secondary school mist provide,- in 'addition, :a more
complete view of the nature, materials, and possibilities of -mathe-
matics, and systematic development sufficiently rigorous that they
may not be haiidicap'ped in iater study. .


In all probability no curriculum can be adequate to the high
purpose the school sets out to achieve, but by continuous re-exami-
nation of content and method in the light of aims and objectives,
continuous progress can be made. Since the teacher-pupil relation-
ship is the heart of the learning process, such advancement is possi-
ble only as individual teachers concentrate their attention upon their
own procedures in an effort to detect and remedy every weakness.
1. Inadequacies in content. The program of the secondary
school developed originally as a plan to prepare students for col-
lege, where in turn they were to be prepared for the learned pro-
fessions. The traditional curriculum was well adapted to its pur-
pose so long as the student body was selected according to this
criterion. In 1880 only about 2.6 percent of the boys and girls be-
tween the ages of fourteen and seventeen were in the high school.
But in 1940, when approximately 60 percent of the children of sec-
ondary school age were attending school, no single curriculum could
satisfy the much wider range of abilities and interests. With the
tremendous influx of the less scholarly type of student, the tradi-
tional curriculum became less and less effective. The change has
been met by offering a wider range of subjects, by not requiring
students to take the "harder" subjects, by omitting "harder" por-
tions of the traditional subjects, and by enriching all subjects with
applications to and illustrations from daily life. This latter policy
will aid in teaching for transfer; recent studies have indicated that
transfer is a process of gradual growth made possible by persistent
applications of generalized concepts and procedures. This is cer-
tainly indicative of the line along which revision must take place,
for if our students are to have the abilities discussed in the early
sections of this bulletin, they must be able to carry the skills and
2 See the Fifteenth Yearbook of the National Council of Teachers of Mathe-
matics, pp. 66.


attitudes of the classroom over into later life. It is not an easy
task, for texts are not written from this point of view, and much
supplementary material must be used.
The practice of not requiring mathematics of all is not in har-
mony with the needs of our present civilization, for as we saw in
opening sections of this bulletin, there is a practically universal
need for training in mathematics. The' practice of diluting exist-
ing courses is equally unsuccessful, for the capable student is then
defrauded of material which could have been stimulating and useful
to him, while the poor student, in whose name the sacrifice has been
made, still finds the material meaningless. The resulting training
in mediocrity is far from inspiring.
A possible solution would seem to be parallel courses, one offer-
ing rigorous training along traditional lines to those able to use it,
and second, to be required of those avoiding the traditional course,
embodying illustrations from and applications to daily life. This
solution has come into rather general use, perhaps without adminis-
trative intention; here in Florida we offer a choice between algebra
and socialized mathematics in the ninth grade and strongly urge a
terminal course in basic mathematics.
2. Inadequacies in method. In an effort to lower the failure
rate, teachers have often put undue emphasis on computational
mathematics and have then found it necessary to resort to large
amounts of drill which in many cases was meaningless. This has
made many a mathematics class a boring period of monotonous repe-
tition. In reaction to this practice, teachers have swung to the
theory of meaning and have given more.attention to the informational
side of mathematics; concepts and processes are made clear to the
pupil and the drill used must have an obvious relation to the newly
learned concepts and processes.
Much study has recently been devoted to the location of items
at the proper maturity level of pupils; more study is needed and
textbooks and courses of study will need to be rewritten with this
newer knowledge as foundation. Perfect adjustment can never be
reached because of the wide range of maturity which exists in every
class. This fact and consideration of the shape of the curve of
forgetting call for spaced learning and the repetition of topics in


several grades, as well as for great effort to care for individual
Grade Seven
Making Sure of Arithmetic, Grade 7, Morton, Gray, Springstun,
and Schaaf (Silver Burdett Company)
Grade Eight
Making Sure of Arithmetic, Grade 8, Morton, Gray, Springstun, and
Schaaf (Silver Burdett Company)
Making Sure of Arithmetic fills the need indicated by its title.
It is excellently adjusted to the understanding of this student level.
Previously learned processes are reviewed, redeveloped, and extended.
Relationships are established by stress on meaning. Active interest is
aroused in the pupil to discover and to correct his own weaknesses.
Readiness for algebra and geometry is established. Adolescent in-
terest in applications to adult life is utilized; e.g., budgets, taxes,
finance. The problems are practical, involving more than mere
manipulation of figures. No additional workbook drill is necessary.
Grade Nine
Mathematics in Action, Book III, Hart and Jahn (D. C. Heath and
This is a general:mathematics textbook. It takes the place of
Socialized General Mathematics. In it arithmetic is reviewed and ex-
tended. Problem material is within the range of pupil interest. Stress
displaced on skill in problem solving rather than on facility in
computation. In addition to arithmetic, some algebra is developed
and beginning geometry is introduced. The text is complete; -no
workbook is needed.
* ;:For pupils who, it is known, will drop out of school at the end of
the niith year Mathematics in Life should probably be used. Although
Mathclhttiu. in Life was adopt, i as a terminal basic course for grades
tenj eleven; and twelve, especially for grade twelve, its: value should
rieverthelesS be received by pupils forced to terminate their school
earliei- :It shouldh be; the :polity of. all scho61s, however, to encourage


all pupils to complete the full program offered by the secondary
Progressive First Algebra, W. W. Hart (D. C. Heath and Company)
This text is by the same author as the text now in use. It has
been carefully brought up to date, however. There is full instruction
on all topics. Equations are constantly applied to each new topic.
Graphs are thoroughly integrated with other topics. The function
concept is thoroughly taught. The page organization simplifies study.

Grade Ten
Progressive Second Algebra, Wells and Hart (D. C. Heath and Com-
pany). See foregoing comment on first year algebra text.
Those students who will terminate their study of mathematics in
the tenth grade or who know they must leave school at the end of the
tenth grade should be encouraged to take basic mathematics the text
for which is Mathematics in Life by Schorling and Clark.

Grades Eleven and Twelve
Plane and Solid Geometry
Modern School Plane Geometry (Revised), Clark, Smith, and
Schorling (World Book Company)
Modern School Solid Geometry, Smith and Clark (World Book
The approach of this series is simple, practical, and interesting to
the student. The content is mathematically sound and complete.
Deductive reasoning is stressed. Critical attitudes and creative think-
ing are developed. New skills are prepared for and old skills are
reviewed. Applications to present-day life are practical; e.g., the
excellent treatment of the geometry of aeronautics.

Plane Trigonometry, Freilich, Shanholt, McCormack (Silver Burdett
The text is written on the high school level. It covers the subject
effectively. Accuracy and clear thinking are developed. The slide
rule is introduced. Use in life situations is emphasized.


Mathematics in Life: Basic Course, Schorling and Clark (World Book
This book has been adopted primarily to serve two purposes:
(1) As a text for twelfth grade students interested in basic mathe-
matics, and (2) as a terminal course in mathematics for those students
not preparing to pursue 'higher mathematics and for those students
who must terminate their education in grades nine, ten, or eleven.
The material in this text is based upon the Report of the Com-
mission on Post-War Planning of the National Council of Teachers
of Mathematics. The text is designed especially as a terminal course
giving mathematical competence. It redevelops rather than merely
reviews essential skills. It applies mathematics to mechanics, voca-
tional work, and science. It does not duplicate any material now
available. This text should adequately meet the needs of the physically
mature but mathematically undeveloped student.
Mathematics is required only in grades seven, eight, and nine.
However, every effort should be made through adequate guidance
to see that every boy or girl has the mathematical training he or she
needs. This means that careful consideration should be given to the
quality and kind of material and experiences included in general
mathematics classes, to the terminal courses in mathematics, and to
the mathematics for pupils preparing to enter selected institutions of
higher learning. It will be helpful for all pupils preparing for col-
lege to find out as early as possible the mathematics required for
entrance into the college of their choice. Further consideration of
requirements is presented in the discussion Mathematics Courses in
Part II of this bulletin.
1. Grade placement of mathematical concepts. No list of mini-
mum essentials will be given for any course, lest that become a max-
imum offering; rather, a wide variety of materials will be suggested,
from which can be made choices suitable to the needs, interests, and
capacities of every youth. The Grade Placement Chart which fol-
lows should suggest ways in which courses may be adjusted to in-
clude those items demanded by the welfare of youth and of society.
In choosing the items to make up a course, a teacher will need the


insight to distinguish between the needs felt by the pupil and those
ascribed to him by adults; the effort to provide for both types will
include an attempt to make each complement and enrich the other,
2. Preliminary consideration. In choosing the topics to be cov-
ered in any course, the teacher must keep in mind the interests and
abilities of the students involved. Whether the problems and illus-
trations have an agricultural or an industrial and commercial con-
text will depend to some extent, upon whether the community is
agricultural or industrial. Keeping in mind the basic interests and
information of the pupil will not only capitalize on his knowledge
but will aid in the transfer of skills learned to out-of-school situa-
tions. In case a majority of the pupils of a given school plan to
attend college, that school is obligated to offer the best possible
college preparatory course. If vocational training is offered, it
should not be too highly specialized, since a choice of vocation made
in the early years of adolescence is seldom final. In any case, the
needs of no boy or girl should be overlooked.
3. Description of courses. It is recommended that mathematics
continue to be required in the seventh, eighth, and ninth grades
and that materials for the courses be taken from the Grade Place-
ment Chart. In the ninth grade there are two possibilities discussed
below; both avoid the common practice of requiring socialized math-
ematics of all students in the ninth grade and thus postponing, the
beginning of algebra to the tenth grade, which practice is to be dis-
couraged. Algebra, geometry, and trigonometry should be offered
and should be elective in the tenth, eleventh, and twelfth grades.
In the twelfth grade a course in review of arithmetic3 should also
be offered; this should probably be required of all students who
have avoided the traditional courses and any others deficient in the
knowledge and skills mentioned earlier in this bulletin.
4. Seventh and eighth grades. The content of the seventh and
eighth grade mathematics courses is indicated in the Grade Place-
ment Chart. In the treatment of arithmetic there is need for ade-
quate emphasis of the social and informational phases as well as of
_the computational phase. Teaching designed to give understanding
3 The State Department of Education already requires this of boys who have
not had three years work in traditional mathematics courses.



1. Experiences, lan-
guage, and ideas.
2. Fundamental
processes with
whole numbers,
fractions, and
decimals, and re-
lated principles.
3. Significant appli-
4. (Optional) Ele-
mentary approx-
imate computa-

1. Experiences, lan-
guage, and ideas.
2. Drawing or con-
structing basic
3. Direct measure-
ment (lengths
and angles).
4. Indirect meas-
urement (areas
and volumes).
5. (Optional) Ap-
plication of ele-
mentary approxi-
mate computa-
6. Related facts and
7. Significant appli-

1. Experiences, lan-
guage, and ideas
2. Fundamental
processes and re-
lated principles
(reviewed and ex-
3. Significant appli-
* cations.
4. (Optional) Ap-
proximate com-
putation (contin-

1. Experiences, lan-
guage, and ideas,
2. Drawing or con-
structing impor-
tant figures.
3. Indirect meas-
4. (Optional) Appli-
cation of elemen-
tary approximate
5. Related facts and
principles (con-
6. Significant appli-

1. Review.an exten-
sion of concepts
and skills.
2. Applications, pref-
erably in connec-
tion with algebra.
3. (Optional) Logar-
ithms and the slide

1. Review and exten-
sion of concepts,
skills, facts, and
2. Applications, pref-
erably in connec-
tion with algebra.
3. (Optional) Intro-
duction to demon-
strative geometry.
(In grades 9 and 10,
algebra and geom-
etry may be close-
ly correlated.)

1. Interpretation of 1. Interpretation of 1. Statistical graphs
simple pictograms statistical graphs. (reviewed and ex-
GRAPHIC or statistical 2. Graphic represen- tended).
REPRESEN- graphs. station of everyday 2. Functional graphs
TATION 2. Graphic represen- statistical data (formulas, y- ax
station of simple (bar graph, line + b, y = ax2).
statistical data graph, circle
3. (Optional) Tab-
ular and graphic re-
presentation of rela-
tionships expressed
__ by simple formulas. _
Note 1. The central theme or core of each year's technical work is indicated
by means of double borders. Mathematical modes of thinking, etc., should be
stressed in all years.
Note 2. In general, no single class should attempt all the optional lines of
work or types of enrichment suggested for each year. Some of the topics not-marked
optional can be deferred or omitted.if local conditions require such modification.
This is especially true of certain of the historical topics suggested.


(Space per-,

I "



1. Review and extension 1. Review and extension. 1. Review and extension.
preferably in connec- 2. Study of the number 2. The number system
tion with applied prob- system. (complex numbers).
lems. 3. Approximate compu- 3. Approximate compu-
2. The use of irrational station. station, including use of
numbers. 4. (Optional) Study and the derivative.
3. Approximate computa- use of the slide rule.
tion. 5. (Optional) Use of cal-
4. (Optional) The use of culating machines.
the slide rule.

1) Review in connection 1. Basic propositions in
(Formal) with trigonometry solid geometry with
transition to formalproperties and mensur-
geometry. ation of solids.
2. Acquisition of skill in 2. Equations of straight
demonstration. line and circle system-
3. Familiarity with facts atically studied.
and propositions, 3. Simple locus problems.
properly organized. 4. (Optional) Introduc-
4. Development of ele- tion to parabola and
mentary spatial in- ellipse.

1. Review and extension, 1. Representation of more 1. Graphic solution of
preferably in connee- complicated statistical equations.
tion with the social data. 2. Representation of com-
studies and science pro- 2. Graphs of linear and plex numbers (either in
grams. quadratic functions. rectangular or in polar
2. (Optional) Graphs of 3. Graphic solutions of coordinates).
simple equations, problems. 3. (Optional) Use of logar-
4. Graphs of trigono- ithmic paper.
metric functions.



1.(Optional) The use (Informal)
and application of 1. The shorthand of (Elementary)
formulasasexpres- algebra (concepts 1. Language and
sions of simple re- and simple tech- ideas (extended).
ALGEBRA nations and of fun- niques). 2. Fundamental
damental rules of 2. Th formula (eval- techniques.
procedure. uation and con- 3. Fundamental
struction). principles.
3. The equation (sim- 4. The functional
pie cases). core of algebra
4. (Optional) Signed (formula, table,
numbers and their equation, graph).
uses. 5. Significant ap-
5. Significant appli- plications. (See
cations. text for details).
(In grades 9 and 10,
algebra and geome-
try may be closely
(Preparatory Work) (Preparatory Work) (Numerical)
1.Scaledrawing (be- 1.Scale drawing 1.Language and
gun). (continued). ideas- '
2. Measurement of 2. Similarity and pro- 2. Necessary skills
TaRIaNOM- lengths and angles. portion. (drawing to, scale,
ETRY .3. Ratio (begun). 3. Out-of-door work using tables of si-
in indirect meas- nes, cosines, tan-
urement. gents).
4. The use of simple 3. Applied problems.
instruments. 4. Approximate com-
putation arising
from use of tables.
5. (Optional) The
slide rule.

1. The development of habits of correctness 1. Continuation of
in computation, measurement, and draw- themodesofthink-
MATHE.MA- in;. and in making vt. rbhl taltrnenrjr. ing outlined for
TICAL .2.The dcvelo-pment of Labits of estimating grades 7 and 8:
MODES OF and checking.. 2. Learning to under-
TH l'N si c, 3. earning to interpret and to analyze ele- stand and to apply
HAnTS, mentary problem situations. relational think-
ATTrrITEs, 4. Learning to prepare neatly and economic- ing (the idea of
TYIPES oP ally arranged written solutions of suit- dependence, of
APPRECIA- able mathematical problems. functional think-
TION 5.The development of an interest in the ing) as a key meth-
study of simple quantitative relation- thod of dealing
ships with the aid of the table, the graph, with quantitative
the formula, and the equation, changes arising in
6. Learningto appreciate theplaceofmathe- nature, inbusiness,
matics in every day life. -- and in everyday



1. Use of algebra in con-
nection with geometric (Intermediate) (Advanced, and Ele-
proofs and work. 1. Review and extension ments of Differential
of basic concepts and Calculus)
techniques. 1. Basic work in the the-
2. Linear functions and ory of equations, in-
equations. cluding determination
3. Quadratic functions of real roots.
and equations. 2. Permutations, combi-
4. Radicals and radical nations, and simple
equations, work in probability.
5. Logarithms and the 3. Differentiation of
slide rule. polynomials.
6. Series (arithmetic, 4. Slopes, maxima and
geometric, binomial), minima, rates of
7-11. (See text.) change, etc.
5. (Optional or as sub-
stitutes) Elements of
finance, statistics.

(Numerical) (Formal) 1. Review.
1. Review and extension. 1. The six trigonometric 2. Radian measure.
2. Functions of 300, 450, functions. 3. Inverse functions.
60. 2. Basic identities. 4. Identities and equations.
3.Significant applied prob- 3. The addition formu- 5. DeMoivre's Theorem.
lems involving use of las.
trigonometric func- 4. Double-angle and (When desirable, topics
tions. half-angle formulas. may be moved from grade
4. The slide rule; 5. Laws of sines, co- 11 to grade 12, and some
5. (Optional) Use of log- sines, tangents, topics above may be
arithms and slide rule. 6. Solution of triangles, omitted.)
7. Components. and re-
8. Simple identities and
9. Field work.

1. Continuation of the 1. Continuation of the modes of thinking suggested
modes of thinking out- for grades 7-10.
lined for grades 7, 8, 2. A more systematic application of functional and
and 9. statistical thinking, not only in mathematics, but
2. Learning to understand also in science, in the social studies, in economics,
and to apply the deduc- and in related fields.
tive type of thinking as 3. The development of greater skill in using deduc-
a method of dealing tive reasoning both in mathematics and in life
with situations involv- situations.
ing set of interdepen- 4. Learning to appreciate more fully both the service
dent concepts and values and the cultural values of mathematics.
closely related proposi-



1. The story of num- 1. The story of the 1. The story of alge-
HISTORY OF bers and numerals. decimal system braic symbolism.
MATHEMA- 2. The story of meas- and of computa- 2. The story of indi-
TICS urement. tion. rect measurement.
2. Early history of 3.(Optional) His-
geometry. tory of signed
numbers and ele-
mentary aspects
of irrational num-

1. Projects (home, 1. Banks and bank- 1. The place of math-
school, commun- ing. matics in the mod-
ity). 2. Income taxes, ern world.
CORRE- 2. The school bank. 3. Family budgets. 2. The mathematics
LATED 3. Simple measure- 4. Installment buy- of business and of
MATHEMA- ment projects. ing. the shop.
TICAL 4. Simple geometric 5. Out-of-door work 3. Graphic devices
PROJECTS designs in nature in measuring used in everyday
AND and art. heights and dis- life.
ACTIVITIES 5. Making mathe- tances. 4. Correlation with
tical source books 6. Making geome- sciences and social
and posters. tric designs or studies.
6. Correlation with posters. 5.Elementaryastron-
centers of interest. 7. Mathematical re- omy ( descrip -
7. Mathematical re- creations. tive).
creations. 6. Mathematical rec-



1. The development of See Note 2.
geometry in Egypt, 1. Systematic development of algebra, as centering
Babylonia, and Greece. around the solution of equations; leading con-
2. Great Greek mathema- tributors.
ticians. 2. Beginning of the modern period, Descartes, New-
3. Pre-Greek mathema- ton, Leibnitz.
tics. 3. Great development of analysis since 1700; leading
4. Mathematical physics and astronomy; leading con-
5. The mathematical discovery of Neptune.
6. The discovery of non-Euclidean geometry.
7. Development of mathematics in America, the in-
fluence of Bowditch, Peirce, etc.
1. Using postulational 1. Calculating machines. 1. Statistics and modem
thinking in life situa- 2. Making simple survey- life.
tions. ing instruments. 2. Mathematics of fi-
2. The geometry of archi- 3. Surveying projects. nance.
tecture, of surveying, 4. Introdction as astron- 3. Elementary work- in
of design, and of related omy (mathematical). mechanics.
fields. 5. Mathematical rec- 4. The mathematicsof the
3. Mathematical rec- relations. telescope.
relations (fallacies and 5. The mathematics need-
the like). ed in the leading pro-
6. Mathematical rec-

'Fifteenth Yearbook of National Council of Teachers of Mathematics, pp.


as well as skills will stress the reasonableness of the answer. If
we can develop .power of judgment and skill in estimating the. re-
sult, the coming generation will not be as dependent as the.present
one on paper .and pencilin every calculation. Grasp of the mean-
ing of numbers, the number system, and number processes will fa-
cilitate mastery of fundamental skills. Practice is needed in the
comparison of large numbers and in the realization of their signifi-
cancc; recognition of the approximate nature of measurement and
the study of significant figures .may well be a part of the study
of measurement and its history. A proper time allotment for in-
tuitive geometry4 must be made in both grades; nothing significant
can be accomplished if:the material is.used merely to fill whatever
time may be left at the enid of the',year. The work or areas andi
volumes will be used with arithmetic ,in both grades and with alge-
bra in the eighth. The algebra in these grades will be limited to!
the understanding of basic :concepts, ithe evaluation of formulas,.
and the solution of very simple equatio.ils' .
5. Ninth grade. Since this is the last year of required mathe-
matics, with the possible exception of a review of arithmetic in the;
twelfth grade, It is the last course that many of our children will
take. That our present system should be improved is demonstrated
by the large number of students graduating or leaving, school with:
an, inadequate 'comeept:.of nuumbber, little appreciationn ,of.. the applic4&'
tions of mathematics, and almost no understanding of the fiifida-
mental ideas of algebra and geometry. The solution of the diffi-
culty may be met in either of two ways: parallel courses or a single
course adaptable to varying situations. In a school so small that
only one section of each subject in the ninth grade is possible, there
is no choice. This single course should be a survey of new material
rather than a repetition of the material of the seventh and eighth
grades; in a class of higher than average ability, it should contain
a large amount of formal algebra, an introduction to the type of
thinking exemplified in demonstrative geometry, and some trigo-
nometry, as well as sufficient arithmetic to retain skills. In a less
capable class, the algebra should be held to the simpler types of

A very helpful article on this subject is "The Teaching of Intuitive Geome-
try" by William Betz in the Eighth Yearbook of The National Council of
Teachers of Mathematics.


equations, the geometry to constructions and, intuitional material,
the trigonometry to .scale drawing and the simplest problems, and
the amount of, arithmetic increased proportionally., Such a course
would help every child to maintain and increase the understandings
and skills developed in previous years;: it would inspire the child
with mathematical talent to further study by giving him..a glimpse
of what is to come; and it would give the pupil who terminates his
study-of mathematics at the ninth grade a wider view of the con-
cepts of the entire field and a better appreciation of the part it has
played and continues to play in our civilization. .
In a larger school, two sections of the survey course with the
different emphases pointed out above might exist concurrently. Or,
the, administration might choose to offer socialized mathematics for
the less scholarly type of student and the traditional algebra course
for the abler students. In this case, great care should be exercised
to see that pupils are placed in the class best suited to their needs
and abilities and not allowed to drift into one or the other by
chance. Of course, if a pupil has already decided which course he
wishes to take, he must be allowed to follow his own choice even if
the adviser feels that he is making a mistake, unless the adviser can,
by showing' him the facts, convince him of the wisdom of the con-
traty course.
The recent adoptions make available material necessary to put into
effect the foregoing suggestions. See list of textbooks on preceding
pages : .
6. Algebra and geometry. When a student has finished the
junior high school he may, if he chooses, terminate his. formal in-
struction in mathematics, unless a review of arithmetic be required
in the twelfth 'grade. If, because of interest and talent, he decides
to pursue his study further, he may elect either algebra, or geometry,
or both,, in either, order. The, algebra course 'which would follow
the proposed ninth grade survey course. would need adaptation, to
the changed.situation, since it is recommended that part of the ninth
year be :devoted to.'arithmetic and :geometry., ,The' resulting de-
crease in the amount of formal algebra,, as well; as the variation due
to the flexibility of the ninth grade course, would. make it necessary
for the teacher to determine the knowledge and: skills possessed -by


the pupils before proceeding with instruction. This is good prac-
tice in any case. Because of the superior educational guidance af-
forded by the ninth grade course, as herein described, the class in
algebra following it will probably contain only students of the usual
ninth grade course, as well as most of what is usually covered in
the present second course.

The courses in junior high school mathematics should have served
the same function for geometry as for algebra; the students electing
it, having had previous work in the subject, should be well suited
by interest and ability to pursue the course with ease and profit.
Under the proposed plan, geometry may either precede or follow
algebra. In either case, there will probably be students who have
had only the algebra presented in the eighth and ninth grades. This
will necessitate some changes in procedure; for example, radicals
will have to be taught in connection with the Pythagorean theorem
and perhaps extensive review of fractions will be needed to facilitate
the work with ratio and proportion. This seems to be an advantage
rather than otherwise, since it helps to dissipate the impression that
algebra and geometry are mutually exclusive; there should be ample
time for such correlations as the revision gets under way, since geo-
metric concepts, relationships, and facts will have been taught
throughout the junior high school.

7. Twelfth grade. The courses in solid geometry and trigonom-
etry which should be offered in the twelfth grade are definitely a
part of specialized knowledge. Because as a rule, only the abler
students elect these subjects and because the courses have been
planned for that type of student, very little revision is needed, ex-
cept that it is always desirable to include as many up-to-date prac-
tical problems and illustrations as possible.

To complete the part of the secondary school in the program of
general education, a review of arithmetic should be offered in the
twelfth grade and required of those students who have not elected
the traditional courses. This policy has been followed in one Flor-
ida high schoolifor a period of about ten years with excellent re-
sults. The recently adopted text Mathematics in Life is suited ad-
mirably to this purpose.


Mathematics for the small high school. An offering differen-
tiated according to needs and abilities of students in a school so
small that only one section of each class is possible, requires no small
effort on the part of each teacher concerned. Unless teachers and
administrators join in a definite effort to overcome the difficulties
involved, the mathematics offered by a school of this size may be in-
adequate to meet the needs of society. In the junior high school,
since mathematics is required in seventh, eighth, and ninth grades,
the problem is one of choosing materials suited to the average stu-
dent, containing elements of value to all, and capable of adaptation
by projects and differentiated assignments, to meet the needs and in-
terests of the groups of greater and of less ability. This variation
of offering is essential if our most gifted are not to be the most
retarded in relation to their capacities.
Since the ninth grade course as described serves the triple func,
tion of contributing to general education, guiding the student in his
decision concerning later work, and initiating specialized training
for those who continue the study of mathematics, the choice of ma-
terials is here a particularly vital problem. In the small high school
where only one section is possible, the interests and development of
the students will determine which of the functions mentioned above
should predominate. Although a teacher will hesitate to say that
one student should pursue the study of senior high school mathe-
matics and another should go no further, the general interests of the
majority will early be apparent; this interest will determine the
emphasis to be given the course. Students who do not fit into the
program determined by the needs of the majority must be cared for
by methods of individual instruction; unless assignments, in such
cases, are differentiated in quality rather than in quantity, the pur-
pose has been, in the main, defeated.
In the senior high school, all mathematics is optional, unless the
course in basic mathematics be required of some or all of the
twelfth grade students, but if the work of the junior high school
has stimulated the children as it should, some (perhaps a third or a
fourth of them) will want to continue their experiences in mathe-
matics. It may take some ingenuity,to arrange a schedule for a
small school which will enable these children to satisfy their needs,


but effort in this' direction will yield valuable returns to society as
'well as to the individuals, for they are the specialists of tomorrow.
Since these students possess real ability, they will be able to fit into
almost any sort of sequence or combination that can be arranged
for them.
In the small senior high school the scheduling of classes suf-
ficiently varied in character to capitalize on the wide interests in-
herent in a group of children may be very difficult. One solution
frequently applied is the alternate offering of such non-sequential
subjects as algebra and geometry. This, however, is not a solution
for every situation. The State Department of Education has author-
ized the teaching of two courses in combination; this plan should
be used with discretion. Its success will be dependent upon a number
of items, the most important of which are the skill, eagerness, and
available time of the teacher and the capacity of the students to do
good work independently. The content of the two courses should
not be too dissimilar, and neither should require constant class dis-
cussion.5 The State Department has limited the number of classes
to be combined to two and the number of students in the combined
classes to fifteen. If a student insists on having a course which can
not, in fairness to others, be included in the schedule, he may be
advised to register with the Extension Division of the University;
the teacher can supervise the student's work, help him, and encour-
age him to complete the undertaking with a minimum addition to the
teaching load. School authorities may well make it a policy that
whenever a student of superior ability requests a course basic to the
achievement of definitely formulated plans, some means must be found
to grant that request.
Mathematics in the larger school. In the high school large
enough to have two or more sections of each of the required courses,
grouping according to some criterion will occur. In the seventh and
eighth grades, where mathematics is a part of the program of general
education, homogeneous grouping may or may not be advisable since
groupings which aid one subject may not provide the same service

"Beginning plane geometry and solid geometry seem not ,to be adapted
Sto combining with other courses, while algebra and trigonometry would work
much better.


ifor another. If practiced, the criterion of* assignment to sections
-may be the intelligence quotient, 'mental age, marks in previous
courses, judgment of teachers, or some combination of these. It is
certain that if assignment to classes is riot made the subject of ad-
ministrative action, some factors beyond mere chance will influence
the grouping; these may be cliques, exigencies of the schedule, or
'division effected by such selective subjects as hone economics, shop,
or Latin. The faculty of a school should make a conscious choice of
the manner in which grouping should take place.
Since specialization begins during the ninth year, the grouping
of students should here be made on the basis of their interests. For
example, if there are to be two sections, the nucleus of the first may
properly be several students who have already decided that their
interests and talents will make further work in mathematics profit-
able; the second section will just as properly be built around a num-
ber of students who are equally certain that they do not wish to
specialize in mathematics. If any member of the faculty feels that
a student in either section has made a choice inconsistent with his
best interests, a conference should be arranged and the situation
thoroughly canvassed from every angle. All possible relevant infor-
mation should be laid before the student; the teacher should not
exert undue pressure to force the decision.
At this point only those students with decided preferences have
'been assigned to sections. By the very nature of this method of
assignment, the first class will probably contain only students of
considerable mathematical talent; the second section will almost
certainly contain a wider range, although students of lower ability
will probably predominate. The remaining students presumably
have little idea as to their probable choice of vocation, and no con-
viction as to the advisability of continuing their training in mathe-
matics. Each student of this group presents an individual problem
in guidance and should be assigned to one of the two sections on the
basis of the best judgment it is possible to make in the light of his
ability, industry, application, and vocational leanings. An ideal
situation might allow transfer from one class to the other for some
specified length of time. During this period, the variation of em-
phasis in the two classes might be pointed out to the students.


This method of grouping will lead inevitably to administrative
difficulties which must be solved on the basis of conditions obtain-
ing in that particular school. Some errors in assignment will, of
course, be made under this system, but they will be less frequent
and less serious than when there is no administrative recognition of
the problem. The next step must be taken by the teacher; he must
choose materials suited to the interests which were the basis for
Cognizance must be taken of the fact that ninth grade students
are not sufficiently mature to make decisions which are irrevocable.
Therefore,. any system must be sufficiently flexible to allow a student
to change his mind. The teacher may regret a given student's
decision to pursue the study of mathematics no further, but such
a decision, if unchanged by friendly discussion of the situation, must
be respected. A more difficult situation arises when a student from
the second section decides to elect further courses; if he is industri-
ous and of ordinary ability,. no obstacles should be put in his way.
If on the other hand, the teacher feels there is little chance of suc-
cess, he should make an effort to dissuade the student; should this
be impossible, the pupil must be allowed to make the trial. Fre-
quently, in an effort to prove the teacher mistaken in his estimate,
such a pupil will achieve outstanding success.
If only two sections of junior high school classes are needed, the
senior high school will probably face the same problems as were
discussed in the previous section. The difficulties in offering a
diversified program may be met in similar ways. The same point
of view is advocated for larger schools. In these larger schools a
moderate degree of homogeneity of interests may be achieved; this
condition should make for efficiency of instruction. As the enroll-
ment increases, the curriculum can be broadened and a wide variety
of subjects offered with greater ease. However, as the number of
students grows larger, the range of interests widens also; therefore,
even in a relatively large school, subjects may be in demand which
strain the available facilities. The suggestions made for the solu-
tion of similar problems in the small school will still obtain.



The program of the school can be better adjusted to the needs
of individual students by (1) careful planning of the offerings of
the school in an attempt to satisfy the demands of general education
for all and for specialized education for the individual; (2) proper
guidance of students, (3) the selection and use of proper materials
within the classroom, (4) the use of effective methods of instruc-
tion, and (5) professional growth on the part of the teacher.
1. Types of materials. The use of subject matter closely re-
lated to the common experiences and knowledge of the students will
stimulate interest and facilitate transfer of training in mathematical
skills and concepts to situations of later use. Care must be taken
that the material is on the proper maturity level for the pupils.
Extensive classroom use of concrete materials will aid in under-
standing, especially in the lower grades.
2. Suggested bibliography for students' library. The following
books have been selected from various lists, from inspection of books
in libraries, and from annotations of books in mathematical journals.
The purpose is not to give a complete list, but to select a few of the
more economical and educational ones that have been shown to be
liked by students. A short annotated list of books covering a va-
riety of topics was thought to be of greater use to mathematics
teachers than the mere naming of a large number. A more com-
prehensive list can be found in the Fifteenth Yearbook and in vari-
ous issues of The Mathematics Teacher.
Abbott, Edwin A., Flatland, Little, Brown and Co., Boston, 1926. 155
-pp. $1.25. Contains an attractive- and thoughtful treatment :of dimen-
sions. Recommended for eighth and ninth grade students. A story of
lands of one, two, and three dimensions.
SAmerican Council on Education.
No. 1, Achievements of Civilization.
S.No. 2, The Story of Numbers, 32 pp.,. $0.10... .
SNo. 3, The Story ot Weights and Measures, $0.10.
No. 4, The Story of Our Calendar, 32 pp., $0.10.
SNo. 5, Pelling Time Throughout the Centuries, 64, pp., $0.20P'' '


American Council on Education, 744.Jackson Place, N. W., Washington,
D. C., 1933.
Ball, W. W. R., Mathematical Recreations and Essays. (7th edition).
Macmillan Co., New York, 1920. 506 pp., $3.50. Valuable.
Breslich, E. R., Excursion in Mathematics. Newson and Co., 1938. 47 pp.,
$1.00. This book in a very interesting way shows applications of basic
facts of geometry in present day life situations. There are twenty large
plates representing three-dimensional objects in two colors to, be viewed
by an accompanying orthoscope.
Dantzig, Tobias, Number, The Language of Science (3rd edition). Mac-
millan Co., New York, \1939. 320 pp., $3.00. Excellent development of
number concepts for the advanced students, and teacher.
Hornung, C. P., Handbook of Designs and Devices, Harper & Brothers,
New York, 1932. 204 pp., $2.50.
Jones, S. I., Mathematical Nuts, The Author, Life and Casualty Bldg.,
Nashville, Tennessee, 1932. 340 pp., $3.50. Contains many interesting
mathematical, problems.
Kasner, Edw., and Newman, James, Mathematics and the Imagination.
Simon and Schuster, New York, 1940. $2.25. Describes the role of the
imagination in the development of mathematics concepts in language
about as simple as possible for the expression of the ideas. For the
teacher and the student of unusual ability.
Sanford, Vera, A Short History of Mathematics, Houghton, Mifflin Co.,
Boston, 1930. Valuable in that the language is simple enough for use
by students of secondary school age.
Shuster, C., and Bedford, F., Field Work in Mathematics, American Book
Co., New York, 1935. 168 pp., $1.20. A worthwhile book to have, describ-
ing the construction and use of measuring instruments.
Smith, D. E., and Ginsburg, J., Numbers and Numerals, Bureau of Pub-
lications, Teachers College, Columbia University, New York, 1937. 56
pp., $0.25. Excellent.
Surveying, Mechanical Drawing, Merit Badge Series, Boy Scouts of
America, 2 Park Ave., New York, N. Y., 1939. 20 each. These two
pamphlets are of a series published in connection with Boy Scout Merit
Badge Scheme. They are written by leading authorities, and contain
useful information for junior or senior high school students.
Periodicals, such as Popular Mechanics, 15, Popular Science, 254, and
Scientific American, 254, have material of mathematical and scientific

3. Materials available in the community. The prerequisites for
making a list of materials available in a community would be ac-
quaintance with the community in question, its people, businesses,
and natural resources, and a wide knowledge of the applications of
mathematical principles. Some one connected with a bank might


explain its services; students might go through the processes of
being identified, writing and endorsing checks, depositing money.
Some schools have found it effective to set up a school bank. The
city budget, the tax systems of city, county, state, and nation, and
'the public services provided by these taxes may be studied and
taxes computed on family property. A local insurance agent would
probably be glad to discuss the protection offered by fire and life
insurance; happenings in the community that show the benefits of
insurance can be cited. Scouting can be related to work in mathe-
matics classes; scout pacing, laying out a tract of land, map read-
ing and map making, judging distance, size, number, height, and
weight are scout tests. This list might be continued to great
length, but at best would be only suggestive and would, of neces-
sity, vary with the community. In using these and other activities
to stimulate interest, care must be taken that the descriptive mate-
rial does not crowd out the mathematical work that it was designed
to supplement.

Description of materials that can be made. Making materials
for classwork is valuable to the extent that the things made (1) can
be used by the students in their work or by the teacher in develop-
ing an understanding or skill, (2) have furnished learning experi-
ences for the students making them, (3) are a source of interest
and motivation, (4) were made without excessive cost, and (5) serve
as inspiration to other teachers. All of the instruments described
in this section except three have been made in classes in Florida
schools; some of the instruments have been described in a bulletin,
Accounts of Learning Experiences from Mathematics Classes, col-
lected by the Florida State Council of the Teachers of Mathematics.
Most of the things described can be made of wood. In a school
where there is a shop it seems advisable for the mathematics teacher
at the beginning of the year to plan the materials to be made and
work out with the shop instructor plans for some students to work
during the school hours or after school.
1. In one Florida school, an end of a slate blackboard in a
classroom was turned into a graph board without impairing
its original use. By means of a straight edge and a pointed
instrument, a square 30 inches on a side was made on the

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board and then marked with horizontal and vertical lines 1"
apart. Each fifth line was made a little wider than the
other four. When these fine scratches were filled with chalk
dust the graph board was ready for use.

2. Circular objects (wooden disks) of various radii may be
added to mathematics collections to be used for calculating
the value of pi.

3. A wooden rectangular solid made up of 1" layers from which
a cubic inch might be extracted is useful in teaching volume.
The arrangement of these layers and the cube are shown
in Figure I. The layers as well as the cubic inch are fastened
together with wooden pegs.
4. An abacus, shown in Figure II, may be made by stringing
wooden beads on parallel wires which are fastened to nails
in a board. Such an instrument may be used when dis-
cussing our number system and ways of counting.
5. A pantograph, shown in Figure III, was made out of strips
of wood 1/2" by 3/16". Strips AC and CD are 18" in length,
while BE and BF are 14" in length. A hole is made at end
D for a small pencil to be inserted. A small needle-like point
projects downward at B, and end A is anchored to a drawing
board by 2 screws through a small block there. The sticks
pivot at points C, B, G, H and A. Points G and H may be
changed by means of two screws, making it possible to vary
the size to which a figure might be enlarged or decreased in
size. With A anchored, D makes an enlarged tracing when
B is moved around the figure.
6. Blackboard spheres for solid geometry have been made by
applying black paint to a 7" globe which might be obtained
from a local store for about 25c.
7. Wooden blocks may be made and assembled that will illus-
trate (a + b)3 = a3 + 3a2b + 3ab2 + b3, or the arithmetic
rule for finding cube root of a number. Figure IV shows
the blocks that were fastened together by pegs to form the
cube'in the picture whose edge is (a + b).




4 ^f
Ji /



8. The different types of triangles may be made out of strips
of wood about 1/" x 3/16". The sides of the triangles may
usually be from 8" to 24" in length. These triangles may
be used in the seventh and eighth grades and also in con-
nection with the theorems about congruent triangles in plane
9. Figure V may be used in connection with theorems relating
to parallel lines being cut by a transversal. CD and EF are
each 12" in length, while GH is 10" long. CE and DF are
strings that may be adjusted. Protractors are mounted with
centers at points A and B. The lines M and M' marked on
GH enable one to read the protractors.
10. Figure VI shows an adjustable quadrilateral with sides made
of strips of wood 1/2" x 3/16" allowed to slip through metal
bands that are fastened to one end of each strip. The
diagonals may be made by sliding a round stick down a
bamboo. All the different types of quadrilaterals may be
formed by adjusting the sides and diagonals.

11. Figure VII may be made of wood or of cardboard blocks
mounted on a 8" x 8" x 1/4" base to be used in proposition 6,
book 7 of solid geometry. Black thread connects the ex-
tended edges of the parallelepipeds to show how one was
projected into the other.
12. The three pyramids in Figure VIII were connected with
wooden pegs as shown. This model is very helpful in visual-
izing proposition 12, book 7, solid geometry.

13. In Figure IX the peg PO was driven into the board MN.
Lines BO, AO, and CO were drawn through 0 on MN. A
rectangular board is fastened so as to rotate around PO.
Line OD is drawn as shown on this board PC. This figure
was used with proposition 2, book 6, solid geometry.

14. Models of many geometric solids may be made from metal,
cardboard, soap, celluloid, clay, and toothpicks. Layouts for
these models may be found in many texts. Figure X shows
a cube made of toothpicks or small sticks. The use of such


A, 7 H-K -





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sticks offers great possibilities for many three-dimensional
figures as well as complicated buildings, etc. A metal cone
and cylinder of equal radii and equal heights, as well as a
metal square pyramid and rectangular solid with equal bases
and equal heights will be useful in eighth grade mathematics
for teaching the facts about volume.
15. Figures XI and XII show a plane table and alidade, two
of the most useful pieces of equipment for mathematics
classes. The dimensions given in the figure are only sug-
gestive. A table about 16" x 20" may be made of some soft
wood such as white pine. Such a board may be used as a
drawing board with a T-square. A tripod may be made
upon which this table can be fastened. Two levels may be
mounted at right angles near some corner D. An alidade
is shown made from two 2" x 1/2" x 1/" strips mounted some
101/2" apart perpendicular to a strip 12" x 3/4" x 1/4". These
two 2" strips have a slot sawed half their lengths. It is
necessary for the line of sight passing through the slots to
be parallel to the edge EF of the base of the alidade.
16. Figures XIII to XV show models of the conic sections.
Figure XIII shows a wooden cone that has been cut so as
to get the four conic sections. The circle is cut perpendicular
to the axis of the cone. The ellipse cuts all elements of the
cone and is not perpendicular to the axis. The parabola is cut
so that the plane of the parabola is parallel to an element
of the cone. The hyperbola, only one branch of which may
be shown on a single cone, is made by passing a plane not
parallel to any element nor cutting all elements in the cone.
In Figure XIV hooks are fastened 1" apart on a stick, and
strings with small weights attached are suspended from the
hooks. The lengths of these strings may be determined from
any large constructed parabola or from the drawing. The
suspended parabola can be turned around to show many
parabolas different from the ones shown in Figures XIV and
XV. A description of many applications of these conics may
be found in "The Mathematics Collection" by Georg Wolff in
The Eighth Yearbook of the National Council'of the Teachers
of Mathematics, pp. 219-227.







,rit Cid view of
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17. Figure XVI shows an astrolabe made of plywood. A is an
erected point which is viewed through a small hole in B (B
being a small opaque rectangle about 1" x 1/"). The line
of sight through A and B is parallel to the 90' line of the
board and perpendicular to the 0" line. By holding this
instrument in a vertical plane, and sighting through B and
A at an object, the plumb-bob hanging from point C will
mark the angle of elevation or depression of an object. This
instrument is easily made and can be used in either junior
or senior high school mathematics classes.

18. Figure XVII shows the view of a sextant that was made by
two students in a trigonometry class. E is a circular board
of plywood with a diameter of 12". A is a peep sight with
a small hole 3/4" above the board. B is a small piece of ply-
wood with a rectangular hole above a fragment of mirror;
as shown in Figure XVIII the line between the mirror and
the hole is 1" above the platform. Line AB, BC, and CX
are lines of sight. C is a mirror mounted on rotating arm
D, pivoted at the center of C and pointing to scale F. This
scale is marked off with center at C and with each degree
marked as two degrees. When the pointer stands on zero it
can be seen that the mirrors B and C are parallel and that
the line CX is parallel to AB. At this setting of the pointer
an object seen through A will appear in the right or inward
end of B as it is reflected from C and it will also appear at
the same time through the left or unsilvered end of B. By
holding the instrument in a vertical plane and sliding the
pointer the top of a tree can be made to coincide with the
bottom, thus giving the angle at your eye between the top
and the bottom of the tree. It can be used to measure hori-
zontal angles, also. When used to measure horizontal angles,
it is sometimes called an "angle mirror."
19. An ellipsograph, shown in Figure XIX, is made of 4 grooved
arms 31/2" in length and at right angles to two other arms.
A and B are small wooden blocks fastened to crossarm C
and allowed to slide in grooves DE and FG respectively. A
section of the groove is shown on the right of G. Chalk is



--I~- ?--I

''gIr,`" ~~1- ~""'"* ~ cc~~r~

t -, --4..

.-, .r? .g
-.-- I


placed in any of the holes at C. Rubber-headed tacks placed
in the bottom of the arms enable the ellipsograph to be held
securely against the blackboard with one hand while the other
hand moves the chalk around.

20. The end, side, and top views of a student-made transit are
shown in Figures XX, XXI, and XXII respectively. This
instrument was built at a cost of less than fifty cents, and
has been used in classwork by seniors, as well as by the
students in the seventh grade. Materials:

One %" x 1%" screw
3 celluloid protractors with radius of 2"
1 brick layer's line level
%" plywood and %" oak
One 1%" x %" machine bolt with wing nut
A equals line level
B & B' equal gun type peep sights
C equals horizontal angle scale
D equals vertical angle scale
E equals %" plywood rotating base
F equals %" oak stationary base
G equals 1%" oak upright
H equals bolt and wing nut
I equals pivoting screw
J equals %" plywood vertical scale holder
K & K' equal pointers for vertical and horizontal scales

Two protractors are placed together to form horizontal scale
(C) and glued on to the stationary base (F). The third
protractor is glued on (J) to form the vertical scale (D).
The level is placed in the center of the table which is nailed
on to (J). This table also holds the sights which are the
same type as the peep sights used on a gun. The hole which
is drilled in (J) to take bolt (H) is in the center of pro-
tractor (D). A wing nut is used on this bolt. The oak
upright (G) is notched out to take vertical scale holder (J).
The oak upright (G) is then nailed to the rotating base (E).
The stationary base (F) is drilled in the center to allow
screw (I) to move freely but not too loosely. Rotating base
(E) is then placed on the horizontal scale (C) and screw (I)

Cr.p vi.)




F :-


(sia. v.\




p o



is placed in base (F) and driven through the center of rotat-
ing base (E) and into the oak upright (G).
21. A demonstration slide rule was constructed of %" oak, as
shown in Figure XXIII. M is %" x 6" x 44". N, 0, and
P are three pieces arranged so that 0 slides between P and N,
which are fastened to M by glue and screws. A hole for the
finger is drilled in each end of O. The A and B scales are
each one meter in length, the numbers from 1 to 10 being
located by finding the logarithm of these numbers and stating
this logarithm as a part of a meter. Since the log of 2 is
.30301, the number 2 is placed .303 meters or 30.3 centimeters
from the initial line, etc. The C, D, and L scales were
similarly located on the slide rule. The rule was first given
four coats of varnish, the scales were applied with India ink,
and then the whole thing was shellacked.
22. Other materials that might be made are linkages, including
angle trisectors; tripod for transit and plane table; copy
of mariner's compass; blackboard instruments; bulletin
boards; display cases; three-dimensional surfaces; models of
ellipsoids and paraboloids; sphere showing some of the parts
such as spherical triangle, hemisphere, spherical segment,
spherical pyramids; models for illustrating the expansion of
(a + b)2 = a2 + 2ab + b2 and the arithmetic rule for find-
ing square root; model of probability curve made by letting a
bag of small shot fall into a pile through nails driven in a
board; models of other solid and plane geometry proposi-
tions; parallel rulers; models to illustrate the Pythagorean
Theorem; three-dimentional blocks for students to draw;
hypsometer; heliometer; sundial; and telescope.


In previous sections, it has been stated that, in the final analysis,
the classroom teacher bears the responsibility for improvement of
SMuch helpful material will be found in the Fifth, Seventh, Eighth, and
Tenth Yearbooks of the National Council of Teachers of Mathematics and
in J. H. Minnick, Teaching Mathematics in the Secondary School, (New
York: Prentice-Hall, Inc.), 1939.


instruction; others may initiate programs, criticize procedures, or
write bulletins, but unless the classroom teacher reads, thinks, and
modifies his own practices, there is little improvement. From that
point of view, these remarks on method are a vital part of this
bulletin. An exhaustive treatment of this subject, however, is be-
yond its scope. The intention is to set up a number of guiding prin-
ciples and to discuss certain materials and procedures in the light
of these principles. Among these principles are:
1. Every activity should be so motivated that the pupils and
the teacher are in harmony as to the goal. Not all purposeful
activity need be student initiated; a skillful teacher will frequently
lead students to accept his purposes and identify them as their own.
Motivation does not aim to take the work out of learning; on the
other hand, its function is to create in students a willingness to
make whatever effort may be necessary to accomplish the accepted
2. Provision for individual differences is an ever present prob-
lem and an obligation of the teacher. In every class, there are
students unable to keep the pace set by the majority and others
who could travel much faster. In order that both types may profit
from experiences in the course, materials must be devised and
assignments differentiated to suit their capacities and interests;
class procedures must be modified to care for their needs. It is
not enough to give differing amounts of work; there must be varia-
tion in quality also. Extra work for the brighter students may well
be done almost entirely independently, by reading and writing.
This is important for this type of pupil, since nearly all later learn-
ing must be done in this way.
3. Certain subjects are better adapted than are others to group
work, but some phases of every course can be found to permit the
class to work together as a whole or in committees. Organization of
this sort is an aid in the solution of the problem of individual dif-
ferences, for in the varied activities of group work, there can be
found some items suited to the capacities of the slowest student, and
others which will challenge the ablest. Opportunities for cooperative
effort should abound in a school which strives to advance the demo-
cratic ideal.


4. The amount of subject matter should be nicely adjusted to the
particular group in question. Understanding and mastery of a
moderate amount of material with ample time for practical illustra-
tions and applications is of far more value to students than mere
exposure to more material than can be assimilated. It is, of course,
as possible to present too little material as too much, but it is much
less likely. Both faults should be avoided.

5. Since, according to the preceding principle, the amount of
material in a course is subject to variation, it is literally the teacher's
first duty to find out what knowledge and skills the students have
when the class is organized. This may be done by tests, standard-
ized or teacher-made, by conference with the previous teacher when
possible, and by continuous observation of the students. Instruction
then starts at whatever level seems indicated by the information.
Remedial work will undoubtedly be necessary for some whose
achievement level is below that of the group as a whole.
6. Whenever the teacher has jurisdiction over the placement of
items in the curriculum, great effort should be made to determine
at what level of maturity each item properly belongs before place-
ment is made. Many studies of this subject have been made and
others are in progress. Results of these studies will give consider-
able help.7

7. Democratic procedure requires that all students be encour-
aged to express any ideas which they may have upon the subject
under consideration, to accept constructive criticism offered by the
teacher -or by fellow students, and to defend those ideas by reason-
able argument. In this connection, it is imperative to refrain from
even the slightest touch of sarcasm; voluntary discussion does not
thrive in an atmosphere of fear. Pupil contributions will be more
frequent if the teacher is not too prompt in supplying the answer
to a question or in volunteering his opinion on a controversial mat-
ter. Frequently, an ordinarily silent child can be induced to talk

Reports on allocation of items of arithmetic will be found in The Twenty-
Ninth Yearbook of the National 'Society for the Study of Education, 1930, and
in The Tenth Yearbook of the National Council of Teachers of Mathematics,
Bureau of Publications, Teachers College, Columbia University, New York,
1936.- .: :


by referring to his hobby and deferring to his opinions; many times,
such a student knows more about the subject in question than the
teacher or any other member of the class. Respect accorded him
by the teacher will cause him to rise in the estimation of his fellows
and often minister to a real need for recognition, especially if the
child is shy, or not apt in his class work.

8. The teacher should recognize his own limitations and not
pretend to knowledge he does not possess. Such pretensions are an
incentive to pupils to catch the teacher in a mistake. Accuracy of
statement and in work at the blackboard is to be sought at all times,
but when a mistake has been made, a simple admission of error is
the only thing consistent with honesty and dignity. It will command
far more respect from the class than any other course and may even
be the cause of a feeling of fellowship on the part of the students.
No pretense of omniscience is necessary; if a question occurs which
the teacher can not answer at the moment, the admission of igno-
rance and the suggestion, "Let's see what we can find out about it,"
may be the key to valuable activity and learning on the part of both
teacher and students. For the sake of the teacher's self-respect
such admissions should not often be necessary in his own field, but
may be fairly frequent in other fields if he is confronted by students
of high ability.

9. Along with the development of knowledge and skills, a stu-
dent should become increasingly conscious of the continuous growth
of mathematics. He should come to realize that ideas of number
may be nearly as old as speech itself, that arithmetic and geometry
are older than history, although the Hindu-Arabic notation and
algebra are comparatively modern inventions. Too frequently,
students have no idea of the growth of the subject; they assume it
has always been just as it is now and can never be changed.

10. The various courses in mathematics are not to be regarded
as masses of unrelated facts and skills to be learned mechanically.
If the meaning of each concept and each process is developed with
care, there should be fewer students who can not work a problem
until told what process to use. "The pupil who performs a series
of exercises because of a conviction in regard to their meaning


that is the result of his own interpretation of the definitional mean-
ings of the expressions is operating on a very different plane from
the pupil who merely follows a prescribed plan."8
11. Alert teachers of mathematics can constantly find quantita-
tive.aspects in the life about them and call these to the attention of
their students. When skillfully done, this is an aid in the transfer
of attitudes and habits of thought to the realm of life out of school
as well as a means of adding interest to the mathematics class.
12. If the teacher will point out the similarity between the
operations of arithmetic and those of algebra, those in algebra will
lose much of their difficulty while those in arithmetic will take on
a richer meaning. Continued stress upon relationships within the
subject will lead to the integration of isolated topics. In algebra and
arithmetic, this will lead to combining rules and to eliminating much
memorization of techniques. In geometry, theorems can be grouped
to accomplish the same purpose. For example, the fact that all of
the theorems concerning the measurement of angles by arcs can
be put into a single statement, providing we assume the are nearer
the vertex to be negative whenever the vertex lies outside the circle,
substitutes a single principle for five separate theorems; this dis-
tinctly makes for economy of effort. In addition, the concept of a
negative arc provides an unsuspected link with algebra. The stress
upon relationships is essential for development of the problem-
solving attitude which is characterized by a search for relevant data
and selection of the appropriate operation.
13. If the aim of critical thinking is to be attained in any meas-
ure as a result of training in mathematics, teaching must be directed
toward it. Insisting upon definition of terms and the statement of
assumptions in situations, mathematical and otherwise, is one step
toward eliminating hasty judgments and the useless arguments
caused by a misunderstanding of terms. If the practices of esti-
mating the answer before working the problem and checking the
answer afterward can be made habitual, a reasonable attitude to-
ward the answer may be established. By the use of examples from
his own reading, the teacher may assist in the carry-over of this

John P. Everett, "Algebra and Mental Perspective," Seventh Yearbook.


attitude to the student's reading of numerical materials, so that he
will take the trouble to check roughly the accuracy of figures which
are given in enough detail to make this possible.
14. Children of secondary school age, whether in the seventh
grade or in the twelfth, find problems easier to understand when
they concern something tangible. In geometry, commercially made
models are a help, but those made by students, even though crude,
are infinitely more valuable. The pages of the book illustrate a
dihedral angle; the back of a tablet, a plane; pencils and rulers,
lines. With such materials and the willing hands of a student or
two to hold things in place, elaborate figures may be extemporized.
Two people and three yardsticks can illustrate the rigidity of the
triangle; the addition of a fourth yardstick shows the quadrilateral
not to be rigid, while the addition of a fifth yardstick as a brace
reaffirms the rigidity of the triangle. Problems and illustrations
from ordinary experience not only aid in transfer, but create in-
terest in the topic under discussion. In this connection, it is im-
portant not to define practical in too narrow a sense; a thing is
practical if it stimulates the child to thought, whether there is any
possibility of earning money from it or not.
15. Teaching for meaning, for ability in critical thinking,
or for power of generalization does not eliminate a need for drill,
but makes use of it in the light of the findings of psychology. After
a process is thoroughly understood, a child needs practice on that
process to fix it in his memory. The definite satisfaction which
comes from doing that which one can do makes such drill meaning-
ful. At short intervals thereafter, that process is included in mixed
drills for the maintaining of skills.


The teacher of mathematics shares the obligation of all teachers
to keep abreast of current thought by reading the magazines and
important books published in the field of education, and to support
efforts toward advancement by membership in state and national
associations of education. He has a similar obligation to keep in-
tellectually active in his own field by reading the magazines devoted


to secondary school mathematics and significant books on that and
allied subjects. A teacher who is vitally interested in his work will
give careful consideration to publications, such as this bulletin,
which attempt to indicate possible methods of attack on the prob-
lem of the improvement of instruction, to determine whatever merit
they may have. Finally, the teacher who earnestly desires to en-
rich his courses and improve his methods will arrange for periodic
attendance at summer school, or for occasional leaves of absence for
more extended advanced study which should include work in mathe-
matics and related fields as well as theoretical courses in education.
"Nothing but study beyond the bare elements of a subject can make
the elements seem simple and natural; nothing else can show them
in their full significance. The teacher's responsibility is not
confined to the details of the lesson of the day, for he should seek
to inspire and interest pupils in things superior to the contents of
the book and the day by day discussion of the class. To pupils he
should appear as a person with experience, knowledge, and capacity
that far exceed the requirements of the moment."9

1. The Mathematics Teacher, National Council of Teachers of Mathe-
matics, 525 W. 120 St., New York. A very helpful publication in the
teaching of secondary school mathematics devoted to methods, enrich-
ment material, and articles concerning items of interest in the field.
Subscription to it is included in the $2.00 membership fee of the
National Council of Teachers of Mathematics.
2. School Science and Mathematics, Central Association of Science and
Mathematics Teachers, Inc., 450 Ahnaip St., Menasha, Wis., $2.50.
Similar to the preceding publication, but dividing the space between
the two fields indicated in the name.
3. American Mathematical Monthly, W. B. Carver, McGraw Hall, Cornell
Univ., Ithaca, N. Y. Price: to members $4.00 (including membership),
to others, $5.00. The content of this magazine is distinctly of a mathe-
matical rather than a pedagogical nature. It should be of particular
interest to teachers planning to take advanced work.

Seventh and Eighth Grades
1. Boyce, G. A., and Beatty, W. W., Mathematics, in Everyday Life,

'Fifteenth Yearbook, op. cit. p. 195.


Inor Publishing Co., 1936. Consists of five units in separate books:
Health, Leisure, Geometry, Finance, and Drill. Excellent. $0.84 each.

Ninth Grade
1. Betz, Wm., Junior Mathematics for Today, Ginn & Co., Boston, 1935.
2. Crandall, Harris, and Seymour, F. E., General Mathematics-A One
Year Course, D. C. Heath, New York, 1937. $1.28.
3. Edgerton, E. I., and Carpenter, P. A., General Mathematics, Allyn &
Bacon, New York, 1937.
4. Nyberg, Joseph A., Survey of High School Mathematics, American
Book Co., New York, 1935. $1.08.
5. Schorling, Raleigh, Clark, J. R., and Smith, R. R., Modern School
Mathematics, Book III, World Book Co., Yonkers-on-Hudson, New
York, 1936. $1.28.
6. Stone, J. C., and Mallory, V. S., Mathematics for Every Day Use,
Benj. H. Sanborn Co., Chicago, 1937. 522 pp. $1.32.
Any of these books is suited to the ninth grade course described in
this chapter.

1. Reichgott, D., and Spiller, L. R., Today's Geometry, Prentice-Hall, Inc.,
New York, 1938. $1.85. Stresses practical applications.
2. Schnell, L. H., and Crawford, M., Clear Thinking-An Approach
Through Plane Geometry,: Harper &*Bro., 1938, New York.. Emphasis
is on reasoning.

Consumer Mathematics
1. Lennes, N. J., Practical Mathematics, Maemillan Co., New York, 1936.
$1.32. This is intended as a ninth grade book; it can be used for a
one-semester course in the twelfth grade, where the majority of stu-
dents have avoided mathematics as much as possible. There is a
good workbook to accompany this text.
2. Nelson, Jacobs, and Burrough, Everyday Problems in Mathematics,
Houghton-Mifflin Co., New York, 1940. Developed for use in the
twelfth grade with those students who do not have great ability in
3. Stone, J. C., and Mallory, V. S., New Higher Arithmetio, Benj. H. San-
born, Chicago, 1938. $1.28. Probably best adapted to a full year's
course in eleventh or twelfth grade.


The Yearbooks of the National Council of Teachers of Mathematics; the
First and Second Yearbooks are out of print. Others may be had from the


National Council, Bureau of Publications, Teachers' College, Columbia Uni-
1. First Yearbook A General Survey of Progress in the Last
(out of print) Twenty-Five Years.
2. Second Yearbook Curriculum Problems in Teaching Mathe-
(out of print) matics.
3. Third Yearbook Selected Topics in the Teaching of Mathe-
4. Fourth Yearbook Significant Changes and Trends in the
Teaching of Mathematics.
5. Fifth Yearbook The Teaching of Geometry.
6. Sixth Yearbook Mathematics in Modern Life.
7. Seventh Yearbook The Teaching of Algebra.
8. Eighth Yearbook Teaching of Mathematics.
9. Ninth Yearbook Functional Thinking.
10. Tenth Yearbook Tr'-whii,g of Arithmetic.
11. Eleventh Yearbook Mathematics in Modern Education.
12. Twelfth Yearbook Approximate Computation.
13. Thirteenth Yearbook The Nature of Proof.
14. Fourteenth Yearbook Training of Mathematics Teachers.
15. Fifteenth Yearbook The Place of Mathematics in Secondary
16. Sixteenth Yearbook Arithmetic in General Education.,
17. Seventeenth Yearbook A Source Book of Mathematical Applications.
18. Bell, E. T., Men of Mathematics, Simon & Schuster, 1937, New York,
19. Christofferson, H. C., Geometry Professionalized for Teachers, Geo.
Banta Publishing Co., Menasha, Wis., 1933.
20. Minnick, J. H., Teaching Mathematics in the Secondary Schools, Pren-
tice-Hall, New York, 1939. $3.00.
21. Smith, D. E., History of Mathematics, Ginn & Co., Boston, 1925. In
two volumes; complete and authoritative.
22. Woodring, Maxie Nave, and Sanford, Vera, Enriched Teaching of
Mathematics in the Junior and Senior High School-A Source Book of
Illustrative and -Supplementary Materials for Teachers of Mathematics,
Bureau of Publications, Teachers' College, Columbia University, New
York, 1938. An excellent source book of free and inexpensive mate-
rials available to mathematics teachers.

1. Kuehn, Martin H., Mathematics for Electricians, McGraw-Hill, New
York, 1930. $1.75.


2. Norris, E. B., and Smith, G. K., Shop Arithmetic, McGraw-Hill, New
York, 1931. $2.00.
3. Wolfe, J. HI., Mueller, A. B., and Mullikin, S. D., Practical Algebra
with Geometric Applications, McGraw-Hill, New York, 1940. Develop-
ed in the Apprentice School of the Ford Motor Co., this book would
convince a skeptical student of the value of algebra and geometry.

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