A STUDY

ISTUDIG‘TBHFICTOFAMGIBIEWCONWNALIZED
WIMOFWICWLCW
IITHEFIFTHAHDSIITHWES

3!
Patricia. mutt. Sprout

A DISSERMIOI
meted to thc sehool of Graduate Studiu
at Michigan Stat. 01:1de in partial
fulfill-nut for flu degree of

DmTOR (I DUCITICI

Schoolotlduution
1962

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ACKNOWEEDGHENTS

The writer wishes to express her appreciation to Dr. Troy L.
Stearns, chairman of her (hidance Connittee, for his interest, advice,
and encouragement in the planning and execution of this study. Special
gratitude is also due to the members of her guidance conittee:

Dr. Harry N. Sundwell, Dr. Carl H. (Ross, and Dr. V1111“ H. Roe who
gave generously of their time for consultation and assistance.

She also wishes to express her thanks to Dr. John R. Myer,
Educational Director of the American Association for the Advancement of
Science, and the nenbers of his staff, also to the Carnegie Corporation
for providing the financial support which made this study possible.
Appreciation is also due to Verl Frans for assistance in the treatment
of the data.

The writer is deeply indebted to the members of the administra-
tive staff of the Lansing Public Schools for their support and ‘
encouragement. To the students and teachers who participated in the
study, the author wishes to express her gratitude for their cooperation
and her pleasure of having had the opportunity to work with then.

mantle:

hinder

Patricia Homtt Spross
candidate for the degree of
DOCTOR a EUCATION

ma exallnatien: lhy 7, 1962, 1:00 9.5, 301: Education Building

Dissertatien: A M of the Effect of a Itangible and Conceptualised
Presentation of Arithetic on Achievement in the am
and Sixth Grades

Outline of studies:

Nor Subject: fle-entary School Curricula
Eiegrqhical Ite-s:

Bern, April 21:, 1913, lashville, mchigan

Undergraduate Studies: Michigan State Universi , East Lansing,
”31318“, 1935.363 19 9'52

Mate Studies: Michigan State University, East Lansing,
maxim, 19520563 1957-62e

lapel—lens: nemtary teacher, Lansing Public Schools, Lansing,

Michigan, 1952—53

Television teacher in ele-entary science and arithnetie,
lensing Public Schools, Lansing, Michigan. Station
um, Michigan State University, East Lansing,

anecial Teacher in elenentary nathenatics, lensing
Public Schools, Lansing, Michigan, 1959-61

Director of nenentary Science and hthenatics, Lansing
Public Schools, lensing, Michigan, 1961-62

tuber of Kappa Delta Pi, Phi Kappa Phi, lichigan Council of Teachers
of Hathesatics, National Council of teachers or hthenaties,
Central Association of Science and hthenaties hachers, National
Science Teachers Association.

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Patricia some Sprose

Al ABSTRACT
Sublitted to the Sehoel er Graduate Studies
of Illehigan State University in partial
fulfill-ant fer the degree of

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This study was concerned with the effect of a tangible and
conceptualised presentation of arithnetio to children in the fifth and
sixth grades as coqared to a routine elassroon presentation.

the regular arithnetio course of stuwofthe school systenwas
divided into coneepts or nethenatical ideas. To these were added enough
additional concepts taken fron pure nathenaties theory to explain the
arithnetic processes regularly taught in these grades. tangible nanipu-
lative item that had cultural significance were used in developing the
concepts whenever possible. The regular arithctic text was used as a
referenoebcokonly. Inadditienthe childrenwere suppliedwith other
reference naterials as they requested then.

My one concept was presented each week in a discussion period in
which the teacher acted as a resource person. Nor points of interest
relating to the topic were identified, and the none of developing these
were planned by the students. The students reacted to the concept by
producing sons tangible iten of their own choosing that they felt would
represent their understanding of the particular nathenatical idea for
theweck. The studentswere not givenanassignnentofprobleu towork
in any text.

Class periods were united to thirty-dive ninutcs per day. There
was no ability grouping, no individual help, no honework, or work at an
other tin. The lessons included no udrill".

mceaweekstudentsweregiventheepportunitytoworkasfaras
possible in a series of problens of grahal difficulty. These were
correctedhythe teacherandreturnedtothen. hiswasdoneinorder
toheeptheehildrenandthepatronsoftheschoolinfonedcftheclass'

progresa

todo as

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self-cor
stractux
the arit
tron abi
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grades I.
grades, :

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progress. No student was ever required to complete these problems, only

to do as many as possible in the class tine.
The tangible and conceptualized method was compared to the routine

self-contained classroom.presentation in which the arithmetic text
structured the program and was worked through as a method of completing
the arithmetic course. The self-contained rooms were not prohibited
from ability grouping, individual help, drill and rote learning methods,
hone work, direct correlation with other subjects, or any other methOd
the teacher chose. These classrooms were informed that they were acting
as control groups for the study.

There were 166 students in the study comprising 8 heterogeneous
classes in two elementary schools. There were two experimental fifth
grades and two fifth grade controls; likewise, two experimental sixth
gnades, and two sixth grade controls. The grades selected were deemed
to be typical of the school population.

The relative effectiveness of the conceptualised and tangible
presentation and the typical routine of the self-contained classroom was
determined by students performance on standardized achievement tests.
These were administered at the beginning, the mid-point and the end of
the school year. Comparisons were made on the students' performance on

achievement tests administered as part of the school routine. These were:
STEP Test - Form A

California.Achievement Test - Elementary Arithmetic Form BB -
Reasoning and Fundamentals

Tests given the students were treated by using the analysis of co-

variance nodel. F ratio showed that there existed a significant difference
between experimental and control groups when measured by the STEP and
California Achievement Test-Elementary Arithmetic-Form.BB, reasoningpr There

was no significant difference found in California Achievement, findsnentals.

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Conclusions: It would appear as a result of the study that it
would be possible for students to achieve essentially the as. progress
heatheticwithannnderstandingotlathelatiealecneeptsandtheir
useastheyeaninaroutinsclassrooaproeednreinwhichroteasthods
oflearningareemloyed. Itwouufurtharappearthatthiseanbe
«connshedisBSdnntesperdqinaheterogeneonsgroupinwhich
thenissoafllitvmupisunoindividnalhelmandnoadditionalwork
outddeoftheBSaisuteclassti-whenthsburdenofproofoflearning
rests autumn. Italsewonldsppearthatitisposdble for
fifthanddxthpadechildrentoanderstandsoftidemtsatheaatical
theoryuwnldflvemeaningtethearitlnetdeoomtentatthefifthand
sixthgradelevel.

Cu

HI- DESCR

TABLE OF CONTENTS

CHAPTER PAGE
I. INTRGJUCTION ............................................o
Need for the study ....................................
Statement of the problem ..............................

Basic considerations aseeeeeeeeeeeeeeoeeeOOOeeeeeeeeOOO

(D -4 Ch \n #4

Definition or terms seesaoeeeeeeeeeeoeeaeeooeeeoeeeesee

Structure Of thO thesis eoeeeeeeeoeoeeeeeeeeeoeeeeeeeee

#3 \0

II. REVIEW OF THE LITERATURE ..........................o......
Review of the literature pertaining to concepts in
arithmetic and methods of teaching them. ............ 11
Literature pertaining to theories of learning .o....... 16
Current literature regarding present practices of
teaching arithmetic ................................ 2b
III. DESCRIPTION OF THE TWO METHODS USED IN THE STUDY ......... 32
The experimental method ............................... 32
The routine classroom.aethod .......................... h6
Iv. ORGiNIZATION as THE INVESTIGATION ........................ 52
Design of Carnegie study .............................. 52
Design of this investigation .......................... 52
Other personnel in the study .......................... 56
Geographic and cultural areas involved ................ 57

Va D‘M‘NDREULTS eeeeOOOOeeoeoeeeeOOOeeeeeeeeeeeeeeeeeeas. 60

E?

RBIUItB Of tOBtB eoooeooeeseeeoeesoeeoeeeoeeeeaeeeoosee
Afl‘li'il 0: data eeeeeeeeeeeoeeeoeeeeeoeeeeeoeooesoseae 60

Conclusions eeeeeeeeoeaaeeesee-eeeeeeeeeee-ooaceases-so 6S

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to

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D e 9 e I I ' e I r l’ v a - I e I a Q o e a Q - I e O
a o o Q l I e O I la Q - O I Q s Q Q d O . e G n O C e
. ' ' O 0 I U - a a e e I 0 e - c e e e e a a a a I e

CHAPTER

VI. m, CWCLUSICNS AND NEGATIONS FOR FURTHER STUD! eee
M .0.C0...........................0...............
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U.“ 0: the Him ”"109“ eoeeooeeeoeeooeooeeeoeoe

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67
67
67

................................

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I.

II.

VII,

VIII.

II.

III,

Cc

AC;

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TABLE

I.

II.

III.

IV.

V.

VI.

VII.

VIII.

I1.

I.

III.

LIST OF TABLES

Concepts Taught in Grades 5 and 6
Weekly Routine Of memental C1888 eeeeeeeeeeoeeeeeeeeO
Design of Sample lesson

Procedures Used in Experimental Class

...................
Summary of Teaching Techniques Used in Experimental and
Routine Classes ......................................
Comparison of Mean Scores on Otis Test Given at Beginning
of Study .............................................
Composition of Experimental and Control Groups

Occupations of Parents of Students in Study as Indicated

on.Form CA 39 Filed in the Two Participating Schools ..

Academic Backgrounds of Parents of Students Participating
in Study‘ .............................................
Analysis of Covariance of sear Test - Form.A Achievement
Scores of Groups Taught by the Experimental and
Control Methods ......................................
Analysis of Covariance of California Arithmetic Test
(Reasoning) - Achievement Scores of Groups Taught By
Experimental and Control Methods ......................
The Analysis of Covariance of California Arithmetic Test
(Fundamentals) - Achievement Scores of Groups Taught

by EXPerimental and Control Method

33
3h
36
38

51

Sh
55

58

59

63

6h

65

TABLE

XIII.

XIV.

Sa

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TABLE PAGE

XIII. Sample LQSSOR eeeeoeeeeo.ooeeeeeeooaeeeeeeeeee.ooeeeeeeee 75

76
77-80

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IV. Scores Of Students eeeeeeoee.oooea.e.oe.eeeeeeeeeoeeoooae

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education:
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the field

cm I
MUCTICI

Improvementintheele-entary carrieulnnhaslongheenapartot
edncational research. Since 1900 each eqhasis has been givento a
consideration of nethods tor iaproving arithnetic programs.1 leaders in
the field of education and in national mathematical organisations have
given mch thought to: (1) factors contributing to a need for an
improved prograa; and, (2) the kind of prograa that will supply these
needs. Price lists the factors which contribute to the need for change
in mathematics ennicnlnm as: (1) Research in aathematica;

(1) Automation; and, (3) Automatic digital computing machines.2 Advances
in mathematics have created mono profound mathematics, greater quantity
or content, introduced new subject areas, and created additional and
more extensive uses of mathematics. The sequential nature of mathematics
makes it imperative that students and teachers possess a thorough know-
ledge of these continuing develop-ants.3

Cultural implications for mathematics have arisen in the
application of nthematics to the development of automation, which has
inturncreatedtheneedtortheproductioaofhighspeedoosputers. In
order to carry out the neehanisatien of antenation, it has becone

‘v

10. T. Essen, 'Arithmetic'. Review of Educational Bsearch
Chester v. Harris, Editor. lee tom W. .e 62.

26. Billy Price, 'Progress in Mathematics and Its Implications for
the Schools.“ The Revolution in School Mathematics. Washington:
latienal cmmx‘m‘rnmamh Pp. 3-5.

3me., p. 2.

 

fa

necessary to design nachines capable of computation at rates far beyond
human abilities. The design, maintenance, and operation of such machines
require the close association of mathematics with the fields of
engineering and ucchanics. Developncnt of such processes have had a
great influence on the way of lich‘
lot only has content in mathematics changed, but areas of emphasis
have changed. While it is still necessary to teach the traditional
subject latter areas, they now receive less emphasis. Specific applica-
tion of sathematics toward increased cnvironnental control mains a thorough
understanding of the properties of functions Just as necessary as a
nsmorisation of these processes. This is brought out by the
Rockefeller report.
. . .wearemovingwithheadlcngspeedintoanswphaseof
man's long struggle to control his environment, a phase beside
which the industrgal revolution mu appear a nodest alteration
of human affairs.
Bomeideacfthescopcandreccncyofthschangeuavbcobtained
by considering . statement from the rim»): Icarbook of the lational
Society for the Study of Education which was published Just nine years
ago.
Arithntic sfiib’its some marked contrasts when compared with
some of the other content areas of education. Unlike chemistry,

physics, and the social sciences, itscon content is not subject to
radical changes due to discoveries. . . .

“mm, Pp. hos.

Manna- Report. The Pursuit of kcellencc Panel Report v
of the Special Studies Project. Garden "_Eitya"‘"51'3"‘nou day. 1958. P. 28.

6G. T. Bushnell, 'Introduction.‘ Fif_______ticth I_c_ar___book 9_f the

National Bocic for the 8 of 1Education Part II. Chicago 2
Eversiw cago‘fiess.l P. I.

-aI;'.*'1“ “1*

_.....—. _—

Further support indicating the inportance of a changing
.nathenatical.pregran.to the cultural develop-ant is offered by Schaat.
The creative language of today is science, and nathenatics
is the alphabet or science. . . . Contemporary'nathenatics is
to be distinguished £ren.all.previous nathenatics in two vital
respects: (1) the intentional.study of abstractions, where the
important considerations are not the things related, but the
relations themselves; and, (2) the relentless experinentation.o£

the very foundations - the foundations of ideas upon ch the
elaborate superstructure of nethenatics is based. . . e

.L consideration of the arithnetic program.needed.in.view of these
changes implies the utilisatien.ot:mathenetics by the culture and, as a
consequence, a change ia.netheds of classreoexpresentation. Suelts in
emphasising the pressure to apply research results to classroo-
nethedology lists three ccnponents or an arithmetic programlthat will
tend to enphasise the social significance of mathenaticsa

1. The essence of nethematics.

2. Conputations and procedures.

world 83' Application to the social, physical, cultural, and economic
O

The inportenee of nunber systems (arithmetic) to the cultural is
also emphasised hy Sueltez

the fact (well accepted) that nedern.science and art could
netwwell functionneithout the hunter systenlin.its present
state of developnent tends to support the concept that the
inwenticn of it is, as it has been called, the greatest
invention or the hunan mind.9

7mm.- L. scheaf, 'Hathematics as . Cultural Heritage. - _‘l'h_e
Arithnetic Teacher 823-5, Jhnuary 1961.

8m A. Suelts, Ii Time for Decision.“ The Arithetic Teacher
8:280, mtObOr, M1.

9Ibid., P. 13.

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The cations planning that is needed to inplenent a nathenatics program
thatwill serve and support: (1) the abstraetnees of nathenatical
invention; (2) the rapid develop-cut of automation; and (3) the
resulting construction of rapid mechanistic conputation is brought out
by Craig:

In the already crowded curriculun any subject should be neat
carefully delineated. Although arithnetic can clain a share of
curriculnn tine as its cultural right, the delineation of that
share of the curriculun should be nost carefully considered.
Thequestionappearstobe, |'Howeanmathenaticsbetreateclin
the curriculnntobe of the nostuse?‘

Although nathenatics as such is a part of the cultural heri-

tage, inthe crowdedcgriculunoftoday, tinespentcnnathenatics

Itwould seathatweneedtonahachoiceastowhat coursethe
arithnetic should pursue: (1) Shall we continue in the way of abstrac-
tion only; (2) can abstraction have utility as a means of enhancing
understanding: (3) is it possible for children in the elementary school
to understand functions and processes as well as the nenorisation of
facts; and (h) can this be «co-pushed in a reasonable anount of class
tin?

The arithnetic progran of the future is pictured by Clark:

e . .arithneticprogramsinthenertdecadewill: (l) he
directed by teachers of high scholarship; (2) be better sup-
ported by the lay public; (3) give greater embasis to learning
by thinking! (h) lake better provision for the range of
abilities in heterogeneous groups of pupils; (5) incorporate
topics not generally found in school curricula; (6) include
experinente with, and eveluation of, newer tools of learning.n

 

”(braid 8 Craig Science for the menen_ta_iz School Teacher
e p __ __ _.__._..'
law fork: (fine and Comm, 1935. P. 33:

uJohn a. mark, 'Ieoking Ahead at Instruction in Arithmetic."
T_he_ Arithnetic Teacher, 8339’» Decenber, 1961.

‘..

It is with these implications in aind that this study originated.

1032 £31; _t_hg_ M' It has been pointed out that: (l) nathcnatics
content is rapidly changing and developing; (2) mathenatics content is
rapidly increasing! (3) there is an urgent necessity for the utilisation
of mathenatics in the control of . rapidly developing teclmological
environnent; and. (1;) these factors imly a change in the content pre-
sentedto students “thenthodaby'hich this is done.

studios involving ntheIatieal research are nun-rou- and attach
any facets of the total nethematieal problem citation.“ However it
would soon that so. atteqt fight be led. to inaugurate an arithnetis
progre- in a typical classroon situation that would incorporate the
inlicetions of such studies.

The sequential nature of nathematics makes a thorough understanding
of previous content necessary in order to master developing natheIatical
innovations.” It has been pointed out that the need to utilise
nethenetics in the control of the environ-ant inpliss understanding on
the part of the student rather than neeorisation. It would seen then
that the elenentary curriculun as the sequential antecedent to this
developing nthenatical progran, could not advantageously be continued
as a neaoriaing, or rote type, progran.

It was the purpose of this study to attempt to design an elenen-
taryconreecfstuwingradesfiveandsixthatwould: (l)bebasedon

unusual, 33. 591., p. 63.
13M“, as £3." Pa 2

a thomuéi
ratios]. 131
content tn“-
telented 5
provide th'
abilities;
life situat
ciples o! 1
d0 Of elas;
In or
t” (1)169
“amenities ;
teachgrva t01
investigate i

a thorough understanding of mathematical theory; as it relates to arith—
metical processes; (2) be consistent as a sequential antecedent to the new
content that is developing at the secondary levels; (3) provide acadendcally
talented students with an arithmetic program equal to their abilities; (1;)
provide the slow learners with arithmetic experiences computable with their
abilities; (5) provide experiences in the application of arithmtic to real-
life situations; (7). be taught in a realistic manner consistent with prin-
ciples of learning; and, (8) be accomplished in thirty-five minutes per
do of class time.

In order to implement a program of this type an attempt was made
to: (1) identify and analyze methods of improving the elementary
mathematics program; (2) investigate methods of reducing the classroom
teacher's total load as it relates to areas of specialization; (3)
investigate a method of teaching elementary mathematics that would
emphasize the understandings of concepts and their application to a
social situation with tangible manipulative materials; and. (h) investigate
the relationship of such teaching-learning procedures to the development
of skill in the manipulation of abstract mathematical symbols.

Statement _o_; 3133 problem. The purpose of this study was to com-
pare two methods of teaching arithmetic in the fifth and sixth grades.
The two methods were designated as those practiced in: (l) the self-
contained (or routine) classroom procedure; and, (2) a tangible and con-
ceptualised presentation of arithmetic as mathematical theory to be applied
in a social situation. In the self-contained classroom, arithmetic was
taught in the routine runner prescribed by the curriculum of the school sys-
tem in which the text structured the arithmetic program. In experimental

rooms the mathematics program was presented as a cultural tool. The two
methods were compared as to the performance of students on selected
standardised achievement tests. These were administered as a part of
the arithmetic program.

In addition the study was concerned with incorporating into the
content of the elementary arithmetic curriculum sufficient mathematical
theory to serve as an adequate explanation of the arithmetical processes
heretofore taught by rote - or memory.

It was the hypothesis of the experimental program that fifth and
sixth grade mathematics presented as a mathematical concept to be used
as a cultural tool will result in: (1) an improved understanding of
mathematical principles; (2) improved ability to manipulate abstract
mathematical symbols and to apply these in a social situation; and,

(3) that these abilities would be exhibited on standardised achievement

tests given as a part of the regular routine of the school.

§_a_s_i_g considerations. The basic considerations pertinent to the
study were as follows:

1. Is it possible to increase understandings of mathematical
concepts by the manipulation of tangible items?

2. Will an increased understanding of mathematical concepts
result in greater mathematical achievemnt by students as measured by
standardised achievement tests?

3. What is the extent which mathematics as a symbolic system my

evolve skill in the manipulation of symbols?
1;. what emphasis should be placed on the understanding of
mathematical concepts as the manipulation of abstract symbols?

Definition 2; Egg. Terms used are defined in appropriate
places in the body of the thesis. However, in order to make clear the
initial presentation, certain terms are defined here. The terms defined
are: arithmetic, mathematical concepts, routine classroom procedure,
and conceptualised presentation.

Arithmetic in this study was used interchangeably with mathe-
matics and elementary mathematics. It also refers to the program used
in the school to develop the part of the curriculum usually referred to
as 'arithmetic". The connotations given to arithmetic in the experi-
mental program are defined in Chapter III.

Mathematical concepts was used to refer to the ideas and general-
isations of mathematics which were taught in connection with the study
of arithmetic. This term was also used to refer to the branches of
arithmetic dealing with denominate numbers, and the areas of mathematics
having to do with measurement. is used in the stuw, the term would
include ideas, generalisations , principles and symbols referring to
the various areas of mathematics.

Routine classroom procedure was the term applied to the process
of arithmetic instruction carried on in the self-contained classroom.

In this study it was referred to as Itho way arithmetic was usually
taught.‘ This isplies a presentation involving: (1) regular assignments
to be completed in the text; (2) following the sequence of the text;

(3) requiring the students to conploto most of the practice and drill
work in the text; (1;) direct correlation with other classes; (5) |'extra"
help for slow learners; (6) additional time other than in class to

complete assignments; (7) drill on 'facts'; and, (8) homework when

 

{200558.17

procedure

represent
teaching 1
single 0pc
an u‘ithme

mS‘U‘eod of 1
not an atte

“WW was 1

of the lath

Spec;

mcessary. This is further developed in the chapter on the teaching
procedures used in the study.

Conceptualised presentation was the term arbitrarily chosen to
represent the presentation of a broad general idea to students as a
teaching process. This was in contrast to a presentation involving a
single operation. Conceptualised presentation included many aspects of

an arithmetic concept and its application in a social situation.

limitation! 2i: _thg m. The study was an evaluation of a
method of teaching arithmetic to fifth and sixth grade students and was
not an attempt to prove that one method was superior to another. The
study was limited to the content, and the understanding of the content,
of the mathematics contained in the course of study of the school system.

Specifically the study was not concerned with:

1. The evaluation of the existing routine classroom presentation.

2. The evaluation of textbooks.

3. The evaluation of the curriculum.

1.. The evaluation of grade placement of mathematical concepts
and/or materials.

The study was specifically concerned with ascertaining if it was
possible to teach the mathematics contained in the curriculum, as
appropriate to grades five and six, as an understanding of concepts

rather than the memorization of facts and processes.

Structure g_f_ _thg thesis. Chapter I contained a review of some of
the developments in mathematics and society that have given rise to the

need for the study. Basic assumptions and terms have been described.

Chapter II will be concerned with a review of the literature that is
pertinent to a conceptualised presentation of arithmetic, literature
pertaining to learning theory that is particularly applicable to the
study, and a review of current literature regarding present practices
efteschimgaritbtieiatheelememtarysehool. ChapterIIIwilleon-v
taimadstailofthe twomsthedsesedinthestndy. ChapterIVwill
oomtaim descriptions of: (1) the organisation of the investigation;
(2) the persons involved in the study; and, (”the cultural and
geographic areas participating in the study. Chapter V presents the
dataandresaltsofthestudy. ChapterVIcontainsasm-aryamd
impli catioms for further stew.

CHAPTEII
REVIWWMLITERATUFE

It was indicated in Chapter I that the literature in regard to
research in the methods and content of the arithmetic program is very
extensive. Itwouldhavebeenbeyondthescope ofthispaperto com-
pletely review all these productions. Consequently this chapter will
concern itself with a review of literature pertinent to: (1) concepts
in arithmetic curricula and methods of teaching them; (2) literature
pertaining to theories of learning that are simificant to this stucu;
and, (3) literature regarding current practices of teaching arithmetic.

might-nun EWEM’Qfimwu"
curricuhgnthodsgftefltg. Itshouldbeemphasisedthatthe
teaching of arithmetical concepts or ideas does not mean an incidental
type of learning situation. m. is brought out by Clark:

Changing conceptions and functions of education, of the
philosopw sf education, have been clearly reflected during
the past several years both in curricula and teaching
methods. . . . There is amiddle groundbetweenthe two
extremes of the now outmoded fonalism of an over-burdened

curri and the established freedom of wholly incidental
learning.

The relation of teaching mathematical concepts with cultural
application is further mhasised by Bredeoake and Groves:
. e . to sumly children with those experiences which lead

torealeoqarehensiomofmumber. . .ehildrenwhofailedto
achieve success in “sum lessons' showed considerable grasp of

 

1"John R. Clark, Arthur 3. Otis, and Caroline Batten, m
Arithmetic Through yerience. Ionkers-on-Hudson: World Book Compaw.

the subject when shopping. . . a scheme which brought to the
classroom '2? of the numerical activities found in daily
1110. e e e

Many, if not most, references dealing with arithmetic instruction
are chiefly concerned with mathematical content, with mch less emphasis
on method of teaching and learning. Banks specifically illustrates
teaching individual and isolated content areas. He does indicate however
thatthechoiceofmaterialintextahasmadearealeffortto'refleot
interests, activities, and experiences of children."1’6

The need of structuring the arithmetic curriculum so that it
contains mathematical concepts is emphasised by Bartung.

. . .morethanasetofthings itmightbeintereetingto

try, or a collection of enrichment materials, or drill materials
. . . It mist provide for a new and deeper understanding of
arithmetic. . . . A few simle ideas provide the foundation
stones upon which arithmetic is built. . . . The ultimate goal
of a modern arithmetic program is that the child be able to
solve problems involving quantitative ideas. To attain this
goal, it is necessary foI him to acquire certain essential
concepts and techniques. 7

The development of arithmetic programs has closely followed the
needs of society. It has progressed from a primitive need for an expres-
sion of quantity, such as a one-to-one relationship, to the coqlicated

processes demanded by current technology.18 Evidence has been presented

 

15:. B'edeoake and 1.13. Grover, Arithmetic in _i____ction. Iondon:
Ummuty or 10mm Pm..e 1956c PO 118.

 

 

16.1. Houston Banks, Is Lad Teac Arithmetic. Boston:
Allyn and Bacon, Incorpora . 9. P.

unsurice L. Hartung, at. 10.54., c the Course Lor Arithmetic.
Chicago: Scott, Foresman andTomp .Pp. 1;-

131mm anger. Smith, History 35. Mathematics. wee Iork. sine
and Comany. 1951. P. 6.

 

I.

in Chapter I to indicate this development. (If particular concern to
this paper are the changes in content and.methodology during the past
century.

Mathematics during the colonial period showed little ingenuity.”
Up to the year 1800 texts were of English origin.and related largely to
the needs of commerce .20 hthodology consisted largely in the dictation
of practical problems by the teacher which were copied by the students.
Texts gays rules for the solution of problems, these were memorized by
the students and applied with a minimum.of emphasis on understanding.
About the first significant change in arithmetic teaching came in 1821,
when Colburn, under the influence of Pestaloszi, broke away from the
deductive method of memorizing rules and applying them. As arithmetic
content continued to incorporate additional mathematics (as it was needed
by the culture) the bulk became so heayy'it was deemed necessary that it
be abridged and enriched. This was undertaken by the Committee of Ten
(1893) and the Committee of Fifteen (1895).”-

In order to reduce the amount of content the theory of “utility"
was applied. In about 1920 arithmetic was organized as a succession
of unit skills. This was followed by drill (the 'law' of exercise) as a
law of learning. In 1930 the National Society for the Study of Education

 

 

 

laflalter 8. Monroe, Development of.Arithmetic as a School& abject,
U.S. Bureau of Education Bulletin, No. 10. ‘Hashington: -Government
Printing Office. 191?. p. 8.

20m".11, a. £155., p. 63.

21
Ibid.

Yearbook paved the way to teaching for meaning.22

As the development of arithmetic programs followed a pattern of
increasing content, it began to be opposed by arithmetic curricula which
emphasised in turn: (1) mental discipline; (2) social utility: (Bk-«1.
life relationships; and. (h) social siss.23 wise lads Io- stun-pt to
shoe that mob of the arithmetic taught during the early part of the
cenhry did not function in life.”1 Consideration of arithmetic as
social utility helped to streamline much of the outmoded content. In
addition to theories of social utility much emphasis has been placed on
an attempt to make it meaningful},5 It is now rather well accepted that
both aims are necessary26 and since content (up to the last few years)
has been pretty well founded, research has dwelt on arrangement.27

this thesis would be incomplete without some reference to
“activity" programs that have been inaugurated. Harap and Mapee
sttsaptsd the lemming of decimals in an activity program, and found
that the upper and lower extremes of the class (according to 1.0.)
acquired some sastery, and that the number of repetitions had nothing

 

2231‘»; De 6h.
231bid., p. 65.

.aCuII-sfise, 'ASurveyofArithmeticProblems Arisingin
Various Occupations“, Elementfl School Journal, 203118-36. 191:9.

25mm11, a. 23:3" pe 65a

“Henry Van Egan, "An Analysis of Meaning in Arithmetic.“
Elementgz School Journal. 103321-29, 395-3103 19149.

27Herbert F. Spitser and Robert L. Burch, 'Methods and Msterisls
in the Teaching of Mathematics." Review 9}: Educational Research.
182337-149; 19h8. "'"" """""'

 

to do with degree of mastery»28 In as such as there was no control
these findings might indicate further study. Willey reported a study of
socisl eJquession intended to lead s child to understandings.” In s
report of an experience curriculum, Williams showed satisfactory gains,
but had no control gratin.”

Bimll reported in 1935.

. s e on the basis of evidence now available, the incidental,
experience approach has not produced a superior substitute for
the more systematic organisation of content, but it may well
provide some insight into supplements to the systematic
program . . . (in an activity program) there are mam
uncontrclable variables . 1 . it is also practically impossible
to duplicate a situstion.3

Since the method of arithmetic instruction described in this study

is concerned with 'drill' teaching techniques, the work of Phillips, 32

Brown,33 and Bromlell3h are significant in which there was some

 

28Henry Harap and Charlotte E. Mapes, "the learning of Decimals

in an Arithmetic Activity Program“. Journal of Educational Research,
29 :686-93 , 1936.

”Ray D. will-y. *1 Study in the Use of Arithmetic in the Schools
of Santa Clara County, California." Journal of Educational Research,
363353.659 19143.

”Catherine M. Williams, 'Arithmetic Learning in an Experience
Curriculum,‘ Educational Research Bulletin, 28:15h-62, 167-683 1919.

3J'Buswell, 32. £11., p. 67

32L. M. Phillips, 'Value of Daily Drill in Arithmetice" Journal
5 Educational chholog. hxlS9-63, 1913.

33.1. 0. Brown, In Investigation of the Value of Drill work in the
Fundamental Operation of Arithmetic, " Journal of; Educational Psychology,
331485-92. 561-703 1912.

3%lliss i. Brownell and Charlotte B. Chesal, "The Effects of
Premature Drill in Third Grade Arithmetic." Journal of Educational
Research. 29 217-28, 1935.

f‘

“I

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m
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l6

indication that drill without understanding any result in poor learning
and learning habits.

Review of w theories 9_f_ siguiicance to this study. It is
not our purpose to investigate all the literature published relative to

theories of learning, but rather to confine this discussion to basic
theories of learning, (as presented by accepted authorities) that are
pertinent to this study. Neither is it the purpose of this dissertation to
evaluate various learning theories, only to investigate such aspects as
would tend to support the techniques which were used in the teaching of
mathematics in this study. In other words, it is our intent to show
that the teaching methods employed were based on some accepted theories
of the learning process. It is further our purpose to point out that a
consideration of some comonly accepted theories of learning would
indicate that a sense-perception (tangible), and conceptualised presen-
tation of arithmetic in a social situation would tend to enhance meaning.

The importance of the process of learning to the tremendous task

of achievement in arithmetic is emphasised by Judd.

. . . childrenare notbornwithamuaber systemaspartof
their physical inheritance e e e the school puts them in contact
with a system of number symbols which is one of the most perfect
creations of thehumanmind e . .inthe shortspanofafew
years, the child becomes smart in the use of a method of
expressing ideas of quantity which cost the race centuries of
effort to invent and perfect.

If arithmetic education is to accomplish such a task as this it

would appear that every effort should be made to do it as effectually as

 

35 ‘
Charles H. Judd, Educational szcholog, New York: Heughton
“nine 1939s p. 2700

L,

17

possible. the fact that mathematics might be considered as an inherent
or instinctive process should not be totally disregarded, however, there
is some support to the theory that behavior in.man is a learned.process,
or, learning is important to education. Instinctive and/or’inherent
determinants hare been shown to have somewhat less significance, though
not to be completely discounted.36

An investigation of much of the literature relating to the
subject matter and.mothods of teaching, approaches the learning process
and its accompanying problems in.a symptomatic rather than a casual
manner.37 That is, reported research tends to analyse isolated problem
areas and suggest methods of attack, but puts less emphasis on.under~
lying principles of the thinking and learning activities of children.
Basic than, to this paper is a consideration not only of learning theory,
but the use to which it could be putlin teaching and learning mathematics.
It has been our assunption that learning theories, or generalisations,
might be employed in predicting or anticipating desirable behavior on the
part of the student in the area of mathematics.

Such a treatment of learning theory seems to inrolve three aspects:
(1) the antecedents within the students background; (2) the results to be
expected from.the application of the “teaching process" on the student,
and (3) resultant behavior. The learning theory employed in this stuq

 

36s. 'renbergen, The St of Instinct. Iondon: Oxford University
Press. 1951. is cited'fi calm, Theories 3; w. New
York: Appleton-Comtury-Crafts, Incorporated. 1953. p. 3.

37Buswell, pp. 33., p. 63-67.

kt

re

\4

18

would need to deal with all of. these areas. All three are ditficult to
describe in that there appears to be little scientific nethod which can
be used to treat then.

The first of these has to do with what is already in the students
lind when exposed to school learning situations. Children think not
necessarily in relationship to the aonentary classroos situation, but in
relationship to this, plus the mental residue contained in their ninds
of flat has gone before. This nental development 'includes the sun total
or nesories, inages, percepts, concepts, and attitudes built up over the
m... 38

The conclusions of Piaget are relevant to a nethodology that
considers the thinking process of children in the learning situation.
This work analyses a situation in which children often I'think" they
understand when the particular understanding is the result of a
'syncretistic perception."39 In other words, children are apt to Jun
to conclusions when presented with instructional naterials that are to
be learned. The supposed understanding is no nore than a quick relating
of the new idea with the child's apperceptive sass. This leads to
Piaget's conclusion:

Syncretistic understanding consists in . . . (a thought

process) that the whole is understood before the parts are

analysed and that the understanding of the detal1i takes
place only as a function of the general ache-e.

 

36David H. Russell, Children's Thinking. New York: Gian and

(Jo-paw, 1956. p. 31;.

39Jean Piaget, in: a e _and Thought 2; _th: Child. New Iork:
Meridian Books, 1955. p. .

 

 

hoIbid" p. 162.

19

In view of this it would seem illogical to teach isolated 'facts'
in subject latter as a prerequisite to either understanding or assimila-
tion of knowledge. This does not intimate that facts or mathematical
ideas, in and of themselves, are unimportant, but rather that they
should be viewed in their totality and their interrelationships.
Carpenter's work is significant in this regard:

We cannot always assume that the perception of the "whole.
gestalt can sosehow suddenly be grasped without eating sure
that the pupil knows the neaning of key concepts that compose
thO gestalt.

This need for an understanding of parts as they relate to wholes
is also brought out by Russell:

Not until eleven years can children understand how three
objects are alike, and not until adolesence can they detect
siniliarities in abstract words. This one example of mental
development suggests that filamentary children should have
many concrete experiences.

The importance of percepts and sensations contributing to a child's
total concept of a situation are further emphasised by Russell who
supports Piaget's conclusion and gives it application.

A child's thinking is based on his experience. His mediate
interpretation of events in his external or internal environment
are his percepts. . . e A percept, an image, or a nenory seldon
exist in isolation. . e e The older distinction that sensory
impression is a lower form of activity than the higher "mental
processes" of abstraction or proqu solving is now considered
to be largely a netaphysical one. Modern research suggests not
a sharp break, but a contim‘snn in the cognitive process fron

 

itll‘enlery Carpenter, I'Gonceptualisation as a Function of
Differential Reinforcement." Social Science and Mathematics.
38:28h-29h. 1951:.

”unseen, 92. $3., p. 59.

 

 

a

20
relatively simple sensations and firceptions to elaborate
aesthetic or creative experiment.

To apply the foregoing to nethodology in arithmetic it needs only
to be rcmnbcred that the subject latter as it is presented to the child
becomes a part of this environnent, or the subject natter is interpreted
by the child in his percepts.

Basic also to this discussion is the function of repetition or
I'drill" nethods, couonly referred to as 'rote' learning. A nethod by
which content is presented to the students by the teacher, either orally
or written, and which the students repeat a sufficient number of tines
until they can reproduce it. Thorndihe’s work is significant to this
kind of nethodology that involves the teacher telling the pupils 'how'
rather than building an understanding for then that tells 'why'. There
is eons indication in Thorndihe's work to show that “response to a com-
nand situation will not result in a waning of the initially nest
frequent connection at the expense of the initially less frequente'u"

If this were applied to a multiple choice problem situation in
which a student may make either a correct or several incorrect answers,
continued repetition of incorrect answers would not necessarily enhance
the correct responses at the expenses of the incorrect answers.”

Further support for a conceptualised, learning and the use of
broad ideas is offered by Carpenter:

 

m’Frhtard L. Thorndike, The Fundamentals _o_f: Le . New York:
Bureau of Publications, Teachers Wis U versity. 1937. p. 18

”me” p. 18

7)..)

21

Functional learning of concepts is more efficient than rote
learning when measured by retfigtion and ability to verbalize
meanings of learned concepts.

In view of the evidence presented in Chapter I indicating that
elementary mathematics education needs now to teach structure and con-
cepts rather than isolated facts, the work of Hargrove which indicates a

structured, rather than an incidental learning program is significant}7
The importance of understanding a number system and its applica-

tion to social situations is a concern of teachers in instructing chil-
dren in ways of solving story problems. The use of a thorough
understanding of a number system, as it applies to the culture is brought

out by Hamilton:

. . . A problem seldom if ever has any numbers associated with it
until we recognise the situation and apply numbers to it. . . . we
have to be able to think well enough to be able to arrange the ele-
ments of a situation and adapt a strategy using models from reality,
concepts, mental imagery, e e .also know enough about numbers to
recognise one or several properties that fit the situation thus
choosing an abstract model in the best mathematical sense.n8

Support for learning in a social situation rather than in
isolation is also offered by McHugh.h9

The learning or behavior to be expected as a result of arithmetic
instruction is basic to this paper. This consideration involves the

 

h6Fenley Carpenter, "The Effect of Different Learning Methods on
Concept Formation." School Science 222 Mathematics, h0:282-285, 1956.

MH. Richard Hargrove, "Proper Euphasis on Science and Mathematics
in the Elementary School.‘ School Science a_ng Mathematics, Maw-91,1960.

l‘BE. W. Hamilton, "About the Articles', The Arithmetic Teacher,
8 3&9, 1961.

MWalter Joseph McHugh, "Pupil Team Learning in Skill Subjects in
Inter-mediate Wes." Dissertation Abstracts,21:lh61, Dec. 1960.

 

t_.

/

‘ua

-JL

22

relationship of mind to the learning process; t, as suggested on
page 17, the use to which the learning (resulting from the instructional
process) shall be put. Guthrie makes such a distinction:

(h‘owth, reproduction, and defense mechanizations are life but
they are not mind. Mind is something more; it is growth and
reproductions and reactions serving these ends plus something
else that common sense might call profiting by experience. . . .
The ability to learn, that is to respond differently to a situ-
ation because of past response to the situation is what
distinguishes those living creatures which canon sense endows
with minds . . . training leaves no observable changes . . . 1%811
must be a mode of behavior which changes with use or practice.

While it might be desirable to find a teaching method that would
assure all the best possible results, such a possibility would seem
improbable. In this light Guthrie continues:

So far as I am aware the only suggestions toward the description
or explanation of the circumstances under which specific changes in
behavior will or will not occur have been made in the form of
association or conditioning . . . when the past of the individual
is used for predicting behavior, we find our predictions are always
in tens of associative learning . . . we can never understand him
(man) but we can understand something of him and know something of
himinterms ofwhatweknowofhumannaturs ingeneralandin
terms of what we know of his past history and the nature of
associative learning.

In contrast to this concept of learning as it relates to the total
past and present experiences of the individual Douay offers a brief sum-
mary of some early aspects of arithmetic instruction:

When number work was first thought to be a necessary part of

instruction of the first two grades, teachers did not know what
to do except to drill their pupils in the abstract combinations.

Investigation later showed most of the time spent this way had
been wasted . . . (the need is) to provide teachers with

 

50 .
E. R. Guthrie, The chholog of Learning. New York: Harper
and Brothers. 1935s Ppfi‘ e _

519g" Pp. 2&3-2u5.

23

sequential materials that will guide the development of
exercises through which children will learn the significanceS2
of numbers rather than the manipulation of abstract symbols.

This importance of learning theory to the teaching of arithmetic
is emphasised by Lankford:

. . . the effective teacher of mathematics encourages
creativity by helping pupils discover the basic laws, or
principles of mathematics 3 he aims for understanding ahead
of skills of operation; and he seeks to five students the
stimulation t comes from accepting and realising worth-
while goals.

In setting forth the important elements in an effective arithmetic

program (h‘ossnickle lists:

1. The nature of the subject is such that it has a cultural
value 3 it is structured; properly taught, it leads to unique
quantitative ways of thinking; and it is basic to the further
study of mathematics.

2. A program for the learning of arithmetic should recognise
such factors as the mental mgiene of the classroom, adequate
records for guidance, provision for optimum individual growth,
use of materials, and ability to read quantitative statements.

3. A specific course in the background of mathematics is
recomended as essential in the training of teachers of
mutiCe

The basic understandings needed, opportunities to create, and
emphasis on meanings rather than drill are also advocated by Jones:

_h

52m J. new, Guiding daggers _1_n_ Arithmetic. Evanston: Row,
Peterson and Company. 1957. PP. IX-l.

53Francis G. Lankford, Jr., 'Implications of the Psychology of
learning for the Teaching of Mathematics," The Growth 93: Mathematical
Ideas Twenty-Fourth Yearbook. National Council of Teachers of
macs. Washington: The Council. 1959. P. 1:05.

5"Foster E. Grossnickle, ”Introduction.“ Instruction in; Arith-
metic Twenty-Fifth Yearbook of the National Council of Teachers of
Me ematics. Washington: The Council. 1960. p. 3.

 

_ I \d'
A .'
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2h

1. The best learning is that in which the learned facts,
concepts, and processes are meaninng to and understood by
the learner.

2. Understanding and meaningfulness are rarely if "all or
none" insights in either the sense of being achieved instan-

taneously or in the sense of embrgging the whole concept and
its implications at any one time.

The foregoing has been presented in support of a method of
teaching arithmetic which would emphasize understandings, insights,
creativity, and cultural utility 5 and which would give no emphasis
to drill methods, repetitious problem solving, and meaningless menorisa-

tion.

Current literature m present practices _i_z_1 EM
arithmetic. To review all the studies regarding present practices in

arithmetic instruction would be far beyond the scope of this study. We
have selected those that appear to be of significance and that are
representative of the general trends in teaching for meaning and under-
standing. Glennon has stated the present condition of arithmetic
research rather well as he points out that ‘there is much research about

what 2 be taught and little concern about what should be taught.56 30
states:

Prior to 1900 the curriculum was detendned by:

 

SSPhillip S. Jones, "The Growth and Development of Mathematical
Ideas in Children," The (Earth 21; Mathematical Ideas, Twenty-Fourth
Yearbook of the National Council of TeaEers of Mathematics. Washington:
The Council, 1959. p. 1.

56Vincent J. Gucnnon, "Editorial? Educational Leadership,
19:35h-56, 1962.

l. The need for mathematical training by society.

2. The need for the subject to be taught as a system.of

3. The present emphasis is on the needs of the child.57

The author continues that all these emphases hare drawbacks,
there is a need for balance in.the curriculum and that ' . . .
modernising the school pgggram.is one way of changing the method
as well as the content.

Further support as to the disorganized state of arithmetic research

is offered by Hartungx

Instructional.programs in arithmetic include a layer of ideas
and practices accumulated through the years. Examined critically
e . . Iran the point of view of learning theory . . . they are
seen to be questionable either on mathematical grounds or on
psychological grounds or both . . . much confusion arises between
means and ends. Computation for its own sake is fruitless . . .
but essential to problem solving. . . . I see great possibilities
for improvement in arithmetic instruction if we are willing to
acknowledge that some of our teaching has been superficial at
best. The remedy is not more drill, but.deeper insight as to
what is really basic arithmetic.

The need for a culturally significant and useful arithmetic
curriculum is brought out by Cook:

Arithmetic is undoubtedly one of the more poorly taught
subjects in the elementary school. Educators may question
whether present demands in mathematics are in focus with
other areas of the curriculum, but we must recognize the
needs of our'youth in the world inhuhich they will live.
A.reappraisa1 of our approach to arithmetic in the elementary
schools, of the coordination with the mathematics program in
the secondary schools, of the preparation of teachers, and

 

57%., Pe 5’4.

58%.! pe SSe

S9Maurice L. Hartung, I'Distinguishing Between Basic and Super-
ficial Ideas in Arithmetic Instruction." :2: Arithmetic Teacher,
6:65-7, 1959.

 

l4

 

26

of the place of inservice programs in the school system is
imperative if 66° are to meet the demands facing us at the

present time.

The need of a program designed to build meaningful mathematical

concepts of cultural and social significance is emphasized by Suelts,61

Madden,62 Flournsy,63 Breuclcner,6’4 and Brommell.6S
Suelts offers a fine “time chart" of the development of arith-

metic instruction and the factors influencing the development of

arithmetic texts :

1850 - 1890 "Faculty" psychology.
‘Training' of the mind.

1900 - 1920 Principles of learning.
Motivation and readiness.
Exercise, effect, threshold of learning,
over-learning.

1920 - 1935 Readiness.
Exercise .
Drill.

1930 - 1950 Progressive education.
Satisfyingness, peer group.
Whole child.

 

6°Raymond F. Cook, ”Improving Arithmetic Instruction,-
National Elementa§z_Principal. 38:37-39; 1959.

5lecn.1. steltz, "Arithmetic in Historical Perspective.'
National Elemental; Principg, 39 :12-163 1959.

62Richard Madden, "HaJor Issues in Teaching Arithmetic.“
National Elementfl School Princi al, 393171;; 1959.

63Frances Flournay, 'Relating Arithmetic to Everydq Life."
National Elementary Principal, 39:29h; 1959.

6I‘Leo J. Breuclcner, “Testing, Diagnosis, and Follow-Up in
Arithmetic.“ National Elementary Principal, 39:33hs 1959.

65mm... A. Broenell, “Arithmetic in 1970.. National Rosana
Principal, 39 :h2-hh3 1959.

 

\L

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27

1950 Child development understandings.
waning, action research.
Discovery learning. 66
Mlltisensory learning.

Suelts continues:
Arithmetic will advance as teuheradevelop an insight into
the subject matter and its significance and as they learn more
and more about the boys and girls they teach and how the human
. minds, W’s, and emotions combine in the behavorial learning
situation.
In considering the characteristics of a good arithmetic program
Suelts advises:

A teaching process through an inductive-deductive cycle (in
which) . . . new concepts are introduced in a socially
significant situation, developed in their mathemagécal sequence,
and returned to socially significant application.

Methods which would teach arithmetic for understanding make it
necessary to present the structure of the number system. This is
emphasised by Brueckner.

It is generally agreed that children must understand the

number system. They met also understand how the number 6
system operates in the performance of number operations. 9
The emphasis on sound mathematics as basic to arithmetic programs

is supported by lundberg70 and Wilsberg.“

 

668ue1tz, 22. _ci_t., Pp. 15-16.

671cm.

681b1d.

69Breuclcner, 132. c_i_t_.

7aflazel Lundberg, "Mathematics in Elementary School."
Educational leadership. 19 8361;4831962.

7114.17 E. Wilsberg, "Freeing Children in Primary Arithmetic.“
Educational leadership, 19:352-63 1962.

 

 

__.- I , _—_._,

28

There are many isolated studies which support teaching methods
employing meaningful application of mathematics, as pointed out at the
beginning of this section. Some of the more outstanding are those of
Stone," DoLcng,73 hanger,7h and Miller,75 in which meaningful methods
generally gave favorable results. Langer emphasized the dependence of
significant technological advances on mathematics.

There have been a few studies recently reported showing the
relationship of time in class to achievement. Daugherty, in a study
conducted in the DesMoines Public Schools, found a fifty minute class
.u‘.‘av'orable.76 Denny77 also showed an increased time allotment as
contributing to greater achievement. ,

Changing methods of teaching arithmetic have pronpted some
studies in regard to evaluation procedures. Breuckner offers six

 

72mm H. Stone, ”Fundamental Issues in the Teaching of
Elementary School mthematicsfl 1113 Arithmetic Teacher, 6:177-793 1959.

73Arthur R. DeLong and Richard M. 01 rk, I'Developing Creativity
in Arithmetic.“ 193 Arithmetic Teacher, 6:208; 1959.

 

7I‘Rudolph E. Langer, “To Hold As't Were the Mirror up to Nature 3
to Show the Very Age and Body of the Time. " The Arithmetic Teacher,
6:289-9h3 1959. .

75 0.11. mller 'How Effective Is the Meaning lbthod?‘ _Th_e_
Arithmetic Teacher, hth-lo; 1957.

76James Iewis Daugherty, "A Study of Achievements in Sixth Grade
Arithmetic in DesMoines Public Schools,‘ doctoral dissertation as
reported in sis of; Research _ii_i_ the Teaching of Mathematics. United
States Dep nt of Health Education and Welfare, Bulletin Number 8.
Uashington: U.S. Office of Education, 1900.

"Robert Ray Denny, u Two-Year Study of an Increased Time Allot-
ment Upon Achievement in Arithmetic in the Intermediate (h'sdes, doctoral
dissertation as reported in w_ of Rese____a_r__ch in th__e_ Teachi_ng__ of
hthematics Uniud States Department'- of—__- Health Education and Welfare,
m No.8 . washington: U.S. Office of Education, 1900.

 

 

(.2

(Q

29

criteria to be considered in a testing program to be used in connection
with meaningful teaching procedures:

1. The selection and clarification of objectives.

2. The determination of the rate of growth.

3. Provision of a basis by which teachers can set up educa-
tional experiences adopted to the needs and ability of the learners.

1:. Motivation and guidance of learning, especially by helping
children to evaluate their own responses and behavior.

5. The location diagnosis and treat-ant of learning
difficulties.

6. Bases for coordinating imrovement programs in related

fields eggh as arithmetic, reading, science, and social
studies.

He adds five basic steps to aid in evaluation:

1. Fomlating general and specific objectives.

2. Defining objectives in terns of pupil behavior.

3. Designing and selecting suitable means of appraisal.

h. Securing a record of behavior and performance.

5. Interpreting and evaluating the information secured. 79

Growing has received sons attention as offering a possible

solution to differences of ability. Larch reports:

Ability grouping as it has ordinarily been used does not solve

the problem of providing for individual differences in arithmtic.

The mere grouping of pupils sith similar characteristics oes
little toward improving the teaching-learning situation.

78mm”, 22s 9.1.2" Fe 33,-“

79mm.

Boflarold larch, 'Inter-Class camping for Arithmetic Instruction-
Critique and Criteria." _Th_e_ Arithmetic Teacher, 8:1:06; 1961.

 

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30

A program of grouping based on individualised textbooks in which
students corrected their own work, progressed at their can speed, and
were instructed during individual teacher-pupil conference periods was
triedbyWhitaker. Success seems to have been measured by the amount of
text covered.81

This review of present practices would be incomplete without
references to the studies being conducted on content, or 111a: it is pas-
sible to teach rather than why it should be taught, as pointed out
previously. The School Mathematics Study Group has prepared materials
for grades four to twelve relating to pure mathematics under the support
of the National Science Foundation. These naterials are being tried in
some areas on an experimental basis. Preliminary reports seem to indi-
cate that students accomplish about the ease achievement in 8.11.8. G.
classes as regular classes. There was no indication given as to tine
students spent in class, teaching methodology, or as to whether these
materials should comrise the total arithmetic program of the
curriculum. 82

Other current experiments include that of Suppes 83 in teaching
“Sets and Numbers“, Stanford University, California, "Geometry in the

 

81mm- L. Hhitaker, W Not Individualise Arithmetic?“ gig

Arithmetic Teacher, 8:11023 1960.
82 I'School Mathematics Study Group,‘ Newsletter No. 10. Island
Stanford University, 1961. See also, Fred J. Weaver, "The School lithe-
matics Study Group on Elementary School Mathematics," Th; Arithmetic
Teacher, 8:32-65; 1959.

83 "Improvement Projects Related to Elementary School Mathematics-
A Selected Listing." Umublished mimeographed material. Lansing:
Michigan Department of Public Instruction, 1962 .

 

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31

Prim ends!" being conducted by Hailey also at Stanford University,8h
the University of Illinois Arithmetic Project'85 and the Syr‘cuse

University"hhdison Project."86

These studies are concerned with
teachingpure mathematical content and not with a total arithmetic
curriculum. They are examples of the 'shat-canébe-taught' studies
pointed out by Glennon.

In contrast to the studies listed above, it was the purpose of
this investigation to design an arithmetic progran.that would take into
consideration the prdblems reviewed in this section: (1) offering a
seaningful.presentation of mathematics; (2) providing for individual
differences; (3) structuring so that it could be accomplished in a
minimum of class time; (h) teaching only the.mathemstical theory that
was needed to adequately explain the arithmetic content of grades five
and six; and. (5) enabling students to show significant achievement on

standardized achievement tests at these grade levels.

 

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851nm.
861nm.

CHAPTER III

DESCRIPTION OF THE TWO METHODS
USED IN THE STUDY

It is not the purpose of this themto present in detail all the
materials used in the teaching of the arithmetic program during the
year of this study. Representative lesson plans appear in the Appendix.
This chapter will be concerned with a descriptive comparison of the
experimental and routine classroom procedures. The experimental
description will contain: (1) the method of presentation of the mathe—
matical concepts taught; and (2) the developmental processes used. The
routine classroom procedure will contain an analysis of interviews with

the teachers of the self-contained, or routine, classroom situations.

The ggperimental £11129. was concerned with concepts to be pre-
sented to the students as important mathematical ideas of social
significance. These were selected arbitrarily by the experimental teacher
as representing the arithmetical materials which the students were
required to cover during the course of the year as prescribed by the
existing curriculum of the school system. Topics from pure mathematics
that would offer sound mathematical explanation for this material were
added. One concept was presented to the students each week in a dis-
cussionperiod. Students were not told £12! to solve problems, but were
encouraged to create and investigate ways of doing this. A list of the
concepts presented and the approximate time spent on each appears in
Table I. The same concepts were presented to both fifth and sixth grades

at the same time during the year in the sequence indicated on the table.

Ill.

)ea

TABLE I

33

CONCEPTS TAUGHT m (moss 5 AND 6

 

TOPIC

Mr Om Number System

Egyptian Number System

Greek Number System

Roman Number System

Hindu Arabic lumber System

Linear Measurement

Square basurement (area)

Dry Measuremnt (volume)

Height

How We Can Add

How We Can Subtract

How We Can Multiply

How We Can Divide

Place Value as Powers of a Base

Other Bases (7 and 12)

Time (and Base 60)

Modular Arithmetic (calendar)

Centuries

Temperature

Averages

Data

Graphs (line)

(h‘aphs (bar )

Graphs (area)

Decimals (as extension of
place value)

Rounding and Estimating

Big Numbers (as powers of 10)

How to Add, Subtract, mltiply
and Divide Fractions)

Cancellation

Accounting

Finding Wholes from Parts

W

APPROXMTE TIE

ltozweeks

1 week
1 week
1 week
1 week
1 week
1 week
1 week
1 week

1 week

1 week

1 week

About 2 to 1; weeks. Taught at
one time all four processes.

1 week

1 week

1 week

 

 

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.3; a

314

Table II indicates the general structure of each weekly lesson.
The teacher acted as a resource person during the class. The text was

used as a reference book. In addition, other reference materials were

supplied as students requested them. When children asked for suggestions
about locating materials, the teacher helped students to find these.
Since much of the pure mathematics needed by the students was not found
in published materials, these were written for the students by the
teacher. These appear in the Appendix to this study.

At all times the student chose the method he would use in pro--
senting evidence of his understanding of the concept for the week.
Students were not given assignments to complete. The only requirement
was that each would demonstrate his understanding of the concept under

consideration.
TABIE II
WEEKII ROUTINE OF EIPERDIENTAL CLASS

I‘DNDAY: Presentation of Concept by teacher in discussion period.
Children discuss: (1) that they already know regarding con-
cepts (2) how to find additional materials; (3) what each
individual can do to produce tangible evidence of his
understanding of concept.

TUESDAY) Children are free to work in groups or individually in the
HEDNESDAI) area under investigation for the week. The experimental
THURSDAY) teacher acts as resource person.

FRIDAI 2 Children are presented problems in a series of increasing
difficulty. These contain selections from the four arith-
metical processes. Children are requested to work as many
as they can during the class time. These are taken up,
corrected by the teacher and generalizations discussed
with the group as a whole.

 

 

The tangible and conceptualized presentation was intentionally de«
signed as a methodology which would attempt to accomplish the existing
arithmetic curriculum in a unimmn of time without the use of rote learning
techniques. The intents was to increase understanding and to decrease the
amount of paper-and-pencil work by both students and teachers. This
seemed desirable because of the recent additions of content courses of

elementary curricula. Chapter II pointed out the relationships between
such a methodoloy and learning theory.

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35

It was also felt that the elementary arithmetic curriculum should
meet the needs of: (l) the student who may leave school at age sixteen,
and; (2) the student who will pursue a career in mathematics. The former
needs a mathematics program to provide him with a cultural tool for daily
living; the later, needs a sound theoretical foundation that will furnish
a sequential background for secondary mathematics. The tangible and con-
ceptional presentation was designed to present mathematical theory inns
simple manner to the whole group. Each child then.took the concept and
produced tangible evidence of his understanding. In this manner, slow
children were permitted to produce tangible evidence at their own level;
academically talented students were permitted to produce evidences
representative of their intellectual level. The burden of proof of underu
standing rested at all times with the student. Details of the mathematical
theory taught are presented in the Appendix.

The procedure for teaching the experimental class followed a very
definite pattern. Since the classes were conducted in two buildings, a
careful time schedule had to be followed. The experimental teacher entered
the building, taught the class, or classes, and left. The regular class-
room teacher stayed in the room and observed the experimental class. This
teacher was prohibited from.teaching any additional arithmetic at any other
time, and from assigning arithmetic as I'seatwork" or for "discipline'.

At the end of the thirty-five minute experimental class period,
the experimental teacher left the room.and travelled to the next school.
This made it impossible to give any extra help to slow learners. It
also prohibited correlation of the arithmetic program.with other subject
matter areas. Students were requested to stop working at the end of the

thirty-five minute class. Table III contains a sample lesson.

 

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TABLE III

DESICN OF SAMPIE IESSON

 

 

H

 

YOUR‘gfl§.NUMBER.SYSTEM

 

 

This lesson is designed to help children understand that a number
system is an orderly sequence of symbols used to express quantity. Often
students become so accustomed to counting and doing arithmetic by rote
that they do not realise how a number system functions. Say to the
children:

“Forget everything that you ever knew about arithmetic. For-
get how to make numbers and how to do problems. Now pretend you
are living many thousands of years ago and that you are in need
of keeping track of something. This is really one of the most
imartant uses of arithmetic. . .writing down amounts of things
by using symbols. We call these numbers and numerals.‘

(Do not go into the 'number and numeral" question at this point. Use
terms the children already understand. This may be introduced as an
outcome of the lesson, however. Point out that it is easier to
manipulate numerals rather than thin s to figure something. For
instance, if a farmer has a large ock of sheep and buys and sells some,
he can simply use a moral to represent the total flock and add and
subtract other numerals to represent the amount he buys and sells. This
is easier than bringing them all together everytime and counting them.)
After developing these or similar ideas continue:

"After you have forgotten all the arithmetic you ever knew,
pretend that you want to figure something out or keep track of
something. Make up a number system and do this in your system."

The children will probably make rather simple symbols such as 2

1. I II II! Just marks are typical of . 1.1
relationship.
2. L b E This type of symbol is really 1.1

relationship also.
3. 5&8 $ <5 Sometimes children will use ideas
like those in the text.
h. :ZE5: :szSL‘16352' symbols like these are vague.
Children need help in seeing that symbols, or a system of
numeration should be 3

1e “”1.

2. convenient
3e 1180M

After they have invented a few symbols, let them work some problems
in their own system. Help them to simplify and improve their system.

_L-I

iv

 

37
TABLE III - (Continued)

moamnmmsasrsm-nfismfim LESSON

1.. Discuss arithmetic as a way of keeping track of things.

2. Introduce the idea that our numbers are a system of symbols
that express amounts.

3. Have children make up their own number system.
h. Let each child work problems in his number system.

5. Discuss how the number systems of the children work. lead
them to conclude that a number system should be :

1. simple
2. useful
3. convenient

(It is best to have children "forget everything that they ever knew about
how to do arithmetic“ in order to do this)

REFERENCE

Smith, David E., and Jekuthiel Ginsburg, Numbers and Numerals.
Washington: National Council of Teachers of 1%th , I9 .

Smith, David 3., History 93 Mathematics. New York: Gin and
comm, 1951e

Batchelor, Julie Forsyth, Conunication: from Cave Uri to
Television. New Iork: Harcourt Brace and World, Inc. I9 .

Hogben, Lancelot, Mathematics in t_hg &. Garden City:
Doubleday and Company, 19557

, The Wonderful World of Mathematics. Garden City:
Doubleday and Company, 1955.

Mueller, Frances J ., Arithmetic Its Structure £22. Concepts.
inglaocd Cliffs : Prentice , Inca E6.

 

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38

Specific procedures of the experimental class are listed in Table IV.

TABLE IV

PROCEDURES USED IN EXPERIMENTAL CLASSES

 

 

1.
2.
3.
h.
S.
6.
7.

8.
9.
10.
11.

12.

 

Class tile lilited to 35 minutes per day.

No ability grouping.

No hone work.

No individual help for 'sloe learners.“

No assignments were given that required students to finish.

No direct coordination with other subjects.

No time spent outside of class on arithmetic such as recess or
after school.

No arithmetic given as I'discipline" or “busy work".

No drill.

Nb ”story'problens' Ire-.text worked.

No attempt use made to work any problems in the text; other than
those used as reference and resource.

Students worked.problees and handed then.in once aueeek only.

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39

In general the class routine was one which: (1) presented an idea
or concept to the students; (2) gave them opportunity to investigate it;
(3) gave them opportunity to react to it by producing some tangible evi-
dence of their understanding of the concept; and, (1;) gave them an idea
of the progress that they would be making were they in a routine text.
This was done by giving them a set of problems once a week in the four
arithmetical processes which were structured in a series of increasing
difficulty. This was felt necessary for the following reasons.

1. Patrons of the school wished to know "what page in the text
their children were on." It was felt necessary to show them that the
students were able to do the routine text materials although they were
not using the text as such.

2. The children wanted to know if the conceptualized program
would retard their regular progress through the grade. They seemd to
wish some evidence of their progress. Doing some problems in the routine
fashion seemed to help them feel they were making progress in the grade.

3. The problems presented once a week differed from the type
contained in the text in that: (1) the four arithmetical processes were
represented; (2) the problems progressed from very simple to very diffi-u
cult ones. Each child did only the problems of which he felt capable;
(3) these problems always contained representative examples from all
proceeding lessons. Samples of these weekly lessons appear in the
Appendix.

As each concept was presented, every effort was made to correlate
and illustrate it with culturally significant items. Fractions were

taught with measures. All four arithmetical processes involving

L10

fractions were taught at the same time. These were presented as experiences
with fractions within the culture rather than as the manipulation of
abstract symbols. After the understanding was built by using tangible
materials, abstract symbols were manipulated in problem situations.

Pages 141 to 145 show a typical lesson in fractions.

LINEAR MEASUREMENT ”l

GRADE 5

Linear (lin e er) measurement is a measurement of line. ‘we say
a e is 6 inches long. ‘we also measure line. or length, in feet,
miles, rods, and.yards. See how many different kinds of linear measure-
ment you can find and list.

'we can use linear measurement to help us understand fractions. This
is a good thing to do because very few of the things that we measure come
in whole inches, whole feet, whole yards, etc.

Take one of the strips of paper that have been given you, place your
ruler on it so that you can draw a line down the middle of the strip. On
the line mark off the inch divisions so that you have a little oak tag

ruler of your own. Like this:

 

11J 11114 J
I2 .3 14 5’6 7 8 9/0 ///

 

 

Do the same thing with the other strip, but cut this one up into
one-inch sections. This will give you a ruler that you can divide up into.
fractionals parts. Place the whole ruler beside the one that is cut up

into inches on your desk like this:

 

 

 

 

 

 

 

 

J. a 5 A. a J.
l 2 3 9 5 6 7 8’ 9 /o // /2
JL—III

 

 

 

I!

142

Do the following things and answer the questions.
1. Divide the cut up inches into two groups. 1/2 foot = 2 inches?
2. Divide the cut up inches into four groups. l/I. foot = 7’ inches?’
3. Divide the cut up inches into three groups. Each of these is a.
third of a foot. 1/3 foot = ‘2 inches?
. .
TherearelZinchesinafoot. Eachinchiel/Dofthewhole
foot. By moving the inch pieces about, answer the following questions.

a s
1. 1/1. ft. = 2/12 it. 3/12 ft. = 2/4 ft.
2. 1/2 ft. = 2/12 ft. 3/1. ft. = 2/2 ft.
3. 1/2 ft. = 2/1. ft. 2/1. ft. = 2/2 rt.

Can you make a list of other statements that are true about the
fractional parts of a foot?

M

Answer these questions using your out up inch pieces to help you.

A B c n
1- 1/2 = 1/4 1/3 = 7/6 3/4 . 7/12 2/2 = ?/12
2- 1/4 = 7/12 1/2 = 7/4 1/6 = 7/12 2/3 = 2/12

Mahasmanyadditionaltruestatementsasyoucan.
sesame

You have found that there are three-twelfths of a foot in one-
fourth of a foot. We can write it this way: 3/12 foot = 1/4 feet, or
just 3/12 3 1/4. Use your inches and see if you can complete these
statements so that they are true.
1. If M. = 3/12, then 3/1. = 2/12
2. 32/34/12, then1/3=:/12
3. Ifl/2=6/12, then 2/222712

J

 

These are more difficult. See if you can do them by thinking
very carefully.

1. If 1/2 = 2/1., then Lug = 2/4. We would read this this. way: "If
one half equals two fiurths, then one and one half halves equal how
many fowrlihs'tm The answer would be three fourths, or 3/4.. Do this
one. '

If 1/2 = 2/4, then 2112 = 2/4.

2. 1: 1/3 = 4/12, then 3.24/2, = 2/12.

3. If 6/8 = 3/1., then 1/83'2 7/4..

4. If 4/12 = 1/3, then 5/12 is how mam twelfth: more than 1/32

Us have a sign in mathematics that means "more than.‘ It is: .

”Less than'I is this signs . You can remember them by thinking that

M is pointed in the direction that you write. M

is pointed backwards to the way you write. Now write the answer to your

problem like this:

If 4/12 = 1/3, then 5/12 is 1/12 1/3.
5. If 1/6 = 2/12, then 3/12 is how many twelfths 1/6?
6. Is 3/12 or 1/6? I
, mama:
You will need to think very carefully to do these. Use your out
upinchesandseeiftheyaretrue.

1. 3/1. foot = 2 1/2 thirds of a foot. (Hint: change both fractions to
twelfths of a foot).

2. 1/2 feet = 1 1/2 thirds of a foot.

3. 2/3 feet = 2 1/1. fom'ths of a foot.

W

 

up.-___ .

 

You have seen that you can write 2 1/2 thirds like this: “/2.
3

Hrite 1 1/2 fourths. If you add like this, then is it true?

 

1.1.12
A
1.112
I.
374
Add:
L112 2.114 3.12
3 2 I,
Lila 2.114. 3.112
3 2 1.

 

 

 

 

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s

Divide :
Label your answer m your remainder if you have one.

A B G

21n.F21n. Bin.ll8in. 21n.|241n.

 

 

D I F G

5.;

 

 

 

61n.|5/.1n. 7in.'1ft.41n. 3In.|2ft. éin.[§£t.6in.

he

The routine 2:; control classroom procedures were established by
interviews with the teachers of these rooms. Pages 1:6 to 50 contain a
sumary of these interviews. The replies of the teachers of the routine
(control) classrooms have been generalized and listed. A summary

appears in the conclusion.
1. How were assignments made?

Control teacher M
Assignments in texts at several levels.
Students required to finish assignment.
Memorising tables, mandatory - drilled.
Homework about 3 times a week.
Text sequence covered.
Talented pupils worked extra problems in advanced texts .

Repetitive problem work for slow learners.

Control teacher M
Assignments in text every day.
Some children ”worked ahead".
Asalgned extra work as "discipline“.
Assigned homework. '

Generally worked through text.

Control teacher No . 3
Assignments in text every day.
Generally covered text.

It?

Control Teacher No. 3 - continued
Students required to finish given.shount‘determined'by teacher.

Tables nemerised.

Control teacher EBarE
Assignments in text every day.
Teacher determined amount to be completed.
Generally covered text.

Followed text sequence.

2. 'Were story probless in text covered?
Control teacher No. 1

Generally covered exeept for slow learners.

Control teacher No. 2

Cowered at least 60% of those in text and made up others.

Control teacher No. 3

Did.nost of the story probless in the text.

Control teacher No. h

'Wbrked nest of the story problems in the text.

3. Did you give individual help?
Control teacher*§g;4l
During odd tiles all during the day.
Set aside 15 minutes twice a week for I'extra“ help.

'Worked with slow learners.

 

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___..contro1 22.1w: Egg—2.

Chve extra help to talented and slow students.

Control teacher No. 3
Helped individuals all day through.

Control teacher No. Q

Chve individual help during the day and set aside 15 minutes per

day extra.for help.

How often were papers sent home?
Control teacher No. 1
At Iarking periods and also other times.

Sent homework assignments home .

Control teacher No. 2

Two or three times a year.

Control teacher No. 3

Twice per week.

Control teacher No. 1;

Each week.

How were students grouped as to ability?

Control teacher No. 1

Students grouped themselves 3 below grade level, at grade level,

and above grade level.

Control teacher No. 2

Some individual cases.

 

f),

(.u

 

6.

7.

Control teacher No: 3

Grouped according to amount of text covered.

Control teacher No. h

All stayed in.same text but progressed at different levels.

'Whatilaterials did students cover?
Control teacher No. 1

Used texts at various ability levels.

Control teacher No. 2

Covered text sequence.

Control teacher No. 3

Covered texts at various ability levels.

Control teacher No. h

Followed text sequence - 2/3 to all of it.
0

How often did.you correct papers?
Control teacher No. 1

At least 3 tiles a week.

Control teacher No. 2

More or less every day.

Control teacher No. 3

Children corrected own.papers every day - teacher once a week.

8.

9.

50

Control teacher No. h

Two times a week, but had children rework problems missed and then
corrected these.
What other learning activities did you use?
Control teacher No. 1

Abacus, flash cards, games, extra papers, teamwork, outside of
class productions, pictures.
Control teacher No. 2
buorised tables by repetitious problems.

Control teacher No. 2
Correlated with science.

Had drill problems.

Control teacher No. I;

Some games, tables (drill)

How mch instructional time was used per day for arithmetic?
Control teacher No. l

1.5 minutes, plus extra help during the day and homework.

Control teacher No. 2

35 ninutes, plus some homework [and 'discipline' problems.

Control teacher He. 2
35 minutes, plus extra help at odd times.

Control teacher No. b

Given extra time to finish.

01

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51

TABLE V

SUMMARY OF TEACHING TECHNIQUES USED
IN EXPERIMENTAL AND ROUTINE CLASSES

 

 

 

EXPERIMENTAL

Assignments were student
detenined

Did not work story problems
in text

Received no individual help

Worked problems to hand in
once a week

Were not grouped

Text did not structure
progra-

Had mamr other learning
activities

Spent 35 minutes per W in
class

 

 

RCIJTINE
Assign-ants teacher determined
Worked lost of story problems
in text
Received individual help

Worked problens to hand in 2 - 5
tines a week

Were partially ability grouped
Text structured program

Learning activities limited to
I'drill" games

Spent over 35 minutes per day
in class

CHAPTER IV
ORGANIZATION OF INVESTIGATION

D_e_s_i_gg c_>_f_ this investigation. This dissertation was carried out
during the course of the "Study in the Use of Special Teachers in Science
and Mathematics in Grades Five and Six a part of the Science Teaching
Program of the American Association for the Advancement of Science“.
Cooperating school systems were: Lansing, Michigan; Cedar Rapids, Iowa;
washington, D.C.3 and Woodford County, Kentuclq. Each school system
hired a teacher who would teach only science or only mathematics. The
purpose of the over-all study was to compare the two methods: (1) special
teachers 3 or, (2) self-contained classroom teachers.

The use of a conceptualized and tangible presentation of arithmetic
was devised as the special method of teaching which would be used in the
Lansing, Michigan experimental center. As each of the four experimental
centers were free to specialize in any way they chose, it was felt that
this in no way either detract from or enhance the study of the American

Association for the Advancement of Science.

Students in the M. The students chosen to participate in the
study were considered as typical of those in the city. Two schools, each
of which contained two fifth and two sixth grades were selected. (he
fifth grade and one sixth grade were taught by the special, or experi-
mental teacher, and the other fifth and sixth gades were taught by the
self-contained, or control room teacher in the routine manner of the

school system.

53

There were, therefore, eight groups in the stw. They were
arbitrarily numbered as groups I to VIII, inclusive. Odd numbered
groups were experimental, even numbers designated the control (self-
contained) groups. No effort was made to match the members of the
experinntal and control classes. They were selected Just as they
exisud in the school during the regular routine of grade placement.
Inthefourfifthandfoursixthgrades, thenwereatotalof89
students in the experimental groups and 77 students in the control
groups that completed the study. They have been treated as a total
universe of experimental and control groups rather than separately.

No effort was made to match the members of the control and
experimental classes. The classes were left as they existed in the
aerial routine of the school. Table VI compares the mean scores of the
groups on the Otis Quick Scoring Test of Mental. Ability given at the

beginning of the study.

Sh

TABLE VI

COK’ARISON OF MEAN TEST SCGLES ON OTIS QUICK SCORING
TEST OF MENTAL ABILITY GIVEN AT BEGINNING 3‘ STUDY

II

 

mu. m
Group I 101.1 Group II 103.1
Group III 107.9 Group Iv 106.1:
Group v 10h.h Group VI 103.9
Group VII 103.9 Group VIII 103.2
mmmmscoarsoummormmums 1011.3
mammscoammonsorcomonmours 101:4

55

Table VII shows the comosition of the experimental and control

 

 

 

 

groups-
TABLE VII
corrosnIon or moors PARTICIPATING IN STUDY
mm
cams mm NUMBER lo: mas 130m
moor mm or soxs or GIRLS Ill mas IN moor
I 6 1h 9 130 - m 23
III 5 11 11 119 - 136 22
v 6 11 13 130 - no 21:
VII 5 __§__ __1_2_ 118 - n7 _2_g_
ram. W
camrs on us 89
99M (ems-comm)
II 6 8 10 131 - 1&9 18
IV 5 5 12 118 - 129 17
v1 6 10 13 128 - lhz 23
VIII 5 I __§_ _y_ m .. 130 $2—
rom. 00:11:01. (moors 31 1:6 77

 

 

 

4-- .

Other personnel in _thg study. The regular classroom teacher

 

remained in the room and observed the experimental lesson at most of the
sessions. It was felt that this would have a certain inservice training
effect. Although no stuck was made of the attitudes of these teachers,
their feelings of anxiety were often expressed to the author. They
indicated that they felt that the conceptualised presentation, the free-
dom which the students enjoyed in planning their own lessons, the fact
that the sequence of the text was not followed, and that no regular
assignments of problems to be completed and handed in would cause the
children to fall behind in their grade level of achievement. It seemed
difficult for them to disengage themselves from the arithmetic program
as taught by the special teacher in their rooms. Pressure and anxiety
became so great in some cases that the regular classroom teacher was
requested to leave the room during the experimental class.

it the beginning of the study, the patrons of the school exhibited
crest anxiety about the experiential prom They Incentive-hex: rm
whatpageofthetextthe classwas on; (2)whytherewerenepapers
worbdandbroughthomebythechildreni (3)1ftheirchildrencouldnot
be mm from the experimental class because 'thsy were learning
nothing“. In order to answer. such questions, group meetings were held
in which the total experimental program was explained to the parents.
They were invited to visit the classes and often did so. Frequently they
asked to come and learn some of the 'new' mathematics with the children.
is the study progressed, the support of the parents appeared to increase.
In many cases they expressed a desire to have the program continue as
part of the regular curriculum. is an outgrowth of the experimental

I.

 

57

arithmetic, an adult education class in 'Arithmetic for Parents' was
inaugurated. This class was taught by the author. The parents showed
keen interest and often expressed the feeling that they could now
“understand what their children were doing.‘

The W _and cultural as involved in the study were
chosen mainly because of their accessability, since it was necessary for
the experimental teacher to travel from one school to the next in less
than thirty minutes. (he of the schools was in an area of the city
which contained factories for manufacturing heavy machinery, the other
participating school was in a residential district containing only local
stores and offices. Table VIII indicates the types of employment listed
on the Form CA 39's in the schools. These should be regarded as an
indication only as these records are not detailed. When both parents

were listed as employed, both occupations were counted.

58

TABLE VIII

CBCUPLTIONS w PARENTS OF STUDENTS IN STUDY AS INDICATED ON
FORM CA 39 FILED IN THE TWO PARTICIPATING SCHOOLS

amorous DISTRICT - moors I: _I__I_, III, __I_v_

SEIF SKILIED AND
EMPLOYED LABORERS ADMINISTRATIVE CIERICAL UNKNCNN
Experimental 2 33 ’ 6 l 2
Control 2 21 11 3 '- 1

'RESIDENTIAL' DISTRICT - CROUPS I, VI, VII, VIII

Experimental 6 ll 23 0 1

Control 1: h 31 2 1

 

 

59

Table II indicates the general academic background of the parents
in the two geographic areas. These listings were taken from the CA 39
records in the participating schools and should be regarded as an
indication only. They do however offer so- idea as to the composition

efthe area.

TABII II

ACADEMIC BACKGIOUND (F PARENTS (F STUDENTS
PARTICIPATING II THE STUDY

 

 

'PLCTGI" DISTRICT
DUCATIGI (F PARENTS II YIARS - moors I, II, III, IV

0-8 9-12 1.2-1.6 “16

 

 

kperimental 101 82s 51 1:
Control 8% 811 7% I:

'MIDHTIAIP DISTRICT
DUCATIGV (F PARENTS IN YEARS - moors V, VI, VII, VIII

Experimental 31 _ 66$ 29$ 1%
Control 71 611 2h} 35

 

CHAPTER V
DATA AND RESULTS

The achievement of the students in this study was evaluated on
the basis of the regular testing program of the school. It was felt
that this would offer some indication as to'whether or not the children
had progressed satisfactorily in.the regular school program. This was
due to the fact that it was an integral part of the study to ascertain
if the arithmetic content required by the curriculum of the school
system.cou1d be accomplished: (l) in a decreased class time: (2) without
drill or rote learning processes; and. (3) With a decreased quantity of
"paper and pencil! problem solving.

In addition, the achievement of the students in this study as
measured by the regular testing program of the school, could be used
to test the assumption.that the 'experiemental' teaching method was equal
to or better than the routine (control) teaching method.

As pointed out above, all students were, at the inception of the
study, given the STEP Test Form.A. The California.Achievement Test -
Arithmetic Form.BB - Reasoning and Fundamentals had been given in the
school routine testing program. They were likewise tested at the
conclusion (the end of the academic year) of the study. These standard
tests were used to measure the relative gain in.achievement of students
after the treatment (tangible and conceptualised method) as opposed to
routine teaching methods (control group).

Mig- gfi t_h_§ 23333. The "before-after” comparison was made by

the use of analysis of Covariance. Iindquist, Taves, and others have

(_

i' C »J

61

demonstrated the superiority of the analysis of variance model over most
other statistical techniques. This model is especially superior to a
standard "differences in the means” model which is based on the (t) dis-
tribution and hence 'breaks down for small samples primarily because the
standard deviations of small samples are not normally distributed."86
‘When testing the hypothesis that the samples were drawn from equally

variable populations, rather than dealing with the difference between the

 

observed standard deviations (t) we deal with the 33312 between corre-
sponding estimates of the true variances as represented by the "variance
ratio" of F or t2. The power and versatility of the analysis of
variance model is demonstrated by the fact that it enables us to deal
with the mean of the variances around the mean within the groups as well
as the variance of the group means.87

‘When dealing with the analysis of "before-after" data normally
the experimental design demands "matched' or equated groups to secure
increased precision. 'This of course means a loss of valuable informer
tion, and loss may sometimes offset any advantage gained by the use
of equated groups."88 This 'loss' can be prevented by the use of
analysis of covariance, an extension of fisher's analysis of variance
model to include regression techniques. Therefore, the analysis of

covariance, in addition to the advantage of the use of Istatistical

 

86E.F. Lindquist. Statistical Analysis in Educational Research.
Boston: Houghton Mifflin campauy. F; . "'

 

 

871b1d., P. 87

881bid., P. 180

\l

62

control" over Rmatched" or equated group experimental control functions
to: (l) utilize regression techniques to statistically matched groups
by cancelling out the effect of initial score differences on final
scores; (2) applies analysis of variance to those adjusted final scores
to determine the significance of the difference between groups after
allowing for initial score differences; and hence, (3) makes assumptions,
as to the characteristics of the data, similar to those implied in the

use of analysis of variance and regression techniques.89

Using the analysis of covariance model the data were analysed to
test the general hypothesis that there are no real differences between
the teaching methods (experimental versus control or routine) and that
any differences in final mean scores (when initial scores are taken into
account) of the methods groups, are due entirely to chance fluctuations
in sampling.

Three specific hypotheses can be tested with this data:

Hypothesis I. No significant difference exists between.the

teaching methods (experimental versus control)
indicated by the achievement gain as measured
by the STEP Test Form A.

Hypothesis II. No significant difference exists between.the

f teaching methods (experimental versus control)

as measured by the California Achievement Test -

Elementary Arithmetic Form BB, Reasoning.

 

89Marvin J. Taves, "The Application of Analysis of Covariance in
Social Science Research,‘I Reprinted from.American Sociological Review.
15:373-381, 1950.

63

mpothesis III. No significant difference exists between the
teaching methods (experimental versus control)
as measured by California Achievement Test -
Elementary Arithmetic Form BB, Fundamentals.

TABIE I

ANALYSIS OF COVARIANCE 0F STEP TEST w Fm A
ACHIEVEMENT SCORES OF GROUPS TAUGIT BY THE
EIPERD‘IEN‘LAL AND C(NTROL METHGJS

 

‘

SUIB 0F SQUARES ERRCRS OF ESTIMATE
Source Degrees AND PRGJUCTS Sun Degrees
Vargition Fregfion 2:2 2. xy 2 ya qufres mated“ 8332
Total 166 11.186 12216 was 3921. 165
Between 1 0 0 ~ 0
Within 165 lhlB6 12 217 lhh32 3665 1614 22
DIFFERENCE FOR TESTING MEANS 259 l 259

_ Mean square difference (betseep)

F Mean square within

-£g-g— - 11.77 significance>.Ol

Degrees of freedom - k " 1

k( N-l)

 

 

Table I shows that the F ratio equals 11.77. This ratio is
significant at the > .01 level. Hypothesis I can therefore be rejected
and interpreted to mean that the achievement gain of the experimental
group over the control group as measured by STEP test can be attributed

to teaching method.

.3

 

6h

TABLE II

ANAHSIS G“ COVARIANCE OF CALIFGINIA ARITHIETIC TEST
(REASONING) - ACHIEVEMENT SCORES OF GROUPS TAUCH'IT
BI EXPERIBENTAL AND CONTROL mmms

 

 

 

*
t

SUIB (IF SQUARES moss OF ESTDIATE
Source Degrees AND PRODUCTS Sun Degrees
of of

- of 1’ Mean
Variation Freedom Z x2 2:. v 2:2 Squares FreZdon Squarez

Total 165 1319?. 115M MN: 11062 165

Between 1 148 119 295

Within 16h 13m 11660 23939 11660 1614 71
omen rm TESTING ADJUSTED MEANS 2702 161; 71

F- Hean square difference (between) _ 2702 _

De eoffreedon-k‘l
gre k(Nv‘-I

 

 

 

The F ratio, as presented in Table II, is 38.05 and is significant
at the ) .01 level. Hypothesis II can be rejected and when based on
the California Arithmetic Test of achievement, the experimental group

shows a significant achievement gain over the control group.

L.

.J.’\

'0

t—fi

65

TABLE III

THE ANALYSIS OF COVARIANCE OF CALIFGiNIA ARITI-D’IETIC TEST
(FUNDAMENTAIS) - ACHIEVEMENT SCORES (F GROUPS
TAUGHT BX EXPERIMENTAL AND CONTROL METHOD

 

 

SUMS OF SQUARES ERRORS 0F ESTIMATE
Source Degrees AND PRODUCTS Sum Degrees
of of 1 of z of Mean
Variation Freedon £12 2: 13* E y2 Squared Freedom Square2

 

Total 165 5661 3852 12800 14551; 165
Between 1 101; 83 66
Within 161; 5557 3769 1273b 10178 16h 62
omen: FOR TESTING ms 0 16h 62
_ Mean square difference (between) . 0 .
F Than square within 55 0

 

Degrees of Freedom II I: - l
klN-U

 

 

The F ratio, as shown in Table III is 0; therefore, Hypothesis III
must be accepted. When neasuring achievement with the California Arith-
metic Test - Fundamentals, the experimental teaching method is not
superior to the control method.

In general, it can be stated that, when Wald andst‘angible
method is used in comparison with routine teaching methods, students
show a statistically significant achievement gain as measured by the
STEP-Fora A and the California Achievement Test - Elementary Arithmetic

Fundamentals -BB - (Reasoning). This supports the “mum underlying

 

\

 

 

66

the experimental teaching method and indicates that a real learning gain
can be accomplished even under the handicaps of limited time allotment

and paper and pencil work over routine teaching.

CHAPTER VI
SUMMARY, CONCLUSIONS AND IMPLICATIONS FOR FURTHER STUDY

Swirl. It has been the purpose of this dissertation to compare
the effect of a tangible and conceptualized method of teaching arithmetic
on achievement in fifth and sixth grades with a self-contained classroom
method. The tangible and conceptualized method was devised to teach the
existing arithmetic curriculum by presenting arithmetical theory within
a culturally significant situation. This was done in order to replace
rote learning processes with a sound mathematical foundation which would
furnish a meaningful arithmetic experience at all levels of academic
ability. It was additionally the purpose of the thesis to show that this
could be accomplished in thirty-five minutes per day. The burden of
proof of understanding rested entirely with the students, who were
required, not to complete an assignment, but to produce some tangible
evidence of their understanding of the concept.

Conclusions. Analysis of the data revealed a significant difference
in favor of the achievement of those students who were taught with the
tangible and conceptualized method in: (1) STEP Arithmetic Test - Form is
(2) California Achievement Test - Elementary Arithmetic Form IE. Reasoning.
There was found to be no significant difference between the experimental
and control groups in performance on California Achievement Test -
Elementary Arithmetic Form BB - Fundamentals.

_ m the basis of these results, it sq be assumd that it is pos-
sible to meet the arithmetic curriculum requirements in grades five and
six by using a tangible and conceptualized presentation which explains
arithmetic rather than by rote-teaching methods. It m further be
assumed that this can be accomplished in 35 minutes of class time per day
without ability grouping, home work, or drill.

Continuation of; fig 53‘ The limitations of this study became
apparent during the progress of the research. The arithmetic program was
presented with a method that did not involve the use of any existing text-
book materials. It would appear that the preparation of such a text would
be indicated so that the study might be repeated by a teacher other than
the author of the materials. These materials were experimental in nature
and developed with the c00peration of the students themselves. There is

 

{V

 

68
a need for further investigation of the contribution which students are
able to make in the structuring of an arithmetic program.

The time spent in class and amount of Qpaper and pencil" work were
very limited. It is the intent of the author to follow the progress of the
students involved in the study through the twelfth grade. This seems indi-
cated in order to investigate the effect of such a limited mathematical
program.on the students performance in the areas of secondary mathematics.

Although it was not a part of the study, the teaching method
employed theories of learning involving the understandings of broad cons
cepts and the application of these concepts by the students to social
problem.situations. Further investigation is needed to evaluate the
worth of such a methodology. ,

Uses 9; Egg materials developed. For this study, the content of
the arithmetic curriculum was divided into approximately forty concepts
or generalizations. These furnished the bases for a.year's program in

arithmetic. Such a program could be adapted to a television.presentation

 

of mathematics. The procedures involved would be:

1. Initial presentation of the arithmetic concept by the
television teacher.

2. Development and investigation of’the concept by the students
and the classroom teacher together.

3. Production by the students of some tangible evidence of their
understanding of the concept.

h. Evaluation and application of the concept to a social.pr0blem
situation by the students.

In addition, it would appear that there is a need to:
1. Investigate other methods of evaluating a study of this nature.

2. Development of instruments to evaluate the understandings of
concepts and their application.by students to problem situations.

3. Development of instruments with which to evaluate the under-
standing of pure mathematical theory that may be possessed by fifth and
sixth grade students.

h. Re-evaluation and re-interpretation of the purposes and goals
of arithmetic programs and their relationship to the culture.

69

LITERATURE CITED
BOOKS

Banks, J. Houston. Learning and TeacEg Arithmetic. Boston: Allyn
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Batchelor, Julie Forsyth, Comnication: tron Cave Writing to
Television. New York: Harcourt Brace and Her , Incorporated, 1953.

 

Bushnell, G.‘r. "Introduction.” Fiftieth Isarbooh_9_f the National
Societ for the 8 of EducationJ Part II. Chicago: University
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Russell, G.T. I'Arithmetic", En cl cdia of Educational Research.
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Clark, John 3., Arthur S. Otis, and Caroline Button. 251...“! Arithmetic
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Craig, Gerald S. Science for _th_g Elementary School Teacher. New York:
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Daugherty, Janss Innis. 'A Study of Achievements in Sixth Grade
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DsMay, m J. Guidi Be ers in Arithmetic. Evanston: Row Peterson
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Dem, Robert Bay. "A Tito-Year Study of an Increased Tine Allotment Upon
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Ch‘ossnickls, Foster E. 'Introduction,‘ Instruction in Arithmetic,
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(hthrie, 13.12. The Psycholog of Learning. New York: Harper and
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(K

a.

70

Hartung, Maurice Le, 33; _a_l_. M1? the Course for Arithmetic.
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Hilgsrd, Ernest R. Theories of Learning. New York: Appleton-Century-

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Hogben, Lancelot. The Wonderful World of Mathematics. Garden City:
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Judd, Charles H. Educational Pszcholoq. New York: Houghton Hifflin,
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Lankford, Francis J. I'Ilplications of the Psychology of learning for
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lindquist, E.I.. Statistical sis in Educational Research. Boston:
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McHugh, Walter Joseph. 'Pupil Teal Learning in Skill Subjects in
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Monroe, Walter 8. Development of Arithmetic as a School SubJect. United
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Muller, Frances J. Arithmetic Its Structure and Conc_ep . mglvood
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Pfiaget, Jean. The e and Mt of the Child. New York:
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Price, G. Bailey. ”Progress in Mathenatics and Its Ilplications for the
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Rockefeller. Report. The Pursuit of Excellence, Panel Report V of the
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re

I.

 

t
‘ J
z
c
t
t
t
r
L
e i
Q
Q
l
t
t
O
‘ t
t J

'x u'ie'hno:

‘ J

A»

71

Smith, David Eugene. History of Mathematics. New York: Ginn and
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Tenbergen, N. The Study of Instinct. London: Oxford University
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Thorndike, Edward L. Theories of learning. New York: Bureau of Publi-
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Smith, David E. Numbers and Numerals. Washington: National Council of
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PERIODICAL LITERATURE

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Brown, J.C. 'An Investigation of the Value of Drill Work in the Funds--
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Brounell, William A. "Arithmetic in 1970, " National Elementary Principal.
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Brosnell, William A. and Charlotte B. Chesal. ”The Effects of Premature
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[1

K4

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72

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Inndberg, Hazel.- ”Mathematics in Elementary School,” Educational
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PAWHLETS

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State College of Washington, 1950.

Inchigan Depart-snt of Public Instruction, ”IIprevenent Projects Related
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nineographed uterial. Lansing: mchigan Department of Public
Instruction.

”School hthenatics Study (h'oup. Newsletter No. 10.” Inland Stanford
University, 1961.

 

 

APPENDIX

75

TABLE XIII

SAMPLE (IF WEEKII PROBLEIVB GIVEN IN GRADE FIVE

ADD:
627 1:5
356 27
a g;
502
__9.
SUBTRACT:
1:5 307
3.1.. .99..
MULTIPLY:
32 1:8
.2. _2.
DIVIDE:
16 / 35

 

 

1/3 1 1/2 1. 1/3 16 us 829 5/8

13 1.5 __2____3 23 26
1/:3 2/; 2.4.4.12. .?__L§

1:009 726 11/12 h/S 1 5/8 1.5 6/8

999. _flLflliiéE—EEL

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77
TABLE IV

SCORES OF STUDENTS IN GROUP I

 

 

CALIFORNIA ACHIEVEMENT TEST

ELEMENTARY ARITMTIC FORM BB STEP ARITHMETIC roan A
Code Reasoning Fundamentals Reasoning Fundamentals
no. Before Before After After Before After
26 ho9 5.h 6.h 5.9 37 39
27 3.6 3.9 h.6 h.5 23 26
28 6.h 6.0 7.h 6.1 36 39
29 he? 5.0 6.5 5.7 33 35
30 6.0 h.8 6.2 5.6 29 26
31 5.1 5.2 5.7 5.3 29 35
32 h.9 5.h 6.5 5.8 32 38
3h h.5 5.1 5.2 5.8 19 28
35 5.1 5oh 7.9 7.0 h5 h?
36 ‘ ho5 3.8 5.9 h.9 17 25
37 5.8 5.0 6.0 5.3 3h 38
39 5.3 h.8 6.5 6.2 to 1:6
to 5.9 5.1. 6.7 6.2 31 u
hl h.6 h.8 h.9 h.6 15 19
h2 5.8 5.h 6.h 5.9 3h h3
h3 3.9 h.h h.6 5.0 12 11
hh 6.1 5.7 7.2 6.7 h3 h5
h5 hez h.2 he? 5e6 19 16
h? h.h 5.0 7.2 6.5 3h 33
h8 h.2 5.h 5.2 h.8 . 26 28
A9 5.h 5.1 6.7 6.0 25 37
50 he3 “.9 he9 5e? 28 33
51 5.9 5.h 7.h 5.8 h2 hl

SCORES OF STUDENTS IN GROUP II

1 6.1 Seh 6.9 6.2 33 33
5 6.1 h.8 6.h 5.5 33 31
6 hel hes 3.7 he? 16 11
8 5e3 he5 he9 5.7 35 23
10 he“ 5.2 6eh h.8 13 22
11 5.9 5.2 5e? Sen 30 35
12 5.6 5.6 6.5 6.2 28 33
13 5.9 he9 5.6 5.7 33 39
1h h.1 h.5 ho9 5.5 23 29
15 5.8 5.6 6.7 6.1 36 37
16 h.2 h.5 5.0 h.7 21 17
19 h.8 5.1 5.9 6.1 39 h6
20 h.0 h.5 h.7 5oh 32 29
_ 21 ‘e6 Ssh 7.2 6.0 ho N6
22 he? 5.7 6.2 6.0 33 36
23 h.8 h.9 6.0 5.9 27 3h
2h 3.7 3.5 h.9 heo 16 17
25 h.9 5.1 5.5 5.0 2h 29

 

 

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78
TABLE IV (Continued)

SCORES OF STUDENTS IN GROUP III

 

 

CALIFORNIA ACHIEVEMENT TEST

EIEMENTARY ARITI-BIETIC FORM BB STEP ARITHMETIC FORM A
Code Reasoning Fundamentals Reasoning Fundamentals
no. Before Before After After Before After
53 5.h 5.h 8.h 7.2 36 38
55 he? 5.1 6e7 5e7 21 32
57 5.5 h.5 5.7 5.1 29 3o
59 6.5 5.7 7.7 6.0 27 38
60 5.5 5.1 7.3 7.6 33 35
61 h.8 5.1 6.3 5.1 2h 28
62 h.6 5.5 6.1 6.3 26 35
6h h.h h.1 6.3 5.9 13 28
65 5.9 h.7 6.5 6.7 25 36
67 3e3 he3 5e3 5.8 10 28
69 h.8 h.5 6.3 b.7 21 22
70 5.9 5.h 7.h 6.7 26 25
72 3.6 h.8 h.2 h.5 13 16
73 5.5 3.8 5.9 5.5 an 28
7h h.2 h.9 5.1 h.2 17 2h
75 h.7 h.8 6.5 6.1 19 17
76 5.0 5.h 6.7 7.3 27 27
77 5.5 5.6 7.1 6.5 32 36
78 6.2 6.h 6.9 6.h 32 uh
79 5.0 he9 7eh 5.1 9 21
80 5.2 5.2 7.h 8.h 23 27
81 2.7 5.0 6.1 5.h 15 26

SCORES OF STUDENTS IN GROUP IV

82 7.1 6.3 5.9 6.1 29 39
83 Ssh Ssh 6.1 5.7 28 32
87 5.5 5.1 6.1 6.7 33 36
88 5.2 5.3 h.5 5.6 15 19
89 6eh 6.2 9.3 9eh hh h6
90 he9 5.2 6.5 6.0 22 32
91 5.5 5.6 5.9 5.6 3h ho
92 h.l 5.3 6.1 5.7 9 18
93 3.5 h.0 hoe 5.1 11 15
9h he7 5.6 h.8 5.2 10 16
97 he2 he7 5.5 5.8 32 16
98 hel 3e5 heh he2 12 ll
102 h.7 5.h hes 5.1 15 19
103 h.9 5.0 6.5 5.6 21 21
10h 6e0 5.5 7.1 6.0 35 hl
105 h.h 5.7 h.l 5.1 13 16
109 6.9 5.6 9.6 5.7 36 h2

 

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x.
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79
TABLE XV (Continued)

SCORES OF STUDENTS IN GROUP V

 

 

CALIFORNIA ACHIEVEMENT TEST

ELEMENTARY ARITHMETIC FORM BB STEP ARITHMETIC FORM A

Reasoning

Fundamentals

Reasoning Fundamentals

Before Before After After Before After
3.0 h.8 5.7 6.h 19 1h
ho? he6 6.5 5.8 35 h5
h.7 h.6 6.5 5.8 25 33
h.6 he3 6.3 6.7 20 3h
5.6 5.0 7.1 6.5 30 37
6.1 5.1 3.7 6.9 hh h6
5.9 5.1 8.h 6.7 hh ho
5.9 5.h 7.6 6.3 32 38
heh he3 5.5 5.6 17 2h
5.1 ho5 8.1 6.3 36 hl
5.6 5.5 8.h 6.7 38 h1
h.9 h.9 7.7 6.3 2h 33
2.7 3.5 5.3 5.3 15 18
he2 3.9 6.5 6eh 22 29
h.3 5.1 6.5 5.9 19 22
6.1 5.1 9.3 7.7 hh h3
5.3 h.3 6.9 6.6 35 h2
hes he7 6.9 6.9 3h 36
h.8 h.3 7.h 6.8 32 37
h.h h.9 6.5 5.9 26 33
h.9 h.7 6.7 6.1 30 38
Sch 5oh 7.1 7.2 30 h3
h.5 he5 6.3 6.5 3h 33
5.9 5.0 5.1 7.1 33 h6

SCORES OF STUDENTS IN GROUP VI
5.7 5.0 3.1 6.h 37 35
5.h h.9 7.7 6.9 37 h5
17.7 5.1. 6.9 7.6 39 he
he3 hes 6.5 6.6 21 31
5.3 Ssh 8.1 6.9 hl 38
5e9 5.6 7.7 7.3 hl hh
6.h 5.8 8.1 7eh 38 h5
5.8 5.9 7.1 7.h 32 38
3.9 3eh h.5 5.2 11 10
6.6 5.8 8.7 8.1 39 hl
6.6 5.3 8.h 7.h hO 37
6.1 5.6 6.3 6.5 38 39 7
ho6 h.7 7.1 6.6 25 28
5.3 h.7 6.7 6.6 2h 2h
5.1 5.5 8.7 7.2 39 h?
5.9 5.3 7.7 7.h hl h?
h.5 hes 5.0 5.9 21 26
heh h.9 6.9 6.2 29 39
5.1 h.h 5.5 6.5 29 35
5.9 5.1 6.9 7.9 h1 h1
17.8 5.1 6.9 6.14 27 33
5.9 5.7 8.7 7.8 37 h2
5.6 6.2 7eh 6.3 31 36

 

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80
TABLE IV (Continued)

SCORES OF STUDENTS IN (BOUP VII

 

CALIFORNIA ACHIEVEMENT TEST

EIEPIENTARI ARITHMETIC FORM BB STEP ARITHMETIC FORM A
Code Reasoning Fundamentals Reasoning Fundamentals
No. Before Before After After Before After
170 3.6 h.6 5.5 S.h 7 15
171 5.7 5.3 7.h 6.1 25 3h
17h 7.2 5.6 7eh 5.8 26 29
177 8.8 h.h 6.1 5.7 20 28
179 5.2 5.6 6.1 5.5 l7 l7
180 7.0 8.7 7.7 6.9 33 35
181 5.0 11.6 6.2 5.2 26 27
183 h.9 5.7 6.5 6.3 2h 23
185 5.9 h.1 6.7 5.6 2h Zn
187 5.h 5.0 7.h 6.2 29 36
189 6.6 5.6 8.h 6.8 36 hl
190 6.1 5.5 5.9 5.6 23 27
192 5.0 5.1 7.1 6.0 25 27
193 6.8 5.8 9.3 6.8 35 33
198 5.0 h.8 6.5 5.8 25 31
196 6.2 5.3 5.3 5.9 28 36
197 5.2 5.6 6.7 5.6 20 28
198 3.8 h.h h.h 5.2 ‘ 1h 12
199 6.6 5.0 6.7 6.h 27 33

SCORES OF STUDENTS IN GROUP VIII

202 6.6 5.5 8.1 8.1 30 37
203 6.2 h.6 5.5 6.8 18 28
295 h.5 he? 7.1 5.5 22 37
206 h.7 3.9 h.7 3.9 17 30
207 6.1 S.h 6.9 6.0 16 37
209 5.9 5.3 6.9 6.2 27 28
210 7.8 5.6 8.7 7.6 38 81
211 5.6 8.8 5.0 5.0 11 2b
213 8.2 h.h 5.0 6.1 10 18
218 3.3 h.0 5.3 5.8 1h 21
216 h.9 5.7 7.7 6.5 2h 33
218 8.7 h.7 6.3 6.1 20 17
219 h.1 5.1 h.7 5.6 17 12
220 6.8 5.5 5.9 5.8 28 30
221 5.0 5.3 6.7 6.0 18 26
222 5.6 8.8 5.7 5.6 13 27
223 6.0 6.0 6.1 6.3 27 30
227 6.2 5.8 7.1 6.7 39 hl

229 3.5 h.6 3.3 h.7 13 2h

«L.

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‘\-

ARITHMETIC THEORY 110cm T0 was
FIVE AND 31: 1s EXPLANATION Foa
CONTENT AREAS FOUND IN
CURRICULUM 11' THESE LEVELS

I. mum

Arithmetic as a Sylbolie Language
(hr m Number System
Suggested Activitiu

ARITHMETIC AS A SYMBOLIC LANGUAGE

Numeration is a way of expressing numbers in words. It is dif-
ficult to figure in written words alone; so over the years, different
peoples have invented different ways of expressing numbers (or quantities)
'with symbols. Many types of symbols have been used throughout the his-
tory of mathematics. It would appear that man has always tried to make
whatever number system he used more simple and convenient. In this sense,
one could say that arithmetic just "grew". It has grown for some six.or
eight thousand years. Many number systems might not be recognized as
such today.

Counting is the foundation of arithmetic. People count in many
ways. For the most part, one can count to the answer in an arithmetic
problem - if one has time enough. Since our mathematics should be simple
and convenient in order to be of the most use, we try to count in bunches
(or groups) rather than by naming every different quantity that we wish
to express. ‘We count in bunches (or groups) or tens. ‘we call this a
base 10 number system. There are other people who count differently.

Some systems are very simple. There are records of one like this:

one (1)
two (2)
some (3)

many (any amount of more than 3)

It makes little difference what these symbols may be or how they
are named. 'we can see that in this kind of a system four could be either
many, or some plus one, or onepplus some, or perhaps just onesome as one
word. This would be something like our number twentybone. ‘When we say
'twenty-one"ws think of an amount, and seldom.stop to think that it is

equal to 2 tens plus 1 one. Other people use other words to express amount.

8h
There are said to be some Eskimos who use a system of naming their
fingers and toes and combining these names into a counting system. In
this way, they count in groups of five instead of ten as we do. They can
do this until they get to twenty and (having run out of fingers and toes)
they are forced to count in groups of twenty. They might figure their

number system.something like this:

1 atauseq 1 finger
2 machdlug 2 fingers
3 pinasut 3 fingers
h sisamat h fingers (hand less 1 finger)
S tadlimat 5 fingers (hand)
6 achfineqpatauseq 1 hand and 1 finger
7 achfineqrmachdlug 1 hand and 2 fingers
8 achfineqepinasut 1 hand and 3 fingers
9 achfineqesisamat 1 hand and h fingers
10 qulit 2 hands
11 achqaneqpatauseq 2 hands and l to.
12 achquanqumachdlug 2 hands and 2 toes
13 achqaneqepinasut 2 hands and 3 toes
1h echqaneqesisamat 2 hands and h toes
15 achfechsaneq 2 hands and 1 foot

16 achfechsaneqyatauseq

l7 achfechsaneqemachdlug
18 achfechsaneqypinasut

l9 achfechsaneqesisamat

20 inuk navdlucho

hands, 1 foot, and 1 toe

hands, 1 foot, and 2 toss
hands, 1 foot, and 3 toes
hands, 1 foot, and h toes
hands and 2 feet

NNMNM

You could make your own number system and do problems in it. You
‘would need to make up some symbols to stand for amounts, and then show
how they could be manipulated in order to keep track of things. The way
the Eskimos have done should give you some ideas.

In the history of our number system, many things have been used to
stand for different quantities. People have used letters like the Romans
and the Greeks, and pictures like the Egyptians. Usually, the symbols had
some meaning or relationship to the quantity that it was to represent,

as in the Eskimo system above.

85
For instance: a nose could stand for one, etc. We might have a system
like this:
Nose stands for 1. (Almost everyone has just one of these.)
Eyes come in.pairs. So, eyes have stood for two.

Sometimes wings were used.

Stopping to draw 2 eyes every time is a bother,

so one eye might come to mean 2 all by itself.
Nose-eye combined could stand for 3.

Since this is rather a nuisance to have to stop

to make these two separately . . . . . .

The symbol for 3 might come to be no more than
something like this. <55

Symbols have grown and changed over hundreds of years in much this
same way. They have grown and changed over the centuries to meet the
needs of a society. As one studies the history of ancient societies and
civilizations, we find a vast variation of the symbols used; but they all
served a particular need for some particular civilization. Many of these
'were adapted by other civilizations and were changed and fitted to their
needs much as we have done with our own Hinku-Arabic system. As we die-
cover and create new forms of energy along with space exploration, we
will probably be adding new symbols to our system to meet the needs of
mathematics. A good illustration of this is the light year. If we
figure that light travels about 186,000 miles a second, we can calculate

the speed that light would travel in a year by:

186,000 x 60 x 60 x zh X 365 - 1 light year

OUR OWN NUMBER SYSTEM

The exact beginnings of our own number system are vague. It is
believed by some historians that it has been in use since the fifth or:
sixth century, B.C. It appears to have been in use for some time by the
nomadic or 'heathen" tribes of Arabia and India before it was recognized
as being superior over the old Greek and Roman systems.

Historians disagree as to the exact time and.place of its origin.
Certain it is that the system.has changed a great deal over the centuries.
The symbols are of Hindu origin. The nomadic Arabians may have picked
them.up in India and carried them to Europe without using them.much. It
probably appeared first without the zero. This would have given it little
advantage over the other older systems. It is thought that the zero may
have come into use about the ninth century. The Hindus hit upon the ingen-
ious idea of place value, or grouping of numbers. In this way, any
amount can be expressed with only 9 digits and the zero to hold an empty
place. It should be kept in mind that this so-called zero could be any
symbol that would hold the place.

This Hindu-Arabic system was known in Europe as early as the
thirteenth century. It was not much used until developments in science
and trade made its computational advantages preferred over the old
existing systems. This was possibly about the sixteenth century.

It is important to remember that over the past 2,000 years or so
our number system.has changed. These changes have been influenced by
the changing needs of science and society. The Arabic and Hindu influences
had.much to do with its early formation. Our system.has grown and changed

with the centuries and it might be expected to continue to do so.

86
SUGGESTED ACTIVITIES

I
Have the children imagine that they have need to "keep track of some-
thing“; but that they know of no number system to use. Ask them to make up
symbols to stand for amounts and then do a problem or two in their system.
II
Make a panarama to show how a shepherd might keep track of a flock
of sheep by using 1 pebble to stand for each sheep.
Do the same thing, but show a 1 to 5 relationship or a l to 10.
(One pebble for'S sheep, or 1 pebble for 10 sheep.)
III
Dramatize a situation of bwing and sellirgin which characters use
primitive ways of counting.
IV
Look up measures uled during Biblical times and produce charts,
booklets, or posters describing them.
V
Make a sign language for quantities and dramatize a situation
involving a sale or exchange in which bargaining takes place.
VI
look up the signs used by auctioneers to indicate amounts.
VII
Make a time line showing the development of different number systems.
Use white wrapping paper and illustrate.
VIII
make a simple abacus or counter and demonstrate it to the class.
A good one can be made with a shirt box and.buttons on strings for
counters. Others can be made with clay. Use the clay for grooves and

small stones for counters. ‘wonderful world of Mathematics by Hogaben has

 

some good material in it. This book should be in your library or office

 

II . PLACE VLLUI

Pattern of Expressing a Nuneral
Use of theses in Place Value
The Passes of a Base (Espenents)

 

 

 

 

PLACE VALUE

We count in bunches or groups of numbers. This makes it possible
to do mathematics with only a few symbols. In our number system we count
in groups of tens. We call this a BASE ten, or we say we have a number
system in Bass Ten. We have nine symbols to stand for the quantities from
one through nine: 1, 2, 3, h, 5, 6, 7, 8, 9, and a zero, 0. We need the
zero because we give numbers a value according to the 23,232 in which they
stand: 3 is equal to 3 ones. If we put the 3 in another place and make
30, we use the zero to literally push the 3 into the second place. It
now stands for 3 tens, or thirty. We can hold the ten's place and put the
three in hundred's place: 300. In each case, we have used the same symbol
to stand for three different amounts. We could go on doing this without
end.

We do not really have a symbol to stand for ten. We simply use a
l and put it in ten's place, using the zero to hold the empty one's place.
We read this number: 10, as 'ten“. This stands for one group of ten.
Ten is our base. The l in the numeral 10 stands for 1 2133s the base of
ten. If the base were something else, this one would stand for that
amount. This is also called the model group of a number system.

If we think about counting it helps us to understand about how the
£933) or model Eoup functions in our number system. It is really unfor-
tunate that we learn to count 23133;: we understand what it means, because
then we are so familiar with the process that we may fail to see the
importance of understanding it. Counting is the very foundation of
arithmetic. Counting with understanding can be a great help in learning
arithmetical processes. It is absolutely necessary to understand it in

order to master arithmetic processes.

 

(I

 

89

We use counting numbers to stand for different quantities. Counting
is adding by ones. Each number has a position in relation to the others.
That is, six always follows five, and comes before seven. We think of
cardinal numbers as the numbers one, two, three, etc. We think of the
ordinal numbers as first, second, third, etc. The number 3, for example,
has a face value of three things. We could put it into a one-to-one rela-
tionship with a group of m things. We could count these and the last
number we name "three" is the name of the number of members of the group.
The number 3 is also the third number we name. This is the ordinal way
of thinking of three. me good way to remember is that “third" expresses
the 9.51.25 or arrangement of the series of numbers.

Counting numbers can be expressed as they relate to the place in

which they stand. This is place value. A number has two values ... a face

 

value and a place value. The number 3 has a value that makes it equal to
m things. It also can be used to stand for 3 tens, 3 hundreds, or any
place that we may choose to use. We express this as though the number
were multiplying the value of the particular place it stands in. For
instance, 333 can be expressed as (3 x 100) + (3 x 10) + (3 x 1). This
is what 333 £93.13 means. It is necessary to think about counting numbers
in this way to understand how a number system really functions. When it
is thought of in this way, it is possible to understand arithmetical
functions in any base. This also furnishes a basis for the process we call
"borrowing" and “carrying".

The number 333 is simply easier to write than (3 x 100) «o- (3 x 10) +
(3 x 1). In writing 333, it is also easy to lose sight of what the number
really is. In order to see this more easily, it is well to indicate how
numbers could be taken apart.

 

 

1“

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7‘

 

 

90

‘We will use parentheses to indicate the multiplication, 3(1) I 3 x 1, etc.

hS - h(10 ’
h6 - h(10) *

99 - 9(10) +
100 - 1(100). o ) 4- 0(1)
150 - 1(100). 5(10) + 0(1)
268 e 2(100)+ 6(10) + 8(1)

1000 - 1(1000) + 0(100) + 0(10) + 0(1)
100k ' 1(1000) + 9(100) + C(10) + h(1)

h579 - h(1000) + 5(100) * 7(10) + 9(1)

.....etc.
It is easy to see when we look at numbers in this fachion that:
(l) the use of place value makes our number system much more convenient;
and, (2) we can.change the model group or base very easily if we change
the value of the places. This will become more apparent as we investigate

the structure of other number systems.

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5!

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USING AN ABACUS TO SHKN PLACE VALUE

Since 10 is our model group, we need to have only nine digits if
we use place value. 0n the other hand, igfiwe have only nine digits, we
are forced to use some other way to express quantities over ten. An
abacus can be used to demonstrate this very well.

Push.up ten ones on the ones column,
counting and writing the numbers as you do:

1, 2’ 3, h, 5, 6’ 7, 8, 9, ? Dee-O.

Now, after we get to 9, we have no more
symbols to express additional quantities, so
we use place value. Push down the 10 ones
beads and push up 1 ten bead on the tens
column. Now write 10. This is: 1(10) +

0(1). The zero indicates and empty column.

Do this again: push up ten ones. 'Ws
have no symbol for tens so push down the ten ones
and push up another ten head. ‘we now have 20,

or 2(10) * 0(l)e

Continue doing this until you have pushed
up all the tens beads. Now push these down and
use 1 one hundred head to stand for 1(130) +
00.0) + 0(1). Again, the zeros stand for empty

columns.

 

 

 

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92

You could continue until you have used up all of the hundred beads.
After using 9 hundred beads, it is easy to see that we have no symbol for
10 hundred unless we go to the next plagg_and, using place value, write
1,000. This is: l(lOOO)+O(10) + 0(1).

If we think carefully about place value, and remember that our
model group, or BASE, is ten, we can make a place value chart that shows
this mathematically rather than by using an abacus. Think first about
place value as we have learned the names for these places we will express
these by using numbers that show how many times ten is multiplied by

itself in order to equal each place.

THE POWERS OF THE BASE

tennmillions
hundred-thousands
ten-thousands
thousands

hundreds
tens
ones

millions

 

Each place is ten times the one to the right of it. For example,
tens are equal to 10 x l, and hundreds are equal to 10 x 10, etc.

Each place is also 1/10 of the place to the left of it. For
example, tens are 1/10 of 100, and hundreds are 1/10 of thousands.

Another way to understand this is to make a chart like the
following one in which the value for each plagg_is expressed as the

number of times ten is multiplied by itself.

 

 

93

l (ones)
10 (tens) 10 x 1
100 (hundreds) 10 x 10
1,000 (thousands) 10 x 10 x 10
10,000 (ten-thousands) 10 x 10 x 10 x 10

100,000 (hundred-thousands) 10 x 10 x 10 x 10 x 10
1,000,000 (millions) 10 x 10 x 10 x 10 x 10 x 10
10,000,000 (ten-millions) 10 x 10 x 10 x 10 x 10 x 10 x 10
We have a more simple m of showing this by using small More
written up to the right of the tens to show how many times ten is multio
plied by itself. We call these emuents. These small figures show the
powers of ten. We would read 102 as I"ten squared" or as 'ten to the
second power.‘ This means 10 x 10 or 100. Expressed in another way
101 x 101 - 102. Here you will see that we can add exponents in order
to multifiy numbers that are expressed exponentially. This is not particu-
larly time saving in the multiplication of small numbers, but it is very
useful in dealing with large ones. It also forms the basis for some very
necessary understandings in regard to place value and the base of a number
system. These ideas are also used in developing the concept of logarithms.
It would be well at this point to consider a chart showing how the

powers of ten can be used to indicate place value.

1 (ones) 10°

10 (tens) 10 x 1 101

100 (hundreds) 10 x 10 102

1,000 (thousands) 10 x 10 x 10 103
10,000 (ten-thousands) 10 x 10 x 10 x 10 10’4
100,000 (hundred-thousands) 10 x 10 x 10 x 10 x 10 105
1,000,000 (millions) 10 x 10 x 10 x 10 x 10 x 10 106

10,000,000 (ten-millions) 10 x 10 x 10 x 10 x 10 x 10 107

Ll

i
\
l\

 

-)

 

 

 

 

9h

We can see why ten to the zero (10°) is equal to 9513 if we do a
problem using the powers of ten. We will start with an example to which
we know the answer:

10 101
x 10 or 1:10

‘106 —m*
We have to get 102 for the answer since 10 x 10 - 100, and 100 is written
with an exponent as 102. Now, if we do a division example, we will see
why 10° is equal to one.
10 101
10 /"10"'0 or 101/10?"
This may also be expressed as a fraction:

2
£1”- 101

Since 100 e 10 - 10, we have to show the answer as ten. (Ten expressed
with an exponent 13 10]”; therefore, we can assume that we would subtract
exponents to divide numbers; or 102 e 101 u. 02"1 or 101.

Since 10 e 10 - 1, we can now show that 101 e 101 - 10°, b96013.
we subtract exponents to divide numbers. In this case, we could show it

as 101 e 101 u. 101‘1 or 10°. 01‘, in other words, ones can be expressed
as 10°. From this, we might conclude that: any number (not a zero)

whose exponent is zero (O) is equal to one, or N0 I 1.

We should now remember that our number system is based on a model
group of 10. Each position in place value can be expressed as a power of
ten. These powers of ten are written as exponents. Knowing these things
we can: (1) write aw number of quantities that we wish to; and, (2) we
can set up a number system in any base other than ten that we may choose.
It is necessary to understand this as a foundation to the mathematics

that is basic to arithmetic.

 

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95

Often in mathematics when we wish to generalize we let letters
stand for numbers. If we let the letter B stand for E base that we may
choose, we can make a chart that will show place value for any model
group or base. This will help to understand just exactly how place value

operates in our familiar base 10 system. We could. compare this with

base 10 e

Base 10 106 105 10h 103 102 101 10°

Any Base B6 B5 8’4 B3 B2 B1 B°

 

F‘s-”— ‘

 

 

III. OTHER SYSTEMS AND BASES

Egyptian Number System
Babylonian Number System
Greek Number System
Roman Number System
Base 7

Suggestions 1‘ or Teaching

THE EGYPTIAN NUMBER SYSTEM

Egypt is one of the oldest civilizations about which we have a
record. From their great tenples and tombs, which they constructed some
four thousand years ago, we have been able to learn a great deal about
their lives. Suprisingly enough, we know quite a bit about their mathe-
matics. They seem to have been a clever folk, quick to invent mathematical
ideas and methods that would help them in their daily lives. An arith-
metic book has been found dating from sheet four thousand years ago.

The Egptisns and the people in the countries around this ares. ap-
pear to have used a kind of picture writing for their words and for the
figures that they made. The Samarians, Babylonians, Arabians, and Hindus,
later the Cheeks and Romans, have all contributed to our mathematical
system. The earliest of these used hieroglyphics, or pictures, to make
their words and numerals. Man appears first to have done his figuring

only by drawing crude pictures‘to represent numbers. For instance:

b for one
D g for two sheep

It is easy to see how something like this could lead to s. hieroglyphic
writing and number system. There is some indication that men also wrote
out words for each number. Instead of writing a symbol like 6 or 7, they
would write out the word, six, and seven. It would be very difficult to
do much figuring with this kind of number system. So, symbols probabe
were invented as a matter of convenience. (he of the oldest of all these
systems is the Emtian. It is quite remarkable for its simplicity and

usefulness. Their figures, like their written words, were hieroglyphics.

 

/‘

(W

98

We have words that illustrate this idea. Water wheel, water shed, water

snake, and the name Waterman might be written in hieroglyphics like this:

water shed /\/ CB

water wheel N ®

water snake N W
“.9.

waterman r» 7\

The Egyptians Just made marks for the number from 1 to 9. It is
easier to read these if the marks are grouped, and this is how they look:
1 2 3 h 5 6 7 8 9

III
II III III IIII III
I II III IIII III III IIII IIII III

In addition to these, they had symbols for a sort of place value

much like ours. It was based on 10.

10 (W heel bone

100 coil of rope

1,000 lotus flower

10,000 bent line

6)
8
i
>
100,000 ? burbot
\9/

1,000,000 )\ man in astonishment

99

We do not know why they picked these particular symbols. It is
easy to imagine that a number like a million would be represented as a
"man in astonishment". He probably was just overcome by such a large
number. This reminds us of the very primitive number system of: I'one,
two, some, and a whole lot.“

The Egyptians had no zero. They had no need of it in this system.
The zero was to come many years later. If you write a few numbers, you

can see that they would not miss a zero.

16 n :i‘. or :1: n
56 88% .‘H or :1: 22m
101 9 I or ‘ 9
1,289 1992222555 or 99 i am 5;;
6,001 iifiil‘i. or . $3111
1,ooo,ooa I? I or I kg,

The Egyptians had a place value of sorts. Yet, it would make no
difference to them if 128, rm .'.'.’1 , was written as {111 9 00 or
n n 9 .117 or if." no 9 , etc. We can see that they used the prin-
ciple of addition in their system. It would make no differenceif they
added 10 «I- 6 to make 16, or 6 + 10 to make 16. This makes it rather
easy to do problems in Egyptian hieroglyphice because all you have to do
is to count the number of times a symbol appears and write it in tens.

We will see how this system Operated in the Roman number system later on.

 

ROMAN NUMBERS

In order to write Roman numbers, we use addition, subtraction,
multiplication, and repetition. When these are understood, reading and
writing these numerals becomes a simple process.

Roman numerals are written with the use of seven (and sometimes
eight) symbols: I V X L C D Maud sometimes 171. This is a very old
number system. Over the centuries there have been changes in the symbols
as we lcnow them today. At the time Roman numerals were first used,
people did not need very large numbers or the symbols to write them. As
the culture of the people developed and their possessions increased, they
had need of a better number system; so they changed the symbols that they
used to fit their particular needs.

Very early Roman numerals were quite different from those of today.
There were several ways to write the same number. Examples of such

numerals have been found in old ruins.

l—XL/h 12$ 9% £29 £90 1%?

EARIX ROMAN NUMERAIS

 

 

There are many interesting books about the development of our number
system. See:
Smith, David Eugene, and Jekuthiel Ginsburg, Numbers and Numerals. The

National Council of Teachers of Mathematics, mmeenth Street, N.W.
Washington, D.C. (35 cents postpaid)

 

Smith, David Eugene, _Nugnber Stories of [gag Age. The National Council of
Teachers of Mathematics, 1201 Sixteenth Street, N.W., Washington, D.C.

 

The National Council also. has fine lists of books on mathematics in many
”0&Se

I‘lldlllllililllll. Ill

 

 

 

 

101
THE USE OF ADDITION

V equals 5 I equals 10 L equals 50

0 equals 100 0 equals 500 M equals 1,000

A line over a numeral is sometimes used to multiply a number by

1,000; 1‘4 would equal one thousand thousand or . million.

VI equals 6 MDCXX equals 1,000 plus 500 plus 100 plus
10 plus 10 (1,620)

VIII equals 8 LIV equals 50 plus 10 plus 5 (65)

III equals 12 DCCC equals 500 plus 100 plus 100 plus
100 or (800)

With the exception of M (1,000), no symbol is repeated more than
three times. When it becomes necessary to repeat a symbol four times, the

next higher one is used and we subtract.

1 2 3 h 5 6 7 8 9 10 30 no
I II III IV V VI VII VIII II x xxx XL
50 80 90 100 200 300 too 500 1,000 1,000,000

L Lm x0 0 00 can on n M M

In order to write several thousands, the symbol M is usually repeated.

assesses-sees.»
MUSEOFSUBTRACTION

Only one gylbol is subtracted in writing Roman numerals. We would

not write 80 as HG because we would be subtracting more than one ten.

IV equals 1: IL equals 1:0 CD equals 1300
II equals 9 :0 equals 90 CM equals 900

As we write Roman numerals now, these are the 221.1 ones that we
use subtraction for. Remember that we subtract to write the numbers with

fours, (1;, ho, 1.00) and the ones with nines (9, 90, 9000).

[j

l

102

'1'le USE OF REPETITION

The only symbols that are repeated are:
I equal to l C equal to 100
I equal to 10 M equal to 1,000

Remember that with the exception of M, which stands for 1,000,

these are only repeated one, two, or three times, never four.

II equals 2 CC equals 200
III equals 3 000 equals 300
II equals 20 mm equals 6,000

************
THE USE OF MULTIPLICATION
In very early times, large numbers were not used a great deal.
The Roman numerals that were written about the beginning of the Christian
era were quite different than ours and very irregular. when it became
necessary to express larger numbers, a bar, _, was drawn over the
number to multiply it by a thousand. This was similar to the Greek M for

expressing larger numbers.

m equals 30 thousand plus 114, or 30,0114
if equals 20 thousand or, 20,000
WIT equals 2 million plus 500 thousand or

2,500,000. The bar over the two M's

makes each of them one thousand thousand.

************
TWO HANDY RULES

I. A smaller number after a larger number is added to the larger number.

II. A smaller number before a larger number is subtracted from the larger

munber.

BASE SEVEN

‘we do not give every separate number a separate name. Instead, we
count in groups of numbers. This makes it possible to count to an in-
finite number with the use of only ten symbols. 'We call this group of
numbers a.model group or the 2222 of the number system. This may be any
size.

For the most part, we use a model group, or base, of ten. Emmy
times we need to have a mathematical system.based on some other model
group, or base, such as 12, 7, 60, etc. we have used some of these model
groups or bases for a long time without realizing it.

we can use a base of 12 in figuring linear measurements. If we
think of these as a base of 12, it makes the addition and subtraction much
more simple.

2 feet 8 inches
+3 feet 6 inches

 

5 feet IE inches
+1 foot ~12 inches (removing 1 model group of 12
3 feet 2 inches and “carrying" it)
or

5 feet 2 inches
-2 feet 6 inches

 

E feet I2+2 inches ("borrow', or regroup, 5 feet
-2 feet 6 incheg_ 2 inches into h feet 1h inches)
2 feet *8 inches (subtraction can now be completed)

In figuring time, we really are using a base of 60.
h hours 50 minutes

 

+2 hours 20 minute§__ (removing 1 model group of 60
6 hours 70 minutes minutes and “carrying" it as
+1 hour -60 minutes 1 hour)

 

7 hoursg_10“minutes

10h
The same operation may be performed for subtraction, we would
regroup or "borrow" 1 model group of 60.

h hours 20 minutes
-1 hour to minutes_

 

3_hours 60:20 minutes (regroup h hours 20 minutes
-1 hour to minutes to 3 hours 60+20 minutes)

 

-—2 hours hO minutes

Science and industry find new uses for bases other than 10. It is
sometimes desirable to organize a system of arithmetic to fit a.particular
situation. In order to be able to symbolise and to understand another
number system.using a different base or model group, it is often well to
first learn to count in the qygtgm. It is extremely important to learn
to 53222, and to understand countigg, in any system'because counting is
the basis for the processes of multiplication, division, addition, and
subtraction. A comparison with our familiar Base Ten will make this
evident.

First, consider the number of symbols needed to write the system.
The model group, or base, to be used dictates how many symbols will compose
the number system. we have nine digits and a zero to write alllour numbers
inbase 10: 1, 2, 3, h, S, 6, 7, 8, 9, ends sero, 0. w. cando thisby
using place value. It would be almost impossible to use a different
number of symbols with a base of 10 and place value as we use it. 'we give
numbers two values to do this: (1) an amount which the number symbolizes
(how many), and (2) an amount which it has because of its place.

If we count, we can see how this functions.

Count: I'1, 2, 3, h, 5. 6, 7, 8, 9 . . . ?????"
When we get to nine, we are forced to use place value simply because we
have no more symbols to use. Our only alternative would be to invent
more symbols. Therefore, we write one, zero (10) and call it ten. Ten

is the name of this place. It is the size of the model group.

x

I
H}
t
O
( (71A
0

 

 

105

The 1 now, because of its place in the numeral, has a value of l_tgn,
rather than Just 1. 0r, expressed in another way, it is equal to l x 10
plus 0 1.1. ‘we would write this more conveniently as 1(10) + 0(1), with
the understanding that the parenthesis indicates that ten is multiplied
by 1. 'Were there a 2 in this place, as in the numeral 29, we would show
this as 2(10) + 0(1). This indicates that the second.place, or tens
place, is now multiplied by 2 and that O ones are added. ‘we simply call
this number "twenty" and, because of learning and habit, it means two
tens to us. It is unfortunate that we have this habit, because it tends
to keep us from understanding what the number really is, and how the system.
actually functions.

Now, if instead of counting in groups of 10, wechange the size of
the group and count in groups of 7, we have changed the 2232) or model
group, of the system. 'We now have, and need, only six.digits and a zero.
That is all that we can use with a base of 7. To understand counting we
need to recall that the values of each place in place value are expressed
as powers of the base. That is, we have ones, tens, hundreds (10 x 10),
thousands (10 x 10,: 10) in Base Ten. Now we have: ones, sevens (7 x 1),
forty-nines (7 x 7), threeshundred-forty-threes (7 x 7.: 7), etc. The
only difference is that we are accustomed to the terms for place value
in a base of 10 and have special names for them. In a base of seven, we
can either make up new names for these places or use terms that indicate
their value in base 10. Making up new names necessitates extra memory
‘work, so it is easier simply to think of them in the terms we know.

Again, countin in the base 7 system will show how the system.func-

tions.

Count: n1, 2, 3, h, 5, 6, . . . 7???

(i

(1

106

'we cannot use 7 because seven is our model group and is now
'written 1 group of sevens and no ones, or 10. In order to avoid mixing
up this system with base 10, in which we call this place "ten", we will
call this "one-oh". Nb might also call it 1 seven,'but this would tend
to be confusing. We have now changed the value of the second place to 7
instead of ten. 'We could show this by indicating the value for the places
as we did before: 10 in base 7 is equal to 1(7) + 3(1); in base 10 it is
equal to 1(10) + 0(1).

It helps to understand a number system.if'you.count in the system
and write out the value for eachgplace. This is true of base ten also,
and helps for real understanding of the system.and how it functions.
First, however, we need to be sure of the place values. .A comparison
with base ten will help to understand base 7. If need be, refer back to
the section on place value to review how the value for each.place is
established by the powers of the base. A comparison of the two bases is

of some help in understanding this.

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Base Ten Base Seven

 

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107

The value for each place is established exactly the same in Base
Ten as in Base Seven. If we think of the value for each place as a
power of the base, it is much simpler. This means the number of times
the base is multiplied by itself. 1c? means 10 x 10; 72 would mean 7 x 7.

Comparing this to base ten, we would have a chart like this:

 

BASE 10 BASE 7
ones 10° ones 7°
tens 10l sevens 71
10 x 1, or (ten ones) 7 x 1, or (seven ones)
hundreds 102 forty—nines 72
10 x 10, or (ten tens) 7 x 7, or (seven sevens)
thousands lO3 three-hundred-forty-threes 73
10.x 10 x 10, or (ten 7 x 7 x 7, or (seven forty-
hundreds) nines)
ten-thousands 10h two-thousand-four-hundred-ones 7h
10 x 10 x 10 x 10, or (ten 7 x.7 x 7 x 7, or (seven three-
thousands) hundred-forty-threes)
hundred-thousands 10S sixteen-thousand-eight- 75
10 x 10 x 10 x 10 x 10, or hundred-sevens
(ten tenOthousands) 7 x.7 x.7 x 7 x 7, or

(seven two-thousand-four-
hundred-ones)

 

 

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108

A COMPARISON OF.PLACE VALUE IN EASE
TEN AND SEVEN

 

106 105 1.011 103 102 101 10°
1,000,000'3 100,000'3 10,00023 1,000'3 100's 1on 1'8

 

 

76 75 ifi 73 72 71 7°
117,6h9's 16,807'3 2,h01's 3h3's h9's 7‘s 1's
3 2 6 O 2 2

Inagine that we have the numeral 326,022. Each digit of this numeral has
a value for the place in which it stands. The places are indicated in the
chart above the numeral. Fbr instance, hundred's place in a base of ten
is only equal to fortyhnine's place in a base of seven. This is reasonable
when we think that the base is smaller. we would show the value of the
above number in this way:

For base ten:

 

326,022 - 3(1oo,ooo) + 2(1o,ooo) + 6(1,000) + 0(100) +
2(10)+ 2(1)

'We read this number “three hundred, twenty-six thousand, twentybtwo". ‘Wb
do not have such names as thousand, etc., for the base of seven. That is
we do not have a name that means to us: “thre-hundred-forty—threes', like
“thousandfl’means ten tens. Therefore, we will use base ten number names
fer the places in Base Seven. This will help to understand the value for
these places.

For base seven:
326.022 - 3(16.807) * 2(2.h01) + 6(3b3) + 0(h9) + 2(7) + 2(1)

we can read this number simply I'three, two, six, zero, two, two,” since
‘we do not have place value names for base seven. 'we would write it with
a little subscript indicating the base like this: 326,0227.

The value for this numeral in base seven is only 57,297 inBase ten, since
the worth of each place is less in.Ease Seven. This is due to the fact
that the model group is (liller.

Since counting may‘be considered as the basis of arithmetic, it would
help to understand another model group if you write out a dart like the
one following. Fill in the missing numbers indicated by . . ., and show
the value for each phoe. Go as far as 100 at least in Base 10. This
will.make a counting chart that can be used later to construct addition
and multiplication tables, with which it is possible to work problems in
the new base.

 

 

 

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109

 

 

EASE 10 BASE 7
1 1
2 2
3 3
h h
S S
6 6
7 10 1 seven + O ones
8 ll 1 seven + 1 one
9 12 1 seven + 2 ones
10 1 ten + 0 ones 13 1 seven + 3 ones
as eeeeeeeeeeeeee O. eeeeoeeeOOOOOOOO
1h 1 ten + b ones 20 2 sevens # 0 ones
15 1 ten + S ones 21 2 sevens + 1 one
a. .eeeoeseeeeeee es 0000000000000...
21 2 tens + 1 one 30 3 sevens e 0 ones
22 2 tons + 2 ones 31 3 sevens + 1 one
00 00000000000000. .0 see-oeeeeeeeeeee
hl h tens e 1 one 56 5 sevens + 6 ones
h2 h tens + 2 ones 60 6 sevens + 0 ones
b3 h tens + 3 ones 61 6 sevens + 1 one
00 00000000000000. .0 00000000000000...
b8 h tens + 8 ones 66 6 sevens + 6 ones
E9 h tens + 9 ones 100 1 forty-nine + 0 sevens + 0 ones
50 5 tens + 0 ones 101 1 forth-nine + 0 sevens + 1 one
00 eeeeeeeeeeeeooo oes OOOOOOOOOOOeOOOOI
101 l hundred + O tens + 1 one 203 2 forty-nines + O sevens + 3 ones
3&2 3 hundreds + h tens + 2 ones 666 6 fortybnines + 6 sevens + 6 ones
3h3 3 hundreds + h tens + 3 ones 1000 l three-hundredpforty-three e
3hh 3 hundreds + h tens + h ones . O forty-nines + O sevens O 0 ones
as. 0000.00.0000000000090000.000 .... oeeoooeaoooeoeoee
2h00 2 thousands + h hundreds + 6666 6 three-hundred-forty-threes +
O tens + 0 ones 6 forty-nines + 6 sevens + 6 ones
PLOl 2 thousands + b hundreds + 10000 1 two-thousand-four-hundred-one
0 thens + 1 one + O three-hundred-forty-three +

0 fortybnines + O ones

 

Reading across on this chart it is possible to see what digits would
represent the same number in each base. For example: h2 in Base Ten
(h tens + 2 ones) is written 60 in Bass Seven (6 sevens and 0 ones);
b3 in Base Ten is written 61 in Base Seven. (6 sevens would give you
h2 in Bass Ten plus one more would make RB.)

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110
The tables on the next page are constructed to shoe ultiplicetion
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III h7167-337
Think: hxb-ZhinBeeekn,butthieieeritten331nBeeeSeven. Write
331ntheepeoefortheenseertehx6. (Thisgiusyou3eevensp103
oneeuhieheqndeZhinBseeTen.)
III. 1372107-1307
Think: mieietheeeleeehz'linneee'ren,buteeveninBeeeSevenis

eritten 39. (or 1 group of seven ones). The probles is done exsctly es in

J

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A _ K ( n 1 r J .
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\ g] i . ..l \‘ ) ~
) J '-
. J I ) - I )
‘ \ f ‘ O ‘| ,.( t ‘ .
I J L ‘r U ' »‘

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{ .
( e
\ e ‘ I" -
. ,
A" d
l
e _x)‘
.
'J J
e “- "L"
3
\\
\ x
A. ‘a
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ea - . ‘
' - _‘x
I- \
I \ v
A. ‘I s
, a J.

Beeeren,bntthe1'snosstsndtorpoupeoteevensinsteedottens.

Theensserh07iseqns1tohsevensor28in3ese10.
I. laxm-IOOinBsse‘ren,end1071107-10071n3ese8even,BUTthe

111

ones in the second exemle etsnd for groups at sevens and forty-nines

insteedottenssndhundredsesinthefiretone.

ADDI'I'Im TABIE IN BASE SEVEN

 

 

e 0 1 2 3 h 6
o o 1 2 3 1. 6
1 z 2 3 h 5 6 1o
2 2 3 h S 6 1o 11
3 3 h 5 6 1o 11 12
h h 5 6 1o 11 12 13
5 5 6 1o 11 12 13 11
6 6 10 11 12 13 1h 15
msIPLIcquu man In BASE sum
1 o 1 2 3 h 5 6
~0 0 o o 0 0 o o
1 o 1 2 3 h 5 6
2 o 2 u 6 11 13 15
3 o 3 6 12 1s 21 21;
u o h 11 15 22 26 33
s o 5 13 21 26 3h 12
6 0 6 15 2h 33 h2 51

I
K _3
\
. .i
\_ _
K S

r

 

112

It is unfortunate that it is often possible to learn to do arithmetic
problem by rote with little or no understanding of the processes involved.
Doing sons siwple problems in another number system helps to understand
how a system really functions, since it is ilpossible to do than without
this knowledge. The following problem are worked in both Base Ten and
BaseSevensothatyoucanseehosslwouldbecarriedassnodelgroup
in the sa- way that a 10 is. Learning to manipulate a nodal group of 1
will sake plain how the-Ease of a systen functions.

 

3132 TEN BASE SEVEN
1- 2; as :7 as
'6 6(1) ‘6'; 6(1)

Since there is no digit with a value over 7, the problen is the sane as
in Bass Ten.

2. I. M1) h? 11(1)
+3 3(1) 237: 3(1)
‘7" 7(1) ‘37 1(7) ¢ 0(1)

How we are forced to I'carry" in the Base Seven problem because 7 is the
Iodel group. The two answers have the same value, but they are in
different bases.

3' +3 3(1) 1:7 3973 1(1)
1 e +
1!" 1(10) + 1(1) 117 1(7) 4» 1(1)

IntheproblenB‘I-B-ll,weareforcedto"csrry'intheBaseTensitu-
stion,butnotinBaseSeven,wedonothavesgroupof7tocarry. The
actualvalue ofthe two answers is, however, the sane.

h. 23 mm 2 3(1) 327 3(7) + 2(1)
.23 MM + 3(1) +327 3(7) + 2(1)
"56" M10) + 6(1) “'65; 6(7) + M1)

The process of addition is the same, except that the nodel group in one
is ten and the other is seven.

5. 19 me) 2 9(1) 257 2(7) + 5(1)
+ 2(10) + M1) +33 3(7) + 3(1)
M10) + 3(1) "57 6(7) + 1(1)

In the case of the problem: 19 4- 21. II 1.3 in Bass ten, we have to “carry.
1 ten. Then the problen is converted into Base Seven, we still have to
“carry- but in this case, we are working with the new base of seven, so
this is our nodel group. The answers have different notation, but the
sale vslue.

 

 

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4!
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Ililt

HWTOWORKAPROBLEMIN BASESEVH

 

 

9112.191
BASE m . use sum
31 3(1o)+ 1(1) 167 1(7) + 6(1)
+25 mm + 5(1) +317 3(7) + 1(1)
19" me) + 9(1) 113? 10(9) + 1(7) . 3(1)

Think: 6+h-10. TeninBsse Seveniswrittenonacounting
chartes13.(0s11this ”one three. sothstyoudonotnixitupwith
Base Ten 'thirteen"). PutdownBand'carry'lm 1.6-7. This
however, really is h sevens plus 3 sevens are 7 sevens, plus the one
that you are “carrying" sakes 8 sevens. fight on a counting chart is
writtenll (call this "one one" soasnottosixitupwithBase Ten
'eleven'.) Put down 11. This gives you the answer, using place value
of: 1019) *1 1(1) '0' 3(1). It is the sane process you would use in base
ten, but the model group is 7.

312 3(100) + M10) + 2(1) 6667 6(h9) + 6(7) + 6(1)
+ 66 6(10) + 6(1) +12 1019) + 2(7) + 3(1)
1158' 1(100) + 0(10) * 8(1) 7 10113) + 1019) + 2(7) 2 2(1)

THINK: 6+3-9, but9inBaseSeveniswrittenMflZ).
PutdownZand'csn-y'lm. 6+2-8. Eightiswrittenwul),
add the 1 seven that you are “carrying“. This sahes M (12). Put
down2and "carry" 1. Inthis case, it'is lforty-nnine. 6+l-7.

Seven in Bee Seven is written 2522 (10). Add the one forty-nine that
you are "carrying". This sakes 912.222 (11). The answer is 1,1227 or
197.3) + 1(10) + 2(3) + 2(1) which would equal 1:09.

\

ADDITION IN BABE SEVEN

These are a series of problens of increasing difficulty in Base Seven.
They could be lileographed and used.with students either for enrichment
or understanding. They night he presented in.such anway that each student
could go as far as possible in the series. The problems have been worked
in Bass Ten.first and the answers indicated. This should make for'lore
(leaning'than.working in Base Seven alone with.no way of conparing'with
has Ten. It is not necessary to indicate the base with a subscript when
it is understood, therefore, these are olitted. Encourage students to
look for the pattern of counting.

 

 

use m 313: smut
1. 5 5
1% 1(10)+ 0(1) 1% 1(7) + 3(1)
2. 6 6
6 6
fi" 1(10) + 2(1) If 1(7) + 5(1)
3e 7 10
10
IE' 100) + 1((1) ‘2"6" 2(7) + 0(1)
h. 8 11
8 11
13“ 1(10) + 6(1) ‘2!" 2(7) + 2(1)
5e 9 12
IE. 12
1(10) + 8(1) ""217 2(7) - 11(1)
6. 10 13
10 13
‘2'0‘ 2(10) + 0(1) 13' 2(7) + 6(1)
7. 11:; 1h
12" 2(10) + 2(1) 'g' 3(7) 4- 1(1)
8. 12 15
12
”511' 2(10) + M1) 1%— 3(7) + 3(1)
9. 13 16
J);— 16
2(10) + 6(1) '33' 3(7) + 5(1)
10. 1h 20
1h 20
'58" 2(10) 4- 8(1) 1175‘ 1(7) + 0(1)
11. 35 50
a; .29—

7(10) + 0(1) 130 1019) + 3(7) 4- 0(1)

 

0..

s
\
--

\r.‘

 

 

 

Hm TO WCBK HILTIPLICATION PROBLEIB IN BASE SEVEN

MSETEII BASESEVEN
l. 8

13 11
7h— 2(10 4- 17(1) - 2h

:3
1'3- 3(7) + 3(1) - 21:

THINK: To convert 8 to Base Seven, eight equals 1 seven plus 1 one. write
one one (11), read it “one one". There is no regrouping necessary because
the ones do not combine to an amount over the base of 7.

'65” 6(10) + 0(1) - 60 m:- 1029) + 1(7) + 1(1) - 60

THINK: FifteeninBaseSevenis equaltoz sevenspluslone. Write 21.
Readit I"also one“. Tomltiply, think: 1:11-11. FourinBaseSeven
is si-ply written 11 because it is not over the node]. group of 7. Write
downh. 1:12 -8. Eight inBase Sevenis equalto lpevenpluslone.
Hritell. Readit "one one”. Theansweris thesaeeaeountasinBase
Ten. The notation is different.

3. 53 1013
3:211 :33
"212’ '513'
1“ 73%:
1272—10000) + 2(100) + 7(10) 3(3113) + mm) + 6(7) + 5(1) -
«2 2(1) - 1272 1272

THINK: Base Ten 53 is written 10!. in Base Seven because 53 has 1 forty-
nine+0sevens+hones. Tonultiply, think: Bxh-lZinBaseTen.
“elveinBaseSeveniewrittenIS. PutdownSandcarrylseven. 310-0.
iddthelseven. Writel. 311-3. Putdown3. Dothcsaleforthe
nextline,andadd.

1. I?) ' x111):
553' 5(100) + 5(10) + “1526 1(313) + 1(19) + 2(7) + 0(1)-
3(1) - 553 553

THINK: In the base seven problee the node]. group is 7, so one sero stands
for 1 seven. The process is the sane as base ten, but the notation is
different. The answer will have the sane value.

5. 17 :33

% 5(10) + 1(1) " 51 IUT 10(9) + 0(7) + 2(1) - 51
THINK: 3 x 3 - 9. (Written in Bass Seven as 12.) Write 2, carry 1 seven.
312-6. (UritteninBaseSevenaséd Addtheluveneaking? sevens.

In Base seven this is written 10. Put down 10. Place value operates here
the sane as in Ree Ten.

MA

116

WHIPIJCATIGI BASE SEVEN

It night be best to do probleas in Base Ten first and then convert then
to Base Seven, and check as indicated.

1.

2.

3.

BISETEN

 

BASE SEVEN

‘35g—3(h9) * 0(7) + 0(1) ' 1h?
33
-3§-'5(7) t 6(1) ' ha
20
IIg—'1(h9) * 1(7) t 0(1) ' 56

006

2136' mm + 1(7) + 0(1) - 126
5
'5156“ 5(19) 2 0(7) + 0(1) - 215

g.-
00

6(19) + 6(7) + 0(1) - 336

1(2101) + 1(313) + 0(19) 2 6(7)
+ 6(1) I 2792

5(2h01) + 2(3h3) * 3(h0) ' 3(7)
+5(1) ' R86h

25355 2(21101) + 0(3113)96 + 60.19) + 0(7)

+ 0(1) -

200
1
11275 MM) + 1(19) + 0(7) + 0(1) - 392

 

117

SUBTRACTION PROBLEM IN BASE SEVEN

Subtraction in has Seven is exactly the sale as in Bass Ten, except that
we would 'borrow', or regroup, sevens instead of tens. If we think about
sons preglens, this is clear.

1. 20 or 1 ten Oth ones
-6 Z tens 4- fl ones
15 6 once

 

Tten +1; ones

It is a listake if we do not think of the 1 ton that we 'borrow' as being
added to the ones in the one's place. 1:. are inclined to write the one
We the sero, in this case, and with little thought silply call it ten.

2. 22 or A l ten 2*10 ones
-6 I tens 4- Z ones
'3' 6 ones

 

Iten +6ones

Thetenonesthatwe 'borrow'areaddedtothe2ones thatarealrsacVin
one's place and this sakes 12. If we do a problen in Bee Seven, we can
see this aore clearly. We must reaesber that we are regrouping in terms
of the base or model group.

3. mm BASESEVEN

2 sevens 14-7 ones (or 8 ones)

 

22 310r1sevens+10nes
-6 -6 6ones
1:5 2sevens+2ones¢16

It 1. easy to ... that it 1. 111303531121 70 me the node]. group (7) to the

ones thatarealreadyintheone‘splaceandnottonlplywritethel
beside the 1 in the one's place and call it eleven. We w do this, if

we have been able to think in the system that 11-6-2. But, until one

thinks in the system, or had aenorised the facts, it is alnost ilpossible

to do this. Thinking of the nodel group as being added when it is "borrowed“
naked it plain that 11 in Bee Seven is ooaposed of 1(7) 0 1(1) or is

W t0 8 in 3‘80 Ten.

Students can gain an understanding of subtraction by naking up probleas ,
converting then to Base Seven, and then working then in both systees.

The probleas elem should be proved according to the above pattern until
the students are very sure of thenselves in the new system.

Long division in has Seven is easily done as mltiple subtraction;
therefore, the subtraction is not developed further here.

.2

'\

 

 

 

118
LONG DIVISION IN BASE SEVEN

It is possible to do division in bee Seven either as multiple subtraction
or in the fore, 2/ S , sometimes called the I'divided--by" fora. For
beginners, it is probably easier to do it as multiple subtration. Following
are sole exanpled. Note that this method carries on the ideas presented
under subtraction.

 

 

 

 

 

 

 

 

BASE TEN BASE SEVEN
1e 7 1‘33 10 1'e3
7/ S2 IO/Ib3 1 forty-nine + 0 sevens + 3 ones
h9 ~10 1 ~ 1 seven t 0 ones
‘3' -63’ 6 sevens + 3 ones
~10 1 ~ 1 seven + 0 ones
.33. sevens + 3 once
~10 1 ~ 1 seven + 0 ones
‘53’ _Ih sevens +‘3—3333
-10 1 ~ 1 seven + 0 ones
.33 3 sevens + 3’ones
~10 1 ~ 1 seven + 0 ones
.23' __2:§evens +*3_3333'
~10 1 ~ 1 seven + 0 ones
'TES I seven + 3Iones
-1() ~ 1 seven + 0 ones
__3 15 1(7)'0(1) 3 ones
Proof: 7 10 l(7)¢0(l)
x7 :10 1(7)*0(1)
3 TEE; 1(19)+0(7)+0(1)
...
'32 ‘1?! 1(19)+0(7)+3(1) - 52 in base ten.
2. 12 r.2 151‘
8/‘T9 n/fw’m
~66 6
101
~66

6
'7 I5" 1(7) . 5(1) - 12

THINK: Take 6 sights away froa 200. You have to
decompose 200 to 6 sevens + 7 ones. Coaplete the
subtraction thinking in groups of 7's.

Proof: 12 15
x8 all
'93 '13
+
‘56 '16;
+ 2
765

THINK: l 1.5 - 5, 1.1 1 - 1, and repeat. Add the remainder. Remember
5 - 2 - 10, or one group of seven. Carry 1 seven. This makes 6 sevens
plus the one you are carrying. Put down the zero and carry again. Yen
now have 1 + l I 2. Your answer is 2(h9) + 0(7) + 0(1).

\Q'

_j.

 

 

DIVISION PROBLEMS IN BASE SEVEN

 

 

BASE TEN BASE SEVEN
1. 11 r. 3 1n r.3
9/ 105‘ 12/721?’
«63 7 ~120 10
39 5
:éé fl -_5.1_ :11.
3 11 3 11 1(7) + u<1) - ll
Proof: ll 1).;
>12 11.2.
99 31
+3 k
102 201
+ 3

555 2(h9) + 0(7) + h(l) ' 102

Remember as you work in base seven, you must either convert each number
to base seven as you go along, or else think in the base.

29 r.3 141 r.3
2. 7 o 10/‘513‘
.19 7 .100 10
E7 313
~h9 7 -100 10
158 213
-u9 7 ~100 10
“E?” 1
-h9 7 -100 10
'TES 13
:_1 1 - 10
3 3 ET h(7) + 1(1) - 29
Pronf: 29 hl
x7 1:10
553 51b
+ 3 + 3
553 E13 u<u9) + 1(7) + 3(1) - 206

In this base seven problem, 10 stands for 1 group of 7 ones instead of
ten as in base ten. Then 10 x 10 =- 100, however, 100 in base seven is
equal to l forty-nine instead of one hundred.

 

L10 r. 10 55 r013
)4. 12/590 15/ 1300
18. m
10 130
11h
Proof : I10 55
x12 x15
"'86“ EOE
ho 55 ,A
1180 12511
10 . 13
1:93 366 1(3143)+3(l§9)+0(7)'0(l}'h90 ,

This problem is done with the "dividedby" method. It is‘more difficult !
because you have to bin]: in the base. ""' ”‘W ' '
* _

120

PLACE VALUE CHART OF OTHER BASE

BASE 135 B5 31* B3 132 31 3°

10 1,000,000'3 100,000's 1o,ooo's 1,000'8 100's 10's 1's

7 117,6149's 16,807 's 2,h01's 3143's h9's 7's 1's
2 611's 32's 16's 8's 11's 2's 1's
3 729's 2h3's 81's 27's 9's 3's 1's
5 15,625's 3,125's 625's 125's 25's 5's 1's

By using place value, we can asks a nusber systen in am base. This helps
us to understand the structure of our own Base Ten, because we are forced
to think of it as it functions as a base or nodel group, not as something
we have learned by rote. This also helps to give us an insight into pure
mathematics.

We have given no bases with a model group larger than ten, became this
would like it necessary to invent extra synbols. However, a base, such

as base twelve would function in the ease way as an that used place value
and a nodel group.

To convert to my base, follow this procedure using the appropriate place
values. To convert 1:69 in base ten to base seven:

1:69
~31;3 l three-hundred-forty-three
TEE

~98 2 forty—nines
"28'

~28 1: sevens
0 ones

THINK: Take the largest place value out of 1469 that you can. This is
31:3. You will have 1 three-hundred-forty-three. Then take out all you
can of the next largest place value. This is 2 forty-nines. The next
digit in the base seven nunber will be 2. Take out u sevesn. You have
used up all the 1:69 but you must hold the one's place with a sero, since
youhave no onesinthisnuseral. Ifyoudidnotputaseroontheend
to hold one's place, this would be only a three place nunber and the
values would be wrong.

After you have subtracted as indicated above, it is possible to sin]:
read down the colusn. This will be the new nuneral in the new base. In
this case, 1,2150: 1(3143) + 2019) 4- M7) 4- 0(1) - 1169.

 

 

1"
1“
In

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l" - '
O -
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... — ' _ 1 -

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V. ‘
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a
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.
'0
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1.

2.

3.

h.

S.

6.
7.

121

SUGGESTIONS FOR TEACHING

Teach counting in the base first. Have students write out the values
for the place value according to the pattern below. Do this for as
far as the base squared at least. In Base Seven, this would be 119
in BOB. Thus

 

BASE m BASE seven
1. 1(1) 1 1(1)
2 2(1) 2 2(1)
3 Bélg 3 3(1)
h h 1 h 11(1)
5 5(1) 5 5(1)
6 6(1) 6 6(1)
7 7(1) 10 1(7) + 0(1)
8 8(1) 11: 1(7) + 1E1)
9 9(1) 12 1(7) + 2 1)
10 1(10) v 0E1) 13 1(7) + 3(1)
11 1(10) + 1 1) 11: 1(7) + 11(1)
12 1(10) * 2(1) e e e CtCe 15 1(7) * 5(1) e e e etc.

liter a counting chart has been lads, use it to lake addition and
ultiplication tables in the base. This helps the student to begin
to THINK in the base.

Use these tables to work silple problems. Prubably the problems
that students lake up the-selves would be best to use. work then
in Bass Ten first, and then convert, compare, and prove the problens
back to base ten as suggested.

Have children prohds a chart or booklet explaining what they under-
stand about another number systen other than ass Ten.

(live oral denonstrations of place value. Tabs place value pockets
andchangsthebasetrontentosonethingelse,andsbowhowtbe
vnlues for the places would have to change.

Let slow children lake simple counting charts in another base.
Hrite a story or develop the idea that a people with twelve fingers

light have a nunber system based on twelve. This would require the
invention of two additional symbols.

IV . ARITIfldETIC PROCESSES
Addition
Subtraction
mitipli cation
Division

Review and Reinforanent Suggestions

 

123
ADDITION

The process of addition for the nest part is usually well established
prior to the grades fcr which this Interial is written: however, sons
attention will be given to it here. Unfortunately, sons students have
learned their addition facts by drill and have little or no real under-

standing of that they are actually doing.

In addition, we are combining groups of irregular counting. Actually,
addition is a natter of regrouping in whatever base you nay be working.

in our system of Base Tn, we group and regroup manbers based in this Eaten.
Forexanple: 5+7-12. wearesayingasubgroup ofSandasubgroup

of 7 are regrouped to form a subgroup of l ten and 2 ones. In addition,

we are regrouping nunbers based on ten. In another base, such as Base

Seven, we would be regrouping nunbers based on 7. While it is not advis-
able to present addition to prinary children in such terms as above, it

is possible to help then to understand that they are dealing with m

of numbers rather than -re facts.

36 3 tens 6 ones
51; tens 1: ones
tens ones

1 ten -10 ones
'—9 tens O ones

Think: In our nunber systen, we have no digit or symbol higher than 9.
Sowe say6+14 givesusltenandOones. We 'carry'lgroupof onesto
the ten's colunn, or place, and this gives us 9 groups of tens plus no
ones. It is easy to see that 8 tens + 10 ones would give us the sans
face amount: however, we could not write the nunsral as §_lg for this
would be read 'eight-hundred-ten'. So, we are forced to combine the
8tensandthe10onestoread9tensorninsty.

Addition is still counting, but it is much easier and faster to count by
Eoups rather than by ones. Instead of the student Just saying 8 + ' -15,
should understand and think of this as I group of tens and 1 group of
S ones. Writing out these problens as we have indicated helps this

prou.le

lgroupofhundreds

ll 1 group of tens OR 367

367 29h

291: II (7 4' 14-11)

“I 150 (60 + 90 II 150)

goo (300 + 200 - 500)
Here, we are regrouping, or “carrying" ones, tons, and hundreds. THINK,
llones regrouptoltenandlone. lbtensrsgrouptelhundredandetens.

The second illustration shows how the problem could be done by silply
adding the value for each place. This would Avoid 'carrying'.

 

 

 

SUBTRACTION

Subtraction can be taught as the inverse process of addition. Instead of
conbining mbgroups as we do in addition, we subtract subgroups. For
exalple: 7 - 5 - 2. We are saying: '1 group of 7 less a subgroup ofS
equals a subgroup of 2". We can apply this to any combination except
where the digit of the subtrahend is greater than the corresponding digit
of the ninuend. Then we lust use decomposition, or I'berrowing", before
we can subtract. 5

(h tens) (3+10 ones)

53 or itensi-Iones
-36 -3tens+60nes
17 Itne+7ones

THINK: We cannot subtract a group of 6 from a group of 3, so we regroup
by deconposing (rearranging) the 5 tens into 14 teas and 10 ones. Now

we regroup the ones by adding the subgroup of 3 ones, to the ten ones we
”borrowed" to make 13 ones from which to subtract the 6 ones of the sub-
trahsnd. The student should understand that this is what he is doing
when he 'borrows' fron the next place and puts a l beside the digit from
which he will subtract. He is increasing the ninuend digit by the snount
of the node]. group of the systen.

(2+lO hundreds)(l+lO tens)
7 thousands 1 hundred I ten 3+lO ones
8323 flthousands+1hundreds+2tens+1ones
-276 -2thousands+7hundredsvjtens+6ones
Tthousands +fhundreds +btens 4- 7 ones

THINK: Wedeconpose the2tenstereadltenand10 ones . . .regrouped,
this gives us 1 ten plus 3 ones or 13 ones. The 3 hundred deconposes to
2 hundred and 10 tens ... regrouped to give us 10 tens lus l ten, or ll
tens. The 8 thousand regroups to read 7 thousand and 15 Endreds, which
regroups to give us 10 hundreds plus 2 hundreds or 12 hundreds.

This process night be explained by showing that the rearrangenent will
still equal the original anount:

7,000 + 1,200 ¢ 110 + 13 - 8,323

Subtraction can also be done by the additive nethod. This is considered
by son to be a better one since it uployes is use of the addition facts
with no new cenbinations to learn. However, the 'talos awq' nethod is
nost conenly used. Essentially it is only a different way of EEK
of the process.

TAKE AWL! ADDITIVE
538 We think: 8-6-7 538 We think: 64-7-0
4126 3-2'? 44.26 2+2-3
.1135 541'? _m h+7e5

MULTIPIICATION

An approach to ultiplication through a thorough understanding of place
value opens many interesting and worthwhile activities that,can be used

to motivate students and increase understanding.
sons different forms of working multiplication problems.

Following are given
315011011“, in.

vestigating these, mu find it worthwhile to invent others.

1. 12 lten+2ones
x3 x3ones
'33 3tenst50nes

 

 

2. l1. lten + hones
I3 xjones
1:2 Ttens-o-lZones
+lten ~lOones
Ttensd- ones
3. 11:
13
'I2' (31h)
*30 (3:10)
'13
1:. 2113
1:211
12 (1413)
160 (hxuo)
300 (hx200)
60 (20x3)
800 (20th)
11000 (20x200)
7337

M
972 ( b x 2113)
533'?

menus

(3 x
(3

(remove l ten and "carry. it)

1211
(20 x 21:3)

I1;
l1:
114
1? (Huh)
30 (10+10+10)
'11?
2113
x211
1:836 (20 x 2113)
12 (h x 3)
160 (14 x 1101
800 (1: x 200)

Such different ways as these can nuke multiplication have a 'new"look and

furnish an excellent means of checking problens.

They may not be used

very successfully if a student does not have a good idea of place value.
These nethods should not be presented to students as sonething that they
are required to menu-I5. The should be the result and natural outcome

of understandins acquired in studying place value and the function of

the number system.

when students are given freedom to invent methods of doing some process
they should be required, however, to prove their invention is sound and
will result in the correct answer.

 

 

3

126
DIVISION

Division, or dividing nunbers, can be thought of as subtracting in a
special way. If we take the problen:

3M—
Wenaywanttoknowhowmanythreesareian, hownuch1/3 ofl2would
be, orhowmanytinsswe cantakeBawayfronl2.
Putting the problen down like this: 3/I2 is Just a_ convenient way of
arranging it so that we can find out what we want to know. The well-
known “box” has absolutely nothing to do with this.

We could also put the problen down like this and find the answer:

-3 1 (three taken away)
-3 l( another three taken away)
-3 1 (another three taken away)
-3 l (the last three taken away)

T
Uehavetakenswayhthrees, orhx3-12. Ittookhthreestouseup
all of the 12. We could find out how eany 3 inch pieces there were in
twelve inches in the sans way; sinply by taking a 12 inch strip of paper
and cutting it up, then count the pieces.

51: inches (the length of the snake in inches)
-1 2 1 (take away one foot of 12 inches)

:% 1 (take away another foot of 12 inches)
:% 1 (take sway another foot of 12 inches)

-12 1 (take away the last mole feet)
—'5 T (there are 6 inches left)

The snake us 1; twelve inches, or 1.: feet, long. There were six inches left
over.

i better way to do this problem would be like this:

if: inches(
- 2 take away 2 twelves)
“'35

12% 12; (take awq 2 nore twelves)

D:-

 

127

The snake was 1.; feet 6 inches long. The children have divided 511 by 12:
or we could say 511 e 12 - 1:, remainder 63 or 12/311, etc. It does not make
a difference how we put it down to work it, if we are careful, we will
get the right answer.

 

It is difficult to do very complicated long division problens with this
kind of a subtractive process unless students have a very good background
in place value. After this has been built up, division done with any of
the processes can become meaningful. Following are some problens of
increasing difficulty. These should demonstrate the process.

lvlfilfll'h 1's5
1.5/1? 15/ 5F

.15 1

;% 1 or
:35 1 -15

~51
‘5'):

A slow student might do the problen this way. (he with more talent night
speed the process up in the following manner.

was! slash

15/ 63
-30 2
‘33
-30 2
‘3 '1?

Sons students would see that 11 x 15 I 60 and know the answer. With this
method, however, there is offered to students of nany abilities an under-
standable way of doing the division process.

of??? Ins/“EST
-60 10 1.60 10
'56 ‘2? 115‘

is

Step before doing a problen and think how mam tines the divisor should
be taken out ... 1 tile, 10 times, 100 times, etc. Do not try to take
out so new time the divisor the multiplication becomes involved. For
instance, it would be foolish to take out 15 tines the divisor. his
would nake the problem nore difficult and invite error.

w~

J
O
;
.
\
n—e
\
w.
. a.
.-
I
—
\
--.-..
\
¢
\
v
\ _
-
\
n
.—
-

 

 

128

15/535-
150

10 (start with 10 tires the divisor)

300 20 (it is easy to see 20 tines the divisor now)
250
150 10 (it would be foolish to use 13 here)

115 3 (this is easier to multiply)
__3

100+200+100+30+5+h'h39 r. 14 (the answer my be put in any

15W convenient and meaningful wq)
(X).

saladé‘é‘ggggu

dodges

...I

mmn There are 1439 fifteens in 650. It 1: possible to find out the
answer sinplybytaldngthenoutuntilyouhaveusedupallthe650. It
would be foolish to take then out 1 at a time. Take them out in groups.

The fern in which the division problen is placed often can obscure the
swam? Such an approach as mltiple subtraction will tend to: (1)
increase understanding; (2) nake the division process easier; and (3)
relate back to the place value and subtraction concepts that the child

It should be understood that sees problems are done nore conveniently
one m, sens another. It would be well to help children understand new
ways and then use whatever way best suits the situation. When a mild
can do this, he is using mathematics intelligently and not gust repeating
a rote, a nsaningless process.

.3 anmt—mmw '11-"

‘nfi‘. -sfl .h“; 'a’. ‘

IV. FRACTIONS

History and Development
Understandings Needed

Methods

 

129
HISTORY AND WHEN!

People think of the anounts of things in two ways: as whole things, and
as parts of things. It would appear that for some time during history,
man did not think much about parts of things. Whatever accounts he kept
and dealings that he had, were done, for the nest part, in whole amounts.
1 way to ‘put down" or express fractions may have been difficult to in-
vent. Ways to add, subtract, multiply and divide fractions were probably
very confusing. We have records of very early fractions that were quite
different fron ours.

The word _f________raction cones from a Renae, or Latin, word, fryere which

neans to break. We have words like fracture that have web the sans
meaning. it first, people seemed to Just express parts of thingsin

aw convenient way. Then, they tried to write fractions in such a wq that
all the numeraters were 1. The Egyptians did this in the follmdng

manner.

Since“) s“neantB, theyinventedasynbolthatneanslpartof
three 3 . $3” steedforl/B. Theyhadaspecial synbolfor

2/3, fl; and for 1/2 I . Other than these two, all fractions
were unit fractions, or fractions with nunerators of 1. This resulted

in sons unusual ways of figuring.

 

 

7 1 1 1 a O O
3.5+F+'3 E 4:11 j’z’n
_ 1 1 o O
é g + E. IJMJ Anuu
8 1 1 Q
15' r + a ' 1:

 

There was no set way of expressing fractions. We believe that sees merchants
I'in the know" who had invented one used it to their own advantage. It is
interesting to try to do this. You must start with the largest fraction
that you nan use, subtract this from the one that you want to express,

use the next largest, etc.

We do such the same thing with money. So, you see, we are not so advanced
after all. Suppose that you buy something for 26¢. You give the clerk
$1.00. You will get 7M change. The clerk will say: "26¢, 27¢. 28¢, 29¢,
30‘: 110‘: 50¢: ”100. Eh. hi. 81"“ you:
, 1 1 1 1
m’rw‘rw‘rw‘m‘m‘r

Your change is equal to 711/100 or a dollar. Therefore:

7h. 1 1 1 1 1 1 1
1w no’roo‘rou‘nofio‘ru’s

--....

~-.--

\1

UNDERSTAND DI GS NEEDED

Fractions may be confusing to children because we tell than that they mean
so many things. We sq:

number you have
name of parts

8
ton- NIH

shows 1 part of two equal parts
or

1

2 shows the relationship between 1 and 2
or 1
2'

meansldividedby2

By the time that a child has been told all of this, it is probable that

he does not know what to think and Just gives up and msmorises everything
that he needs to know about fractions in order to ''pass". It would seem
well at this point to make clear some of the basic understandings that a
child needs, and to show some of the ways that these may be demonstrated.

Few of the things that we deal with in our daily lives come in wholes.
We are always using parts of things. This is also true fer children.
Fractions can be put in the tangible context of a child 's life, using the
things that he knows. These can be used to help hin understand the
reality and usefulness 'al fractions. What he sees and can touch, he will
rensmber better. I[here are new things that can be used in this way to
help with fractions.

When fractions are approached through their historical development, we

see that the narmer of writing them is Just one of convenience. Fractions
can be Just a logical, convenient way of figuring with parts of things.
This brings us to the first understanding.

1. Fractions are Parts of Mugs

This can be shown graphically or with things drawn from children's
environment.

@51 [31:1 7,) I
12 of 1 of t 1/2 of 1/2

 

 

 

 

 

 

This also can be demonstrated. Use liquid measures and show the
following facts. Demonstrate them, list then, make graphs of them, and
then draw conclusions.

1"

 

 

131

FACT GRAPHIC

2 dupe equal 1 pint

Conclusion: 1 cup " 1/2 P“ 13+ CY: B

1/2 cup .- 1/1, pt. cup 4- cup - pint

 

eeeee OtCe

FACT GRAPHIC
2 cups equal 1 pint

2pintsequallquart [=3 +6: .

51st cant: -

 

 

 

 

 

Conclusion: 1 c II 1/1: qt.
2 c - 2/1: qt. (or 1/2 qt.)
3 c - B/h Qte

or 3 O ' 1'142 01 I qt.

These can be carried as far as the talents of students permit. Slow
children may not go very far. More gifted ones will be able to draw
quite advanced conclusions. Introducing fractions in this manner, at
any grade level, puts the symbolination and the concrete illustration
on the child's level of thinking. The next step would be to do simle
problems, and then more difficult ones with deneninate numbers. Below
are some illustrations.

lpte 2 c. 1/2 gt. 1 1/2 8 0
+1 pt. 1 c. 152 gt. 1 152 gal.
pt. c. 2 Qt. - 1 Q‘o 2 2 8‘1. - 3 881.
+1 Eta .2 Ce
Pt. cs
After children have demonstrated these facts, drain conclusions from the
facts, made graphics of the facts, and worked problems regarding them,

they may see reasons for writing fractions as we do, the logic behind
reducing fractions, and ways of combining them.

 

3 1,1
.I'..
1
:J
3.)
:)K
.— “)K
A..
3
1
j , A
I
1
(
O
0)} e
a} .
e~‘ e
e1—e

 

132

It is possible to help children make up very interesting displm or
graphics developing any of the facts that they use. They could demonstrate

that:
1: quarts . 1 gallon

8 pints - 1 gallon
16 cups '- 1 gallon

(61 n (n [Ln

 

 

1&1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

qt. qt. qt. qt. - 1 gale
111 I [I [U 1 111111111
2 pt. 2 pt. 2 pt. 2 pt. ' 1 gal.

F 1 , 1
EC, E1 1% LL. l 1 d
1: c. 14 c. 1: c. 1: c. - 1 gal.

Conclusion fron the above: 1 pint - 1/8 gallon
1 cup I'1/16 gallon
2 pints - 2/8 gallon I 1/1; gallon, etc.

He can also use a number line to show the sequence of fractions. It is
possible to count in fractions. This helps to put them in a logical
arrangement for children. Children could make any kind of a nnnber line
for demonstrating fractions that was at their own particular level of

thinking. Below are suggestions.

 

 

 

 

0 142 242 342 1:42 542 ...

0 1(2 2/271 3/2-1 1/2 h/2-2 5/2-2 1/2 ...

0 142 1 1 142 2 2'1/2

... 5/'2 h/'2 342 242 142 (3
2/142 2 1 1/2 1 142 o

 

Aw logical sequence makes a good number line for fractions. Counting for-
wards, backwards, and by reducing helps children to see how fractions re-
late. This is also excellent to help them see reduction of fractions.

 

 

1 I
II 0 1
A V 0
V
J .
O
. y‘
n
l
O
x.
x
) \ Y! 'x
\ \
o
c
. .
_
a
o >
.
. . _
1 a
I y ,
v . c \
. . —. V II . . II
r‘ . I.
_
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. .
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n e
u e
.l .
. . .
\ __ .
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. . i . l I .
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_ \
, . , y a _ -
_. z 1
‘
x;
. Z a
6. w
. e , y
’
Q
I
r e
v
e
.

133

A number line is also a fine help in teaching children that there are
proper and improper fractions, while numbers and nixed numbers. This
can be shown with things drawn hen the child's environnent and with
graphics also. Below are some illustrations.

 

 

 

 

 

proper [17/2 I )
$2232: HQ #3 ITJ.

l + 1/2 . 3/2

To show that all tractions can be changed to higher terns use neasures and
demonstrate a use a nunber line like the one below. These dan be lads of
any convenient material. Oak tag is pod and can be kept in desks.
Wrapping paper makes a good one to put up for the whole class to use. This
also takes a fine project for a group of students to do.

-
‘
‘

142 2(2 3(2 M2 szz 642 742 8(2 942 10/3
I *r T w r * 7 * 1

V T

2. A Fraction is a Ratio

This is difficult for sons children to understand. Perhpas a problen will
help to denonstrate this idea. Again, this is putting the traction con-
cept into a concrete situation drawn from the child's experience.

John has 5 narbles, 2 are blue. What
part are blue?

Solution: Let each marble represent 1/5 of the total. Therefore, 2 larbles,
or 2/5 are blue.

THINK: l Iarble - 1/5
*1 urble - l
Turbles- 2

 

3. Fractions are Couposed of Other Nunbers

Just like whole numbers, fractions are comosed of other numbers. The
number 6 is node of: l+l+l+lel+l and 2+2+2 and 3+3, etc. 14/5 is lads of
2/5 + 2/5, or 1/5 :- 1/5 + 1/5 4- 1/5, etc.

Also, h/S has the sale value asggauy of these combinations. This is a
fine wq to help children see ideas that relate to lowest col-non denomina-
tors and to reducing. Opportunities to express fractions in as Harv
different ways as one can are good. Another method that might lead to
understanding would be to do problems in fractions as the Egyptians did.
Below are some illustrations. Relenber that it makes it easier to start

with the largest fraction that you can.

Directions :

1. Express the following as unit fractions.
6/8 - 1/2 + 1/8 4- 1/8
14/9 - 1/3 + 1/9

2. Express the following as unit fractions starting with the
first fraction given.

ll/2-l/164-
3/10 '1/5 +

Ihny such activities can be nade up by students and teachers to give leaning
to lowest common denominators and reducing.

h. A Fraction is a Division hample

This is further developed in the section on division of fractions. It
should be emphasised that division can be expressed in any convenient
way, such as:

3/ 11 h o 3 - W3
h. Fractions can be Added, Subtracted, mltiplied and Divided

This needs to be lads plain with concrete eaterials and with sylbols. It
needs to be illustrated that unless this were true fractions would be of
no use to the culture. It would be interesting for children to see if
they could get along without fractions for a period of time, and then
giveareportonhowtheynadeout.

135

5. Fractions can be Estimated

Just like whole nunbers, it is possible to estimate fractions. Children
need practice in this. They should have opportunities to practice, such
as doing demonstrations, saking charts and comparisons, etc.

Such expressions as: 6/3 - 7 > 1
3/5 - 2 ) 1/19
1/2 ' 7 < 6/8, OtCe

should be encouraged and explored. These can be suited to each child's
talent. Children should be encouraged to denonstrate their reasons and
the-ethodtheyuseto arrive atananswer.

METHODS WITH FRACTIONS

While it is fairly easy to visualise and to denonstrate addition and sub-
traction of fractions, this is here difficult to do with multiplication
and division. Sonstissa the Isthod for these processes is e-phasised so
web that the meaning is obscured. Concentrating on the nethod until it
is nenorised nay nake it possible to 'do the problem" but it helps very
little with understandings and lasting learning.

Division and lultiplication of fractions are closely related. They will
be presented together here as a way of "thinking about fractions. rather
than a process. There are several Imethods" popular for teaching these.
If the understandings behind these are mastered, the nethod as such,
should develop of itself as a convenient way of doing the Ianipulation.
This section will be divided into: (1) Understandings Basic to Division
of Fractions, and (2) The Methods of Doing Division of Fractions. These
will be accowpanied with suggestions for classrooa activities.

1. Basic Understandings

1. Children need to understand the relationship between multiplication
and division. They should know that:

3/12— neans also: 1 13-12

when a child does the first problen, or ones like it, he should also
understand the second one and the w the two relate. Then

B/hel-3/h,and 7 31-3/1:

would be the next step in understanding the process with fractions.
This helps to nake answers more neaningful.

 

 

136

2. Children need to be able to change whole numbers and mixed where to
improper fractions. If they have acquired the basic understanding
that fractions are parts of things, this can be demonstrated with

tangible items and then carried to sywbols. They should be led to
see that:

‘ll/21h'I ?
11/2+11/2+11/2e
orwithanuwberlinecrmler:

------ 1 1/2-----g--.1 1/2---—-.g----1 1/2----;

 

f T— r I T f I IV I T

1 2 3 h 5

0.0.. OCH

 

r T T T 1

1/2 2/2 3/2 1:72 5;2 6/r2 7}2 8}2 9/2 10/2

AAMAA
vvwv

 

 

or they can see that the problen could be done Just like multiplica-
tion of whole numbers.

3

71312 $113)

1 2 1/2 of 3)
H 1’52 (total of (113) + (1/2 x 3)

or 1 1/2 - 3/2, so we have, finally:
3/213/1'9/2-h1/2

Children need experiences with each problen. They need to see
it in the abstra and the concrete. The need to be able to find
several ways of doing the same problen. Tine might better be spent
doing the sale problen lam ways and understanding it, than by doing
nary probleas without understanding.

Children should be encouraged to invent reasonable, loncal, and
convenient was of doing problen: like this . They nay nake reports
or do deaonstrations, or nake charts and graphs to explain their thinking.

3. How to wultiply common fractions and nixed numbers may be approached
in much the sans manner as the above. Helping children to see the
problen in new neaningful ways is better than dwelling on one method
“ fiche

3/hxll/2-3/hoflplus3/hofl/2wouldbeeasytoseeifinches

 

 

were used.
3/h of 1/2
13/1‘ d 1". L :----:g: ‘
TIFIITTI: ruifcuarvr
1 2 : I i

S , l

 

 

—¢—.—--——

 

137

Children can count the eighths and see that they will have 9/8 or

1 l/2. They can sse that 3/1: - 6/8. Half of 3/1: or 6/8 is then 3/8.
So, they will have 6/8 + 3/8 or 9/8 for the answer. Children need to
be encouraged to think problens through logically.

To help in understanding that wultiplic ation by fractions results in a
sneller nuaber than the multiplicand, introduce the expression taken so
tins. Use this in the place of ultiplied by, or tiles. Help
n understand that:
ll/ZxB/h- IeusB/htakenoneandone-half (ll/2) tiles.

This way be shown in a sequence or nuaber line relationship.

lsB/h- neansB/htalnenltine
2:3/h- neansB/htahenZtiaes
BxB/h- neansB/htakenBtiaes,etc.

Now the half way points no be shown:
le/h- neansB/htakenltine
1 1/2 x 3/lt'I neans 3/h taken 1 and 1/2 tines
2xB/h- neans3/htaken2tiaes
2 1/2 x 3/h - neans B/h tabs 2 and l/2 tines, etc.

The problen nay be done Just as they have done nultiplication with whole
mmbers in the past.

3/h
x l 2
(l x 3/h)
31,3 (1/2 x 3/h)
4- (total of the two partial products)
or
9/8 (coabining with a con-on denominator)

Such investigations as these, lead to understandings and pronote insights
that are often missed in rote situations. They are neaningful because
they are based on things that the child alrefl knows.

1;. Children. need to understand division as a recess in order to under-
stand division and lultiplication of fractions.

 

 

h therearehthreesinl2
3/12- leans therearehgroupsofthreeinl2
Bombesubtiactedfroaufourtiaes
23-12

35““ HELL-11:41?

_A'A

”id-L in: 35.7-} ’I

 

 

—- -~-.

S.

138
In the same way:

6/8 e 3/8 - 2 neans there are 2 three-eighths in 6/8
there are 2 groups of 3/8's in 6/8

01‘
6/8 _ we) um (1/8)
We) we) we)

3/8 can be subtracted twice from 6/8
2 x 3/8 I 63/8

Finding relationships and ways to express them such as these helps
children to see that the answer 2 is not wrong or foolish, although
it may first seem that way to then.

The fact that fractions can be written in different foms, all having
the same value should be plain to children. This comes easily from
the basic understanding that “fractions are parts of things'. is
illustrated on page 131, 1 cup is equal to l/l.6 of a gallon. Taking
this as a starting point, children can make true statements, such as:

lcup-l/logallon
lpint-2cups

therefore, 2 cups - 2/16 of a gallon

1 pint equals 1/8 of a gallon
2 cups equal 1 pint

therefore,
1 pint equals 2/16 of a gallon

Children should be encouraged to nake their own observations, record
them and draw conclusions. The lesson in the appendix on the fractional
or combo ruler helps children to see these relationships.

Making tables of equal fractions helps children to see this. Start
with something that they know.

ya - 2/h. we, 8/16, etc.
1/3 II 2/6, 14/12, etc.

Enchurage the children to see that they can do this with tangible ob-
Jects and with symbols. Then encourage them to draw the conclusion
that they are smltiplying the fraction, top and Bottomty the same
number. After they have discovered this, help then see that the reason
this multiplication does not change the value of the fractions is that
n/n is always equal to one.

2 x l . 2 2 2 l

'2' x 2 - H 2 equals 1, therefore II - 2' because the fraction has been
multiplied by the equivalent of 1. This leads directly to the dis-
covery of lowest comnon denominators.

6.

7.

8.

139

Children need to have help in finding a lowest common denominator and
help in a wq to put it down so that it makes sense to them and can
be read. Many children in the fifth and sixth grades have difficulty
with fractions because their small muscle development has not reached
a point where they can write them. If children have been taught that
n/n - 1, they night he encouraged to write a problem involving lowest
comon denominators like the following:

E- IVh

It should be emphasised that in every case they are mltiplying by
the equivalent of 1.

In line with the above, they should see how to change a fraction to
higher terns. They should see this graphically, tangibly, and then
With symbols.

312-6
313-3

THINK; 2/5 is equal to how new fiMenths? If I multiply the
denominator by 3, that will give as fifteenths. I can multiply the
numerator then by 3 and I will have 6 fifteenths. This is all right
to do because 3/3 is equal to l. Ihltiplying a number by 1 does not
change its value.

In order to shorten difficult multiplication processes, children should
know how to cancel. However, they should know that cancelling is the
same as dividing the fraction, or fracti or; by the equivalent of l or

n/n-
2

3 1.
3*5‘2

1
fxgxé- (unsung)

- (dividing by {a

W
N
m
H
”II-4

 

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1.

11.0
UNDERSTANDINGS BASIC T0 DIVISION OF FRACTIONS

Divid a number less than one will yield an answer greater than
ONE. This 031d Be thoroughly understood or the quoti—ent in a
mision example is quite meaningless. Following are some suggestions
as to how it may be approached.

Show division in a sequence of examples.

Another way to show this would be to indicate the number of halves or
fourths in three. This can be done with symbols or a number line.

3e%- l-§

...:
I
MIN

to] MIN

1
3 I
If a youngster knows “division as a process“ as listed in the basic

requiremnts, he cannow see that3el/2 canaeanhowyhalvesin
“twee! .tCO

 

 

 

3e%-T I 2 3

o123hg6
£222 2

If the problem is one of the type: 3 e 2/h -, we have much the same
situation, except that the fourths are tahen two at a time. If the
youngsters have been introduced to this phase, they can see that now
the question asks how many groups of taro-fourths are there in three?
This can be shown graphically and symbolically also:

 

2.

_u (2 2)
H' m E)

H

(2 2)
1 '§'m*n>
10 Iii-(12702)
3. He n+1; +§+12z+12i—(6two-fourths)
or there are 6 two-fourths in 3. The question is one of putting a label

on the answer. It might be well for a tine to have children label
their answers for fraction problems in division.

3: 2/h - onto-fourths

logically, new students can be lead to see that we have to find out
how-_agfourthsthereareinBandthentake them2 atatime. This
meanswehavetomlti-EIbeyFaHdividebyi, orINVERTand
MILTIPII.

 

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3:14-12 thenumberoffcurthsinB-
2' ‘2' 6(2/h'.) takentwo at atims.

The same thing can be done with a numberline and the problem shown
graphically.

3 , 2 u ‘34: I I I T r’ I I I I I I
H 0 l 2 3

 

This is a logical approach to division of fractions. We simly have
a convenient way of putting down the process. Children should have
new experiences in “thinking about" what the process means. They

should be encouraged to I'find out for themselves and to experiment".

Another me____th__od for doing division of fractions is called the common
denofinator me__th__od. Children frequently discover this for themseres.
ET; change the —-fractions to a lowest common denominator and divide
as with whole numbers.

3e2-
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532%”
or 6times

I fourths W

This is a reasonable and understandable method for children that has
an explanation lying within their previous experiences with fractions.

 

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A method that might be called the reciprocal method is possible to
teach if children have mastered the idea that nfn - l: and that, for
this reason, you can multiply both terms of a fraction by the same
number without changing its value.

The difficulty in understanding this method is sometimes the dif-
ficulty of understanding the term reciprocal. Reciprocal refers to
N0 numbers. (1‘ TWO NUMBERS WHICH ARE MULTIPLIED TOGETHER SO THAT A
PRODUCT 0F 1 IS OBTAINED ARE RECIPRmAIS.

3x2-l 311/3-1 (Bandl/Bareredprocals)
1/2 x 7 - l l/2 x 2/1 - 1, etc.( 1/2 and 2/1 are reciprocals)
We can state our division example as a fraction (complex fraction) and

show this inamanner thatwill stillbewithin the context of the
previous experiences the children have had with fractions.

30%-

3
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I

If we multiply both terms by the same nuaher, we are multiplying by l
and this will not change the value of the fraction.

,If we can make the denofinator of this cgplex fraction 1, we will have

the answer.

Therefore, multiply both term by the reciprocal of the denominator
and the denominator will then be 1.

 

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I I 1 I 2:33,,12“12
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We have shown this in a round-about method. Now we can conclude,
though, that the reciprocal of the divisor will be the divisor inverted,
this .111 make the difisor 1 {five multiply ton: firms Wthe ac on
by this reciprocal. So, all we have to do is to invert the divisor

in the first place and multiply. This method gives a reason for
inversion. Sometimes the reason is too difficult for all but a few
youngsters. The important thing to remember is to give them a great
many experiences, and new ways of thinking logically about fraction
division. Without doubt, students and teachers can think of many
additional ways of demonstrating the process. Time should be taken

to investigate these. It will be well spent in developing interest

in mathematics and real understanding.

 

V. DECIHAI. FRACTIONS

Decimal and Cannon Fractions
in htension of Our Number System
Changing Common Fractions to Decimal Fraction

lhh

Because decimal fractions all have denominators with powers of ten, we
can leave the denominators off and write them as whole numbers if we can

findawaytodoit.

In order to do this, we put a 0 after the whole number and call all the

numbers after the decimal point-decimal fractions. The first place after

the decimal point is tenths' place, the next, hundredths', etc., Just as
the places for the whole numbers.

WE CALL THIS WAY CF WRITING DECIHAI. FRACTIONS AS AN EXTENSION OF we
NUMBERSYSTEH. WEHAVE JUST EXTENDED THE SISTEMHCRDE TOWRITETBE

DECIHLL FRACTIONS.

 

fillions

hundred-thousands

I ten-thousands

thousands
tens

ne
tenths
hundredths
thousandths

ten-thousandths

 

 

 

 

 

 

 

 

 

 

 

 

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Reading decimal fractions is comparatively simple if we refer to the
chart. The next problem is how to change all fractions to decimal
fractions. To approach this logically and within the context of what
the children know about fractions and lowest comon denominators, it
better than presenting a rule to be memorized.

Some common fractions can be changed to decimal fractions with little
or no margin of error. Some examples will demonstrate this.

1 7 l 3
{'16 Z'IU'C
5:1“: 2931.29
3:2 10 50x2 100
However:
1_ 2 1_ 2
II 1'5 1: gm
2x1_ 2 gx1_ 25
211; I5 11? 155

 

Inthefirstexample,wehaveachoiceofmultiplyinghby2l/21n
order to make the decimal come out even. This is foolish, since the
purpose of decimals is to make them easy to use. So, we carry the
fraction one more place and it comes out even. (Mr error would be
less if it came out 1/2 of a hundredth than if it came out 1/2 of a
tenth.

WE ARE MULTIPLYING BOTH TERIB OF THESE FRACTIONS BY THEDUIVALENT
OF 1 , JUST AS WE DID WITH LOWEST comes DENOHINATGRS BEFORE.

In order to find out what to multiply the common fraction denomina-

tor by, THINK: I'tvlhat do I multiply h by to get ten‘:"' Or, IWhat do

I multiply h by to get 100?", etc. A problem like the following, is
more difficult. Students may actually need to perform the division

in order to find out what number to use as a multiplier.

3. ? 6 611%. 18
IE ICC 16/135 3 166'
Itmaybenecessaryforchildrentodivideloobylointhiscase

before they can complete the equation. 18/100 is as close as we
can come with whole numbers.

 

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